Properties

Label 825.2.c.g.199.1
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-0.854638 + 0.854638i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.g.199.6

$q$-expansion

\(f(q)\) \(=\) \(q-2.70928i q^{2} -1.00000i q^{3} -5.34017 q^{4} -2.70928 q^{6} +1.07838i q^{7} +9.04945i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-2.70928i q^{2} -1.00000i q^{3} -5.34017 q^{4} -2.70928 q^{6} +1.07838i q^{7} +9.04945i q^{8} -1.00000 q^{9} +1.00000 q^{11} +5.34017i q^{12} +4.34017i q^{13} +2.92162 q^{14} +13.8371 q^{16} +7.75872i q^{17} +2.70928i q^{18} -5.26180 q^{19} +1.07838 q^{21} -2.70928i q^{22} +2.15676i q^{23} +9.04945 q^{24} +11.7587 q^{26} +1.00000i q^{27} -5.75872i q^{28} -1.41855 q^{29} -4.68035 q^{31} -19.3896i q^{32} -1.00000i q^{33} +21.0205 q^{34} +5.34017 q^{36} -2.00000i q^{37} +14.2557i q^{38} +4.34017 q^{39} -9.41855 q^{41} -2.92162i q^{42} -7.60197i q^{43} -5.34017 q^{44} +5.84324 q^{46} +4.68035i q^{47} -13.8371i q^{48} +5.83710 q^{49} +7.75872 q^{51} -23.1773i q^{52} -0.156755i q^{53} +2.70928 q^{54} -9.75872 q^{56} +5.26180i q^{57} +3.84324i q^{58} -6.15676 q^{59} -4.15676 q^{61} +12.6803i q^{62} -1.07838i q^{63} -24.8576 q^{64} -2.70928 q^{66} -8.68035i q^{67} -41.4329i q^{68} +2.15676 q^{69} -4.68035 q^{71} -9.04945i q^{72} +10.4969i q^{73} -5.41855 q^{74} +28.0989 q^{76} +1.07838i q^{77} -11.7587i q^{78} +8.09890 q^{79} +1.00000 q^{81} +25.5174i q^{82} +11.0205i q^{83} -5.75872 q^{84} -20.5958 q^{86} +1.41855i q^{87} +9.04945i q^{88} +12.8371 q^{89} -4.68035 q^{91} -11.5174i q^{92} +4.68035i q^{93} +12.6803 q^{94} -19.3896 q^{96} +14.6803i q^{97} -15.8143i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.70928i − 1.91575i −0.287190 0.957873i \(-0.592721\pi\)
0.287190 0.957873i \(-0.407279\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −5.34017 −2.67009
\(5\) 0 0
\(6\) −2.70928 −1.10606
\(7\) 1.07838i 0.407588i 0.979014 + 0.203794i \(0.0653274\pi\)
−0.979014 + 0.203794i \(0.934673\pi\)
\(8\) 9.04945i 3.19946i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 5.34017i 1.54158i
\(13\) 4.34017i 1.20375i 0.798591 + 0.601874i \(0.205579\pi\)
−0.798591 + 0.601874i \(0.794421\pi\)
\(14\) 2.92162 0.780836
\(15\) 0 0
\(16\) 13.8371 3.45928
\(17\) 7.75872i 1.88177i 0.338730 + 0.940883i \(0.390003\pi\)
−0.338730 + 0.940883i \(0.609997\pi\)
\(18\) 2.70928i 0.638582i
\(19\) −5.26180 −1.20714 −0.603569 0.797311i \(-0.706255\pi\)
−0.603569 + 0.797311i \(0.706255\pi\)
