Properties

Label 405.2.a.i.1.1
Level $405$
Weight $2$
Character 405.1
Self dual yes
Analytic conductor $3.234$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.51414 q^{2} +4.32088 q^{4} -1.00000 q^{5} -0.514137 q^{7} -5.83502 q^{8} +O(q^{10})\) \(q-2.51414 q^{2} +4.32088 q^{4} -1.00000 q^{5} -0.514137 q^{7} -5.83502 q^{8} +2.51414 q^{10} +3.32088 q^{11} -1.32088 q^{13} +1.29261 q^{14} +6.02827 q^{16} +3.32088 q^{17} -1.32088 q^{19} -4.32088 q^{20} -8.34916 q^{22} -4.12763 q^{23} +1.00000 q^{25} +3.32088 q^{26} -2.22153 q^{28} +1.38650 q^{29} +8.73566 q^{31} -3.48586 q^{32} -8.34916 q^{34} +0.514137 q^{35} +0.292611 q^{37} +3.32088 q^{38} +5.83502 q^{40} +11.3492 q^{41} +10.3492 q^{43} +14.3492 q^{44} +10.3774 q^{46} +4.86330 q^{47} -6.73566 q^{49} -2.51414 q^{50} -5.70739 q^{52} +5.02827 q^{53} -3.32088 q^{55} +3.00000 q^{56} -3.48586 q^{58} +5.02827 q^{59} +7.34916 q^{61} -21.9627 q^{62} -3.29261 q^{64} +1.32088 q^{65} +9.44852 q^{67} +14.3492 q^{68} -1.29261 q^{70} -8.99093 q^{71} +6.05655 q^{73} -0.735663 q^{74} -5.70739 q^{76} -1.70739 q^{77} -8.05655 q^{79} -6.02827 q^{80} -28.5333 q^{82} -1.54241 q^{83} -3.32088 q^{85} -26.0192 q^{86} -19.3774 q^{88} +3.00000 q^{89} +0.679116 q^{91} -17.8350 q^{92} -12.2270 q^{94} +1.32088 q^{95} -12.2553 q^{97} +16.9344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8} + q^{10} + 2 q^{11} + 4 q^{13} + 9 q^{14} + 5 q^{16} + 2 q^{17} + 4 q^{19} - 5 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 2 q^{26} + 5 q^{28} + 7 q^{29} + 8 q^{31} - 17 q^{32} - 4 q^{34} - 5 q^{35} + 6 q^{37} + 2 q^{38} + 3 q^{40} + 13 q^{41} + 10 q^{43} + 22 q^{44} - 3 q^{46} - 13 q^{47} - 2 q^{49} - q^{50} - 12 q^{52} + 2 q^{53} - 2 q^{55} + 9 q^{56} - 17 q^{58} + 2 q^{59} + q^{61} - 42 q^{62} - 15 q^{64} - 4 q^{65} + 11 q^{67} + 22 q^{68} - 9 q^{70} + 10 q^{71} - 8 q^{73} + 16 q^{74} - 12 q^{76} + 2 q^{79} - 5 q^{80} - 29 q^{82} + 15 q^{83} - 2 q^{85} - 28 q^{86} - 24 q^{88} + 9 q^{89} + 10 q^{91} - 39 q^{92} - 31 q^{94} - 4 q^{95} - 18 q^{97} + 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.51414 −1.77776 −0.888882 0.458137i \(-0.848517\pi\)
−0.888882 + 0.458137i \(0.848517\pi\)
\(3\) 0 0
\(4\) 4.32088 2.16044
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.514137 −0.194325 −0.0971627 0.995269i \(-0.530977\pi\)
−0.0971627 + 0.995269i \(0.530977\pi\)
\(8\) −5.83502 −2.06299
\(9\) 0 0
\(10\) 2.51414 0.795040
\(11\) 3.32088 1.00128 0.500642 0.865654i \(-0.333097\pi\)
0.500642 + 0.865654i \(0.333097\pi\)
\(12\) 0 0
\(13\) −1.32088 −0.366347 −0.183174 0.983081i \(-0.558637\pi\)
−0.183174 + 0.983081i \(0.558637\pi\)
\(14\) 1.29261 0.345465
\(15\) 0 0
\(16\) 6.02827 1.50707
\(17\) 3.32088 0.805433 0.402716 0.915325i \(-0.368066\pi\)
0.402716 + 0.915325i \(0.368066\pi\)
\(18\) 0 0
\(19\) −1.32088 −0.303032 −0.151516 0.988455i \(-0.548415\pi\)
−0.151516 + 0.988455i \(0.548415\pi\)
\(20\) −4.32088 −0.966179
\(21\) 0 0
\(22\) −8.34916 −1.78005
\(23\) −4.12763 −0.860671 −0.430335 0.902669i \(-0.641605\pi\)
−0.430335 + 0.902669i \(0.641605\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.32088 0.651279
\(27\) 0 0
\(28\) −2.22153 −0.419829
\(29\) 1.38650 0.257467 0.128734 0.991679i \(-0.458909\pi\)
0.128734 + 0.991679i \(0.458909\pi\)
\(30\) 0 0
\(31\) 8.73566 1.56897 0.784486 0.620147i \(-0.212927\pi\)
0.784486 + 0.620147i \(0.212927\pi\)
\(32\) −3.48586 −0.616219
\(33\) 0 0
\(34\) −8.34916 −1.43187
\(35\) 0.514137 0.0869050
\(36\) 0 0
\(37\) 0.292611 0.0481049 0.0240524 0.999711i \(-0.492343\pi\)
0.0240524 + 0.999711i \(0.492343\pi\)
\(38\) 3.32088 0.538719
\(39\) 0 0
\(40\) 5.83502 0.922598
\(41\) 11.3492 1.77244 0.886220 0.463264i \(-0.153322\pi\)
0.886220 + 0.463264i \(0.153322\pi\)
\(42\) 0 0
\(43\) 10.3492 1.57823 0.789116 0.614244i \(-0.210539\pi\)
0.789116 + 0.614244i \(0.210539\pi\)
\(44\) 14.3492 2.16322
\(45\) 0 0
\(46\) 10.3774 1.53007
\(47\) 4.86330 0.709385 0.354692 0.934983i \(-0.384586\pi\)
0.354692 + 0.934983i \(0.384586\pi\)
\(48\) 0 0
\(49\) −6.73566 −0.962238
\(50\) −2.51414 −0.355553
\(51\) 0 0
\(52\) −5.70739 −0.791472
\(53\) 5.02827 0.690687 0.345343 0.938476i \(-0.387762\pi\)
0.345343 + 0.938476i \(0.387762\pi\)
\(54\) 0 0
\(55\) −3.32088 −0.447788
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −3.48586 −0.457716
\(59\) 5.02827 0.654625 0.327313 0.944916i \(-0.393857\pi\)
0.327313 + 0.944916i \(0.393857\pi\)
\(60\) 0 0
\(61\) 7.34916 0.940963 0.470482 0.882410i \(-0.344080\pi\)
0.470482 + 0.882410i \(0.344080\pi\)
\(62\) −21.9627 −2.78926
\(63\) 0 0
\(64\) −3.29261 −0.411576
\(65\) 1.32088 0.163836
\(66\) 0 0
\(67\) 9.44852 1.15432 0.577160 0.816631i \(-0.304161\pi\)
