Properties

Label 2025.2.a.n.1.1
Level $2025$
Weight $2$
Character 2025.1
Self dual yes
Analytic conductor $16.170$
Analytic rank $1$
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: \(1\)
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 \(\chi\) \(=\) 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 \(-0.848517\pi\)
−0.888882 + 0.458137i \(0.848517\pi\)
\(3\) 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 \(-0.666903\pi\)
−0.500642 + 0.865654i \(0.666903\pi\)
\(12\) 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 \(-0.368066\pi\)
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 \(0.548415\pi\)
\(20\) 0 0
\(21\) 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
\(25\) 0 0
\(26\) −3.32088 −0.651279
\(27\) 0 0
\(28\) 2.22153 0.419829
\(29\) −1.38650 −0.257467 −0.128734 0.991679i \(-0.541091\pi\)
−0.128734 + 0.991679i \(0.541091\pi\)
\(30\) 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 \(-0.846678\pi\)
−0.886220 + 0.463264i \(0.846678\pi\)
\(42\) 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 \(-0.384586\pi\)
0.354692 + 0.934983i \(0.384586\pi\)
\(48\) 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 \(-0.387762\pi\)
0.345343 + 0.938476i \(0.387762\pi\)
\(54\) 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 \(-0.606143\pi\)
−0.327313 + 0.944916i \(0.606143\pi\)
\(60\) 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 \(-0.320871\pi\)
0.533513 + 0.845792i \(0.320871\pi\)
\(72\) 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 \(-0.526977\pi\)
−0.0846508 + 0.996411i \(0.526977\pi\)
\(84\) 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 \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 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 \(-0.302816\pi\)
0.580606 + 0.814184i \(0.302816\pi\)
\(102\) 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 \(-0.528848\pi\)
−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 \(0.414399\pi\)
\(110\) 0 0
\(111\) 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
\(115\) 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 \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 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 \(0.389818\pi\)
\(140\) 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 \(-0.242568\pi\)
0.723422 + 0.690407i \(0.242568\pi\)
\(150\) 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 \(-0.576666\pi\)
−0.238530 + 0.971135i \(0.576666\pi\)
\(168\) 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 \(-0.605823\pi\)
−0.326361 + 0.945245i \(0.605823\pi\)
\(174\) 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 \(-0.512680\pi\)
−0.0398241 + 0.999207i \(0.512680\pi\)
\(180\) 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 \(-0.709902\pi\)
−0.612664 + 0.790343i \(0.709902\pi\)
\(192\) 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 \(-0.330450\pi\)
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 \(0.838538\pi\)
\(200\) 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 \(-0.535152\pi\)
−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 \(0.815301\pi\)
\(230\) 0 0
\(231\) 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
\(235\) 0 0
\(236\) −21.7266 −1.41428
\(237\) 0 0
\(238\) −4.29261 −0.278249
\(239\) −4.19872 −0.271592 −0.135796 0.990737i \(-0.543359\pi\)
−0.135796 + 0.990737i \(0.543359\pi\)
\(240\) 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 \(-0.569648\pi\)
−0.217062 + 0.976158i \(0.569648\pi\)
\(252\) 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
\(260\) 0 0
\(261\) 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
\(265\) 0 0
\(266\) 1.70739 0.104687
\(267\) 0 0
\(268\) −40.8259 −2.49384
\(269\) 9.92345 0.605044 0.302522 0.953142i \(-0.402172\pi\)
0.302522 + 0.953142i \(0.402172\pi\)
\(270\) 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 \(-0.653498\pi\)
−0.463754 + 0.885964i \(0.653498\pi\)
\(282\) 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 \(-0.487189\pi\)
0.0402352 + 0.999190i \(0.487189\pi\)