\(20\) 0 0
\(21\) 1.07838 0.235321
\(22\) − 2.70928i − 0.577619i
\(23\) 2.15676i 0.449715i 0.974392 + 0.224857i \(0.0721916\pi\)
−0.974392 + 0.224857i \(0.927808\pi\)
\(24\) 9.04945 1.84721
\(25\) 0 0
\(26\) 11.7587 2.30608
\(27\) 1.00000i 0.192450i
\(28\) − 5.75872i − 1.08830i
\(29\) −1.41855 −0.263418 −0.131709 0.991288i \(-0.542046\pi\)
−0.131709 + 0.991288i \(0.542046\pi\)
\(30\) 0 0
\(31\) −4.68035 −0.840615 −0.420307 0.907382i \(-0.638078\pi\)
−0.420307 + 0.907382i \(0.638078\pi\)
\(32\) − 19.3896i − 3.42763i
\(33\) − 1.00000i − 0.174078i
\(34\) 21.0205 3.60499
\(35\) 0 0
\(36\) 5.34017 0.890029
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 14.2557i 2.31257i
\(39\) 4.34017 0.694984
\(40\) 0 0
\(41\) −9.41855 −1.47093 −0.735465 0.677562i \(-0.763036\pi\)
−0.735465 + 0.677562i \(0.763036\pi\)
\(42\) − 2.92162i − 0.450816i
\(43\) − 7.60197i − 1.15929i −0.814869 0.579645i \(-0.803191\pi\)
0.814869 0.579645i \(-0.196809\pi\)
\(44\) −5.34017 −0.805061
\(45\) 0 0
\(46\) 5.84324 0.861539
\(47\) 4.68035i 0.682699i 0.939937 + 0.341349i \(0.110884\pi\)
−0.939937 + 0.341349i \(0.889116\pi\)
\(48\) − 13.8371i − 1.99721i
\(49\) 5.83710 0.833872
\(50\) 0 0
\(51\) 7.75872 1.08644
\(52\) − 23.1773i − 3.21411i
\(53\) − 0.156755i − 0.0215320i −0.999942 0.0107660i \(-0.996573\pi\)
0.999942 0.0107660i \(-0.00342699\pi\)
\(54\) 2.70928 0.368686
\(55\) 0 0
\(56\) −9.75872 −1.30406
\(57\) 5.26180i 0.696942i
\(58\) 3.84324i 0.504643i
\(59\) −6.15676 −0.801541 −0.400771 0.916178i \(-0.631258\pi\)
−0.400771 + 0.916178i \(0.631258\pi\)
\(60\) 0 0
\(61\) −4.15676 −0.532218 −0.266109 0.963943i \(-0.585738\pi\)
−0.266109 + 0.963943i \(0.585738\pi\)
\(62\) 12.6803i 1.61041i
\(63\) − 1.07838i − 0.135863i
\(64\) −24.8576 −3.10720
\(65\) 0 0
\(66\) −2.70928 −0.333489
\(67\) − 8.68035i − 1.06047i −0.847850 0.530237i \(-0.822103\pi\)
0.847850 0.530237i \(-0.177897\pi\)
\(68\) − 41.4329i − 5.02448i
\(69\) 2.15676 0.259643
\(70\) 0 0
\(71\) −4.68035 −0.555455 −0.277727 0.960660i \(-0.589581\pi\)
−0.277727 + 0.960660i \(0.589581\pi\)
\(72\) − 9.04945i − 1.06649i
\(73\) 10.4969i 1.22857i 0.789083 + 0.614286i \(0.210556\pi\)
−0.789083 + 0.614286i \(0.789444\pi\)
\(74\) −5.41855 −0.629894
\(75\) 0 0
\(76\) 28.0989 3.22316
\(77\) 1.07838i 0.122893i
\(78\) − 11.7587i − 1.33141i
\(79\) 8.09890 0.911197 0.455599 0.890185i \(-0.349425\pi\)
0.455599 + 0.890185i \(0.349425\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 25.5174i 2.81793i