0.577160 + 0.816631i \(0.304161\pi\)
\(68\) 14.3492 1.74009
\(69\) 0 0
\(70\) −1.29261 −0.154497
\(71\) −8.99093 −1.06703 −0.533513 0.845792i \(-0.679129\pi\)
−0.533513 + 0.845792i \(0.679129\pi\)
\(72\) 0 0
\(73\) 6.05655 0.708865 0.354433 0.935082i \(-0.384674\pi\)
0.354433 + 0.935082i \(0.384674\pi\)
\(74\) −0.735663 −0.0855191
\(75\) 0 0
\(76\) −5.70739 −0.654682
\(77\) −1.70739 −0.194575
\(78\) 0 0
\(79\) −8.05655 −0.906432 −0.453216 0.891401i \(-0.649723\pi\)
−0.453216 + 0.891401i \(0.649723\pi\)
\(80\) −6.02827 −0.673982
\(81\) 0 0
\(82\) −28.5333 −3.15098
\(83\) −1.54241 −0.169302 −0.0846508 0.996411i \(-0.526977\pi\)
−0.0846508 + 0.996411i \(0.526977\pi\)
\(84\) 0 0
\(85\) −3.32088 −0.360200
\(86\) −26.0192 −2.80572
\(87\) 0 0
\(88\) −19.3774 −2.06564
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 0.679116 0.0711906
\(92\) −17.8350 −1.85943
\(93\) 0 0
\(94\) −12.2270 −1.26112
\(95\) 1.32088 0.135520
\(96\) 0 0
\(97\) −12.2553 −1.24433 −0.622167 0.782885i \(-0.713747\pi\)
−0.622167 + 0.782885i \(0.713747\pi\)
\(98\) 16.9344 1.71063
\(99\) 0 0
\(100\) 4.32088 0.432088
\(101\) −11.6700 −1.16121 −0.580606 0.814184i \(-0.697184\pi\)
−0.580606 + 0.814184i \(0.697184\pi\)
\(102\) 0 0
\(103\) 0.292611 0.0288318 0.0144159 0.999896i \(-0.495411\pi\)
0.0144159 + 0.999896i \(0.495411\pi\)
\(104\) 7.70739 0.755772
\(105\) 0 0
\(106\) −12.6418 −1.22788
\(107\) −1.87237 −0.181009 −0.0905043 0.995896i \(-0.528848\pi\)
−0.0905043 + 0.995896i \(0.528848\pi\)
\(108\) 0 0
\(109\) 5.54787 0.531390 0.265695 0.964057i \(-0.414399\pi\)
0.265695 + 0.964057i \(0.414399\pi\)
\(110\) 8.34916 0.796061
\(111\) 0 0
\(112\) −3.09936 −0.292862
\(113\) −7.80128 −0.733883 −0.366942 0.930244i \(-0.619595\pi\)
−0.366942 + 0.930244i \(0.619595\pi\)
\(114\) 0 0
\(115\) 4.12763 0.384904
\(116\) 5.99093 0.556244
\(117\) 0 0
\(118\) −12.6418 −1.16377
\(119\) −1.70739 −0.156516
\(120\) 0 0
\(121\) 0.0282739 0.00257035
\(122\) −18.4768 −1.67281
\(123\) 0 0
\(124\) 37.7458 3.38967
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 17.8916 1.58762 0.793810 0.608166i \(-0.208094\pi\)
0.793810 + 0.608166i \(0.208094\pi\)
\(128\) 15.2498 1.34790
\(129\) 0 0
\(130\) −3.32088 −0.291261
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 0.679116 0.0588868
\(134\) −23.7549 −2.05211
\(135\) 0 0
\(136\) −19.3774 −1.66160
\(137\) −5.67004 −0.484424 −0.242212 0.970223i \(-0.577873\pi\)
−0.242212 + 0.970223i \(0.577873\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 2.22153 0.187753
\(141\) 0 0
\(142\) 22.6044 1.89692
\(143\) −4.38650 −0.366818
\(144\) 0 0
\(145\) −1.38650 −0.115143
\(146\) −15.2270 −1.26019
\(147\) 0 0
\(148\) 1.26434 0.103928
\(149\) −17.6610 −1.44684 −0.723422 0.690407i \(-0.757432\pi\)
−0.723422 + 0.690407i \(0.757432\pi\)
\(150\) 0 0
\(151\) 1.26434 0.102890 0.0514451 0.998676i \(-0.483617\pi\)
0.0514451 + 0.998676i \(0.483617\pi\)
\(152\) 7.70739 0.625152
\(153\) 0 0
\(154\) 4.29261 0.345908
\(155\) −8.73566 −0.701665
\(156\) 0 0
\(157\) −15.6700 −1.25061 −0.625303 0.780382i \(-0.715025\pi\)
−0.625303 + 0.780382i \(0.715025\pi\)
\(158\) 20.2553 1.61142
\(159\) 0 0
\(160\) 3.48586 0.275582
\(161\) 2.12217 0.167250
\(162\) 0 0
\(163\) 15.7074 1.23030 0.615149 0.788411i \(-0.289096\pi\)
0.615149 + 0.788411i \(0.289096\pi\)
\(164\) 49.0384 3.82926
\(165\) 0 0
\(166\) 3.87783 0.300978
\(167\) −6.16498 −0.477060 −0.238530 0.971135i \(-0.576666\pi\)
−0.238530 + 0.971135i \(0.576666\pi\)
\(168\) 0 0
\(169\) −11.2553 −0.865790
\(170\) 8.34916 0.640351
\(171\) 0 0
\(172\) 44.7175 3.40968
\(173\) −8.58522 −0.652722 −0.326361 0.945245i \(-0.605823\pi\)
−0.326361 + 0.945245i \(0.605823\pi\)
\(174\) 0 0
\(175\) −0.514137 −0.0388651
\(176\) 20.0192 1.50900
\(177\) 0 0
\(178\) −7.54241 −0.565328
\(179\) 1.06562 0.0796482 0.0398241 0.999207i \(-0.487320\pi\)
0.0398241 + 0.999207i \(0.487320\pi\)
\(180\) 0 0
\(181\) −12.6700 −0.941757 −0.470878 0.882198i \(-0.656063\pi\)
−0.470878 + 0.882198i \(0.656063\pi\)
\(182\) −1.70739 −0.126560
\(183\) 0 0
\(184\) 24.0848 1.77556
\(185\) −0.292611 −0.0215132
\(186\) 0 0
\(187\) 11.0283 0.806467
\(188\) 21.0137 1.53258
\(189\) 0 0
\(190\) −3.32088 −0.240922
\(191\) 16.9344 1.22533 0.612664 0.790343i \(-0.290098\pi\)
0.612664 + 0.790343i \(0.290098\pi\)
\(192\) 0 0
\(193\) 26.7175 1.92317 0.961585 0.274509i \(-0.0885153\pi\)
0.961585 + 0.274509i \(0.0885153\pi\)
\(194\) 30.8114 2.21213
\(195\) 0 0
\(196\) −29.1040 −2.07886
\(197\) 14.2553 1.01565 0.507823 0.861462i \(-0.330450\pi\)
0.507823 + 0.861462i \(0.330450\pi\)
\(198\) 0 0
\(199\) −24.6610 −1.74817 −0.874085 0.485773i \(-0.838538\pi\)
−0.874085 + 0.485773i \(0.838538\pi\)
