Properties

Label 2025.2.b.l.649.6
Level $2025$
Weight $2$
Character 2025.649
Analytic conductor $16.170$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5089536.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.6
Root \(1.66044 - 1.66044i\) of defining polynomial
Character \(\chi\) \(=\) 2025.649
Dual form 2025.2.b.l.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.51414i q^{2} -4.32088 q^{4} -0.514137i q^{7} -5.83502i q^{8} +O(q^{10})\) \(q+2.51414i q^{2} -4.32088 q^{4} -0.514137i q^{7} -5.83502i q^{8} -3.32088 q^{11} +1.32088i q^{13} +1.29261 q^{14} +6.02827 q^{16} -3.32088i q^{17} +1.32088 q^{19} -8.34916i q^{22} -4.12763i q^{23} -3.32088 q^{26} +2.22153i q^{28} +1.38650 q^{29} +8.73566 q^{31} +3.48586i q^{32} +8.34916 q^{34} +0.292611i q^{37} +3.32088i q^{38} -11.3492 q^{41} -10.3492i q^{43} +14.3492 q^{44} +10.3774 q^{46} -4.86330i q^{47} +6.73566 q^{49} -5.70739i q^{52} +5.02827i q^{53} -3.00000 q^{56} +3.48586i q^{58} +5.02827 q^{59} +7.34916 q^{61} +21.9627i q^{62} +3.29261 q^{64} +9.44852i q^{67} +14.3492i q^{68} +8.99093 q^{71} -6.05655i q^{73} -0.735663 q^{74} -5.70739 q^{76} +1.70739i q^{77} +8.05655 q^{79} -28.5333i q^{82} -1.54241i q^{83} +26.0192 q^{86} +19.3774i q^{88} +3.00000 q^{89} +0.679116 q^{91} +17.8350i q^{92} +12.2270 q^{94} -12.2553i q^{97} +16.9344i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} - 4 q^{11} + 18 q^{14} + 10 q^{16} - 8 q^{19} - 4 q^{26} + 14 q^{29} + 16 q^{31} + 8 q^{34} - 26 q^{41} + 44 q^{44} - 6 q^{46} + 4 q^{49} - 18 q^{56} + 4 q^{59} + 2 q^{61} + 30 q^{64} - 20 q^{71} + 32 q^{74} - 24 q^{76} - 4 q^{79} + 56 q^{86} + 18 q^{89} + 20 q^{91} + 62 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.51414i 1.77776i 0.458137 + 0.888882i \(0.348517\pi\)
−0.458137 + 0.888882i \(0.651483\pi\)
\(3\) 0 0
\(4\) −4.32088 −2.16044
\(5\) 0 0
\(6\) 0 0
\(7\) − 0.514137i − 0.194325i −0.995269 0.0971627i \(-0.969023\pi\)
0.995269 0.0971627i \(-0.0309767\pi\)
\(8\) − 5.83502i − 2.06299i
\(9\) 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
\(13\) 1.32088i 0.366347i 0.983081 + 0.183174i \(0.0586371\pi\)
−0.983081 + 0.183174i \(0.941363\pi\)
\(14\) 1.29261 0.345465
\(15\) 0 0
\(16\) 6.02827 1.50707
\(17\) − 3.32088i − 0.805433i −0.915325 0.402716i \(-0.868066\pi\)
0.915325 0.402716i \(-0.131934\pi\)
\(18\) 0 0
\(19\) 1.32088 0.303032 0.151516 0.988455i \(-0.451585\pi\)
0.151516 + 0.988455i \(0.451585\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 8.34916i − 1.78005i
\(23\) − 4.12763i − 0.860671i −0.902669 0.430335i \(-0.858395\pi\)
0.902669 0.430335i \(-0.141605\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.32088 −0.651279
\(27\) 0 0
\(28\) 2.22153i 0.419829i
\(29\) 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 \(0.212927\pi\)
\(32\) 3.48586i 0.616219i
\(33\) 0 0
\(34\) 8.34916 1.43187
\(35\) 0 0
\(36\) 0 0
\(37\) 0.292611i 0.0481049i 0.999711 + 0.0240524i \(0.00765687\pi\)
−0.999711 + 0.0240524i \(0.992343\pi\)
\(38\) 3.32088i 0.538719i
\(39\) 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
\(43\) − 10.3492i − 1.57823i −0.614244 0.789116i \(-0.710539\pi\)
0.614244 0.789116i \(-0.289461\pi\)
\(44\) 14.3492 2.16322
\(45\) 0 0
\(46\) 10.3774 1.53007
\(47\) − 4.86330i − 0.709385i −0.934983 0.354692i \(-0.884586\pi\)
0.934983 0.354692i \(-0.115414\pi\)
\(48\) 0 0
\(49\) 6.73566 0.962238
\(50\) 0 0
\(51\) 0 0
\(52\) − 5.70739i − 0.791472i
\(53\) 5.02827i 0.690687i 0.938476 + 0.345343i \(0.112238\pi\)
−0.938476 + 0.345343i \(0.887762\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) 3.48586i 0.457716i
\(59\) 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 \(0.344080\pi\)
\(62\) 21.9627i 2.78926i
\(63\) 0 0
\(64\) 3.29261 0.411576
\(65\) 0 0
\(66\) 0 0
\(67\) 9.44852i 1.15432i 0.816631 + 0.577160i \(0.195839\pi\)
−0.816631 + 0.577160i \(0.804161\pi\)
\(68\) 14.3492i 1.74009i
\(69\) 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
\(73\) − 6.05655i − 0.708865i −0.935082 0.354433i \(-0.884674\pi\)
0.935082 0.354433i \(-0.115326\pi\)
\(74\) −0.735663 −0.0855191
\(75\) 0 0
\(76\) −5.70739 −0.654682