\(83\) 11.0205i 1.20966i 0.796355 + 0.604830i \(0.206759\pi\)
−0.796355 + 0.604830i \(0.793241\pi\)
\(84\) −5.75872 −0.628328
\(85\) 0 0
\(86\) −20.5958 −2.22090
\(87\) 1.41855i 0.152085i
\(88\) 9.04945i 0.964674i
\(89\) 12.8371 1.36073 0.680365 0.732873i \(-0.261821\pi\)
0.680365 + 0.732873i \(0.261821\pi\)
\(90\) 0 0
\(91\) −4.68035 −0.490634
\(92\) − 11.5174i − 1.20078i
\(93\) 4.68035i 0.485329i
\(94\) 12.6803 1.30788
\(95\) 0 0
\(96\) −19.3896 −1.97894
\(97\) 14.6803i 1.49056i 0.666750 + 0.745282i \(0.267685\pi\)
−0.666750 + 0.745282i \(0.732315\pi\)
\(98\) − 15.8143i − 1.59749i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −15.5753 −1.54980 −0.774900 0.632083i \(-0.782200\pi\)
−0.774900 + 0.632083i \(0.782200\pi\)
\(102\) − 21.0205i − 2.08134i
\(103\) − 6.83710i − 0.673680i −0.941562 0.336840i \(-0.890642\pi\)
0.941562 0.336840i \(-0.109358\pi\)
\(104\) −39.2762 −3.85135
\(105\) 0 0
\(106\) −0.424694 −0.0412499
\(107\) 6.34017i 0.612928i 0.951882 + 0.306464i \(0.0991458\pi\)
−0.951882 + 0.306464i \(0.900854\pi\)
\(108\) − 5.34017i − 0.513858i
\(109\) −2.31351 −0.221594 −0.110797 0.993843i \(-0.535340\pi\)
−0.110797 + 0.993843i \(0.535340\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 14.9216i 1.40996i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 14.2557 1.33516
\(115\) 0 0
\(116\) 7.57531 0.703350
\(117\) − 4.34017i − 0.401249i
\(118\) 16.6803i 1.53555i
\(119\) −8.36683 −0.766987
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 11.2618i 1.01960i
\(123\) 9.41855i 0.849242i
\(124\) 24.9939 2.24451
\(125\) 0 0
\(126\) −2.92162 −0.260279
\(127\) − 2.24128i − 0.198881i −0.995044 0.0994406i \(-0.968295\pi\)
0.995044 0.0994406i \(-0.0317053\pi\)
\(128\) 28.5669i 2.52498i
\(129\) −7.60197 −0.669316
\(130\) 0 0
\(131\) 8.68035 0.758405 0.379203 0.925314i \(-0.376198\pi\)
0.379203 + 0.925314i \(0.376198\pi\)
\(132\) 5.34017i 0.464802i
\(133\) − 5.67420i − 0.492016i
\(134\) −23.5174 −2.03160
\(135\) 0 0
\(136\) −70.2122 −6.02064
\(137\) − 15.3607i − 1.31235i −0.754608 0.656176i \(-0.772173\pi\)
0.754608 0.656176i \(-0.227827\pi\)
\(138\) − 5.84324i − 0.497410i
\(139\) −8.58145 −0.727869 −0.363935 0.931425i \(-0.618567\pi\)
−0.363935 + 0.931425i \(0.618567\pi\)
\(140\) 0 0
\(141\) 4.68035 0.394156
\(142\) 12.6803i 1.06411i
\(143\) 4.34017i 0.362943i
\(144\) −13.8371 −1.15309
\(145\) 0 0
\(146\) 28.4391 2.35363
