Properties

Label 1925.2.b.d.1849.1
Level $1925$
Weight $2$
Character 1925.1849
Analytic conductor $15.371$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1925.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.3712023891\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1925.1849
Dual form 1925.2.b.d.1849.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +2.00000i q^{12} +4.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} -4.00000i q^{17} +1.00000i q^{18} -2.00000 q^{21} -1.00000i q^{22} -4.00000i q^{23} +6.00000 q^{24} +4.00000 q^{26} +4.00000i q^{27} +1.00000i q^{28} +6.00000 q^{29} +10.0000 q^{31} -5.00000i q^{32} +2.00000i q^{33} -4.00000 q^{34} -1.00000 q^{36} +6.00000i q^{37} -8.00000 q^{39} +4.00000 q^{41} +2.00000i q^{42} +12.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} +10.0000i q^{47} -2.00000i q^{48} -1.00000 q^{49} +8.00000 q^{51} +4.00000i q^{52} -6.00000i q^{53} +4.00000 q^{54} +3.00000 q^{56} -6.00000i q^{58} -2.00000 q^{59} -10.0000i q^{62} -1.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} -8.00000i q^{67} -4.00000i q^{68} +8.00000 q^{69} -12.0000 q^{71} +3.00000i q^{72} -8.00000i q^{73} +6.00000 q^{74} +1.00000i q^{77} +8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} -4.00000i q^{82} -2.00000 q^{84} +12.0000 q^{86} +12.0000i q^{87} -3.00000i q^{88} +6.00000 q^{89} -4.00000 q^{91} -4.00000i q^{92} +20.0000i q^{93} +10.0000 q^{94} +10.0000 q^{96} +10.0000i q^{97} +1.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{4} + 4q^{6} - 2q^{9} + 2q^{11} + 2q^{14} - 2q^{16} - 4q^{21} + 12q^{24} + 8q^{26} + 12q^{29} + 20q^{31} - 8q^{34} - 2q^{36} - 16q^{39} + 8q^{41} + 2q^{44} - 8q^{46} - 2q^{49} + 16q^{51} + 8q^{54} + 6q^{56} - 4q^{59} - 14q^{64} + 4q^{66} + 16q^{69} - 24q^{71} + 12q^{74} - 16q^{79} - 22q^{81} - 4q^{84} + 24q^{86} + 12q^{89} - 8q^{91} + 20q^{94} + 20q^{96} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 −0.935414 0.353553i \(-0.884973\pi\)
0.935414 0.353553i \(-0.115027\pi\)
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 1.00000i 0.377964i
\(8\) − 3.00000i − 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.00000i 0.577350i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 4.00000i − 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) − 1.00000i − 0.213201i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 6.00000 1.22474
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 4.00000i 0.769800i
\(28\) 1.00000i 0.188982i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 2.00000i 0.348155i
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 10.0000i 1.45865i 0.684167 + 0.729325i \(0.260166\pi\)
−0.684167 + 0.729325i \(0.739834\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 4.00000i 0.554700i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) − 10.0000i − 1.27000i
\(63\) − 1.00000i − 0.125988i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 3.00000i 0.353553i
\(73\) − 8.00000i − 0.936329i −0.883641 0.468165i \(-0.844915\pi\)
0.883641 0.468165i \(-0.155085\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000i 0.113961i
\(78\) 8.00000i 0.905822i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) − 4.00000i − 0.441726i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 12.0000i 1.28654i
\(88\) − 3.00000i − 0.319801i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) − 4.00000i − 0.417029i
\(93\) 20.0000i 2.07390i
\(94\) 10.0000 1.03142
\(95\) 0 0
\(96\) 10.0000 1.02062
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 1.00000i 0.101015i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) − 8.00000i − 0.792118i
\(103\) 14.0000i 1.37946i 0.724066 + 0.689730i \(0.242271\pi\)
−0.724066 + 0.689730i \(0.757729\pi\)
\(104\) 12.0000 1.17670
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 4.00000i 0.384900i
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) − 1.00000i − 0.0944911i
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) − 4.00000i − 0.369800i
