Properties

Label 2025.2.a.o.1.3
Level $2025$
Weight $2$
Character 2025.1
Self dual yes
Analytic conductor $16.170$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.1697064093\)
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 1.3
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 2025.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.51414 q^{2} +4.32088 q^{4} +0.514137 q^{7} +5.83502 q^{8} +O(q^{10})\) \(q+2.51414 q^{2} +4.32088 q^{4} +0.514137 q^{7} +5.83502 q^{8} +3.32088 q^{11} +1.32088 q^{13} +1.29261 q^{14} +6.02827 q^{16} -3.32088 q^{17} -1.32088 q^{19} +8.34916 q^{22} +4.12763 q^{23} +3.32088 q^{26} +2.22153 q^{28} +1.38650 q^{29} +8.73566 q^{31} +3.48586 q^{32} -8.34916 q^{34} -0.292611 q^{37} -3.32088 q^{38} +11.3492 q^{41} -10.3492 q^{43} +14.3492 q^{44} +10.3774 q^{46} -4.86330 q^{47} -6.73566 q^{49} +5.70739 q^{52} -5.02827 q^{53} +3.00000 q^{56} +3.48586 q^{58} +5.02827 q^{59} +7.34916 q^{61} +21.9627 q^{62} -3.29261 q^{64} -9.44852 q^{67} -14.3492 q^{68} -8.99093 q^{71} -6.05655 q^{73} -0.735663 q^{74} -5.70739 q^{76} +1.70739 q^{77} -8.05655 q^{79} +28.5333 q^{82} +1.54241 q^{83} -26.0192 q^{86} +19.3774 q^{88} +3.00000 q^{89} +0.679116 q^{91} +17.8350 q^{92} -12.2270 q^{94} +12.2553 q^{97} -16.9344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 5 q^{4} - 5 q^{7} + 3 q^{8} + 2 q^{11} - 4 q^{13} + 9 q^{14} + 5 q^{16} - 2 q^{17} + 4 q^{19} + 4 q^{22} + 3 q^{23} + 2 q^{26} - 5 q^{28} + 7 q^{29} + 8 q^{31} + 17 q^{32} - 4 q^{34} - 6 q^{37} - 2 q^{38} + 13 q^{41} - 10 q^{43} + 22 q^{44} - 3 q^{46} + 13 q^{47} - 2 q^{49} + 12 q^{52} - 2 q^{53} + 9 q^{56} + 17 q^{58} + 2 q^{59} + q^{61} + 42 q^{62} - 15 q^{64} - 11 q^{67} - 22 q^{68} + 10 q^{71} + 8 q^{73} + 16 q^{74} - 12 q^{76} + 2 q^{79} + 29 q^{82} - 15 q^{83} - 28 q^{86} + 24 q^{88} + 9 q^{89} + 10 q^{91} + 39 q^{92} - 31 q^{94} + 18 q^{97} - 40 q^{98}+O(q^{100}) \) 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 \(-0.151483\pi\)
0.888882 + 0.458137i \(0.151483\pi\)
\(3\) 0 0
\(4\) 4.32088 2.16044
\(5\) 0 0
\(6\) 0 0
\(7\) 0.514137 0.194325 0.0971627 0.995269i \(-0.469023\pi\)
0.0971627 + 0.995269i \(0.469023\pi\)
\(8\) 5.83502 2.06299
\(9\) 0 0
\(10\) 0 0
\(11\) 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 \(-0.441363\pi\)
0.183174 + 0.983081i \(0.441363\pi\)
\(14\) 1.29261 0.345465
\(15\) 0 0
\(16\) 6.02827 1.50707
\(17\) −3.32088 −0.805433 −0.402716 0.915325i \(-0.631934\pi\)
−0.402716 + 0.915325i \(0.631934\pi\)
\(18\) 0 0
\(19\) −1.32088 −0.303032 −0.151516 0.988455i \(-0.548415\pi\)
−0.151516 + 0.988455i \(0.548415\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 8.34916 1.78005
\(23\) 4.12763 0.860671 0.430335 0.902669i \(-0.358395\pi\)
0.430335 + 0.902669i \(0.358395\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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 0
\(36\) 0 0
\(37\) −0.292611 −0.0481049 −0.0240524 0.999711i \(-0.507657\pi\)
−0.0240524 + 0.999711i \(0.507657\pi\)