\(77\) 1.70739i 0.194575i
\(78\) 0 0
\(79\) 8.05655 0.906432 0.453216 0.891401i \(-0.350277\pi\)
0.453216 + 0.891401i \(0.350277\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 28.5333i − 3.15098i
\(83\) − 1.54241i − 0.169302i −0.996411 0.0846508i \(-0.973023\pi\)
0.996411 0.0846508i \(-0.0269775\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 26.0192 2.80572
\(87\) 0 0
\(88\) 19.3774i 2.06564i
\(89\) 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
\(92\) 17.8350i 1.85943i
\(93\) 0 0
\(94\) 12.2270 1.26112
\(95\) 0 0
\(96\) 0 0
\(97\) − 12.2553i − 1.24433i −0.782885 0.622167i \(-0.786253\pi\)
0.782885 0.622167i \(-0.213747\pi\)
\(98\) 16.9344i 1.71063i
\(99\) 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
\(103\) − 0.292611i − 0.0288318i −0.999896 0.0144159i \(-0.995411\pi\)
0.999896 0.0144159i \(-0.00458888\pi\)
\(104\) 7.70739 0.755772
\(105\) 0 0
\(106\) −12.6418 −1.22788
\(107\) 1.87237i 0.181009i 0.995896 + 0.0905043i \(0.0288479\pi\)
−0.995896 + 0.0905043i \(0.971152\pi\)
\(108\) 0 0
\(109\) −5.54787 −0.531390 −0.265695 0.964057i \(-0.585601\pi\)
−0.265695 + 0.964057i \(0.585601\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 3.09936i − 0.292862i
\(113\) − 7.80128i − 0.733883i −0.930244 0.366942i \(-0.880405\pi\)
0.930244 0.366942i \(-0.119595\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −5.99093 −0.556244
\(117\) 0 0
\(118\) 12.6418i 1.16377i
\(119\) −1.70739 −0.156516
\(120\) 0 0
\(121\) 0.0282739 0.00257035
\(122\) 18.4768i 1.67281i
\(123\) 0 0
\(124\) −37.7458 −3.38967
\(125\) 0 0
\(126\) 0 0
\(127\) 17.8916i 1.58762i 0.608166 + 0.793810i \(0.291906\pi\)
−0.608166 + 0.793810i \(0.708094\pi\)
\(128\) 15.2498i 1.34790i
\(129\) 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
\(133\) − 0.679116i − 0.0588868i
\(134\) −23.7549 −2.05211
\(135\) 0 0
\(136\) −19.3774 −1.66160
\(137\) 5.67004i 0.484424i 0.970223 + 0.242212i \(0.0778730\pi\)
−0.970223 + 0.242212i \(0.922127\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 22.6044i 1.89692i
\(143\) − 4.38650i − 0.366818i
\(144\) 0 0
\(145\) 0 0
\(146\) 15.2270 1.26019
\(147\) 0 0
\(148\) − 1.26434i − 0.103928i
\(149\) −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 \(0.483617\pi\)
\(152\) − 7.70739i − 0.625152i
\(153\) 0 0
\(154\) −4.29261 −0.345908
\(155\) 0 0
\(156\) 0 0
\(157\) − 15.6700i − 1.25061i −0.780382 0.625303i \(-0.784975\pi\)
0.780382 0.625303i \(-0.215025\pi\)
\(158\) 20.2553i 1.61142i
\(159\) 0 0
\(160\) 0 0
\(161\) −2.12217 −0.167250
\(162\) 0 0
\(163\) − 15.7074i − 1.23030i −0.788411 0.615149i \(-0.789096\pi\)
0.788411 0.615149i \(-0.210904\pi\)
\(164\) 49.0384 3.82926
\(165\) 0 0
\(166\) 3.87783 0.300978
\(167\) 6.16498i 0.477060i 0.971135 + 0.238530i \(0.0766656\pi\)
−0.971135 + 0.238530i \(0.923334\pi\)
\(168\) 0 0
\(169\) 11.2553 0.865790
\(170\) 0 0
\(171\) 0 0
\(172\) 44.7175i 3.40968i
\(173\) − 8.58522i − 0.652722i −0.945245 0.326361i \(-0.894177\pi\)
0.945245 0.326361i \(-0.105823\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −20.0192 −1.50900
\(177\) 0 0
\(178\) 7.54241i 0.565328i
\(179\) 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 \(0.656063\pi\)
\(182\) 1.70739i 0.126560i
\(183\) 0 0
\(184\) −24.0848 −1.77556
\(185\) 0 0
\(186\) 0 0
\(187\) 11.0283i 0.806467i
\(188\) 21.0137i 1.53258i
\(189\) 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
\(193\) − 26.7175i − 1.92317i −0.274509 0.961585i \(-0.588515\pi\)
0.274509 0.961585i \(-0.411485\pi\)
\(194\) 30.8114 2.21213
\(195\) 0 0
\(196\) −29.1040 −2.07886
\(197\) − 14.2553i − 1.01565i −0.861462 0.507823i \(-0.830450\pi\)
0.861462 0.507823i \(-0.169550\pi\)
\(198\) 0 0
\(199\) 24.6610 1.74817 0.874085 0.485773i \(-0.161462\pi\)
0.874085 + 0.485773i \(0.161462\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 29.3401i 2.06436i
\(203\) − 0.712853i − 0.0500325i
\(204\) 0 0
\(205\) 0 0
\(206\) 0.735663 0.0512561
\(207\) 0 0
\(208\) 7.96265i 0.552111i
\(209\) −4.38650 −0.303421
\(210\) 0 0
\(211\) −5.37743 −0.370198 −0.185099 0.982720i \(-0.559261\pi\)
−0.185099 + 0.982720i \(0.559261\pi\)