\(118\) 2.00000i 0.184115i
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 8.00000i 0.721336i
\(124\) 10.0000 0.898027
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) − 3.00000i − 0.265165i
\(129\) −24.0000 −2.11308
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) 10.0000i 0.854358i 0.904167 + 0.427179i \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) − 8.00000i − 0.681005i
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −20.0000 −1.68430
\(142\) 12.0000i 1.00702i
\(143\) 4.00000i 0.334497i
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −8.00000 −0.662085
\(147\) − 2.00000i − 0.164957i
\(148\) 6.00000i 0.493197i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 4.00000i 0.323381i
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −8.00000 −0.640513
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 11.0000i 0.864242i
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 6.00000i 0.462910i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 12.0000i 0.914991i
\(173\) 12.0000i 0.912343i 0.889892 + 0.456172i \(0.150780\pi\)
−0.889892 + 0.456172i \(0.849220\pi\)
\(174\) 12.0000 0.909718
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) − 4.00000i − 0.300658i
\(178\) − 6.00000i − 0.449719i
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 4.00000i 0.296500i
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) 20.0000 1.46647
\(187\) − 4.00000i − 0.292509i
\(188\) 10.0000i 0.729325i
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) − 14.0000i − 1.01036i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 1.00000i 0.0710669i
\(199\) 18.0000 1.27599 0.637993 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 4.00000i 0.281439i
\(203\) 6.00000i 0.421117i
\(204\) 8.00000 0.560112
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) 4.00000i 0.278019i
\(208\) − 4.00000i − 0.277350i
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) − 24.0000i − 1.64445i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 12.0000 0.816497
\(217\) 10.0000i 0.678844i
\(218\) − 14.0000i − 0.948200i
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) 16.0000 1.07628
\(222\) 12.0000i 0.805387i
\(223\) 22.0000i 1.47323i 0.676313 + 0.736614i \(0.263577\pi\)
−0.676313 + 0.736614i \(0.736423\pi\)
\(224\) 5.00000 0.334077
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) − 18.0000i − 1.18176i
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −2.00000 −0.130189
\(237\) − 16.0000i − 1.03931i
\(238\) − 4.00000i − 0.259281i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) − 1.00000i − 0.0642824i
\(243\) − 10.0000i − 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) 0 0
\(248\) − 30.0000i − 1.90500i
\(249\) 0 0
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) − 4.00000i − 0.251478i
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 14.0000i 0.873296i 0.899632 + 0.436648i \(0.143834\pi\)
−0.899632 + 0.436648i \(0.856166\pi\)
\(258\) 24.0000i 1.49417i
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) − 12.0000i − 0.741362i
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) − 8.00000i − 0.488678i
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 4.00000i 0.242536i
\(273\) − 8.00000i − 0.484182i
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) − 8.00000i − 0.479808i
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 20.0000i 1.19098i
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 4.00000i 0.236113i
\(288\) 5.00000i 0.294628i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −20.0000 −1.17242
\(292\) − 8.00000i − 0.468165i
\(293\) − 24.0000i − 1.40209i −0.713115 0.701047i \(-0.752716\pi\)
0.713115 0.701047i \(-0.247284\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 18.0000 1.04623
\(297\) 4.00000i 0.232104i
\(298\) − 10.0000i − 0.579284i
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 16.0000i 0.920697i