\(38\) −3.32088 −0.538719
\(39\) 0 0
\(40\) 0 0
\(41\) 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 \(-0.789461\pi\)
−0.789116 + 0.614244i \(0.789461\pi\)
\(44\) 14.3492 2.16322
\(45\) 0 0
\(46\) 10.3774 1.53007
\(47\) −4.86330 −0.709385 −0.354692 0.934983i \(-0.615414\pi\)
−0.354692 + 0.934983i \(0.615414\pi\)
\(48\) 0 0
\(49\) −6.73566 −0.962238
\(50\) 0 0
\(51\) 0 0
\(52\) 5.70739 0.791472
\(53\) −5.02827 −0.690687 −0.345343 0.938476i \(-0.612238\pi\)
−0.345343 + 0.938476i \(0.612238\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(65\) 0 0
\(66\) 0 0
\(67\) −9.44852 −1.15432 −0.577160 0.816631i \(-0.695839\pi\)
−0.577160 + 0.816631i \(0.695839\pi\)
\(68\) −14.3492 −1.74009
\(69\) 0 0
\(70\) 0 0
\(71\) −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 \(-0.615326\pi\)
−0.354433 + 0.935082i \(0.615326\pi\)
\(74\) −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 \(0.649723\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 28.5333 3.15098
\(83\) 1.54241 0.169302 0.0846508 0.996411i \(-0.473023\pi\)
0.0846508 + 0.996411i \(0.473023\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −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
\(95\) 0 0
\(96\) 0 0
\(97\) 12.2553 1.24433 0.622167 0.782885i \(-0.286253\pi\)
0.622167 + 0.782885i \(0.286253\pi\)
\(98\) −16.9344 −1.71063
\(99\) 0 0
\(100\) 0 0
\(101\) −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 \(-0.504589\pi\)
−0.0144159 + 0.999896i \(0.504589\pi\)
\(104\) 7.70739 0.755772
\(105\) 0 0
\(106\) −12.6418 −1.22788
\(107\) 1.87237 0.181009 0.0905043 0.995896i \(-0.471152\pi\)
0.0905043 + 0.995896i \(0.471152\pi\)
\(108\) 0 0
\(109\) 5.54787 0.531390 0.265695 0.964057i \(-0.414399\pi\)
0.265695 + 0.964057i \(0.414399\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.09936 0.292862
\(113\) 7.80128 0.733883 0.366942 0.930244i \(-0.380405\pi\)
0.366942 + 0.930244i \(0.380405\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) −17.8916 −1.58762 −0.793810 0.608166i \(-0.791906\pi\)
−0.793810 + 0.608166i \(0.791906\pi\)
\(128\) −15.2498 −1.34790
\(129\) 0 0
\(130\) 0 0
\(131\) 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 \(-0.422127\pi\)
0.242212 + 0.970223i \(0.422127\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −22.6044 −1.89692
\(143\) 4.38650 0.366818
\(144\) 0 0
\(145\) 0 0
\(146\) −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
\(155\) 0 0
\(156\) 0 0
\(157\) 15.6700 1.25061 0.625303 0.780382i \(-0.284975\pi\)
0.625303 + 0.780382i \(0.284975\pi\)
\(158\) −20.2553 −1.61142
\(159\) 0 0
\(160\) 0 0
\(161\) 2.12217 0.167250
\(162\) 0 0
\(163\) −15.7074 −1.23030 −0.615149 0.788411i \(-0.710904\pi\)
−0.615149 + 0.788411i \(0.710904\pi\)
\(164\) 49.0384 3.82926
\(165\) 0 0
\(166\) 3.87783 0.300978
\(167\) 6.16498 0.477060 0.238530 0.971135i \(-0.423334\pi\)
0.238530 + 0.971135i \(0.423334\pi\)
\(168\) 0 0
\(169\) −11.2553 −0.865790
\(170\) 0 0
\(171\) 0 0
\(172\) −44.7175 −3.40968