\(212\) − 21.7266i − 1.49219i
\(213\) 0 0
\(214\) −4.70739 −0.321791
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.49133i − 0.304891i
\(218\) − 13.9481i − 0.944686i
\(219\) 0 0
\(220\) 0 0
\(221\) 4.38650 0.295068
\(222\) 0 0
\(223\) − 8.66458i − 0.580223i −0.956993 0.290112i \(-0.906308\pi\)
0.956993 0.290112i \(-0.0936924\pi\)
\(224\) 1.79221 0.119747
\(225\) 0 0
\(226\) 19.6135 1.30467
\(227\) 3.32088i 0.220415i 0.993909 + 0.110207i \(0.0351515\pi\)
−0.993909 + 0.110207i \(0.964848\pi\)
\(228\) 0 0
\(229\) 25.3118 1.67265 0.836326 0.548233i \(-0.184699\pi\)
0.836326 + 0.548233i \(0.184699\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 8.09029i − 0.531153i
\(233\) − 27.6327i − 1.81028i −0.425116 0.905139i \(-0.639767\pi\)
0.425116 0.905139i \(-0.360233\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −21.7266 −1.41428
\(237\) 0 0
\(238\) − 4.29261i − 0.278249i
\(239\) 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 \(0.462964\pi\)
\(242\) 0.0710844i 0.00456948i
\(243\) 0 0
\(244\) −31.7549 −2.03290
\(245\) 0 0
\(246\) 0 0
\(247\) 1.74474i 0.111015i
\(248\) − 50.9728i − 3.23677i
\(249\) 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
\(253\) 13.7074i 0.861776i
\(254\) −44.9819 −2.82241
\(255\) 0 0
\(256\) −31.7549 −1.98468
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) 0.150442 0.00934801
\(260\) 0 0
\(261\) 0 0
\(262\) − 15.0848i − 0.931943i
\(263\) − 6.23606i − 0.384532i −0.981343 0.192266i \(-0.938416\pi\)
0.981343 0.192266i \(-0.0615837\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.70739 0.104687
\(267\) 0 0
\(268\) − 40.8259i − 2.49384i
\(269\) −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 \(0.435712\pi\)
\(272\) − 20.0192i − 1.21384i
\(273\) 0 0
\(274\) −14.2553 −0.861192
\(275\) 0 0
\(276\) 0 0
\(277\) 22.6610i 1.36157i 0.732485 + 0.680783i \(0.238360\pi\)
−0.732485 + 0.680783i \(0.761640\pi\)
\(278\) − 20.1131i − 1.20630i
\(279\) 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
\(283\) − 0.645378i − 0.0383637i −0.999816 0.0191819i \(-0.993894\pi\)
0.999816 0.0191819i \(-0.00610615\pi\)
\(284\) −38.8488 −2.30525
\(285\) 0 0
\(286\) 11.0283 0.652116
\(287\) 5.83502i 0.344430i
\(288\) 0 0
\(289\) 5.97173 0.351278
\(290\) 0 0
\(291\) 0 0
\(292\) 26.1696i 1.53146i
\(293\) 1.37743i 0.0804704i 0.999190 + 0.0402352i \(0.0128107\pi\)
−0.999190 + 0.0402352i \(0.987189\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.70739 0.0992400
\(297\) 0 0
\(298\) − 44.4021i − 2.57214i
\(299\) 5.45213 0.315305
\(300\) 0 0
\(301\) −5.32088 −0.306691
\(302\) 3.17872i 0.182915i
\(303\) 0 0
\(304\) 7.96265 0.456689
\(305\) 0 0
\(306\) 0 0
\(307\) − 7.98546i − 0.455754i −0.973690 0.227877i \(-0.926822\pi\)
0.973690 0.227877i \(-0.0731785\pi\)
\(308\) − 7.37743i − 0.420368i
\(309\) 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
\(313\) 24.5369i 1.38691i 0.720500 + 0.693455i \(0.243912\pi\)
−0.720500 + 0.693455i \(0.756088\pi\)
\(314\) 39.3966 2.22328
\(315\) 0 0
\(316\) −34.8114 −1.95829
\(317\) − 20.3492i − 1.14292i −0.820629 0.571461i \(-0.806377\pi\)
0.820629 0.571461i \(-0.193623\pi\)
\(318\) 0 0
\(319\) −4.60442 −0.257798
\(320\) 0 0
\(321\) 0 0
\(322\) − 5.33542i − 0.297331i
\(323\) − 4.38650i − 0.244072i
\(324\) 0 0
\(325\) 0 0
\(326\) 39.4905 2.18718
\(327\) 0 0
\(328\) 66.2226i 3.65653i
\(329\) −2.50040 −0.137851
\(330\) 0 0
\(331\) 16.4431 0.903792 0.451896 0.892071i \(-0.350748\pi\)
0.451896 + 0.892071i \(0.350748\pi\)
\(332\) 6.66458i 0.365766i
\(333\) 0 0
\(334\) −15.4996 −0.848100
\(335\) 0 0
\(336\) 0 0
\(337\) 4.89703i 0.266758i 0.991065 + 0.133379i \(0.0425828\pi\)
−0.991065 + 0.133379i \(0.957417\pi\)
\(338\) 28.2973i 1.53917i
\(339\) 0 0
\(340\) 0 0
\(341\) −29.0101 −1.57099
\(342\) 0 0
\(343\) − 7.06201i − 0.381313i
\(344\) −60.3876 −3.25588
\(345\) 0 0
\(346\) 21.5844 1.16039
\(347\) 22.2745i 1.19576i 0.801587 + 0.597878i \(0.203989\pi\)
−0.801587 + 0.597878i \(0.796011\pi\)
\(348\) 0 0
\(349\) 2.94345 0.157559 0.0787797 0.996892i \(-0.474898\pi\)
0.0787797 + 0.996892i \(0.474898\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 11.5761i − 0.617011i