\(303\) − 8.00000i − 0.459588i
\(304\) 0 0
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) − 20.0000i − 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 1.00000i 0.0569803i
\(309\) −28.0000 −1.59286
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 24.0000i 1.35873i
\(313\) 2.00000i 0.113047i 0.998401 + 0.0565233i \(0.0180015\pi\)
−0.998401 + 0.0565233i \(0.981998\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) − 12.0000i − 0.672927i
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) − 4.00000i − 0.222911i
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) 28.0000i 1.54840i
\(328\) − 12.0000i − 0.662589i
\(329\) −10.0000 −0.551318
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) − 6.00000i − 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) 3.00000i 0.163178i
\(339\) −36.0000 −1.95525
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 36.0000 1.94099
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) − 4.00000i − 0.214731i −0.994220 0.107366i \(-0.965758\pi\)
0.994220 0.107366i \(-0.0342415\pi\)
\(348\) 12.0000i 0.643268i
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) − 5.00000i − 0.266501i
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 8.00000i 0.423405i
\(358\) 12.0000i 0.634220i
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 10.0000i − 0.525588i
\(363\) 2.00000i 0.104973i
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) − 22.0000i − 1.14839i −0.818718 0.574195i \(-0.805315\pi\)
0.818718 0.574195i \(-0.194685\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 20.0000i 1.03695i
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 30.0000 1.54713
\(377\) 24.0000i 1.23606i
\(378\) 4.00000i 0.205738i
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) − 8.00000i − 0.409316i
\(383\) 2.00000i 0.102195i 0.998694 + 0.0510976i \(0.0162720\pi\)
−0.998694 + 0.0510976i \(0.983728\pi\)
\(384\) 6.00000 0.306186
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) − 12.0000i − 0.609994i
\(388\) 10.0000i 0.507673i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 3.00000i 0.151523i
\(393\) 24.0000i 1.21064i
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) − 22.0000i − 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) − 18.0000i − 0.902258i
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) − 16.0000i − 0.798007i
\(403\) 40.0000i 1.99254i
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 6.00000i 0.297409i
\(408\) − 24.0000i − 1.18818i
\(409\) −24.0000 −1.18672 −0.593362 0.804936i \(-0.702200\pi\)
−0.593362 + 0.804936i \(0.702200\pi\)
\(410\) 0 0
\(411\) −20.0000 −0.986527
\(412\) 14.0000i 0.689730i
\(413\) − 2.00000i − 0.0984136i
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 20.0000 0.980581
\(417\) 16.0000i 0.783523i
\(418\) 0 0
\(419\) −2.00000 −0.0977064 −0.0488532 0.998806i \(-0.515557\pi\)
−0.0488532 + 0.998806i \(0.515557\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 12.0000i 0.584151i
\(423\) − 10.0000i − 0.486217i
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) −24.0000 −1.16280
\(427\) 0 0
\(428\) − 12.0000i − 0.580042i
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) − 26.0000i − 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) 10.0000 0.480015
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) − 16.0000i − 0.764510i
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) − 16.0000i − 0.761042i
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) −12.0000 −0.569495
\(445\) 0 0
\(446\) 22.0000 1.04173
\(447\) 20.0000i 0.945968i
\(448\) − 7.00000i − 0.330719i
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 18.0000i 0.846649i
\(453\) − 32.0000i − 1.50349i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) − 18.0000i − 0.842004i −0.907060 0.421002i \(-0.861678\pi\)
0.907060 0.421002i \(-0.138322\pi\)
\(458\) 18.0000i 0.841085i
\(459\) 16.0000 0.746816
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 2.00000i 0.0930484i