\(200\) −5.83502 −0.412598
\(201\) 0 0
\(202\) 29.3401 2.06436
\(203\) −0.712853 −0.0500325
\(204\) 0 0
\(205\) −11.3492 −0.792660
\(206\) −0.735663 −0.0512561
\(207\) 0 0
\(208\) −7.96265 −0.552111
\(209\) −4.38650 −0.303421
\(210\) 0 0
\(211\) −5.37743 −0.370198 −0.185099 0.982720i \(-0.559261\pi\)
−0.185099 + 0.982720i \(0.559261\pi\)
\(212\) 21.7266 1.49219
\(213\) 0 0
\(214\) 4.70739 0.321791
\(215\) −10.3492 −0.705807
\(216\) 0 0
\(217\) −4.49133 −0.304891
\(218\) −13.9481 −0.944686
\(219\) 0 0
\(220\) −14.3492 −0.967420
\(221\) −4.38650 −0.295068
\(222\) 0 0
\(223\) 8.66458 0.580223 0.290112 0.956993i \(-0.406308\pi\)
0.290112 + 0.956993i \(0.406308\pi\)
\(224\) 1.79221 0.119747
\(225\) 0 0
\(226\) 19.6135 1.30467
\(227\) −3.32088 −0.220415 −0.110207 0.993909i \(-0.535152\pi\)
−0.110207 + 0.993909i \(0.535152\pi\)
\(228\) 0 0
\(229\) −25.3118 −1.67265 −0.836326 0.548233i \(-0.815301\pi\)
−0.836326 + 0.548233i \(0.815301\pi\)
\(230\) −10.3774 −0.684268
\(231\) 0 0
\(232\) −8.09029 −0.531153
\(233\) −27.6327 −1.81028 −0.905139 0.425116i \(-0.860233\pi\)
−0.905139 + 0.425116i \(0.860233\pi\)
\(234\) 0 0
\(235\) −4.86330 −0.317246
\(236\) 21.7266 1.41428
\(237\) 0 0
\(238\) 4.29261 0.278249
\(239\) 4.19872 0.271592 0.135796 0.990737i \(-0.456641\pi\)
0.135796 + 0.990737i \(0.456641\pi\)
\(240\) 0 0
\(241\) 3.60442 0.232181 0.116091 0.993239i \(-0.462964\pi\)
0.116091 + 0.993239i \(0.462964\pi\)
\(242\) −0.0710844 −0.00456948
\(243\) 0 0
\(244\) 31.7549 2.03290
\(245\) 6.73566 0.430326
\(246\) 0 0
\(247\) 1.74474 0.111015
\(248\) −50.9728 −3.23677
\(249\) 0 0
\(250\) 2.51414 0.159008
\(251\) 6.87783 0.434125 0.217062 0.976158i \(-0.430352\pi\)
0.217062 + 0.976158i \(0.430352\pi\)
\(252\) 0 0
\(253\) −13.7074 −0.861776
\(254\) −44.9819 −2.82241
\(255\) 0 0
\(256\) −31.7549 −1.98468
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −0.150442 −0.00934801
\(260\) 5.70739 0.353957
\(261\) 0 0
\(262\) −15.0848 −0.931943
\(263\) −6.23606 −0.384532 −0.192266 0.981343i \(-0.561584\pi\)
−0.192266 + 0.981343i \(0.561584\pi\)
\(264\) 0 0
\(265\) −5.02827 −0.308884
\(266\) −1.70739 −0.104687
\(267\) 0 0
\(268\) 40.8259 2.49384
\(269\) −9.92345 −0.605044 −0.302522 0.953142i \(-0.597828\pi\)
−0.302522 + 0.953142i \(0.597828\pi\)
\(270\) 0 0
\(271\) 6.60442 0.401190 0.200595 0.979674i \(-0.435712\pi\)
0.200595 + 0.979674i \(0.435712\pi\)
\(272\) 20.0192 1.21384
\(273\) 0 0
\(274\) 14.2553 0.861192
\(275\) 3.32088 0.200257
\(276\) 0 0
\(277\) 22.6610 1.36157 0.680783 0.732485i \(-0.261640\pi\)
0.680783 + 0.732485i \(0.261640\pi\)
\(278\) −20.1131 −1.20630
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) 15.5479 0.927508 0.463754 0.885964i \(-0.346502\pi\)
0.463754 + 0.885964i \(0.346502\pi\)
\(282\) 0 0
\(283\) 0.645378 0.0383637 0.0191819 0.999816i \(-0.493894\pi\)
0.0191819 + 0.999816i \(0.493894\pi\)
\(284\) −38.8488 −2.30525
\(285\) 0 0
\(286\) 11.0283 0.652116
\(287\) −5.83502 −0.344430
\(288\) 0 0
\(289\) −5.97173 −0.351278
\(290\) 3.48586 0.204697
\(291\) 0 0
\(292\) 26.1696 1.53146
\(293\) 1.37743 0.0804704 0.0402352 0.999190i \(-0.487189\pi\)
0.0402352 + 0.999190i \(0.487189\pi\)
\(294\) 0 0
\(295\) −5.02827 −0.292757
\(296\) −1.70739 −0.0992400
\(297\) 0 0
\(298\) 44.4021 2.57214
\(299\) 5.45213 0.315305
\(300\) 0 0
\(301\) −5.32088 −0.306691
\(302\) −3.17872 −0.182915
\(303\) 0 0
\(304\) −7.96265 −0.456689
\(305\) −7.34916 −0.420812
\(306\) 0 0
\(307\) −7.98546 −0.455754 −0.227877 0.973690i \(-0.573178\pi\)
−0.227877 + 0.973690i \(0.573178\pi\)
\(308\) −7.37743 −0.420368
\(309\) 0 0
\(310\) 21.9627 1.24739
\(311\) −9.63270 −0.546220 −0.273110 0.961983i \(-0.588052\pi\)
−0.273110 + 0.961983i \(0.588052\pi\)
\(312\) 0 0
\(313\) −24.5369 −1.38691 −0.693455 0.720500i \(-0.743912\pi\)
−0.693455 + 0.720500i \(0.743912\pi\)
\(314\) 39.3966 2.22328
\(315\) 0 0
\(316\) −34.8114 −1.95829
\(317\) 20.3492 1.14292 0.571461 0.820629i \(-0.306377\pi\)
0.571461 + 0.820629i \(0.306377\pi\)
\(318\) 0 0
\(319\) 4.60442 0.257798
\(320\) 3.29261 0.184063
\(321\) 0 0
\(322\) −5.33542 −0.297331
\(323\) −4.38650 −0.244072
\(324\) 0 0
\(325\) −1.32088 −0.0732695
\(326\) −39.4905 −2.18718
\(327\) 0 0
\(328\) −66.2226 −3.65653
\(329\) −2.50040 −0.137851
\(330\) 0 0
\(331\) 16.4431 0.903792 0.451896 0.892071i \(-0.350748\pi\)
0.451896 + 0.892071i \(0.350748\pi\)
\(332\) −6.66458 −0.365766
\(333\) 0 0
\(334\) 15.4996 0.848100
\(335\) −9.44852 −0.516228
\(336\) 0 0
\(337\) 4.89703 0.266758 0.133379 0.991065i \(-0.457417\pi\)
0.133379 + 0.991065i \(0.457417\pi\)
\(338\) 28.2973 1.53917
\(339\) 0 0
\(340\) −14.3492 −0.778192
\(341\) 29.0101 1.57099
\(342\) 0 0