\(294\) 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 \(-0.411948\pi\)
0.273110 + 0.961983i \(0.411948\pi\)
\(312\) 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 \(-0.306377\pi\)
0.571461 + 0.820629i \(0.306377\pi\)
\(318\) 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 \(-0.703989\pi\)
−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 \(0.525102\pi\)
\(350\) 0 0
\(351\) 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
\(355\) 0 0
\(356\) −12.9627 −0.687019
\(357\) 0 0
\(358\) 2.67912 0.141596
\(359\) −31.8770 −1.68241 −0.841203 0.540720i \(-0.818152\pi\)
−0.841203 + 0.540720i \(0.818152\pi\)
\(360\) 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 \(-0.563092\pi\)
−0.196915 + 0.980421i \(0.563092\pi\)
\(384\) 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 \(-0.714682\pi\)
−0.624464 + 0.781054i \(0.714682\pi\)
\(390\) 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 \(-0.652837\pi\)
−0.461913 + 0.886925i \(0.652837\pi\)
\(402\) 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 \(-0.799929\pi\)
−0.808886 + 0.587966i \(0.799929\pi\)
\(420\) 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 \(-0.789335\pi\)
−0.788873 + 0.614556i \(0.789335\pi\)
\(432\) 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 \(-0.256255\pi\)
0.693076 + 0.720865i \(0.256255\pi\)
\(444\) 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 \(-0.352256\pi\)
0.447666 + 0.894201i \(0.352256\pi\)
\(450\) 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 \(-0.467136\pi\)
0.103063 + 0.994675i \(0.467136\pi\)
\(462\) 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 \(-0.307321\pi\)
0.569026 + 0.822320i \(0.307321\pi\)
\(468\) 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 \(-0.230868\pi\)
0.748304 + 0.663356i \(0.230868\pi\)
\(480\) 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 \(-0.394332\pi\)
0.325903 + 0.945403i \(0.394332\pi\)
\(492\) 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 \(-0.462178\pi\)
0.118543 + 0.992949i \(0.462178\pi\)
\(504\) 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 \(-0.367637\pi\)
0.403949 + 0.914782i \(0.367637\pi\)
\(510\) 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 \(-0.158268\pi\)
0.878915 + 0.476978i \(0.158268\pi\)
\(522\) 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 \(-0.619739\pi\)
−0.367362 + 0.930078i \(0.619739\pi\)
\(558\) 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 \(-0.588290\pi\)
−0.273827 + 0.961779i \(0.588290\pi\)
\(564\) 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 \(-0.613694\pi\)
−0.349635 + 0.936886i \(0.613694\pi\)
\(570\) 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 \(-0.697133\pi\)
−0.580476 + 0.814277i \(0.697133\pi\)
\(588\) 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 \(-0.439650\pi\)
0.188462 + 0.982080i \(0.439650\pi\)
\(594\) 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 \(-0.722286\pi\)
−0.642942 + 0.765915i \(0.722286\pi\)
\(600\) 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 \(-0.452242\pi\)
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 \(0.445047\pi\)
\(620\) 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 \(-0.500837\pi\)
−0.00262849 + 0.999997i \(0.500837\pi\)
\(642\) 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 \(-0.865394\pi\)
−0.911913 + 0.410385i \(0.865394\pi\)
\(648\) 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 \(-0.752361\pi\)
−0.712333 + 0.701842i \(0.752361\pi\)
\(654\) 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 \(-0.378391\pi\)
0.372821 + 0.927903i \(0.378391\pi\)
\(660\) 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 \(-0.408184\pi\)
0.284465 + 0.958686i \(0.408184\pi\)
\(678\) 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 \(-0.469794\pi\)
0.0947515 + 0.995501i \(0.469794\pi\)
\(684\) 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 \(-0.686995\pi\)
−0.554251 + 0.832349i \(0.686995\pi\)
\(702\) 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 \(-0.590760\pi\)
−0.281284 + 0.959624i \(0.590760\pi\)