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) − 30.0000i − 1.38823i −0.719862 0.694117i \(-0.755795\pi\)
0.719862 0.694117i \(-0.244205\pi\)
\(468\) − 4.00000i − 0.184900i
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 28.0000 1.29017
\(472\) 6.00000i 0.276172i
\(473\) 12.0000i 0.551761i
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 20.0000i 0.910975i
\(483\) 8.00000i 0.364013i
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 28.0000i 1.26880i 0.773004 + 0.634401i \(0.218753\pi\)
−0.773004 + 0.634401i \(0.781247\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 8.00000i 0.360668i
\(493\) − 24.0000i − 1.08091i
\(494\) 0 0
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) − 12.0000i − 0.538274i
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.00000i 0.0892644i
\(503\) 4.00000i 0.178351i 0.996016 + 0.0891756i \(0.0284232\pi\)
−0.996016 + 0.0891756i \(0.971577\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) − 6.00000i − 0.266469i
\(508\) − 8.00000i − 0.354943i
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 11.0000i 0.486136i
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 0 0
\(516\) −24.0000 −1.05654
\(517\) 10.0000i 0.439799i
\(518\) 6.00000i 0.263625i
\(519\) −24.0000 −1.05348
\(520\) 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
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) − 40.0000i − 1.74243i
\(528\) − 2.00000i − 0.0870388i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 2.00000 0.0867926
\(532\) 0 0
\(533\) 16.0000i 0.693037i
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) −24.0000 −1.03664
\(537\) − 24.0000i − 1.03568i
\(538\) 10.0000i 0.431131i
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 4.00000i 0.171815i
\(543\) 20.0000i 0.858282i
\(544\) −20.0000 −0.857493
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 28.0000i 1.19719i 0.801050 + 0.598597i \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 10.0000i 0.427179i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) − 24.0000i − 1.02151i
\(553\) − 8.00000i − 0.340195i
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) − 22.0000i − 0.932170i −0.884740 0.466085i \(-0.845664\pi\)
0.884740 0.466085i \(-0.154336\pi\)
\(558\) 10.0000i 0.423334i
\(559\) −48.0000 −2.03018
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) − 6.00000i − 0.253095i
\(563\) − 32.0000i − 1.34864i −0.738440 0.674320i \(-0.764437\pi\)
0.738440 0.674320i \(-0.235563\pi\)
\(564\) −20.0000 −0.842152
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) − 11.0000i − 0.461957i
\(568\) 36.0000i 1.51053i
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 4.00000i 0.167248i
\(573\) 16.0000i 0.668410i
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 18.0000i 0.749350i 0.927156 + 0.374675i \(0.122246\pi\)
−0.927156 + 0.374675i \(0.877754\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) 28.0000 1.16364
\(580\) 0 0
\(581\) 0 0
\(582\) 20.0000i 0.829027i
\(583\) − 6.00000i − 0.248495i
\(584\) −24.0000 −0.993127
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) 2.00000i 0.0825488i 0.999148 + 0.0412744i \(0.0131418\pi\)
−0.999148 + 0.0412744i \(0.986858\pi\)
\(588\) − 2.00000i − 0.0824786i
\(589\) 0 0
\(590\) 0 0
\(591\) 44.0000 1.80992
\(592\) − 6.00000i − 0.246598i
\(593\) 32.0000i 1.31408i 0.753855 + 0.657041i \(0.228192\pi\)
−0.753855 + 0.657041i \(0.771808\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 36.0000i 1.47338i
\(598\) − 16.0000i − 0.654289i
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 12.0000i 0.489083i
\(603\) 8.00000i 0.325785i
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) −8.00000 −0.324978
\(607\) 40.0000i 1.62355i 0.583970 + 0.811775i \(0.301498\pi\)
−0.583970 + 0.811775i \(0.698502\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) −40.0000 −1.61823
\(612\) 4.00000i 0.161690i
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 28.0000i 1.12633i