\(343\) 7.06201 0.381313
\(344\) −60.3876 −3.25588
\(345\) 0 0
\(346\) 21.5844 1.16039
\(347\) −22.2745 −1.19576 −0.597878 0.801587i \(-0.703989\pi\)
−0.597878 + 0.801587i \(0.703989\pi\)
\(348\) 0 0
\(349\) −2.94345 −0.157559 −0.0787797 0.996892i \(-0.525102\pi\)
−0.0787797 + 0.996892i \(0.525102\pi\)
\(350\) 1.29261 0.0690929
\(351\) 0 0
\(352\) −11.5761 −0.617011
\(353\) −18.8296 −1.00220 −0.501098 0.865390i \(-0.667070\pi\)
−0.501098 + 0.865390i \(0.667070\pi\)
\(354\) 0 0
\(355\) 8.99093 0.477189
\(356\) 12.9627 0.687019
\(357\) 0 0
\(358\) −2.67912 −0.141596
\(359\) 31.8770 1.68241 0.841203 0.540720i \(-0.181848\pi\)
0.841203 + 0.540720i \(0.181848\pi\)
\(360\) 0 0
\(361\) −17.2553 −0.908172
\(362\) 31.8542 1.67422
\(363\) 0 0
\(364\) 2.93438 0.153803
\(365\) −6.05655 −0.317014
\(366\) 0 0
\(367\) −18.3492 −0.957818 −0.478909 0.877864i \(-0.658968\pi\)
−0.478909 + 0.877864i \(0.658968\pi\)
\(368\) −24.8825 −1.29709
\(369\) 0 0
\(370\) 0.735663 0.0382453
\(371\) −2.58522 −0.134218
\(372\) 0 0
\(373\) −2.19872 −0.113845 −0.0569226 0.998379i \(-0.518129\pi\)
−0.0569226 + 0.998379i \(0.518129\pi\)
\(374\) −27.7266 −1.43371
\(375\) 0 0
\(376\) −28.3774 −1.46345
\(377\) −1.83141 −0.0943226
\(378\) 0 0
\(379\) 15.4713 0.794709 0.397354 0.917665i \(-0.369928\pi\)
0.397354 + 0.917665i \(0.369928\pi\)
\(380\) 5.70739 0.292783
\(381\) 0 0
\(382\) −42.5753 −2.17834
\(383\) −7.70739 −0.393829 −0.196915 0.980421i \(-0.563092\pi\)
−0.196915 + 0.980421i \(0.563092\pi\)
\(384\) 0 0
\(385\) 1.70739 0.0870166
\(386\) −67.1715 −3.41894
\(387\) 0 0
\(388\) −52.9536 −2.68831
\(389\) 24.6327 1.24893 0.624464 0.781054i \(-0.285318\pi\)
0.624464 + 0.781054i \(0.285318\pi\)
\(390\) 0 0
\(391\) −13.7074 −0.693212
\(392\) 39.3027 1.98509
\(393\) 0 0
\(394\) −35.8397 −1.80558
\(395\) 8.05655 0.405369
\(396\) 0 0
\(397\) −6.77301 −0.339928 −0.169964 0.985450i \(-0.554365\pi\)
−0.169964 + 0.985450i \(0.554365\pi\)
\(398\) 62.0011 3.10783
\(399\) 0 0
\(400\) 6.02827 0.301414
\(401\) 18.4996 0.923826 0.461913 0.886925i \(-0.347163\pi\)
0.461913 + 0.886925i \(0.347163\pi\)
\(402\) 0 0
\(403\) −11.5388 −0.574789
\(404\) −50.4249 −2.50873
\(405\) 0 0
\(406\) 1.79221 0.0889459
\(407\) 0.971726 0.0481667
\(408\) 0 0
\(409\) −13.4148 −0.663318 −0.331659 0.943399i \(-0.607608\pi\)
−0.331659 + 0.943399i \(0.607608\pi\)
\(410\) 28.5333 1.40916
\(411\) 0 0
\(412\) 1.26434 0.0622894
\(413\) −2.58522 −0.127210
\(414\) 0 0
\(415\) 1.54241 0.0757140
\(416\) 4.60442 0.225750
\(417\) 0 0
\(418\) 11.0283 0.539411
\(419\) 33.1150 1.61777 0.808886 0.587966i \(-0.200071\pi\)
0.808886 + 0.587966i \(0.200071\pi\)
\(420\) 0 0
\(421\) −14.6983 −0.716352 −0.358176 0.933654i \(-0.616601\pi\)
−0.358176 + 0.933654i \(0.616601\pi\)
\(422\) 13.5196 0.658124
\(423\) 0 0
\(424\) −29.3401 −1.42488
\(425\) 3.32088 0.161087
\(426\) 0 0
\(427\) −3.77847 −0.182853
\(428\) −8.09029 −0.391059
\(429\) 0 0
\(430\) 26.0192 1.25476
\(431\) 32.7549 1.57775 0.788873 0.614556i \(-0.210665\pi\)
0.788873 + 0.614556i \(0.210665\pi\)
\(432\) 0 0
\(433\) −11.8314 −0.568581 −0.284291 0.958738i \(-0.591758\pi\)
−0.284291 + 0.958738i \(0.591758\pi\)
\(434\) 11.2918 0.542024
\(435\) 0 0
\(436\) 23.9717 1.14804
\(437\) 5.45213 0.260811
\(438\) 0 0
\(439\) 8.31181 0.396701 0.198351 0.980131i \(-0.436442\pi\)
0.198351 + 0.980131i \(0.436442\pi\)
\(440\) 19.3774 0.923783
\(441\) 0 0
\(442\) 11.0283 0.524561
\(443\) 29.1751 1.38615 0.693076 0.720865i \(-0.256255\pi\)
0.693076 + 0.720865i \(0.256255\pi\)
\(444\) 0 0
\(445\) −3.00000 −0.142214
\(446\) −21.7839 −1.03150
\(447\) 0 0
\(448\) 1.69285 0.0799798
\(449\) −18.9717 −0.895331 −0.447666 0.894201i \(-0.647744\pi\)
−0.447666 + 0.894201i \(0.647744\pi\)
\(450\) 0 0
\(451\) 37.6892 1.77472
\(452\) −33.7084 −1.58551
\(453\) 0 0
\(454\) 8.34916 0.391845
\(455\) −0.679116 −0.0318374
\(456\) 0 0
\(457\) −23.2353 −1.08690 −0.543450 0.839442i \(-0.682882\pi\)
−0.543450 + 0.839442i \(0.682882\pi\)
\(458\) 63.6374 2.97358
\(459\) 0 0
\(460\) 17.8350 0.831562
\(461\) −4.42571 −0.206126 −0.103063 0.994675i \(-0.532864\pi\)
−0.103063 + 0.994675i \(0.532864\pi\)
\(462\) 0 0
\(463\) −19.5087 −0.906645 −0.453322 0.891347i \(-0.649761\pi\)
−0.453322 + 0.891347i \(0.649761\pi\)
\(464\) 8.35823 0.388021
\(465\) 0 0
\(466\) 69.4724 3.21825
\(467\) 24.5935 1.13805 0.569026 0.822320i \(-0.307321\pi\)
0.569026 + 0.822320i \(0.307321\pi\)
\(468\) 0 0
\(469\) −4.85783 −0.224314
\(470\) 12.2270 0.563989
\(471\) 0 0
\(472\) −29.3401 −1.35049
\(473\) 34.3684 1.58026
\(474\) 0 0
\(475\) −1.32088 −0.0606063
\(476\) −7.37743 −0.338144
\(477\) 0 0
\(478\) −10.5561 −0.482827
\(479\) −32.7549 −1.49661 −0.748304 0.663356i \(-0.769132\pi\)