\(147\) − 5.83710i − 0.481436i
\(148\) 10.6803i 0.877919i
\(149\) 18.0989 1.48272 0.741360 0.671108i \(-0.234181\pi\)
0.741360 + 0.671108i \(0.234181\pi\)
\(150\) 0 0
\(151\) 22.9360 1.86651 0.933253 0.359221i \(-0.116958\pi\)
0.933253 + 0.359221i \(0.116958\pi\)
\(152\) − 47.6163i − 3.86220i
\(153\) − 7.75872i − 0.627256i
\(154\) 2.92162 0.235431
\(155\) 0 0
\(156\) −23.1773 −1.85567
\(157\) − 10.9939i − 0.877405i −0.898632 0.438703i \(-0.855438\pi\)
0.898632 0.438703i \(-0.144562\pi\)
\(158\) − 21.9421i − 1.74562i
\(159\) −0.156755 −0.0124315
\(160\) 0 0
\(161\) −2.32580 −0.183298
\(162\) − 2.70928i − 0.212861i
\(163\) 6.52359i 0.510967i 0.966813 + 0.255484i \(0.0822347\pi\)
−0.966813 + 0.255484i \(0.917765\pi\)
\(164\) 50.2967 3.92751
\(165\) 0 0
\(166\) 29.8576 2.31740
\(167\) 1.97334i 0.152701i 0.997081 + 0.0763507i \(0.0243269\pi\)
−0.997081 + 0.0763507i \(0.975673\pi\)
\(168\) 9.75872i 0.752902i
\(169\) −5.83710 −0.449008
\(170\) 0 0
\(171\) 5.26180 0.402380
\(172\) 40.5958i 3.09540i
\(173\) − 3.75872i − 0.285770i −0.989739 0.142885i \(-0.954362\pi\)
0.989739 0.142885i \(-0.0456380\pi\)
\(174\) 3.84324 0.291356
\(175\) 0 0
\(176\) 13.8371 1.04301
\(177\) 6.15676i 0.462770i
\(178\) − 34.7792i − 2.60681i
\(179\) −15.1506 −1.13241 −0.566205 0.824264i \(-0.691589\pi\)
−0.566205 + 0.824264i \(0.691589\pi\)
\(180\) 0 0
\(181\) 4.83710 0.359539 0.179769 0.983709i \(-0.442465\pi\)
0.179769 + 0.983709i \(0.442465\pi\)
\(182\) 12.6803i 0.939930i
\(183\) 4.15676i 0.307276i
\(184\) −19.5174 −1.43885
\(185\) 0 0
\(186\) 12.6803 0.929768
\(187\) 7.75872i 0.567374i
\(188\) − 24.9939i − 1.82286i
\(189\) −1.07838 −0.0784404
\(190\) 0 0
\(191\) 2.52359 0.182601 0.0913003 0.995823i \(-0.470898\pi\)
0.0913003 + 0.995823i \(0.470898\pi\)
\(192\) 24.8576i 1.79394i
\(193\) − 0.0266620i − 0.00191917i −1.00000 0.000959586i \(-0.999695\pi\)
1.00000 0.000959586i \(-0.000305446\pi\)
\(194\) 39.7731 2.85554
\(195\) 0 0
\(196\) −31.1711 −2.22651
\(197\) 21.1194i 1.50470i 0.658766 + 0.752348i \(0.271079\pi\)
−0.658766 + 0.752348i \(0.728921\pi\)
\(198\) 2.70928i 0.192540i
\(199\) −10.5236 −0.745998 −0.372999 0.927832i \(-0.621670\pi\)
−0.372999 + 0.927832i \(0.621670\pi\)
\(200\) 0 0
\(201\) −8.68035 −0.612264
\(202\) 42.1978i 2.96903i
\(203\) − 1.52973i − 0.107366i
\(204\) −41.4329 −2.90089
\(205\) 0 0
\(206\) −18.5236 −1.29060
\(207\) − 2.15676i − 0.149905i
\(208\) 60.0554i 4.16409i
\(209\) −5.26180 −0.363966
\(210\) 0 0
