Properties

Label 2025.2.b.m.649.1
Level $2025$
Weight $2$
Character 2025.649
Analytic conductor $16.170$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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 649.1
Root \(1.66044 + 1.66044i\) of defining polynomial
Character \(\chi\) \(=\) 2025.649
Dual form 2025.2.b.m.649.6

$q$-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 \(-0.651483\pi\)
0.458137 0.888882i \(-0.348517\pi\)
\(3\) 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 \(-0.333097\pi\)
0.500642 + 0.865654i \(0.333097\pi\)
\(12\) 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 \(0.131934\pi\)
−0.915325 + 0.402716i \(0.868066\pi\)
\(18\) 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 \(0.141605\pi\)
−0.902669 + 0.430335i \(0.858395\pi\)
\(24\) 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 \(-0.541091\pi\)
−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 \(0.212927\pi\)
\(32\) − 3.48586i − 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 \(-0.153322\pi\)
0.886220 + 0.463264i \(0.153322\pi\)
\(42\) 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 \(0.115414\pi\)
−0.934983 + 0.354692i \(0.884586\pi\)
\(48\) 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 \(-0.887762\pi\)
0.938476 0.345343i \(-0.112238\pi\)
\(54\) 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 \(-0.606143\pi\)
−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 \(0.344080\pi\)
\(62\) − 21.9627i − 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 \(-0.679129\pi\)
−0.533513 + 0.845792i \(0.679129\pi\)
\(72\) 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 \(0.0269775\pi\)
−0.996411 + 0.0846508i \(0.973023\pi\)
\(84\) 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 \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 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 \(-0.697184\pi\)
−0.580606 + 0.814184i \(0.697184\pi\)
\(102\) 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 \(-0.971152\pi\)
0.995896 0.0905043i \(-0.0288479\pi\)
\(108\) 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 \(0.119595\pi\)
−0.930244 + 0.366942i \(0.880405\pi\)
\(114\) 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 \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 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 \(-0.922127\pi\)
0.970223 0.242212i \(-0.0778730\pi\)
\(138\) 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 \(-0.242568\pi\)
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 \(0.483617\pi\)
\(152\) 7.70739i 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 \(-0.923334\pi\)
0.971135 0.238530i \(-0.0766656\pi\)
\(168\) 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 \(0.105823\pi\)
−0.945245 + 0.326361i \(0.894177\pi\)
\(174\) 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 \(-0.512680\pi\)
−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 \(0.656063\pi\)
\(182\) − 1.70739i − 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 \(-0.290098\pi\)
0.612664 + 0.790343i \(0.290098\pi\)
\(192\) 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 \(0.169550\pi\)
−0.861462 + 0.507823i \(0.830450\pi\)
\(198\) 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 \(-0.964848\pi\)
0.993909 0.110207i \(-0.0351515\pi\)
\(228\) 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 \(0.360233\pi\)
−0.425116 + 0.905139i \(0.639767\pi\)
\(234\) 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 \(-0.543359\pi\)
−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 \(0.462964\pi\)
\(242\) − 0.0710844i − 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 \(-0.430352\pi\)
0.217062 + 0.976158i \(0.430352\pi\)
\(252\) 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 \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 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 \(0.0615837\pi\)
−0.981343 + 0.192266i \(0.938416\pi\)
\(264\) 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 \(-0.402172\pi\)
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 \(0.435712\pi\)
\(272\) 20.0192i 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 \(-0.346502\pi\)
0.463754 + 0.885964i \(0.346502\pi\)
\(282\) 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 \(-0.987189\pi\)