−0.748304 + 0.663356i \(0.769132\pi\)
\(480\) 0 0
\(481\) −0.386505 −0.0176231
\(482\) −9.06201 −0.412763
\(483\) 0 0
\(484\) 0.122168 0.00555309
\(485\) 12.2553 0.556483
\(486\) 0 0
\(487\) −6.03735 −0.273578 −0.136789 0.990600i \(-0.543678\pi\)
−0.136789 + 0.990600i \(0.543678\pi\)
\(488\) −42.8825 −1.94120
\(489\) 0 0
\(490\) −16.9344 −0.765017
\(491\) −14.4431 −0.651806 −0.325903 0.945403i \(-0.605668\pi\)
−0.325903 + 0.945403i \(0.605668\pi\)
\(492\) 0 0
\(493\) 4.60442 0.207373
\(494\) −4.38650 −0.197358
\(495\) 0 0
\(496\) 52.6610 2.36455
\(497\) 4.62257 0.207351
\(498\) 0 0
\(499\) 20.9717 0.938823 0.469412 0.882979i \(-0.344466\pi\)
0.469412 + 0.882979i \(0.344466\pi\)
\(500\) −4.32088 −0.193236
\(501\) 0 0
\(502\) −17.2918 −0.771771
\(503\) 5.31728 0.237086 0.118543 0.992949i \(-0.462178\pi\)
0.118543 + 0.992949i \(0.462178\pi\)
\(504\) 0 0
\(505\) 11.6700 0.519310
\(506\) 34.4623 1.53203
\(507\) 0 0
\(508\) 77.3074 3.42996
\(509\) −18.2270 −0.807897 −0.403949 0.914782i \(-0.632363\pi\)
−0.403949 + 0.914782i \(0.632363\pi\)
\(510\) 0 0
\(511\) −3.11389 −0.137751
\(512\) 49.3365 2.18038
\(513\) 0 0
\(514\) −45.2545 −1.99609
\(515\) −0.292611 −0.0128940
\(516\) 0 0
\(517\) 16.1504 0.710296
\(518\) 0.378232 0.0166185
\(519\) 0 0
\(520\) −7.70739 −0.337991
\(521\) −40.1232 −1.75783 −0.878915 0.476978i \(-0.841732\pi\)
−0.878915 + 0.476978i \(0.841732\pi\)
\(522\) 0 0
\(523\) 18.9873 0.830257 0.415129 0.909763i \(-0.363737\pi\)
0.415129 + 0.909763i \(0.363737\pi\)
\(524\) 25.9253 1.13255
\(525\) 0 0
\(526\) 15.6783 0.683607
\(527\) 29.0101 1.26370
\(528\) 0 0
\(529\) −5.96265 −0.259246
\(530\) 12.6418 0.549123
\(531\) 0 0
\(532\) 2.93438 0.127221
\(533\) −14.9909 −0.649329
\(534\) 0 0
\(535\) 1.87237 0.0809495
\(536\) −55.1323 −2.38135
\(537\) 0 0
\(538\) 24.9489 1.07562
\(539\) −22.3684 −0.963473
\(540\) 0 0
\(541\) 16.5279 0.710589 0.355294 0.934754i \(-0.384381\pi\)
0.355294 + 0.934754i \(0.384381\pi\)
\(542\) −16.6044 −0.713221
\(543\) 0 0
\(544\) −11.5761 −0.496323
\(545\) −5.54787 −0.237645
\(546\) 0 0
\(547\) 17.6737 0.755671 0.377835 0.925873i \(-0.376669\pi\)
0.377835 + 0.925873i \(0.376669\pi\)
\(548\) −24.4996 −1.04657
\(549\) 0 0
\(550\) −8.34916 −0.356009
\(551\) −1.83141 −0.0780208
\(552\) 0 0
\(553\) 4.14217 0.176143
\(554\) −56.9728 −2.42054
\(555\) 0 0
\(556\) 34.5671 1.46597
\(557\) −17.3401 −0.734723 −0.367362 0.930078i \(-0.619739\pi\)
−0.367362 + 0.930078i \(0.619739\pi\)
\(558\) 0 0
\(559\) −13.6700 −0.578181
\(560\) 3.09936 0.130972
\(561\) 0 0
\(562\) −39.0895 −1.64889
\(563\) −12.9945 −0.547654 −0.273827 0.961779i \(-0.588290\pi\)
−0.273827 + 0.961779i \(0.588290\pi\)
\(564\) 0 0
\(565\) 7.80128 0.328202
\(566\) −1.62257 −0.0682016
\(567\) 0 0
\(568\) 52.4623 2.20127
\(569\) 16.6802 0.699269 0.349635 0.936886i \(-0.386306\pi\)
0.349635 + 0.936886i \(0.386306\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −18.9536 −0.792489
\(573\) 0 0
\(574\) 14.6700 0.612316
\(575\) −4.12763 −0.172134
\(576\) 0 0
\(577\) −23.5953 −0.982287 −0.491144 0.871079i \(-0.663421\pi\)
−0.491144 + 0.871079i \(0.663421\pi\)
\(578\) 15.0137 0.624489
\(579\) 0 0
\(580\) −5.99093 −0.248760
\(581\) 0.793010 0.0328996
\(582\) 0 0
\(583\) 16.6983 0.691574
\(584\) −35.3401 −1.46238
\(585\) 0 0
\(586\) −3.46305 −0.143057
\(587\) −28.1276 −1.16095 −0.580476 0.814277i \(-0.697133\pi\)
−0.580476 + 0.814277i \(0.697133\pi\)
\(588\) 0 0
\(589\) −11.5388 −0.475448
\(590\) 12.6418 0.520453
\(591\) 0 0
\(592\) 1.76394 0.0724974
\(593\) 9.17872 0.376925 0.188462 0.982080i \(-0.439650\pi\)
0.188462 + 0.982080i \(0.439650\pi\)
\(594\) 0 0
\(595\) 1.70739 0.0699961
\(596\) −76.3110 −3.12582
\(597\) 0 0
\(598\) −13.7074 −0.560537
\(599\) 31.4713 1.28588 0.642942 0.765915i \(-0.277714\pi\)
0.642942 + 0.765915i \(0.277714\pi\)
\(600\) 0 0
\(601\) −29.2654 −1.19376 −0.596880 0.802330i \(-0.703593\pi\)
−0.596880 + 0.802330i \(0.703593\pi\)
\(602\) 13.3774 0.545223
\(603\) 0 0
\(604\) 5.46305 0.222288
\(605\) −0.0282739 −0.00114950
\(606\) 0 0
\(607\) −44.2034 −1.79416 −0.897080 0.441868i \(-0.854316\pi\)
−0.897080 + 0.441868i \(0.854316\pi\)
\(608\) 4.60442 0.186734
\(609\) 0 0
\(610\) 18.4768 0.748103
\(611\) −6.42385 −0.259881
\(612\) 0 0
\(613\) −35.1715 −1.42056 −0.710282 0.703918i \(-0.751432\pi\)
−0.710282 + 0.703918i \(0.751432\pi\)
\(614\) 20.0765 0.810224
\(615\) 0 0
\(616\) 9.96265 0.401407
\(617\) 7.42571 0.298948 0.149474 0.988766i \(-0.452242\pi\)
0.149474 + 0.988766i \(0.452242\pi\)
\(618\) 0 0
\(619\) 8.54787 0.343568 0.171784 0.985135i \(-0.445047\pi\)
0.171784 + 0.985135i \(0.445047\pi\)
\(620\) −37.7458 −1.51591