\(353\) − 18.8296i − 1.00220i −0.865390 0.501098i \(-0.832930\pi\)
0.865390 0.501098i \(-0.167070\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −12.9627 −0.687019
\(357\) 0 0
\(358\) 2.67912i 0.141596i
\(359\) 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
\(362\) − 31.8542i − 1.67422i
\(363\) 0 0
\(364\) −2.93438 −0.153803
\(365\) 0 0
\(366\) 0 0
\(367\) − 18.3492i − 0.957818i −0.877864 0.478909i \(-0.841032\pi\)
0.877864 0.478909i \(-0.158968\pi\)
\(368\) − 24.8825i − 1.29709i
\(369\) 0 0
\(370\) 0 0
\(371\) 2.58522 0.134218
\(372\) 0 0
\(373\) 2.19872i 0.113845i 0.998379 + 0.0569226i \(0.0181288\pi\)
−0.998379 + 0.0569226i \(0.981871\pi\)
\(374\) −27.7266 −1.43371
\(375\) 0 0
\(376\) −28.3774 −1.46345
\(377\) 1.83141i 0.0943226i
\(378\) 0 0
\(379\) −15.4713 −0.794709 −0.397354 0.917665i \(-0.630072\pi\)
−0.397354 + 0.917665i \(0.630072\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 42.5753i − 2.17834i
\(383\) − 7.70739i − 0.393829i −0.980421 0.196915i \(-0.936908\pi\)
0.980421 0.196915i \(-0.0630922\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 67.1715 3.41894
\(387\) 0 0
\(388\) 52.9536i 2.68831i
\(389\) 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
\(392\) − 39.3027i − 1.98509i
\(393\) 0 0
\(394\) 35.8397 1.80558
\(395\) 0 0
\(396\) 0 0
\(397\) − 6.77301i − 0.339928i −0.985450 0.169964i \(-0.945635\pi\)
0.985450 0.169964i \(-0.0543651\pi\)
\(398\) 62.0011i 3.10783i
\(399\) 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
\(403\) 11.5388i 0.574789i
\(404\) −50.4249 −2.50873
\(405\) 0 0
\(406\) 1.79221 0.0889459
\(407\) − 0.971726i − 0.0481667i
\(408\) 0 0
\(409\) 13.4148 0.663318 0.331659 0.943399i \(-0.392392\pi\)
0.331659 + 0.943399i \(0.392392\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.26434i 0.0622894i
\(413\) − 2.58522i − 0.127210i
\(414\) 0 0
\(415\) 0 0
\(416\) −4.60442 −0.225750
\(417\) 0 0
\(418\) − 11.0283i − 0.539411i
\(419\) 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 \(0.616601\pi\)
\(422\) − 13.5196i − 0.658124i
\(423\) 0 0
\(424\) 29.3401 1.42488
\(425\) 0 0
\(426\) 0 0
\(427\) − 3.77847i − 0.182853i
\(428\) − 8.09029i − 0.391059i
\(429\) 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
\(433\) 11.8314i 0.568581i 0.958738 + 0.284291i \(0.0917581\pi\)
−0.958738 + 0.284291i \(0.908242\pi\)
\(434\) 11.2918 0.542024
\(435\) 0 0
\(436\) 23.9717 1.14804
\(437\) − 5.45213i − 0.260811i
\(438\) 0 0
\(439\) −8.31181 −0.396701 −0.198351 0.980131i \(-0.563558\pi\)
−0.198351 + 0.980131i \(0.563558\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 11.0283i 0.524561i
\(443\) 29.1751i 1.38615i 0.720865 + 0.693076i \(0.243745\pi\)
−0.720865 + 0.693076i \(0.756255\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 21.7839 1.03150
\(447\) 0 0
\(448\) − 1.69285i − 0.0799798i
\(449\) −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
\(452\) 33.7084i 1.58551i
\(453\) 0 0
\(454\) −8.34916 −0.391845
\(455\) 0 0
\(456\) 0 0
\(457\) − 23.2353i − 1.08690i −0.839442 0.543450i \(-0.817118\pi\)
0.839442 0.543450i \(-0.182882\pi\)
\(458\) 63.6374i 2.97358i
\(459\) 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
\(463\) 19.5087i 0.906645i 0.891347 + 0.453322i \(0.149761\pi\)
−0.891347 + 0.453322i \(0.850239\pi\)
\(464\) 8.35823 0.388021
\(465\) 0 0
\(466\) 69.4724 3.21825
\(467\) − 24.5935i − 1.13805i −0.822320 0.569026i \(-0.807321\pi\)
0.822320 0.569026i \(-0.192679\pi\)
\(468\) 0 0
\(469\) 4.85783 0.224314
\(470\) 0 0
\(471\) 0 0
\(472\) − 29.3401i − 1.35049i
\(473\) 34.3684i 1.58026i
\(474\) 0 0
\(475\) 0 0
\(476\) 7.37743 0.338144
\(477\) 0 0
\(478\) 10.5561i 0.482827i
\(479\) −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
\(482\) 9.06201i 0.412763i
\(483\) 0 0
\(484\) −0.122168 −0.00555309
\(485\) 0 0
\(486\) 0 0
\(487\) − 6.03735i − 0.273578i −0.990600 0.136789i \(-0.956322\pi\)
0.990600 0.136789i \(-0.0436783\pi\)
\(488\) − 42.8825i − 1.94120i
\(489\) 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
\(493\) − 4.60442i − 0.207373i
\(494\) −4.38650 −0.197358
\(495\) 0 0