0.999190 0.0402352i \(-0.0128107\pi\)
\(294\) 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 \(-0.588052\pi\)
−0.273110 + 0.961983i \(0.588052\pi\)
\(312\) 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 \(0.193623\pi\)
−0.820629 + 0.571461i \(0.806377\pi\)
\(318\) 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 \(-0.796011\pi\)
0.801587 0.597878i \(-0.203989\pi\)
\(348\) 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 \(0.167070\pi\)
−0.865390 + 0.501098i \(0.832930\pi\)
\(354\) 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 \(-0.818152\pi\)
−0.841203 + 0.540720i \(0.818152\pi\)
\(360\) 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 \(0.0630922\pi\)
−0.980421 + 0.196915i \(0.936908\pi\)
\(384\) 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 \(-0.714682\pi\)
−0.624464 + 0.781054i \(0.714682\pi\)
\(390\) 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 \(-0.347163\pi\)
0.461913 + 0.886925i \(0.347163\pi\)
\(402\) 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 \(-0.799929\pi\)
−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 \(0.616601\pi\)
\(422\) 13.5196i 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 \(-0.210665\pi\)
0.788873 + 0.614556i \(0.210665\pi\)
\(432\) 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 \(-0.756255\pi\)
0.720865 0.693076i \(-0.243745\pi\)
\(444\) 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 \(-0.352256\pi\)
0.447666 + 0.894201i \(0.352256\pi\)
\(450\) 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 \(-0.532864\pi\)
−0.103063 + 0.994675i \(0.532864\pi\)
\(462\) 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 \(0.192679\pi\)
−0.822320 + 0.569026i \(0.807321\pi\)
\(468\) 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 \(-0.230868\pi\)
0.748304 + 0.663356i \(0.230868\pi\)
\(480\) 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 \(-0.605668\pi\)
−0.325903 + 0.945403i \(0.605668\pi\)
\(492\) 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 \(-0.962178\pi\)
0.992949 0.118543i \(-0.0378223\pi\)
\(504\) 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 \(-0.367637\pi\)
0.403949 + 0.914782i \(0.367637\pi\)
\(510\) 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 \(-0.841732\pi\)
−0.878915 + 0.476978i \(0.841732\pi\)
\(522\) 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 \(-0.880261\pi\)
0.930078 0.367362i \(-0.119739\pi\)
\(558\) 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 \(0.0882896\pi\)
−0.961779 + 0.273827i \(0.911710\pi\)
\(564\) 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 \(-0.613694\pi\)
−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 \(0.362561\pi\)
\(572\) − 18.9536i − 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 \(-0.802867\pi\)
0.814277 0.580476i \(-0.197133\pi\)
\(588\) 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 \(-0.939650\pi\)
0.982080 0.188462i \(-0.0603503\pi\)
\(594\) 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 \(-0.722286\pi\)
−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 \(0.703593\pi\)
\(602\) 13.3774i 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 \(0.0477580\pi\)
−0.988766 + 0.149474i \(0.952242\pi\)
\(618\) 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 \(-0.499163\pi\)
0.00262849 + 0.999997i \(0.499163\pi\)
\(642\) 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 \(-0.634606\pi\)
0.410385 0.911913i \(-0.365394\pi\)
\(648\) 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 \(0.252361\pi\)
−0.701842 + 0.712333i \(0.747639\pi\)
\(654\) 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 \(-0.378391\pi\)
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 \(0.217193\pi\)
\(662\) − 41.3401i − 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 \(0.0918161\pi\)
−0.958686 + 0.284465i \(0.908184\pi\)
\(678\) 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 \(-0.969794\pi\)
0.995501 0.0947515i \(-0.0302057\pi\)
\(684\) 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 \(-0.313005\pi\)
0.554251 + 0.832349i \(0.313005\pi\)
\(702\) 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 \(-0.590760\pi\)
−0.281284 + 0.959624i \(0.590760\pi\)
\(720\) 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 \(0.347329\pi\)