\(621\) 0 0
\(622\) 24.2179 0.971050
\(623\) −1.54241 −0.0617954
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 61.6892 2.46560
\(627\) 0 0
\(628\) −67.7084 −2.70186
\(629\) 0.971726 0.0387453
\(630\) 0 0
\(631\) −2.36836 −0.0942829 −0.0471415 0.998888i \(-0.515011\pi\)
−0.0471415 + 0.998888i \(0.515011\pi\)
\(632\) 47.0101 1.86996
\(633\) 0 0
\(634\) −51.1606 −2.03185
\(635\) −17.8916 −0.710005
\(636\) 0 0
\(637\) 8.89703 0.352513
\(638\) −11.5761 −0.458304
\(639\) 0 0
\(640\) −15.2498 −0.602801
\(641\) 0.133096 0.00525698 0.00262849 0.999997i \(-0.499163\pi\)
0.00262849 + 0.999997i \(0.499163\pi\)
\(642\) 0 0
\(643\) −22.6464 −0.893088 −0.446544 0.894762i \(-0.647345\pi\)
−0.446544 + 0.894762i \(0.647345\pi\)
\(644\) 9.16964 0.361335
\(645\) 0 0
\(646\) 11.0283 0.433902
\(647\) −46.3912 −1.82383 −0.911913 0.410385i \(-0.865394\pi\)
−0.911913 + 0.410385i \(0.865394\pi\)
\(648\) 0 0
\(649\) 16.6983 0.655466
\(650\) 3.32088 0.130256
\(651\) 0 0
\(652\) 67.8698 2.65799
\(653\) −36.4057 −1.42467 −0.712333 0.701842i \(-0.752361\pi\)
−0.712333 + 0.701842i \(0.752361\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) 68.4158 2.67119
\(657\) 0 0
\(658\) 6.28635 0.245067
\(659\) −19.1414 −0.745642 −0.372821 0.927903i \(-0.621609\pi\)
−0.372821 + 0.927903i \(0.621609\pi\)
\(660\) 0 0
\(661\) 39.9072 1.55221 0.776104 0.630605i \(-0.217193\pi\)
0.776104 + 0.630605i \(0.217193\pi\)
\(662\) −41.3401 −1.60673
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) −0.679116 −0.0263350
\(666\) 0 0
\(667\) −5.72298 −0.221595
\(668\) −26.6382 −1.03066
\(669\) 0 0
\(670\) 23.7549 0.917730
\(671\) 24.4057 0.942172
\(672\) 0 0
\(673\) 23.6508 0.911673 0.455836 0.890064i \(-0.349340\pi\)
0.455836 + 0.890064i \(0.349340\pi\)
\(674\) −12.3118 −0.474233
\(675\) 0 0
\(676\) −48.6327 −1.87049
\(677\) 14.8031 0.568931 0.284465 0.958686i \(-0.408184\pi\)
0.284465 + 0.958686i \(0.408184\pi\)
\(678\) 0 0
\(679\) 6.30088 0.241806
\(680\) 19.3774 0.743091
\(681\) 0 0
\(682\) −72.9354 −2.79284
\(683\) 4.95252 0.189503 0.0947515 0.995501i \(-0.469794\pi\)
0.0947515 + 0.995501i \(0.469794\pi\)
\(684\) 0 0
\(685\) 5.67004 0.216641
\(686\) −17.7549 −0.677884
\(687\) 0 0
\(688\) 62.3876 2.37850
\(689\) −6.64177 −0.253031
\(690\) 0 0
\(691\) −19.2088 −0.730739 −0.365369 0.930863i \(-0.619057\pi\)
−0.365369 + 0.930863i \(0.619057\pi\)
\(692\) −37.0957 −1.41017
\(693\) 0 0
\(694\) 56.0011 2.12577
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) 37.6892 1.42758
\(698\) 7.40024 0.280103
\(699\) 0 0
\(700\) −2.22153 −0.0839658
\(701\) 29.3492 1.10850 0.554251 0.832349i \(-0.313005\pi\)
0.554251 + 0.832349i \(0.313005\pi\)
\(702\) 0 0
\(703\) −0.386505 −0.0145773
\(704\) −10.9344 −0.412105
\(705\) 0 0
\(706\) 47.3401 1.78167
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 38.7266 1.45441 0.727204 0.686422i \(-0.240820\pi\)
0.727204 + 0.686422i \(0.240820\pi\)
\(710\) −22.6044 −0.848329
\(711\) 0 0
\(712\) −17.5051 −0.656030
\(713\) −36.0576 −1.35037
\(714\) 0 0
\(715\) 4.38650 0.164046
\(716\) 4.60442 0.172075
\(717\) 0 0
\(718\) −80.1432 −2.99092
\(719\) 15.0848 0.562569 0.281284 0.959624i \(-0.409240\pi\)
0.281284 + 0.959624i \(0.409240\pi\)
\(720\) 0 0
\(721\) −0.150442 −0.00560275
\(722\) 43.3821 1.61451
\(723\) 0 0
\(724\) −54.7458 −2.03461
\(725\) 1.38650 0.0514935
\(726\) 0 0
\(727\) 12.3455 0.457871 0.228936 0.973442i \(-0.426475\pi\)
0.228936 + 0.973442i \(0.426475\pi\)
\(728\) −3.96265 −0.146866
\(729\) 0 0
\(730\) 15.2270 0.563576
\(731\) 34.3684 1.27116
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 46.1323 1.70277
\(735\) 0 0
\(736\) 14.3884 0.530362
\(737\) 31.3774 1.15580
\(738\) 0 0
\(739\) 29.7266 1.09351 0.546755 0.837293i \(-0.315863\pi\)
0.546755 + 0.837293i \(0.315863\pi\)
\(740\) −1.26434 −0.0464779
\(741\) 0 0
\(742\) 6.49960 0.238608
\(743\) −48.3648 −1.77433 −0.887165 0.461452i \(-0.847329\pi\)
−0.887165 + 0.461452i \(0.847329\pi\)
\(744\) 0 0
\(745\) 17.6610 0.647048
\(746\) 5.52787 0.202390
\(747\) 0 0
\(748\) 47.6519 1.74233
\(749\) 0.962653 0.0351746
\(750\) 0 0
\(751\) −31.8205 −1.16115 −0.580573 0.814208i \(-0.697171\pi\)
−0.580573 + 0.814208i \(0.697171\pi\)
\(752\) 29.3173 1.06909
\(753\) 0 0
\(754\) 4.60442 0.167683
\(755\) −1.26434 −0.0460139
\(756\) 0 0
\(757\) 4.94531 0.179740 0.0898701 0.995953i \(-0.471355\pi\)
0.0898701 + 0.995953i \(0.471355\pi\)
\(758\) −38.8970 −1.41280
\(759\) 0 0
\(760\) −7.70739 −0.279576
\(761\) −35.4249 −1.28415 −0.642076 0.766641i \(-0.721927\pi\)
−0.642076 + 0.766641i \(0.721927\pi\)
\(762\) 0 0
\(763\) −2.85237 −0.103263
\(764\) 73.1715 2.64725
\(765\) 0 0
\(766\) 19.3774 0.700135