\(211\) 9.57531 0.659191 0.329596 0.944122i \(-0.393088\pi\)
0.329596 + 0.944122i \(0.393088\pi\)
\(212\) 0.837101i 0.0574924i
\(213\) 4.68035i 0.320692i
\(214\) 17.1773 1.17421
\(215\) 0 0
\(216\) −9.04945 −0.615737
\(217\) − 5.04718i − 0.342625i
\(218\) 6.26794i 0.424518i
\(219\) 10.4969 0.709317
\(220\) 0 0
\(221\) −33.6742 −2.26517
\(222\) 5.41855i 0.363669i
\(223\) 2.15676i 0.144427i 0.997389 + 0.0722135i \(0.0230063\pi\)
−0.997389 + 0.0722135i \(0.976994\pi\)
\(224\) 20.9093 1.39706
\(225\) 0 0
\(226\) 16.2557 1.08131
\(227\) 9.65983i 0.641145i 0.947224 + 0.320573i \(0.103875\pi\)
−0.947224 + 0.320573i \(0.896125\pi\)
\(228\) − 28.0989i − 1.86089i
\(229\) 3.36069 0.222081 0.111040 0.993816i \(-0.464582\pi\)
0.111040 + 0.993816i \(0.464582\pi\)
\(230\) 0 0
\(231\) 1.07838 0.0709520
\(232\) − 12.8371i − 0.842797i
\(233\) 2.39803i 0.157100i 0.996910 + 0.0785501i \(0.0250291\pi\)
−0.996910 + 0.0785501i \(0.974971\pi\)
\(234\) −11.7587 −0.768692
\(235\) 0 0
\(236\) 32.8781 2.14018
\(237\) − 8.09890i − 0.526080i
\(238\) 22.6681i 1.46935i
\(239\) 7.20394 0.465984 0.232992 0.972479i \(-0.425148\pi\)
0.232992 + 0.972479i \(0.425148\pi\)
\(240\) 0 0
\(241\) −5.20394 −0.335215 −0.167608 0.985854i \(-0.553604\pi\)
−0.167608 + 0.985854i \(0.553604\pi\)
\(242\) − 2.70928i − 0.174159i
\(243\) − 1.00000i − 0.0641500i
\(244\) 22.1978 1.42107
\(245\) 0 0
\(246\) 25.5174 1.62693
\(247\) − 22.8371i − 1.45309i
\(248\) − 42.3545i − 2.68952i
\(249\) 11.0205 0.698397
\(250\) 0 0
\(251\) 15.3197 0.966968 0.483484 0.875353i \(-0.339371\pi\)
0.483484 + 0.875353i \(0.339371\pi\)
\(252\) 5.75872i 0.362765i
\(253\) 2.15676i 0.135594i
\(254\) −6.07223 −0.381006
\(255\) 0 0
\(256\) 27.6803 1.73002
\(257\) 4.15676i 0.259291i 0.991560 + 0.129646i \(0.0413840\pi\)
−0.991560 + 0.129646i \(0.958616\pi\)
\(258\) 20.5958i 1.28224i
\(259\) 2.15676 0.134014
\(260\) 0 0
\(261\) 1.41855 0.0878061
\(262\) − 23.5174i − 1.45291i
\(263\) 18.7070i 1.15352i 0.816912 + 0.576762i \(0.195684\pi\)
−0.816912 + 0.576762i \(0.804316\pi\)
\(264\) 9.04945 0.556955
\(265\) 0 0
\(266\) −15.3730 −0.942578
\(267\) − 12.8371i − 0.785618i
\(268\) 46.3545i 2.83155i
\(269\) −23.3607 −1.42433 −0.712163 0.702014i \(-0.752284\pi\)
−0.712163 + 0.702014i \(0.752284\pi\)
\(270\) 0 0
\(271\) −5.57531 −0.338676 −0.169338 0.985558i \(-0.554163\pi\)
−0.169338 + 0.985558i \(0.554163\pi\)
\(272\) 107.358i 6.50955i
\(273\) 4.68035i 0.283267i