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 18.0000i 0.721734i
\(623\) 6.00000i 0.240385i
\(624\) 8.00000 0.320256
\(625\) 0 0
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) − 14.0000i − 0.558661i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 24.0000i 0.954669i
\(633\) − 24.0000i − 0.953914i
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) − 4.00000i − 0.158486i
\(638\) − 6.00000i − 0.237542i
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) − 24.0000i − 0.947204i
\(643\) 14.0000i 0.552106i 0.961142 + 0.276053i \(0.0890266\pi\)
−0.961142 + 0.276053i \(0.910973\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) 22.0000i 0.864909i 0.901656 + 0.432455i \(0.142352\pi\)
−0.901656 + 0.432455i \(0.857648\pi\)
\(648\) 33.0000i 1.29636i
\(649\) −2.00000 −0.0785069
\(650\) 0 0
\(651\) −20.0000 −0.783862
\(652\) − 8.00000i − 0.313304i
\(653\) − 26.0000i − 1.01746i −0.860927 0.508729i \(-0.830115\pi\)
0.860927 0.508729i \(-0.169885\pi\)
\(654\) 28.0000 1.09489
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) 8.00000i 0.312110i
\(658\) 10.0000i 0.389841i
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 32.0000i 1.24278i
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) − 24.0000i − 0.929284i
\(668\) 0 0
\(669\) −44.0000 −1.70114
\(670\) 0 0
\(671\) 0 0
\(672\) 10.0000i 0.385758i
\(673\) − 34.0000i − 1.31060i −0.755367 0.655302i \(-0.772541\pi\)
0.755367 0.655302i \(-0.227459\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) 12.0000i 0.461197i 0.973049 + 0.230599i \(0.0740685\pi\)
−0.973049 + 0.230599i \(0.925932\pi\)
\(678\) 36.0000i 1.38257i
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) − 10.0000i − 0.382920i
\(683\) − 4.00000i − 0.153056i −0.997067 0.0765279i \(-0.975617\pi\)
0.997067 0.0765279i \(-0.0243834\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) − 36.0000i − 1.37349i
\(688\) − 12.0000i − 0.457496i
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −46.0000 −1.74992 −0.874961 0.484193i \(-0.839113\pi\)
−0.874961 + 0.484193i \(0.839113\pi\)
\(692\) 12.0000i 0.456172i
\(693\) − 1.00000i − 0.0379869i
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 36.0000 1.36458
\(697\) − 16.0000i − 0.606043i
\(698\) 0 0
\(699\) 36.0000 1.36165
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 16.0000i 0.603881i
\(703\) 0 0
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) − 4.00000i − 0.150435i
\(708\) − 4.00000i − 0.150329i
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) − 18.0000i − 0.674579i
\(713\) − 40.0000i − 1.49801i
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 16.0000i 0.597115i
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 19.0000i 0.707107i
\(723\) − 40.0000i − 1.48762i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) − 18.0000i − 0.667583i −0.942647 0.333792i \(-0.891672\pi\)
0.942647 0.333792i \(-0.108328\pi\)
\(728\) 12.0000i 0.444750i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) −22.0000 −0.812035
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) − 8.00000i − 0.294684i
\(738\) 4.00000i 0.147242i
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 6.00000i − 0.220267i
\(743\) − 8.00000i − 0.293492i −0.989174 0.146746i \(-0.953120\pi\)
0.989174 0.146746i \(-0.0468799\pi\)
\(744\) 60.0000 2.19971
\(745\) 0 0
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) − 4.00000i − 0.146254i
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) − 10.0000i − 0.364662i
\(753\) − 4.00000i − 0.145768i
\(754\) 24.0000 0.874028
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) − 8.00000i − 0.290573i
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) − 16.0000i − 0.579619i
\(763\) 14.0000i 0.506834i
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 2.00000 0.0722629
\(767\) − 8.00000i − 0.288863i
\(768\) − 34.0000i − 1.22687i
\(769\) −32.0000 −1.15395 −0.576975 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) −28.0000 −1.00840