\(767\) −6.64177 −0.239820
\(768\) 0 0
\(769\) 49.4249 1.78231 0.891154 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(770\) −4.29261 −0.154695
\(771\) 0 0
\(772\) 115.443 4.15490
\(773\) −12.6599 −0.455345 −0.227673 0.973738i \(-0.573112\pi\)
−0.227673 + 0.973738i \(0.573112\pi\)
\(774\) 0 0
\(775\) 8.73566 0.313794
\(776\) 71.5097 2.56705
\(777\) 0 0
\(778\) −61.9300 −2.22030
\(779\) −14.9909 −0.537106
\(780\) 0 0
\(781\) −29.8578 −1.06840
\(782\) 34.4623 1.23237
\(783\) 0 0
\(784\) −40.6044 −1.45016
\(785\) 15.6700 0.559288
\(786\) 0 0
\(787\) 30.9344 1.10269 0.551346 0.834277i \(-0.314115\pi\)
0.551346 + 0.834277i \(0.314115\pi\)
\(788\) 61.5953 2.19424
\(789\) 0 0
\(790\) −20.2553 −0.720650
\(791\) 4.01093 0.142612
\(792\) 0 0
\(793\) −9.70739 −0.344720
\(794\) 17.0283 0.604311
\(795\) 0 0
\(796\) −106.557 −3.77682
\(797\) 30.5935 1.08368 0.541839 0.840483i \(-0.317728\pi\)
0.541839 + 0.840483i \(0.317728\pi\)
\(798\) 0 0
\(799\) 16.1504 0.571362
\(800\) −3.48586 −0.123244
\(801\) 0 0
\(802\) −46.5105 −1.64234
\(803\) 20.1131 0.709776
\(804\) 0 0
\(805\) −2.12217 −0.0747966
\(806\) 29.0101 1.02184
\(807\) 0 0
\(808\) 68.0950 2.39557
\(809\) 2.89703 0.101854 0.0509271 0.998702i \(-0.483782\pi\)
0.0509271 + 0.998702i \(0.483782\pi\)
\(810\) 0 0
\(811\) −14.8861 −0.522722 −0.261361 0.965241i \(-0.584171\pi\)
−0.261361 + 0.965241i \(0.584171\pi\)
\(812\) −3.08016 −0.108092
\(813\) 0 0
\(814\) −2.44305 −0.0856289
\(815\) −15.7074 −0.550206
\(816\) 0 0
\(817\) −13.6700 −0.478254
\(818\) 33.7266 1.17922
\(819\) 0 0
\(820\) −49.0384 −1.71250
\(821\) 8.95173 0.312417 0.156209 0.987724i \(-0.450073\pi\)
0.156209 + 0.987724i \(0.450073\pi\)
\(822\) 0 0
\(823\) 2.99454 0.104383 0.0521915 0.998637i \(-0.483379\pi\)
0.0521915 + 0.998637i \(0.483379\pi\)
\(824\) −1.70739 −0.0594797
\(825\) 0 0
\(826\) 6.49960 0.226150
\(827\) −31.9663 −1.11158 −0.555788 0.831324i \(-0.687583\pi\)
−0.555788 + 0.831324i \(0.687583\pi\)
\(828\) 0 0
\(829\) 22.7458 0.789994 0.394997 0.918682i \(-0.370746\pi\)
0.394997 + 0.918682i \(0.370746\pi\)
\(830\) −3.87783 −0.134602
\(831\) 0 0
\(832\) 4.34916 0.150780
\(833\) −22.3684 −0.775018
\(834\) 0 0
\(835\) 6.16498 0.213348
\(836\) −18.9536 −0.655523
\(837\) 0 0
\(838\) −83.2555 −2.87601
\(839\) 23.2643 0.803174 0.401587 0.915821i \(-0.368459\pi\)
0.401587 + 0.915821i \(0.368459\pi\)
\(840\) 0 0
\(841\) −27.0776 −0.933710
\(842\) 36.9536 1.27350
\(843\) 0 0
\(844\) −23.2353 −0.799791
\(845\) 11.2553 0.387193
\(846\) 0 0
\(847\) −0.0145366 −0.000499485 0
\(848\) 30.3118 1.04091
\(849\) 0 0
\(850\) −8.34916 −0.286374
\(851\) −1.20779 −0.0414025
\(852\) 0 0
\(853\) 10.9909 0.376322 0.188161 0.982138i \(-0.439747\pi\)
0.188161 + 0.982138i \(0.439747\pi\)
\(854\) 9.49960 0.325070
\(855\) 0 0
\(856\) 10.9253 0.373419
\(857\) 16.1504 0.551689 0.275844 0.961202i \(-0.411043\pi\)
0.275844 + 0.961202i \(0.411043\pi\)
\(858\) 0 0
\(859\) 28.5188 0.973049 0.486524 0.873667i \(-0.338264\pi\)
0.486524 + 0.873667i \(0.338264\pi\)
\(860\) −44.7175 −1.52485
\(861\) 0 0
\(862\) −82.3502 −2.80486
\(863\) −12.2890 −0.418322 −0.209161 0.977881i \(-0.567073\pi\)
−0.209161 + 0.977881i \(0.567073\pi\)
\(864\) 0 0
\(865\) 8.58522 0.291906
\(866\) 29.7458 1.01080
\(867\) 0 0
\(868\) −19.4065 −0.658700
\(869\) −26.7549 −0.907597
\(870\) 0 0
\(871\) −12.4804 −0.422882
\(872\) −32.3720 −1.09625
\(873\) 0 0
\(874\) −13.7074 −0.463659
\(875\) 0.514137 0.0173810
\(876\) 0 0
\(877\) −39.7002 −1.34058 −0.670290 0.742099i \(-0.733830\pi\)
−0.670290 + 0.742099i \(0.733830\pi\)
\(878\) −20.8970 −0.705241
\(879\) 0 0
\(880\) −20.0192 −0.674847
\(881\) −32.1040 −1.08161 −0.540806 0.841147i \(-0.681881\pi\)
−0.540806 + 0.841147i \(0.681881\pi\)
\(882\) 0 0
\(883\) 13.5051 0.454482 0.227241 0.973839i \(-0.427030\pi\)
0.227241 + 0.973839i \(0.427030\pi\)
\(884\) −18.9536 −0.637478
\(885\) 0 0
\(886\) −73.3502 −2.46425
\(887\) −35.1222 −1.17929 −0.589643 0.807664i \(-0.700732\pi\)
−0.589643 + 0.807664i \(0.700732\pi\)
\(888\) 0 0
\(889\) −9.19872 −0.308515
\(890\) 7.54241 0.252822
\(891\) 0 0
\(892\) 37.4386 1.25354
\(893\) −6.42385 −0.214966
\(894\) 0 0
\(895\) −1.06562 −0.0356198
\(896\) −7.84049 −0.261932
\(897\) 0 0
\(898\) 47.6975 1.59169
\(899\) 12.1120 0.403959
\(900\) 0 0
\(901\) 16.6983 0.556302
\(902\) −94.7559 −3.15503
\(903\) 0 0
\(904\) 45.5207 1.51399
\(905\) 12.6700 0.421166
\(906\) 0 0
\(907\) 15.1186 0.502004 0.251002 0.967987i \(-0.419240\pi\)
0.251002 + 0.967987i \(0.419240\pi\)
\(908\) −14.3492 −0.476194
\(909\) 0 0
\(910\) 1.70739 0.0565994
\(911\) −52.5561 −1.74126 −0.870631 0.491936i \(-0.836289\pi\)
−0.870631 + 0.491936i \(0.836289\pi\)
\(912\) 0 0