\(173\) 8.58522 0.652722 0.326361 0.945245i \(-0.394177\pi\)
0.326361 + 0.945245i \(0.394177\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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 0
\(186\) 0 0
\(187\) −11.0283 −0.806467
\(188\) −21.0137 −1.53258
\(189\) 0 0
\(190\) 0 0
\(191\) 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 \(-0.911485\pi\)
−0.961585 + 0.274509i \(0.911485\pi\)
\(194\) 30.8114 2.21213
\(195\) 0 0
\(196\) −29.1040 −2.07886
\(197\) −14.2553 −1.01565 −0.507823 0.861462i \(-0.669550\pi\)
−0.507823 + 0.861462i \(0.669550\pi\)
\(198\) 0 0
\(199\) −24.6610 −1.74817 −0.874085 0.485773i \(-0.838538\pi\)
−0.874085 + 0.485773i \(0.838538\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −29.3401 −2.06436
\(203\) 0.712853 0.0500325
\(204\) 0 0
\(205\) 0 0
\(206\) −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
\(215\) 0 0
\(216\) 0 0
\(217\) 4.49133 0.304891
\(218\) 13.9481 0.944686
\(219\) 0 0
\(220\) 0 0
\(221\) −4.38650 −0.295068
\(222\) 0 0
\(223\) −8.66458 −0.580223 −0.290112 0.956993i \(-0.593692\pi\)
−0.290112 + 0.956993i \(0.593692\pi\)
\(224\) 1.79221 0.119747
\(225\) 0 0
\(226\) 19.6135 1.30467
\(227\) 3.32088 0.220415 0.110207 0.993909i \(-0.464848\pi\)
0.110207 + 0.993909i \(0.464848\pi\)
\(228\) 0 0
\(229\) −25.3118 −1.67265 −0.836326 0.548233i \(-0.815301\pi\)
−0.836326 + 0.548233i \(0.815301\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.09029 0.531153
\(233\) 27.6327 1.81028 0.905139 0.425116i \(-0.139767\pi\)
0.905139 + 0.425116i \(0.139767\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) −1.74474 −0.111015
\(248\) 50.9728 3.23677
\(249\) 0 0
\(250\) 0 0
\(251\) 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 \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −0.150442 −0.00934801
\(260\) 0 0
\(261\) 0 0
\(262\) 15.0848 0.931943
\(263\) 6.23606 0.384532 0.192266 0.981343i \(-0.438416\pi\)
0.192266 + 0.981343i \(0.438416\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −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
\(275\) 0 0
\(276\) 0 0
\(277\) −22.6610 −1.36157 −0.680783 0.732485i \(-0.738360\pi\)
−0.680783 + 0.732485i \(0.738360\pi\)
\(278\) 20.1131 1.20630
\(279\) 0 0
\(280\) 0 0
\(281\) 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 \(-0.506106\pi\)
−0.0191819 + 0.999816i \(0.506106\pi\)
\(284\) −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
\(290\) 0 0
\(291\) 0 0
\(292\) −26.1696 −1.53146
\(293\) −1.37743 −0.0804704 −0.0402352 0.999190i \(-0.512811\pi\)
−0.0402352 + 0.999190i \(0.512811\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −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
\(305\) 0 0
\(306\) 0 0
\(307\) 7.98546 0.455754 0.227877 0.973690i \(-0.426822\pi\)
0.227877 + 0.973690i \(0.426822\pi\)
\(308\) 7.37743 0.420368
\(309\) 0 0
\(310\) 0 0
\(311\) −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 \(-0.256088\pi\)
0.693455 + 0.720500i \(0.256088\pi\)
\(314\) 39.3966 2.22328