\(274\) −41.6163 −2.51414
\(275\) 0 0
\(276\) −11.5174 −0.693269
\(277\) − 26.0144i − 1.56305i −0.623872 0.781526i \(-0.714442\pi\)
0.623872 0.781526i \(-0.285558\pi\)
\(278\) 23.2495i 1.39441i
\(279\) 4.68035 0.280205
\(280\) 0 0
\(281\) −9.41855 −0.561864 −0.280932 0.959728i \(-0.590643\pi\)
−0.280932 + 0.959728i \(0.590643\pi\)
\(282\) − 12.6803i − 0.755104i
\(283\) − 14.2413i − 0.846556i −0.906000 0.423278i \(-0.860879\pi\)
0.906000 0.423278i \(-0.139121\pi\)
\(284\) 24.9939 1.48311
\(285\) 0 0
\(286\) 11.7587 0.695308
\(287\) − 10.1568i − 0.599534i
\(288\) 19.3896i 1.14254i
\(289\) −43.1978 −2.54105
\(290\) 0 0
\(291\) 14.6803 0.860577
\(292\) − 56.0554i − 3.28039i
\(293\) 15.7587i 0.920634i 0.887754 + 0.460317i \(0.152264\pi\)
−0.887754 + 0.460317i \(0.847736\pi\)
\(294\) −15.8143 −0.922310
\(295\) 0 0
\(296\) 18.0989 1.05198
\(297\) 1.00000i 0.0580259i
\(298\) − 49.0349i − 2.84052i
\(299\) −9.36069 −0.541343
\(300\) 0 0
\(301\) 8.19779 0.472513
\(302\) − 62.1399i − 3.57575i
\(303\) 15.5753i 0.894778i
\(304\) −72.8080 −4.17582
\(305\) 0 0
\(306\) −21.0205 −1.20166
\(307\) − 18.9216i − 1.07991i −0.841693 0.539957i \(-0.818440\pi\)
0.841693 0.539957i \(-0.181560\pi\)
\(308\) − 5.75872i − 0.328134i
\(309\) −6.83710 −0.388949
\(310\) 0 0
\(311\) −20.8781 −1.18389 −0.591945 0.805978i \(-0.701640\pi\)
−0.591945 + 0.805978i \(0.701640\pi\)
\(312\) 39.2762i 2.22358i
\(313\) − 6.31351i − 0.356861i −0.983953 0.178430i \(-0.942898\pi\)
0.983953 0.178430i \(-0.0571019\pi\)
\(314\) −29.7854 −1.68089
\(315\) 0 0
\(316\) −43.2495 −2.43297
\(317\) 31.3607i 1.76139i 0.473682 + 0.880696i \(0.342925\pi\)
−0.473682 + 0.880696i \(0.657075\pi\)
\(318\) 0.424694i 0.0238156i
\(319\) −1.41855 −0.0794236
\(320\) 0 0
\(321\) 6.34017 0.353874
\(322\) 6.30122i 0.351154i
\(323\) − 40.8248i − 2.27155i
\(324\) −5.34017 −0.296676
\(325\) 0 0
\(326\) 17.6742 0.978884
\(327\) 2.31351i 0.127937i
\(328\) − 85.2327i − 4.70619i
\(329\) −5.04718 −0.278260
\(330\) 0 0
\(331\) 19.2039 1.05554 0.527772 0.849386i \(-0.323028\pi\)
0.527772 + 0.849386i \(0.323028\pi\)
\(332\) − 58.8515i − 3.22989i
\(333\) 2.00000i 0.109599i
\(334\) 5.34632 0.292537
\(335\) 0 0
\(336\) 14.9216 0.814041
\(337\) 13.5031i 0.735559i 0.929913 + 0.367780i \(0.119882\pi\)
−0.929913 + 0.367780i \(0.880118\pi\)
\(338\) 15.8143i 0.860185i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −4.68035 −0.253455
\(342\) − 14.2557i − 0.770857i
\(343\) 13.8432i 0.747465i