\(913\) −5.12217 −0.169519
\(914\) 58.4166 1.93225
\(915\) 0 0
\(916\) −109.369 −3.61367
\(917\) −3.08482 −0.101870
\(918\) 0 0
\(919\) −54.5489 −1.79940 −0.899702 0.436505i \(-0.856216\pi\)
−0.899702 + 0.436505i \(0.856216\pi\)
\(920\) −24.0848 −0.794053
\(921\) 0 0
\(922\) 11.1268 0.366443
\(923\) 11.8760 0.390903
\(924\) 0 0
\(925\) 0.292611 0.00962098
\(926\) 49.0475 1.61180
\(927\) 0 0
\(928\) −4.83317 −0.158656
\(929\) −20.3793 −0.668623 −0.334311 0.942463i \(-0.608504\pi\)
−0.334311 + 0.942463i \(0.608504\pi\)
\(930\) 0 0
\(931\) 8.89703 0.291588
\(932\) −119.398 −3.91100
\(933\) 0 0
\(934\) −61.8314 −2.02319
\(935\) −11.0283 −0.360663
\(936\) 0 0
\(937\) 49.1979 1.60723 0.803613 0.595152i \(-0.202908\pi\)
0.803613 + 0.595152i \(0.202908\pi\)
\(938\) 12.2133 0.398777
\(939\) 0 0
\(940\) −21.0137 −0.685393
\(941\) 23.2371 0.757508 0.378754 0.925497i \(-0.376353\pi\)
0.378754 + 0.925497i \(0.376353\pi\)
\(942\) 0 0
\(943\) −46.8452 −1.52549
\(944\) 30.3118 0.986565
\(945\) 0 0
\(946\) −86.4068 −2.80933
\(947\) −37.1642 −1.20767 −0.603837 0.797108i \(-0.706362\pi\)
−0.603837 + 0.797108i \(0.706362\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 3.32088 0.107744
\(951\) 0 0
\(952\) 9.96265 0.322891
\(953\) 23.5761 0.763706 0.381853 0.924223i \(-0.375286\pi\)
0.381853 + 0.924223i \(0.375286\pi\)
\(954\) 0 0
\(955\) −16.9344 −0.547984
\(956\) 18.1422 0.586760
\(957\) 0 0
\(958\) 82.3502 2.66061
\(959\) 2.91518 0.0941360
\(960\) 0 0
\(961\) 45.3118 1.46167
\(962\) 0.971726 0.0313297
\(963\) 0 0
\(964\) 15.5743 0.501614
\(965\) −26.7175 −0.860067
\(966\) 0 0
\(967\) 8.38290 0.269576 0.134788 0.990874i \(-0.456965\pi\)
0.134788 + 0.990874i \(0.456965\pi\)
\(968\) −0.164979 −0.00530261
\(969\) 0 0
\(970\) −30.8114 −0.989295
\(971\) −13.2078 −0.423858 −0.211929 0.977285i \(-0.567975\pi\)
−0.211929 + 0.977285i \(0.567975\pi\)
\(972\) 0 0
\(973\) −4.11310 −0.131860
\(974\) 15.1787 0.486357
\(975\) 0 0
\(976\) 44.3027 1.41810
\(977\) 14.3310 0.458490 0.229245 0.973369i \(-0.426374\pi\)
0.229245 + 0.973369i \(0.426374\pi\)
\(978\) 0 0
\(979\) 9.96265 0.318408
\(980\) 29.1040 0.929694
\(981\) 0 0
\(982\) 36.3118 1.15876
\(983\) 32.3082 1.03047 0.515236 0.857048i \(-0.327704\pi\)
0.515236 + 0.857048i \(0.327704\pi\)
\(984\) 0 0
\(985\) −14.2553 −0.454210
\(986\) −11.5761 −0.368660
\(987\) 0 0
\(988\) 7.53880 0.239841
\(989\) −42.7175 −1.35834
\(990\) 0 0
\(991\) −39.6700 −1.26016 −0.630080 0.776530i \(-0.716978\pi\)
−0.630080 + 0.776530i \(0.716978\pi\)
\(992\) −30.4513 −0.966831
\(993\) 0 0
\(994\) −11.6218 −0.368620
\(995\) 24.6610 0.781805
\(996\) 0 0
\(997\) 38.6874 1.22524 0.612621 0.790377i \(-0.290115\pi\)
0.612621 + 0.790377i \(0.290115\pi\)
\(998\) −52.7258 −1.66901
\(999\) 0 0
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 405.2.a.i.1.1 3
3.2 odd 2 405.2.a.j.1.3 3
4.3 odd 2 6480.2.a.bs.1.3 3
5.2 odd 4 2025.2.b.m.649.1 6
5.3 odd 4 2025.2.b.m.649.6 6
5.4 even 2 2025.2.a.o.1.3 3
9.2 odd 6 45.2.e.b.31.1 yes 6
9.4 even 3 135.2.e.b.46.3 6
9.5 odd 6 45.2.e.b.16.1 6
9.7 even 3 135.2.e.b.91.3 6
12.11 even 2 6480.2.a.bv.1.3 3
15.2 even 4 2025.2.b.l.649.6 6
15.8 even 4 2025.2.b.l.649.1 6
15.14 odd 2 2025.2.a.n.1.1 3
36.7 odd 6 2160.2.q.k.1441.1 6
36.11 even 6 720.2.q.i.481.1 6
36.23 even 6 720.2.q.i.241.1 6
36.31 odd 6 2160.2.q.k.721.1 6
45.2 even 12 225.2.k.b.49.6 12
45.4 even 6 675.2.e.b.451.1 6
45.7 odd 12 675.2.k.b.199.1 12
45.13 odd 12 675.2.k.b.424.1 12
45.14 odd 6 225.2.e.b.151.3 6
45.22 odd 12 675.2.k.b.424.6 12
45.23 even 12 225.2.k.b.124.6 12
45.29 odd 6 225.2.e.b.76.3 6
45.32 even 12 225.2.k.b.124.1 12
45.34 even 6 675.2.e.b.226.1 6
45.38 even 12 225.2.k.b.49.1 12
45.43 odd 12 675.2.k.b.199.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.1 6 9.5 odd 6
45.2.e.b.31.1 yes 6 9.2 odd 6
135.2.e.b.46.3 6 9.4 even 3
135.2.e.b.91.3 6 9.7 even 3
225.2.e.b.76.3 6 45.29 odd 6
225.2.e.b.151.3 6 45.14 odd 6
225.2.k.b.49.1 12 45.38 even 12
225.2.k.b.49.6 12 45.2 even 12
225.2.k.b.124.1 12 45.32 even 12
225.2.k.b.124.6 12 45.23 even 12
405.2.a.i.1.1 3 1.1 even 1 trivial
405.2.a.j.1.3 3 3.2 odd 2
675.2.e.b.226.1 6 45.34 even 6
675.2.e.b.451.1 6 45.4 even 6
675.2.k.b.199.1 12 45.7 odd 12
675.2.k.b.199.6 12 45.43 odd 12
675.2.k.b.424.1 12 45.13 odd 12
675.2.k.b.424.6 12 45.22 odd 12
720.2.q.i.241.1 6 36.23 even 6
720.2.q.i.481.1 6 36.11 even 6
2025.2.a.n.1.1 3 15.14 odd 2
2025.2.a.o.1.3 3 5.4 even 2
2025.2.b.l.649.1 6 15.8 even 4
2025.2.b.l.649.6 6 15.2 even 4
2025.2.b.m.649.1 6 5.2 odd 4
2025.2.b.m.649.6 6 5.3 odd 4
2160.2.q.k.721.1 6 36.31 odd 6
2160.2.q.k.1441.1 6 36.7 odd 6
6480.2.a.bs.1.3 3 4.3 odd 2
6480.2.a.bv.1.3 3 12.11 even 2