\(315\) 0 0
\(316\) −34.8114 −1.95829
\(317\) −20.3492 −1.14292 −0.571461 0.820629i \(-0.693623\pi\)
−0.571461 + 0.820629i \(0.693623\pi\)
\(318\) 0 0
\(319\) 4.60442 0.257798
\(320\) 0 0
\(321\) 0 0
\(322\) 5.33542 0.297331
\(323\) 4.38650 0.244072
\(324\) 0 0
\(325\) 0 0
\(326\) −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
\(335\) 0 0
\(336\) 0 0
\(337\) −4.89703 −0.266758 −0.133379 0.991065i \(-0.542583\pi\)
−0.133379 + 0.991065i \(0.542583\pi\)
\(338\) −28.2973 −1.53917
\(339\) 0 0
\(340\) 0 0
\(341\) 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 \(-0.296011\pi\)
0.597878 + 0.801587i \(0.296011\pi\)
\(348\) 0 0
\(349\) −2.94345 −0.157559 −0.0787797 0.996892i \(-0.525102\pi\)
−0.0787797 + 0.996892i \(0.525102\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 11.5761 0.617011
\(353\) 18.8296 1.00220 0.501098 0.865390i \(-0.332930\pi\)
0.501098 + 0.865390i \(0.332930\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) 18.3492 0.957818 0.478909 0.877864i \(-0.341032\pi\)
0.478909 + 0.877864i \(0.341032\pi\)
\(368\) 24.8825 1.29709
\(369\) 0 0
\(370\) 0 0
\(371\) −2.58522 −0.134218
\(372\) 0 0
\(373\) 2.19872 0.113845 0.0569226 0.998379i \(-0.481871\pi\)
0.0569226 + 0.998379i \(0.481871\pi\)
\(374\) −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 \(0.369928\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 42.5753 2.17834
\(383\) 7.70739 0.393829 0.196915 0.980421i \(-0.436908\pi\)
0.196915 + 0.980421i \(0.436908\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −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
\(395\) 0 0
\(396\) 0 0
\(397\) 6.77301 0.339928 0.169964 0.985450i \(-0.445635\pi\)
0.169964 + 0.985450i \(0.445635\pi\)
\(398\) −62.0011 −3.10783
\(399\) 0 0
\(400\) 0 0
\(401\) 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 \(0.607608\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.26434 −0.0622894
\(413\) 2.58522 0.127210
\(414\) 0 0
\(415\) 0 0
\(416\) 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
\(425\) 0 0
\(426\) 0 0
\(427\) 3.77847 0.182853
\(428\) 8.09029 0.391059
\(429\) 0 0
\(430\) 0 0
\(431\) 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 \(-0.408242\pi\)
0.284291 + 0.958738i \(0.408242\pi\)
\(434\) 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 \(0.436442\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −11.0283 −0.524561
\(443\) −29.1751 −1.38615 −0.693076 0.720865i \(-0.743745\pi\)
−0.693076 + 0.720865i \(0.743745\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −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 0
\(456\) 0 0
\(457\) 23.2353 1.08690 0.543450 0.839442i \(-0.317118\pi\)
0.543450 + 0.839442i \(0.317118\pi\)
\(458\) −63.6374 −2.97358
\(459\) 0 0
\(460\) 0 0
\(461\) −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 \(-0.350239\pi\)
0.453322 + 0.891347i \(0.350239\pi\)
\(464\) 8.35823 0.388021
\(465\) 0 0
\(466\) 69.4724 3.21825
\(467\) −24.5935 −1.13805 −0.569026 0.822320i \(-0.692679\pi\)