\(344\) 68.7936 3.70910
\(345\) 0 0
\(346\) −10.1834 −0.547464
\(347\) 6.34017i 0.340358i 0.985413 + 0.170179i \(0.0544346\pi\)
−0.985413 + 0.170179i \(0.945565\pi\)
\(348\) − 7.57531i − 0.406079i
\(349\) −16.1568 −0.864851 −0.432426 0.901670i \(-0.642342\pi\)
−0.432426 + 0.901670i \(0.642342\pi\)
\(350\) 0 0
\(351\) −4.34017 −0.231661
\(352\) − 19.3896i − 1.03347i
\(353\) 13.2039i 0.702775i 0.936230 + 0.351387i \(0.114290\pi\)
−0.936230 + 0.351387i \(0.885710\pi\)
\(354\) 16.6803 0.886550
\(355\) 0 0
\(356\) −68.5523 −3.63327
\(357\) 8.36683i 0.442820i
\(358\) 41.0472i 2.16941i
\(359\) −3.31965 −0.175205 −0.0876023 0.996156i \(-0.527920\pi\)
−0.0876023 + 0.996156i \(0.527920\pi\)
\(360\) 0 0
\(361\) 8.68649 0.457184
\(362\) − 13.1050i − 0.688786i
\(363\) − 1.00000i − 0.0524864i
\(364\) 24.9939 1.31003
\(365\) 0 0
\(366\) 11.2618 0.588663
\(367\) − 36.1445i − 1.88673i −0.331762 0.943363i \(-0.607643\pi\)
0.331762 0.943363i \(-0.392357\pi\)
\(368\) 29.8432i 1.55569i
\(369\) 9.41855 0.490310
\(370\) 0 0
\(371\) 0.169042 0.00877620
\(372\) − 24.9939i − 1.29587i
\(373\) 2.81044i 0.145519i 0.997350 + 0.0727595i \(0.0231806\pi\)
−0.997350 + 0.0727595i \(0.976819\pi\)
\(374\) 21.0205 1.08695
\(375\) 0 0
\(376\) −42.3545 −2.18427
\(377\) − 6.15676i − 0.317089i
\(378\) 2.92162i 0.150272i
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −2.24128 −0.114824
\(382\) − 6.83710i − 0.349817i
\(383\) − 33.5585i − 1.71476i −0.514685 0.857379i \(-0.672091\pi\)
0.514685 0.857379i \(-0.327909\pi\)
\(384\) 28.5669 1.45780
\(385\) 0 0
\(386\) −0.0722347 −0.00367665
\(387\) 7.60197i 0.386430i
\(388\) − 78.3956i − 3.97993i
\(389\) −12.8371 −0.650867 −0.325433 0.945565i \(-0.605510\pi\)
−0.325433 + 0.945565i \(0.605510\pi\)
\(390\) 0 0
\(391\) −16.7337 −0.846258
\(392\) 52.8225i 2.66794i
\(393\) − 8.68035i − 0.437866i
\(394\) 57.2183 2.88262
\(395\) 0 0
\(396\) 5.34017 0.268354
\(397\) − 5.31965i − 0.266986i −0.991050 0.133493i \(-0.957381\pi\)
0.991050 0.133493i \(-0.0426193\pi\)
\(398\) 28.5113i 1.42914i
\(399\) −5.67420 −0.284065
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 23.5174i 1.17294i
\(403\) − 20.3135i − 1.01189i
\(404\) 83.1748 4.13810
\(405\) 0 0
\(406\) −4.14447 −0.205687
\(407\) − 2.00000i − 0.0991363i
\(408\) 70.2122i 3.47602i
\(409\) −26.1978 −1.29540 −0.647699 0.761897i \(-0.724268\pi\)
−0.647699 + 0.761897i \(0.724268\pi\)
\(410\) 0 0
\(411\) −15.3607 −0.757687