\(772\) − 14.0000i − 0.503871i
\(773\) − 30.0000i − 1.07903i −0.841978 0.539513i \(-0.818609\pi\)
0.841978 0.539513i \(-0.181391\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 30.0000 1.07694
\(777\) − 12.0000i − 0.430498i
\(778\) 6.00000i 0.215110i
\(779\) 0 0
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 16.0000i 0.572159i
\(783\) 24.0000i 0.857690i
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 24.0000 0.856052
\(787\) − 16.0000i − 0.570338i −0.958477 0.285169i \(-0.907950\pi\)
0.958477 0.285169i \(-0.0920498\pi\)
\(788\) − 22.0000i − 0.783718i
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 3.00000i 0.106600i
\(793\) 0 0
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) − 14.0000i − 0.495905i −0.968772 0.247953i \(-0.920242\pi\)
0.968772 0.247953i \(-0.0797578\pi\)
\(798\) 0 0
\(799\) 40.0000 1.41510
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 22.0000i 0.776847i
\(803\) − 8.00000i − 0.282314i
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 40.0000 1.40894
\(807\) − 20.0000i − 0.704033i
\(808\) 12.0000i 0.422159i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 6.00000i 0.210559i
\(813\) − 8.00000i − 0.280572i
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) 0 0
\(818\) 24.0000i 0.839140i
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 20.0000i 0.697580i
\(823\) 24.0000i 0.836587i 0.908312 + 0.418294i \(0.137372\pi\)
−0.908312 + 0.418294i \(0.862628\pi\)
\(824\) 42.0000 1.46314
\(825\) 0 0
\(826\) −2.00000 −0.0695889
\(827\) 28.0000i 0.973655i 0.873498 + 0.486828i \(0.161846\pi\)
−0.873498 + 0.486828i \(0.838154\pi\)
\(828\) 4.00000i 0.139010i
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) − 28.0000i − 0.970725i
\(833\) 4.00000i 0.138592i
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) 0 0
\(837\) 40.0000i 1.38260i
\(838\) 2.00000i 0.0690889i
\(839\) −34.0000 −1.17381 −0.586905 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 14.0000i 0.482472i
\(843\) 12.0000i 0.413302i
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) −10.0000 −0.343807
\(847\) 1.00000i 0.0343604i
\(848\) 6.00000i 0.206041i
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) − 24.0000i − 0.822226i
\(853\) 44.0000i 1.50653i 0.657716 + 0.753266i \(0.271523\pi\)
−0.657716 + 0.753266i \(0.728477\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) − 56.0000i − 1.91292i −0.291858 0.956462i \(-0.594273\pi\)
0.291858 0.956462i \(-0.405727\pi\)
\(858\) 8.00000i 0.273115i
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 20.0000 0.680414
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) 2.00000i 0.0679236i
\(868\) 10.0000i 0.339422i
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) − 42.0000i − 1.42230i
\(873\) − 10.0000i − 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) 16.0000 0.540590
\(877\) − 42.0000i − 1.41824i −0.705088 0.709120i \(-0.749093\pi\)
0.705088 0.709120i \(-0.250907\pi\)
\(878\) − 20.0000i − 0.674967i
\(879\) 48.0000 1.61900
\(880\) 0 0
\(881\) −34.0000 −1.14549 −0.572745 0.819734i \(-0.694121\pi\)
−0.572745 + 0.819734i \(0.694121\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) 28.0000i 0.942275i 0.882060 + 0.471138i \(0.156156\pi\)
−0.882060 + 0.471138i \(0.843844\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 28.0000i 0.940148i 0.882627 + 0.470074i \(0.155773\pi\)
−0.882627 + 0.470074i \(0.844227\pi\)
\(888\) 36.0000i 1.20808i
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) 22.0000i 0.736614i
\(893\) 0 0
\(894\) 20.0000 0.668900
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 32.0000i 1.06845i
\(898\) − 10.0000i − 0.333704i
\(899\) 60.0000 2.00111
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) − 4.00000i − 0.133185i
\(903\) − 24.0000i − 0.798670i
\(904\) 54.0000 1.79601
\(905\) 0 0
\(906\) −32.0000 −1.06313
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −18.0000 −0.594737
\(917\) 12.0000i 0.396275i