\(496\) 52.6610 2.36455
\(497\) − 4.62257i − 0.207351i
\(498\) 0 0
\(499\) −20.9717 −0.938823 −0.469412 0.882979i \(-0.655534\pi\)
−0.469412 + 0.882979i \(0.655534\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 17.2918i − 0.771771i
\(503\) 5.31728i 0.237086i 0.992949 + 0.118543i \(0.0378223\pi\)
−0.992949 + 0.118543i \(0.962178\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −34.4623 −1.53203
\(507\) 0 0
\(508\) − 77.3074i − 3.42996i
\(509\) −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
\(512\) − 49.3365i − 2.18038i
\(513\) 0 0
\(514\) 45.2545 1.99609
\(515\) 0 0
\(516\) 0 0
\(517\) 16.1504i 0.710296i
\(518\) 0.378232i 0.0166185i
\(519\) 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
\(523\) − 18.9873i − 0.830257i −0.909763 0.415129i \(-0.863737\pi\)
0.909763 0.415129i \(-0.136263\pi\)
\(524\) 25.9253 1.13255
\(525\) 0 0
\(526\) 15.6783 0.683607
\(527\) − 29.0101i − 1.26370i
\(528\) 0 0
\(529\) 5.96265 0.259246
\(530\) 0 0
\(531\) 0 0
\(532\) 2.93438i 0.127221i
\(533\) − 14.9909i − 0.649329i
\(534\) 0 0
\(535\) 0 0
\(536\) 55.1323 2.38135
\(537\) 0 0
\(538\) − 24.9489i − 1.07562i
\(539\) −22.3684 −0.963473
\(540\) 0 0
\(541\) 16.5279 0.710589 0.355294 0.934754i \(-0.384381\pi\)
0.355294 + 0.934754i \(0.384381\pi\)
\(542\) 16.6044i 0.713221i
\(543\) 0 0
\(544\) 11.5761 0.496323
\(545\) 0 0
\(546\) 0 0
\(547\) 17.6737i 0.755671i 0.925873 + 0.377835i \(0.123331\pi\)
−0.925873 + 0.377835i \(0.876669\pi\)
\(548\) − 24.4996i − 1.04657i
\(549\) 0 0
\(550\) 0 0
\(551\) 1.83141 0.0780208
\(552\) 0 0
\(553\) − 4.14217i − 0.176143i
\(554\) −56.9728 −2.42054
\(555\) 0 0
\(556\) 34.5671 1.46597
\(557\) 17.3401i 0.734723i 0.930078 + 0.367362i \(0.119739\pi\)
−0.930078 + 0.367362i \(0.880261\pi\)
\(558\) 0 0
\(559\) 13.6700 0.578181
\(560\) 0 0
\(561\) 0 0
\(562\) − 39.0895i − 1.64889i
\(563\) − 12.9945i − 0.547654i −0.961779 0.273827i \(-0.911710\pi\)
0.961779 0.273827i \(-0.0882896\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.62257 0.0682016
\(567\) 0 0
\(568\) − 52.4623i − 2.20127i
\(569\) 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 \(0.362561\pi\)
\(572\) 18.9536i 0.792489i
\(573\) 0 0
\(574\) −14.6700 −0.612316
\(575\) 0 0
\(576\) 0 0
\(577\) − 23.5953i − 0.982287i −0.871079 0.491144i \(-0.836579\pi\)
0.871079 0.491144i \(-0.163421\pi\)
\(578\) 15.0137i 0.624489i
\(579\) 0 0
\(580\) 0 0
\(581\) −0.793010 −0.0328996
\(582\) 0 0
\(583\) − 16.6983i − 0.691574i
\(584\) −35.3401 −1.46238
\(585\) 0 0
\(586\) −3.46305 −0.143057
\(587\) 28.1276i 1.16095i 0.814277 + 0.580476i \(0.197133\pi\)
−0.814277 + 0.580476i \(0.802867\pi\)
\(588\) 0 0
\(589\) 11.5388 0.475448
\(590\) 0 0
\(591\) 0 0
\(592\) 1.76394i 0.0724974i
\(593\) 9.17872i 0.376925i 0.982080 + 0.188462i \(0.0603503\pi\)
−0.982080 + 0.188462i \(0.939650\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 76.3110 3.12582
\(597\) 0 0
\(598\) 13.7074i 0.560537i
\(599\) 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 \(0.703593\pi\)
\(602\) − 13.3774i − 0.545223i
\(603\) 0 0
\(604\) −5.46305 −0.222288
\(605\) 0 0
\(606\) 0 0
\(607\) − 44.2034i − 1.79416i −0.441868 0.897080i \(-0.645684\pi\)
0.441868 0.897080i \(-0.354316\pi\)
\(608\) 4.60442i 0.186734i
\(609\) 0 0
\(610\) 0 0
\(611\) 6.42385 0.259881
\(612\) 0 0
\(613\) 35.1715i 1.42056i 0.703918 + 0.710282i \(0.251432\pi\)
−0.703918 + 0.710282i \(0.748568\pi\)
\(614\) 20.0765 0.810224
\(615\) 0 0
\(616\) 9.96265 0.401407
\(617\) − 7.42571i − 0.298948i −0.988766 0.149474i \(-0.952242\pi\)
0.988766 0.149474i \(-0.0477580\pi\)
\(618\) 0 0
\(619\) −8.54787 −0.343568 −0.171784 0.985135i \(-0.554953\pi\)
−0.171784 + 0.985135i \(0.554953\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 24.2179i 0.971050i
\(623\) − 1.54241i − 0.0617954i
\(624\) 0 0
\(625\) 0 0
\(626\) −61.6892 −2.46560
\(627\) 0 0
\(628\) 67.7084i 2.70186i
\(629\) 0.971726 0.0387453
\(630\) 0 0
\(631\) −2.36836 −0.0942829 −0.0471415 0.998888i \(-0.515011\pi\)
−0.0471415 + 0.998888i \(0.515011\pi\)
\(632\) − 47.0101i − 1.86996i
\(633\) 0 0
\(634\) 51.1606 2.03185
\(635\) 0 0
\(636\) 0 0