−0.569026 + 0.822320i \(0.692679\pi\)
\(468\) 0 0
\(469\) −4.85783 −0.224314
\(470\) 0 0
\(471\) 0 0
\(472\) 29.3401 1.35049
\(473\) −34.3684 −1.58026
\(474\) 0 0
\(475\) 0 0
\(476\) −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
\(485\) 0 0
\(486\) 0 0
\(487\) 6.03735 0.273578 0.136789 0.990600i \(-0.456322\pi\)
0.136789 + 0.990600i \(0.456322\pi\)
\(488\) 42.8825 1.94120
\(489\) 0 0
\(490\) 0 0
\(491\) −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 \(0.344466\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 17.2918 0.771771
\(503\) −5.31728 −0.237086 −0.118543 0.992949i \(-0.537822\pi\)
−0.118543 + 0.992949i \(0.537822\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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 0
\(516\) 0 0
\(517\) −16.1504 −0.710296
\(518\) −0.378232 −0.0166185
\(519\) 0 0
\(520\) 0 0
\(521\) −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 \(-0.636263\pi\)
−0.415129 + 0.909763i \(0.636263\pi\)
\(524\) 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
\(530\) 0 0
\(531\) 0 0
\(532\) −2.93438 −0.127221
\(533\) 14.9909 0.649329
\(534\) 0 0
\(535\) 0 0
\(536\) −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
\(545\) 0 0
\(546\) 0 0
\(547\) −17.6737 −0.755671 −0.377835 0.925873i \(-0.623331\pi\)
−0.377835 + 0.925873i \(0.623331\pi\)
\(548\) 24.4996 1.04657
\(549\) 0 0
\(550\) 0 0
\(551\) −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 \(-0.380261\pi\)
0.367362 + 0.930078i \(0.380261\pi\)
\(558\) 0 0
\(559\) −13.6700 −0.578181
\(560\) 0 0
\(561\) 0 0
\(562\) 39.0895 1.64889
\(563\) 12.9945 0.547654 0.273827 0.961779i \(-0.411710\pi\)
0.273827 + 0.961779i \(0.411710\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −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
\(575\) 0 0
\(576\) 0 0
\(577\) 23.5953 0.982287 0.491144 0.871079i \(-0.336579\pi\)
0.491144 + 0.871079i \(0.336579\pi\)
\(578\) −15.0137 −0.624489
\(579\) 0 0
\(580\) 0 0
\(581\) 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 \(-0.302867\pi\)
0.580476 + 0.814277i \(0.302867\pi\)
\(588\) 0 0
\(589\) −11.5388 −0.475448
\(590\) 0 0
\(591\) 0 0
\(592\) −1.76394 −0.0724974
\(593\) −9.17872 −0.376925 −0.188462 0.982080i \(-0.560350\pi\)
−0.188462 + 0.982080i \(0.560350\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −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 0
\(606\) 0 0
\(607\) 44.2034 1.79416 0.897080 0.441868i \(-0.145684\pi\)
0.897080 + 0.441868i \(0.145684\pi\)
\(608\) −4.60442 −0.186734
\(609\) 0 0
\(610\) 0 0
\(611\) −6.42385 −0.259881
\(612\) 0 0
\(613\) 35.1715 1.42056 0.710282 0.703918i \(-0.248568\pi\)
0.710282 + 0.703918i \(0.248568\pi\)
\(614\) 20.0765 0.810224
\(615\) 0 0
\(616\) 9.96265 0.401407
\(617\) −7.42571 −0.298948 −0.149474 0.988766i \(-0.547758\pi\)
−0.149474 + 0.988766i \(0.547758\pi\)
\(618\) 0 0
\(619\) 8.54787 0.343568 0.171784 0.985135i \(-0.445047\pi\)