−0.461452 + 0.887165i \(0.652671\pi\)
\(744\) 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 \(-0.721927\pi\)
−0.642076 + 0.766641i \(0.721927\pi\)
\(762\) 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 \(0.0731116\pi\)
−0.973738 + 0.227673i \(0.926888\pi\)
\(774\) 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 \(0.182272\pi\)
−0.840483 + 0.541839i \(0.817728\pi\)
\(798\) 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 \(-0.516218\pi\)
−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 \(0.584171\pi\)
\(812\) − 3.08016i − 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 \(-0.450073\pi\)
0.156209 + 0.987724i \(0.450073\pi\)
\(822\) 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 \(-0.812417\pi\)
0.831324 0.555788i \(-0.187583\pi\)
\(828\) 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 \(-0.631541\pi\)
−0.401587 + 0.915821i \(0.631541\pi\)
\(840\) 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 \(0.0889574\pi\)
−0.961202 + 0.275844i \(0.911043\pi\)
\(858\) 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 \(0.0670734\pi\)
−0.977881 + 0.209161i \(0.932927\pi\)
\(864\) 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 \(-0.681881\pi\)
−0.540806 + 0.841147i \(0.681881\pi\)
\(882\) 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 \(-0.799268\pi\)
0.807664 0.589643i \(-0.200732\pi\)
\(888\) 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 \(-0.836289\pi\)
−0.870631 + 0.491936i \(0.836289\pi\)
\(912\) 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 \(-0.391496\pi\)
0.334311 + 0.942463i \(0.391496\pi\)
\(930\) 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 \(-0.376353\pi\)
0.378754 + 0.925497i \(0.376353\pi\)
\(942\) 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 \(-0.793638\pi\)
0.797108 0.603837i \(-0.206362\pi\)
\(948\) 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 \(-0.875286\pi\)
0.924223 0.381853i \(-0.124714\pi\)
\(954\) 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 \(-0.567975\pi\)
−0.211929 + 0.977285i \(0.567975\pi\)
\(972\) 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 \(0.0736257\pi\)
−0.973369 + 0.229245i \(0.926374\pi\)
\(978\) 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 \(-0.827704\pi\)
0.857048 0.515236i \(-0.172296\pi\)
\(984\) 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.m.649.1 6
3.2 odd 2 2025.2.b.l.649.6 6
5.2 odd 4 2025.2.a.o.1.3 3
5.3 odd 4 405.2.a.i.1.1 3
5.4 even 2 inner 2025.2.b.m.649.6 6
9.2 odd 6 225.2.k.b.49.6 12
9.4 even 3 675.2.k.b.424.6 12
9.5 odd 6 225.2.k.b.124.1 12
9.7 even 3 675.2.k.b.199.1 12
15.2 even 4 2025.2.a.n.1.1 3
15.8 even 4 405.2.a.j.1.3 3
15.14 odd 2 2025.2.b.l.649.1 6
20.3 even 4 6480.2.a.bs.1.3 3
45.2 even 12 225.2.e.b.76.3 6
45.4 even 6 675.2.k.b.424.1 12
45.7 odd 12 675.2.e.b.226.1 6
45.13 odd 12 135.2.e.b.46.3 6
45.14 odd 6 225.2.k.b.124.6 12
45.22 odd 12 675.2.e.b.451.1 6
45.23 even 12 45.2.e.b.16.1 6
45.29 odd 6 225.2.k.b.49.1 12
45.32 even 12 225.2.e.b.151.3 6
45.34 even 6 675.2.k.b.199.6 12
45.38 even 12 45.2.e.b.31.1 yes 6
45.43 odd 12 135.2.e.b.91.3 6
60.23 odd 4 6480.2.a.bv.1.3 3
180.23 odd 12 720.2.q.i.241.1 6
180.43 even 12 2160.2.q.k.1441.1 6
180.83 odd 12 720.2.q.i.481.1 6
180.103 even 12 2160.2.q.k.721.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.1 6 45.23 even 12
45.2.e.b.31.1 yes 6 45.38 even 12
135.2.e.b.46.3 6 45.13 odd 12
135.2.e.b.91.3 6 45.43 odd 12
225.2.e.b.76.3 6 45.2 even 12
225.2.e.b.151.3 6 45.32 even 12
225.2.k.b.49.1 12 45.29 odd 6
225.2.k.b.49.6 12 9.2 odd 6
225.2.k.b.124.1 12 9.5 odd 6
225.2.k.b.124.6 12 45.14 odd 6
405.2.a.i.1.1 3 5.3 odd 4
405.2.a.j.1.3 3 15.8 even 4
675.2.e.b.226.1 6 45.7 odd 12
675.2.e.b.451.1 6 45.22 odd 12
675.2.k.b.199.1 12 9.7 even 3
675.2.k.b.199.6 12 45.34 even 6
675.2.k.b.424.1 12 45.4 even 6
675.2.k.b.424.6 12 9.4 even 3
720.2.q.i.241.1 6 180.23 odd 12
720.2.q.i.481.1 6 180.83 odd 12
2025.2.a.n.1.1 3 15.2 even 4
2025.2.a.o.1.3 3 5.2 odd 4
2025.2.b.l.649.1 6 15.14 odd 2
2025.2.b.l.649.6 6 3.2 odd 2
2025.2.b.m.649.1 6 1.1 even 1 trivial
2025.2.b.m.649.6 6 5.4 even 2 inner
2160.2.q.k.721.1 6 180.103 even 12
2160.2.q.k.1441.1 6 180.43 even 12
6480.2.a.bs.1.3 3 20.3 even 4
6480.2.a.bv.1.3 3 60.23 odd 4