\(637\) 8.89703i 0.352513i
\(638\) − 11.5761i − 0.458304i
\(639\) 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
\(643\) 22.6464i 0.893088i 0.894762 + 0.446544i \(0.147345\pi\)
−0.894762 + 0.446544i \(0.852655\pi\)
\(644\) 9.16964 0.361335
\(645\) 0 0
\(646\) 11.0283 0.433902
\(647\) 46.3912i 1.82383i 0.410385 + 0.911913i \(0.365394\pi\)
−0.410385 + 0.911913i \(0.634606\pi\)
\(648\) 0 0
\(649\) −16.6983 −0.655466
\(650\) 0 0
\(651\) 0 0
\(652\) 67.8698i 2.65799i
\(653\) − 36.4057i − 1.42467i −0.701842 0.712333i \(-0.747639\pi\)
0.701842 0.712333i \(-0.252361\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −68.4158 −2.67119
\(657\) 0 0
\(658\) − 6.28635i − 0.245067i
\(659\) −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 \(0.217193\pi\)
\(662\) 41.3401i 1.60673i
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) 0 0
\(667\) − 5.72298i − 0.221595i
\(668\) − 26.6382i − 1.03066i
\(669\) 0 0
\(670\) 0 0
\(671\) −24.4057 −0.942172
\(672\) 0 0
\(673\) − 23.6508i − 0.911673i −0.890064 0.455836i \(-0.849340\pi\)
0.890064 0.455836i \(-0.150660\pi\)
\(674\) −12.3118 −0.474233
\(675\) 0 0
\(676\) −48.6327 −1.87049
\(677\) − 14.8031i − 0.568931i −0.958686 0.284465i \(-0.908184\pi\)
0.958686 0.284465i \(-0.0918161\pi\)
\(678\) 0 0
\(679\) −6.30088 −0.241806
\(680\) 0 0
\(681\) 0 0
\(682\) − 72.9354i − 2.79284i
\(683\) 4.95252i 0.189503i 0.995501 + 0.0947515i \(0.0302057\pi\)
−0.995501 + 0.0947515i \(0.969794\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 17.7549 0.677884
\(687\) 0 0
\(688\) − 62.3876i − 2.37850i
\(689\) −6.64177 −0.253031
\(690\) 0 0
\(691\) −19.2088 −0.730739 −0.365369 0.930863i \(-0.619057\pi\)
−0.365369 + 0.930863i \(0.619057\pi\)
\(692\) 37.0957i 1.41017i
\(693\) 0 0
\(694\) −56.0011 −2.12577
\(695\) 0 0
\(696\) 0 0
\(697\) 37.6892i 1.42758i
\(698\) 7.40024i 0.280103i
\(699\) 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
\(703\) 0.386505i 0.0145773i
\(704\) −10.9344 −0.412105
\(705\) 0 0
\(706\) 47.3401 1.78167
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) −38.7266 −1.45441 −0.727204 0.686422i \(-0.759180\pi\)
−0.727204 + 0.686422i \(0.759180\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 17.5051i − 0.656030i
\(713\) − 36.0576i − 1.35037i
\(714\) 0 0
\(715\) 0 0
\(716\) −4.60442 −0.172075
\(717\) 0 0
\(718\) 80.1432i 2.99092i
\(719\) 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
\(722\) − 43.3821i − 1.61451i
\(723\) 0 0
\(724\) 54.7458 2.03461
\(725\) 0 0
\(726\) 0 0
\(727\) 12.3455i 0.457871i 0.973442 + 0.228936i \(0.0735245\pi\)
−0.973442 + 0.228936i \(0.926475\pi\)
\(728\) − 3.96265i − 0.146866i
\(729\) 0 0
\(730\) 0 0
\(731\) −34.3684 −1.27116
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 46.1323 1.70277
\(735\) 0 0
\(736\) 14.3884 0.530362
\(737\) − 31.3774i − 1.15580i
\(738\) 0 0
\(739\) −29.7266 −1.09351 −0.546755 0.837293i \(-0.684137\pi\)
−0.546755 + 0.837293i \(0.684137\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.49960i 0.238608i
\(743\) − 48.3648i − 1.77433i −0.461452 0.887165i \(-0.652671\pi\)
0.461452 0.887165i \(-0.347329\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −5.52787 −0.202390
\(747\) 0 0
\(748\) − 47.6519i − 1.74233i
\(749\) 0.962653 0.0351746
\(750\) 0 0
\(751\) −31.8205 −1.16115 −0.580573 0.814208i \(-0.697171\pi\)
−0.580573 + 0.814208i \(0.697171\pi\)
\(752\) − 29.3173i − 1.06909i
\(753\) 0 0
\(754\) −4.60442 −0.167683
\(755\) 0 0
\(756\) 0 0
\(757\) 4.94531i 0.179740i 0.995953 + 0.0898701i \(0.0286452\pi\)
−0.995953 + 0.0898701i \(0.971355\pi\)
\(758\) − 38.8970i − 1.41280i
\(759\) 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
\(763\) 2.85237i 0.103263i
\(764\) 73.1715 2.64725
\(765\) 0 0
\(766\) 19.3774 0.700135
\(767\) 6.64177i 0.239820i
\(768\) 0 0
\(769\) −49.4249 −1.78231 −0.891154 0.453701i \(-0.850103\pi\)
−0.891154 + 0.453701i \(0.850103\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 115.443i 4.15490i
\(773\) − 12.6599i − 0.455345i −0.973738 0.227673i \(-0.926888\pi\)
0.973738 0.227673i \(-0.0731116\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −71.5097 −2.56705