0.171784 + 0.985135i \(0.445047\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.2179 −0.971050
\(623\) 1.54241 0.0617954
\(624\) 0 0
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) 0 0
\(637\) −8.89703 −0.352513
\(638\) 11.5761 0.458304
\(639\) 0 0
\(640\) 0 0
\(641\) 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 \(-0.352655\pi\)
0.446544 + 0.894762i \(0.352655\pi\)
\(644\) 9.16964 0.361335
\(645\) 0 0
\(646\) 11.0283 0.433902
\(647\) 46.3912 1.82383 0.911913 0.410385i \(-0.134606\pi\)
0.911913 + 0.410385i \(0.134606\pi\)
\(648\) 0 0
\(649\) 16.6983 0.655466
\(650\) 0 0
\(651\) 0 0
\(652\) −67.8698 −2.65799
\(653\) 36.4057 1.42467 0.712333 0.701842i \(-0.247639\pi\)
0.712333 + 0.701842i \(0.247639\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 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 0
\(666\) 0 0
\(667\) 5.72298 0.221595
\(668\) 26.6382 1.03066
\(669\) 0 0
\(670\) 0 0
\(671\) 24.4057 0.942172
\(672\) 0 0
\(673\) −23.6508 −0.911673 −0.455836 0.890064i \(-0.650660\pi\)
−0.455836 + 0.890064i \(0.650660\pi\)
\(674\) −12.3118 −0.474233
\(675\) 0 0
\(676\) −48.6327 −1.87049
\(677\) −14.8031 −0.568931 −0.284465 0.958686i \(-0.591816\pi\)
−0.284465 + 0.958686i \(0.591816\pi\)
\(678\) 0 0
\(679\) 6.30088 0.241806
\(680\) 0 0
\(681\) 0 0
\(682\) 72.9354 2.79284
\(683\) −4.95252 −0.189503 −0.0947515 0.995501i \(-0.530206\pi\)
−0.0947515 + 0.995501i \(0.530206\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −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
\(695\) 0 0
\(696\) 0 0
\(697\) −37.6892 −1.42758
\(698\) −7.40024 −0.280103
\(699\) 0 0
\(700\) 0 0
\(701\) 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 \(0.240820\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 17.5051 0.656030
\(713\) 36.0576 1.35037
\(714\) 0 0
\(715\) 0 0
\(716\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) −12.3455 −0.457871 −0.228936 0.973442i \(-0.573525\pi\)
−0.228936 + 0.973442i \(0.573525\pi\)
\(728\) 3.96265 0.146866
\(729\) 0 0
\(730\) 0 0
\(731\) 34.3684 1.27116
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 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 \(0.315863\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −6.49960 −0.238608
\(743\) 48.3648 1.77433 0.887165 0.461452i \(-0.152671\pi\)
0.887165 + 0.461452i \(0.152671\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) −4.94531 −0.179740 −0.0898701 0.995953i \(-0.528645\pi\)
−0.0898701 + 0.995953i \(0.528645\pi\)
\(758\) 38.8970 1.41280
\(759\) 0 0
\(760\) 0 0
\(761\) −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 \(0.149897\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −115.443 −4.15490
\(773\) 12.6599 0.455345 0.227673 0.973738i \(-0.426888\pi\)
0.227673 + 0.973738i \(0.426888\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) −30.9344 −1.10269 −0.551346 0.834277i \(-0.685885\pi\)
−0.551346 + 0.834277i \(0.685885\pi\)
\(788\) −61.5953 −2.19424
\(789\) 0 0
\(790\) 0 0
\(791\) 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 \(-0.682272\pi\)