\(918\) − 16.0000i − 0.528079i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 40.0000 1.31804
\(922\) 0 0
\(923\) − 48.0000i − 1.57994i
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) − 14.0000i − 0.459820i
\(928\) − 30.0000i − 0.984798i
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 18.0000i − 0.589610i
\(933\) − 36.0000i − 1.17859i
\(934\) −30.0000 −0.981630
\(935\) 0 0
\(936\) −12.0000 −0.392232
\(937\) 16.0000i 0.522697i 0.965244 + 0.261349i \(0.0841672\pi\)
−0.965244 + 0.261349i \(0.915833\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) − 28.0000i − 0.912289i
\(943\) − 16.0000i − 0.521032i
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) − 16.0000i − 0.519656i
\(949\) 32.0000 1.03876
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) − 12.0000i − 0.388922i
\(953\) 34.0000i 1.10137i 0.834714 + 0.550684i \(0.185633\pi\)
−0.834714 + 0.550684i \(0.814367\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) 12.0000i 0.387905i
\(958\) − 4.00000i − 0.129234i
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 24.0000i 0.773791i
\(963\) 12.0000i 0.386695i
\(964\) −20.0000 −0.644157
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) − 40.0000i − 1.28631i −0.765735 0.643157i \(-0.777624\pi\)
0.765735 0.643157i \(-0.222376\pi\)
\(968\) − 3.00000i − 0.0964237i
\(969\) 0 0
\(970\) 0 0
\(971\) 14.0000 0.449281 0.224641 0.974442i \(-0.427879\pi\)
0.224641 + 0.974442i \(0.427879\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) 8.00000i 0.256468i
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) 0 0
\(977\) − 42.0000i − 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) − 16.0000i − 0.511624i
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) − 28.0000i − 0.893516i
\(983\) 54.0000i 1.72233i 0.508323 + 0.861166i \(0.330265\pi\)
−0.508323 + 0.861166i \(0.669735\pi\)
\(984\) 24.0000 0.765092
\(985\) 0 0
\(986\) −24.0000 −0.764316
\(987\) − 20.0000i − 0.636607i
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) − 50.0000i − 1.58750i
\(993\) − 40.0000i − 1.26936i
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) − 20.0000i − 0.633406i −0.948525 0.316703i \(-0.897424\pi\)
0.948525 0.316703i \(-0.102576\pi\)
\(998\) − 16.0000i − 0.506471i
\(999\) −24.0000 −0.759326
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 1925.2.b.d.1849.1 2
5.2 odd 4 77.2.a.c.1.1 1
5.3 odd 4 1925.2.a.c.1.1 1
5.4 even 2 inner 1925.2.b.d.1849.2 2
15.2 even 4 693.2.a.a.1.1 1
20.7 even 4 1232.2.a.a.1.1 1
35.2 odd 12 539.2.e.a.67.1 2
35.12 even 12 539.2.e.b.67.1 2
35.17 even 12 539.2.e.b.177.1 2
35.27 even 4 539.2.a.d.1.1 1
35.32 odd 12 539.2.e.a.177.1 2
40.27 even 4 4928.2.a.bi.1.1 1
40.37 odd 4 4928.2.a.g.1.1 1
55.2 even 20 847.2.f.k.323.1 4
55.7 even 20 847.2.f.k.148.1 4
55.17 even 20 847.2.f.k.729.1 4
55.27 odd 20 847.2.f.e.729.1 4
55.32 even 4 847.2.a.a.1.1 1
55.37 odd 20 847.2.f.e.148.1 4
55.42 odd 20 847.2.f.e.323.1 4
55.47 odd 20 847.2.f.e.372.1 4
55.52 even 20 847.2.f.k.372.1 4
105.62 odd 4 4851.2.a.a.1.1 1
140.27 odd 4 8624.2.a.bc.1.1 1
165.32 odd 4 7623.2.a.n.1.1 1
385.307 odd 4 5929.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.c.1.1 1 5.2 odd 4
539.2.a.d.1.1 1 35.27 even 4
539.2.e.a.67.1 2 35.2 odd 12
539.2.e.a.177.1 2 35.32 odd 12
539.2.e.b.67.1 2 35.12 even 12
539.2.e.b.177.1 2 35.17 even 12
693.2.a.a.1.1 1 15.2 even 4
847.2.a.a.1.1 1 55.32 even 4
847.2.f.e.148.1 4 55.37 odd 20
847.2.f.e.323.1 4 55.42 odd 20
847.2.f.e.372.1 4 55.47 odd 20
847.2.f.e.729.1 4 55.27 odd 20
847.2.f.k.148.1 4 55.7 even 20
847.2.f.k.323.1 4 55.2 even 20
847.2.f.k.372.1 4 55.52 even 20
847.2.f.k.729.1 4 55.17 even 20
1232.2.a.a.1.1 1 20.7 even 4
1925.2.a.c.1.1 1 5.3 odd 4
1925.2.b.d.1849.1 2 1.1 even 1 trivial
1925.2.b.d.1849.2 2 5.4 even 2 inner
4851.2.a.a.1.1 1 105.62 odd 4
4928.2.a.g.1.1 1 40.37 odd 4
4928.2.a.bi.1.1 1 40.27 even 4
5929.2.a.b.1.1 1 385.307 odd 4
7623.2.a.n.1.1 1 165.32 odd 4
8624.2.a.bc.1.1 1 140.27 odd 4