\(777\) 0 0
\(778\) 61.9300i 2.22030i
\(779\) −14.9909 −0.537106
\(780\) 0 0
\(781\) −29.8578 −1.06840
\(782\) − 34.4623i − 1.23237i
\(783\) 0 0
\(784\) 40.6044 1.45016
\(785\) 0 0
\(786\) 0 0
\(787\) 30.9344i 1.10269i 0.834277 + 0.551346i \(0.185885\pi\)
−0.834277 + 0.551346i \(0.814115\pi\)
\(788\) 61.5953i 2.19424i
\(789\) 0 0
\(790\) 0 0
\(791\) −4.01093 −0.142612
\(792\) 0 0
\(793\) 9.70739i 0.344720i
\(794\) 17.0283 0.604311
\(795\) 0 0
\(796\) −106.557 −3.77682
\(797\) − 30.5935i − 1.08368i −0.840483 0.541839i \(-0.817728\pi\)
0.840483 0.541839i \(-0.182272\pi\)
\(798\) 0 0
\(799\) −16.1504 −0.571362
\(800\) 0 0
\(801\) 0 0
\(802\) − 46.5105i − 1.64234i
\(803\) 20.1131i 0.709776i
\(804\) 0 0
\(805\) 0 0
\(806\) −29.0101 −1.02184
\(807\) 0 0
\(808\) − 68.0950i − 2.39557i
\(809\) 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 \(0.584171\pi\)
\(812\) 3.08016i 0.108092i
\(813\) 0 0
\(814\) 2.44305 0.0856289
\(815\) 0 0
\(816\) 0 0
\(817\) − 13.6700i − 0.478254i
\(818\) 33.7266i 1.17922i
\(819\) 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
\(823\) − 2.99454i − 0.104383i −0.998637 0.0521915i \(-0.983379\pi\)
0.998637 0.0521915i \(-0.0166206\pi\)
\(824\) −1.70739 −0.0594797
\(825\) 0 0
\(826\) 6.49960 0.226150
\(827\) 31.9663i 1.11158i 0.831324 + 0.555788i \(0.187583\pi\)
−0.831324 + 0.555788i \(0.812417\pi\)
\(828\) 0 0
\(829\) −22.7458 −0.789994 −0.394997 0.918682i \(-0.629254\pi\)
−0.394997 + 0.918682i \(0.629254\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4.34916i 0.150780i
\(833\) − 22.3684i − 0.775018i
\(834\) 0 0
\(835\) 0 0
\(836\) 18.9536 0.655523
\(837\) 0 0
\(838\) 83.2555i 2.87601i
\(839\) 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
\(842\) − 36.9536i − 1.27350i
\(843\) 0 0
\(844\) 23.2353 0.799791
\(845\) 0 0
\(846\) 0 0
\(847\) − 0.0145366i 0 0.000499485i
\(848\) 30.3118i 1.04091i
\(849\) 0 0
\(850\) 0 0
\(851\) 1.20779 0.0414025
\(852\) 0 0
\(853\) − 10.9909i − 0.376322i −0.982138 0.188161i \(-0.939747\pi\)
0.982138 0.188161i \(-0.0602527\pi\)
\(854\) 9.49960 0.325070
\(855\) 0 0
\(856\) 10.9253 0.373419
\(857\) − 16.1504i − 0.551689i −0.961202 0.275844i \(-0.911043\pi\)
0.961202 0.275844i \(-0.0889574\pi\)
\(858\) 0 0
\(859\) −28.5188 −0.973049 −0.486524 0.873667i \(-0.661736\pi\)
−0.486524 + 0.873667i \(0.661736\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 82.3502i − 2.80486i
\(863\) − 12.2890i − 0.418322i −0.977881 0.209161i \(-0.932927\pi\)
0.977881 0.209161i \(-0.0670734\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −29.7458 −1.01080
\(867\) 0 0
\(868\) 19.4065i 0.658700i
\(869\) −26.7549 −0.907597
\(870\) 0 0
\(871\) −12.4804 −0.422882
\(872\) 32.3720i 1.09625i
\(873\) 0 0
\(874\) 13.7074 0.463659
\(875\) 0 0
\(876\) 0 0
\(877\) − 39.7002i − 1.34058i −0.742099 0.670290i \(-0.766170\pi\)
0.742099 0.670290i \(-0.233830\pi\)
\(878\) − 20.8970i − 0.705241i
\(879\) 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
\(883\) − 13.5051i − 0.454482i −0.973839 0.227241i \(-0.927030\pi\)
0.973839 0.227241i \(-0.0729704\pi\)
\(884\) −18.9536 −0.637478
\(885\) 0 0
\(886\) −73.3502 −2.46425
\(887\) 35.1222i 1.17929i 0.807664 + 0.589643i \(0.200732\pi\)
−0.807664 + 0.589643i \(0.799268\pi\)
\(888\) 0 0
\(889\) 9.19872 0.308515
\(890\) 0 0
\(891\) 0 0
\(892\) 37.4386i 1.25354i
\(893\) − 6.42385i − 0.214966i
\(894\) 0 0
\(895\) 0 0
\(896\) 7.84049 0.261932
\(897\) 0 0
\(898\) − 47.6975i − 1.59169i
\(899\) 12.1120 0.403959
\(900\) 0 0
\(901\) 16.6983 0.556302
\(902\) 94.7559i 3.15503i
\(903\) 0 0
\(904\) −45.5207 −1.51399
\(905\) 0 0
\(906\) 0 0
\(907\) 15.1186i 0.502004i 0.967987 + 0.251002i \(0.0807600\pi\)
−0.967987 + 0.251002i \(0.919240\pi\)
\(908\) − 14.3492i − 0.476194i
\(909\) 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
\(913\) 5.12217i 0.169519i
\(914\) 58.4166 1.93225
\(915\) 0 0
\(916\) −109.369 −3.61367
\(917\) 3.08482i 0.101870i
\(918\) 0 0
\(919\) 54.5489 1.79940 0.899702 0.436505i \(-0.143784\pi\)
0.899702 + 0.436505i \(0.143784\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 11.1268i 0.366443i