−0.541839 + 0.840483i \(0.682272\pi\)
\(798\) 0 0
\(799\) 16.1504 0.571362
\(800\) 0 0
\(801\) 0 0
\(802\) 46.5105 1.64234
\(803\) −20.1131 −0.709776
\(804\) 0 0
\(805\) 0 0
\(806\) 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
\(815\) 0 0
\(816\) 0 0
\(817\) 13.6700 0.478254
\(818\) −33.7266 −1.17922
\(819\) 0 0
\(820\) 0 0
\(821\) 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 \(-0.516621\pi\)
−0.0521915 + 0.998637i \(0.516621\pi\)
\(824\) −1.70739 −0.0594797
\(825\) 0 0
\(826\) 6.49960 0.226150
\(827\) 31.9663 1.11158 0.555788 0.831324i \(-0.312417\pi\)
0.555788 + 0.831324i \(0.312417\pi\)
\(828\) 0 0
\(829\) 22.7458 0.789994 0.394997 0.918682i \(-0.370746\pi\)
0.394997 + 0.918682i \(0.370746\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.34916 −0.150780
\(833\) 22.3684 0.775018
\(834\) 0 0
\(835\) 0 0
\(836\) −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
\(845\) 0 0
\(846\) 0 0
\(847\) 0.0145366 0.000499485 0
\(848\) −30.3118 −1.04091
\(849\) 0 0
\(850\) 0 0
\(851\) −1.20779 −0.0414025
\(852\) 0 0
\(853\) −10.9909 −0.376322 −0.188161 0.982138i \(-0.560253\pi\)
−0.188161 + 0.982138i \(0.560253\pi\)
\(854\) 9.49960 0.325070
\(855\) 0 0
\(856\) 10.9253 0.373419
\(857\) −16.1504 −0.551689 −0.275844 0.961202i \(-0.588957\pi\)
−0.275844 + 0.961202i \(0.588957\pi\)
\(858\) 0 0
\(859\) 28.5188 0.973049 0.486524 0.873667i \(-0.338264\pi\)
0.486524 + 0.873667i \(0.338264\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 82.3502 2.80486
\(863\) 12.2890 0.418322 0.209161 0.977881i \(-0.432927\pi\)
0.209161 + 0.977881i \(0.432927\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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 0
\(876\) 0 0
\(877\) 39.7002 1.34058 0.670290 0.742099i \(-0.266170\pi\)
0.670290 + 0.742099i \(0.266170\pi\)
\(878\) 20.8970 0.705241
\(879\) 0 0
\(880\) 0 0
\(881\) −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 \(-0.572970\pi\)
−0.227241 + 0.973839i \(0.572970\pi\)
\(884\) −18.9536 −0.637478
\(885\) 0 0
\(886\) −73.3502 −2.46425
\(887\) 35.1222 1.17929 0.589643 0.807664i \(-0.299268\pi\)
0.589643 + 0.807664i \(0.299268\pi\)
\(888\) 0 0
\(889\) −9.19872 −0.308515
\(890\) 0 0
\(891\) 0 0
\(892\) −37.4386 −1.25354
\(893\) 6.42385 0.214966
\(894\) 0 0
\(895\) 0 0
\(896\) −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
\(905\) 0 0
\(906\) 0 0
\(907\) −15.1186 −0.502004 −0.251002 0.967987i \(-0.580760\pi\)
−0.251002 + 0.967987i \(0.580760\pi\)
\(908\) 14.3492 0.476194
\(909\) 0 0
\(910\) 0 0
\(911\) −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 \(0.856216\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −11.1268 −0.366443
\(923\) −11.8760 −0.390903
\(924\) 0 0
\(925\) 0 0
\(926\) 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
\(935\) 0 0
\(936\) 0 0
\(937\) −49.1979 −1.60723 −0.803613 0.595152i \(-0.797092\pi\)
−0.803613 + 0.595152i \(0.797092\pi\)
\(938\) −12.2133 −0.398777