\(720\) 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 \(-0.847329\pi\)
−0.887165 + 0.461452i \(0.847329\pi\)
\(744\) 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 \(-0.278073\pi\)
0.642076 + 0.766641i \(0.278073\pi\)
\(762\) 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 \(-0.573112\pi\)
−0.227673 + 0.973738i \(0.573112\pi\)
\(774\) 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 \(-0.317728\pi\)
0.541839 + 0.840483i \(0.317728\pi\)
\(798\) 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 \(-0.516218\pi\)
−0.0509271 + 0.998702i \(0.516218\pi\)
\(810\) 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 \(-0.549927\pi\)
−0.156209 + 0.987724i \(0.549927\pi\)
\(822\) 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 \(-0.687583\pi\)
−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 \(0.370746\pi\)
\(830\) 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 \(-0.631541\pi\)
−0.401587 + 0.915821i \(0.631541\pi\)
\(840\) 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 \(-0.411043\pi\)
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 \(0.338264\pi\)
\(860\) 0 0
\(861\) 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
\(865\) 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 \(-0.318119\pi\)
0.540806 + 0.841147i \(0.318119\pi\)
\(882\) 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 \(-0.700732\pi\)
−0.589643 + 0.807664i \(0.700732\pi\)
\(888\) 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 \(-0.163711\pi\)
0.870631 + 0.491936i \(0.163711\pi\)
\(912\) 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 \(-0.391496\pi\)
0.334311 + 0.942463i \(0.391496\pi\)
\(930\) 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 \(-0.623647\pi\)
−0.378754 + 0.925497i \(0.623647\pi\)
\(942\) 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
\(950\) 0 0
\(951\) 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
\(955\) 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 \(-0.432025\pi\)
0.211929 + 0.977285i \(0.432025\pi\)
\(972\) 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
\(980\) 0 0
\(981\) 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
\(985\) 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.n.1.1 3
3.2 odd 2 2025.2.a.o.1.3 3
5.2 odd 4 2025.2.b.l.649.1 6
5.3 odd 4 2025.2.b.l.649.6 6
5.4 even 2 405.2.a.j.1.3 3
9.2 odd 6 675.2.e.b.226.1 6
9.4 even 3 225.2.e.b.151.3 6
9.5 odd 6 675.2.e.b.451.1 6
9.7 even 3 225.2.e.b.76.3 6
15.2 even 4 2025.2.b.m.649.6 6
15.8 even 4 2025.2.b.m.649.1 6
15.14 odd 2 405.2.a.i.1.1 3
20.19 odd 2 6480.2.a.bv.1.3 3
45.2 even 12 675.2.k.b.199.6 12
45.4 even 6 45.2.e.b.16.1 6
45.7 odd 12 225.2.k.b.49.1 12
45.13 odd 12 225.2.k.b.124.1 12
45.14 odd 6 135.2.e.b.46.3 6
45.22 odd 12 225.2.k.b.124.6 12
45.23 even 12 675.2.k.b.424.6 12
45.29 odd 6 135.2.e.b.91.3 6
45.32 even 12 675.2.k.b.424.1 12
45.34 even 6 45.2.e.b.31.1 yes 6
45.38 even 12 675.2.k.b.199.1 12
45.43 odd 12 225.2.k.b.49.6 12
60.59 even 2 6480.2.a.bs.1.3 3
180.59 even 6 2160.2.q.k.721.1 6
180.79 odd 6 720.2.q.i.481.1 6
180.119 even 6 2160.2.q.k.1441.1 6
180.139 odd 6 720.2.q.i.241.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.1 6 45.4 even 6
45.2.e.b.31.1 yes 6 45.34 even 6
135.2.e.b.46.3 6 45.14 odd 6
135.2.e.b.91.3 6 45.29 odd 6
225.2.e.b.76.3 6 9.7 even 3
225.2.e.b.151.3 6 9.4 even 3
225.2.k.b.49.1 12 45.7 odd 12
225.2.k.b.49.6 12 45.43 odd 12
225.2.k.b.124.1 12 45.13 odd 12
225.2.k.b.124.6 12 45.22 odd 12
405.2.a.i.1.1 3 15.14 odd 2
405.2.a.j.1.3 3 5.4 even 2
675.2.e.b.226.1 6 9.2 odd 6
675.2.e.b.451.1 6 9.5 odd 6
675.2.k.b.199.1 12 45.38 even 12
675.2.k.b.199.6 12 45.2 even 12
675.2.k.b.424.1 12 45.32 even 12
675.2.k.b.424.6 12 45.23 even 12
720.2.q.i.241.1 6 180.139 odd 6
720.2.q.i.481.1 6 180.79 odd 6
2025.2.a.n.1.1 3 1.1 even 1 trivial
2025.2.a.o.1.3 3 3.2 odd 2
2025.2.b.l.649.1 6 5.2 odd 4
2025.2.b.l.649.6 6 5.3 odd 4
2025.2.b.m.649.1 6 15.8 even 4
2025.2.b.m.649.6 6 15.2 even 4
2160.2.q.k.721.1 6 180.59 even 6
2160.2.q.k.1441.1 6 180.119 even 6
6480.2.a.bs.1.3 3 60.59 even 2
6480.2.a.bv.1.3 3 20.19 odd 2