\(923\) 11.8760i 0.390903i
\(924\) 0 0
\(925\) 0 0
\(926\) −49.0475 −1.61180
\(927\) 0 0
\(928\) 4.83317i 0.158656i
\(929\) −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
\(932\) 119.398i 3.91100i
\(933\) 0 0
\(934\) 61.8314 2.02319
\(935\) 0 0
\(936\) 0 0
\(937\) 49.1979i 1.60723i 0.595152 + 0.803613i \(0.297092\pi\)
−0.595152 + 0.803613i \(0.702908\pi\)
\(938\) 12.2133i 0.398777i
\(939\) 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
\(943\) 46.8452i 1.52549i
\(944\) 30.3118 0.986565
\(945\) 0 0
\(946\) −86.4068 −2.80933
\(947\) 37.1642i 1.20767i 0.797108 + 0.603837i \(0.206362\pi\)
−0.797108 + 0.603837i \(0.793638\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) 9.96265i 0.322891i
\(953\) 23.5761i 0.763706i 0.924223 + 0.381853i \(0.124714\pi\)
−0.924223 + 0.381853i \(0.875286\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −18.1422 −0.586760
\(957\) 0 0
\(958\) − 82.3502i − 2.66061i
\(959\) 2.91518 0.0941360
\(960\) 0 0
\(961\) 45.3118 1.46167
\(962\) − 0.971726i − 0.0313297i
\(963\) 0 0
\(964\) −15.5743 −0.501614
\(965\) 0 0
\(966\) 0 0
\(967\) 8.38290i 0.269576i 0.990874 + 0.134788i \(0.0430353\pi\)
−0.990874 + 0.134788i \(0.956965\pi\)
\(968\) − 0.164979i − 0.00530261i
\(969\) 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
\(973\) 4.11310i 0.131860i
\(974\) 15.1787 0.486357
\(975\) 0 0
\(976\) 44.3027 1.41810
\(977\) − 14.3310i − 0.458490i −0.973369 0.229245i \(-0.926374\pi\)
0.973369 0.229245i \(-0.0736257\pi\)
\(978\) 0 0
\(979\) −9.96265 −0.318408
\(980\) 0 0
\(981\) 0 0
\(982\) 36.3118i 1.15876i
\(983\) 32.3082i 1.03047i 0.857048 + 0.515236i \(0.172296\pi\)
−0.857048 + 0.515236i \(0.827704\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 11.5761 0.368660
\(987\) 0 0
\(988\) − 7.53880i − 0.239841i
\(989\) −42.7175 −1.35834
\(990\) 0 0
\(991\) −39.6700 −1.26016 −0.630080 0.776530i \(-0.716978\pi\)
−0.630080 + 0.776530i \(0.716978\pi\)
\(992\) 30.4513i 0.966831i
\(993\) 0 0
\(994\) 11.6218 0.368620
\(995\) 0 0
\(996\) 0 0
\(997\) 38.6874i 1.22524i 0.790377 + 0.612621i \(0.209885\pi\)
−0.790377 + 0.612621i \(0.790115\pi\)
\(998\) − 52.7258i − 1.66901i
\(999\) 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.b.l.649.6 6
3.2 odd 2 2025.2.b.m.649.1 6
5.2 odd 4 2025.2.a.n.1.1 3
5.3 odd 4 405.2.a.j.1.3 3
5.4 even 2 inner 2025.2.b.l.649.1 6
9.2 odd 6 675.2.k.b.199.1 12
9.4 even 3 225.2.k.b.124.1 12
9.5 odd 6 675.2.k.b.424.6 12
9.7 even 3 225.2.k.b.49.6 12
15.2 even 4 2025.2.a.o.1.3 3
15.8 even 4 405.2.a.i.1.1 3
15.14 odd 2 2025.2.b.m.649.6 6
20.3 even 4 6480.2.a.bv.1.3 3
45.2 even 12 675.2.e.b.226.1 6
45.4 even 6 225.2.k.b.124.6 12
45.7 odd 12 225.2.e.b.76.3 6
45.13 odd 12 45.2.e.b.16.1 6
45.14 odd 6 675.2.k.b.424.1 12
45.22 odd 12 225.2.e.b.151.3 6
45.23 even 12 135.2.e.b.46.3 6
45.29 odd 6 675.2.k.b.199.6 12
45.32 even 12 675.2.e.b.451.1 6
45.34 even 6 225.2.k.b.49.1 12
45.38 even 12 135.2.e.b.91.3 6
45.43 odd 12 45.2.e.b.31.1 yes 6
60.23 odd 4 6480.2.a.bs.1.3 3
180.23 odd 12 2160.2.q.k.721.1 6
180.43 even 12 720.2.q.i.481.1 6
180.83 odd 12 2160.2.q.k.1441.1 6
180.103 even 12 720.2.q.i.241.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.1 6 45.13 odd 12
45.2.e.b.31.1 yes 6 45.43 odd 12
135.2.e.b.46.3 6 45.23 even 12
135.2.e.b.91.3 6 45.38 even 12
225.2.e.b.76.3 6 45.7 odd 12
225.2.e.b.151.3 6 45.22 odd 12
225.2.k.b.49.1 12 45.34 even 6
225.2.k.b.49.6 12 9.7 even 3
225.2.k.b.124.1 12 9.4 even 3
225.2.k.b.124.6 12 45.4 even 6
405.2.a.i.1.1 3 15.8 even 4
405.2.a.j.1.3 3 5.3 odd 4
675.2.e.b.226.1 6 45.2 even 12
675.2.e.b.451.1 6 45.32 even 12
675.2.k.b.199.1 12 9.2 odd 6
675.2.k.b.199.6 12 45.29 odd 6
675.2.k.b.424.1 12 45.14 odd 6
675.2.k.b.424.6 12 9.5 odd 6
720.2.q.i.241.1 6 180.103 even 12
720.2.q.i.481.1 6 180.43 even 12
2025.2.a.n.1.1 3 5.2 odd 4
2025.2.a.o.1.3 3 15.2 even 4
2025.2.b.l.649.1 6 5.4 even 2 inner
2025.2.b.l.649.6 6 1.1 even 1 trivial
2025.2.b.m.649.1 6 3.2 odd 2
2025.2.b.m.649.6 6 15.14 odd 2
2160.2.q.k.721.1 6 180.23 odd 12
2160.2.q.k.1441.1 6 180.83 odd 12
6480.2.a.bs.1.3 3 60.23 odd 4
6480.2.a.bv.1.3 3 20.3 even 4