\(939\) 0 0
\(940\) 0 0
\(941\) 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 \(-0.293638\pi\)
0.603837 + 0.797108i \(0.293638\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) −9.96265 −0.322891
\(953\) −23.5761 −0.763706 −0.381853 0.924223i \(-0.624714\pi\)
−0.381853 + 0.924223i \(0.624714\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) −8.38290 −0.269576 −0.134788 0.990874i \(-0.543035\pi\)
−0.134788 + 0.990874i \(0.543035\pi\)
\(968\) 0.164979 0.00530261
\(969\) 0 0
\(970\) 0 0
\(971\) −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 \(-0.573626\pi\)
−0.229245 + 0.973369i \(0.573626\pi\)
\(978\) 0 0
\(979\) 9.96265 0.318408
\(980\) 0 0
\(981\) 0 0
\(982\) −36.3118 −1.15876
\(983\) −32.3082 −1.03047 −0.515236 0.857048i \(-0.672296\pi\)
−0.515236 + 0.857048i \(0.672296\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −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
\(995\) 0 0
\(996\) 0 0
\(997\) −38.6874 −1.22524 −0.612621 0.790377i \(-0.709885\pi\)
−0.612621 + 0.790377i \(0.709885\pi\)
\(998\) 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 2025.2.a.o.1.3 3
3.2 odd 2 2025.2.a.n.1.1 3
5.2 odd 4 2025.2.b.m.649.6 6
5.3 odd 4 2025.2.b.m.649.1 6
5.4 even 2 405.2.a.i.1.1 3
9.2 odd 6 225.2.e.b.76.3 6
9.4 even 3 675.2.e.b.451.1 6
9.5 odd 6 225.2.e.b.151.3 6
9.7 even 3 675.2.e.b.226.1 6
15.2 even 4 2025.2.b.l.649.1 6
15.8 even 4 2025.2.b.l.649.6 6
15.14 odd 2 405.2.a.j.1.3 3
20.19 odd 2 6480.2.a.bs.1.3 3
45.2 even 12 225.2.k.b.49.1 12
45.4 even 6 135.2.e.b.46.3 6
45.7 odd 12 675.2.k.b.199.6 12
45.13 odd 12 675.2.k.b.424.6 12
45.14 odd 6 45.2.e.b.16.1 6
45.22 odd 12 675.2.k.b.424.1 12
45.23 even 12 225.2.k.b.124.1 12
45.29 odd 6 45.2.e.b.31.1 yes 6
45.32 even 12 225.2.k.b.124.6 12
45.34 even 6 135.2.e.b.91.3 6
45.38 even 12 225.2.k.b.49.6 12
45.43 odd 12 675.2.k.b.199.1 12
60.59 even 2 6480.2.a.bv.1.3 3
180.59 even 6 720.2.q.i.241.1 6
180.79 odd 6 2160.2.q.k.1441.1 6
180.119 even 6 720.2.q.i.481.1 6
180.139 odd 6 2160.2.q.k.721.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.1 6 45.14 odd 6
45.2.e.b.31.1 yes 6 45.29 odd 6
135.2.e.b.46.3 6 45.4 even 6
135.2.e.b.91.3 6 45.34 even 6
225.2.e.b.76.3 6 9.2 odd 6
225.2.e.b.151.3 6 9.5 odd 6
225.2.k.b.49.1 12 45.2 even 12
225.2.k.b.49.6 12 45.38 even 12
225.2.k.b.124.1 12 45.23 even 12
225.2.k.b.124.6 12 45.32 even 12
405.2.a.i.1.1 3 5.4 even 2
405.2.a.j.1.3 3 15.14 odd 2
675.2.e.b.226.1 6 9.7 even 3
675.2.e.b.451.1 6 9.4 even 3
675.2.k.b.199.1 12 45.43 odd 12
675.2.k.b.199.6 12 45.7 odd 12
675.2.k.b.424.1 12 45.22 odd 12
675.2.k.b.424.6 12 45.13 odd 12
720.2.q.i.241.1 6 180.59 even 6
720.2.q.i.481.1 6 180.119 even 6
2025.2.a.n.1.1 3 3.2 odd 2
2025.2.a.o.1.3 3 1.1 even 1 trivial
2025.2.b.l.649.1 6 15.2 even 4
2025.2.b.l.649.6 6 15.8 even 4
2025.2.b.m.649.1 6 5.3 odd 4
2025.2.b.m.649.6 6 5.2 odd 4
2160.2.q.k.721.1 6 180.139 odd 6
2160.2.q.k.1441.1 6 180.79 odd 6
6480.2.a.bs.1.3 3 20.19 odd 2
6480.2.a.bv.1.3 3 60.59 even 2