\(412\) 36.5113i 1.79878i
\(413\) − 6.63931i − 0.326699i
\(414\) −5.84324 −0.287180
\(415\) 0 0
\(416\) 84.1543 4.12600
\(417\) 8.58145i 0.420235i
\(418\) 14.2557i 0.697267i
\(419\) 2.83710 0.138601 0.0693007 0.997596i \(-0.477923\pi\)
0.0693007 + 0.997596i \(0.477923\pi\)
\(420\) 0 0
\(421\) 11.4764 0.559326 0.279663 0.960098i \(-0.409777\pi\)
0.279663 + 0.960098i \(0.409777\pi\)
\(422\) − 25.9421i − 1.26284i
\(423\) − 4.68035i − 0.227566i
\(424\) 1.41855 0.0688909
\(425\) 0 0
\(426\) 12.6803 0.614365
\(427\) − 4.48255i − 0.216926i
\(428\) − 33.8576i − 1.63657i
\(429\) 4.34017 0.209546
\(430\) 0 0
\(431\) 23.5708 1.13536 0.567682 0.823248i \(-0.307840\pi\)
0.567682 + 0.823248i \(0.307840\pi\)
\(432\) 13.8371i 0.665738i
\(433\) 14.9939i 0.720559i 0.932844 + 0.360279i \(0.117319\pi\)
−0.932844 + 0.360279i \(0.882681\pi\)
\(434\) −13.6742 −0.656383
\(435\) 0 0
\(436\) 12.3545 0.591676
\(437\) − 11.3484i − 0.542868i
\(438\) − 28.4391i − 1.35887i
\(439\) −4.77924 −0.228101 −0.114050 0.993475i \(-0.536383\pi\)
−0.114050 + 0.993475i \(0.536383\pi\)
\(440\) 0 0
\(441\) −5.83710 −0.277957
\(442\) 91.2327i 4.33950i
\(443\) − 20.1978i − 0.959626i −0.877371 0.479813i \(-0.840704\pi\)
0.877371 0.479813i \(-0.159296\pi\)
\(444\) 10.6803 0.506867
\(445\) 0 0
\(446\) 5.84324 0.276686
\(447\) − 18.0989i − 0.856048i
\(448\) − 26.8059i − 1.26646i
\(449\) 21.5708 1.01799 0.508994 0.860770i \(-0.330018\pi\)
0.508994 + 0.860770i \(0.330018\pi\)
\(450\) 0 0
\(451\) −9.41855 −0.443502
\(452\) − 32.0410i − 1.50708i
\(453\) − 22.9360i − 1.07763i
\(454\) 26.1711 1.22827
\(455\) 0 0
\(456\) −47.6163 −2.22984
\(457\) 28.1711i 1.31779i 0.752235 + 0.658895i \(0.228976\pi\)
−0.752235 + 0.658895i \(0.771024\pi\)
\(458\) − 9.10504i − 0.425451i
\(459\) −7.75872 −0.362146
\(460\) 0 0
\(461\) −1.47187 −0.0685520 −0.0342760 0.999412i \(-0.510913\pi\)
−0.0342760 + 0.999412i \(0.510913\pi\)
\(462\) − 2.92162i − 0.135926i
\(463\) 23.2039i 1.07838i 0.842185 + 0.539189i \(0.181269\pi\)
−0.842185 + 0.539189i \(0.818731\pi\)
\(464\) −19.6286 −0.911236
\(465\) 0 0
\(466\) 6.49693 0.300964
\(467\) 14.1568i 0.655097i 0.944834 + 0.327548i \(0.106222\pi\)
−0.944834 + 0.327548i \(0.893778\pi\)
\(468\) 23.1773i 1.07137i
\(469\) 9.36069 0.432237
\(470\) 0 0
\(471\) −10.9939 −0.506570
\(472\) − 55.7152i − 2.56450i
\(473\) − 7.60197i − 0.349539i
\(474\) −21.9421 −1.00784
\(475\) 0 0
\(476\) 44.6803 2.04792
\(477\) 0.156755i 0.00717734i
\(478\) − 19.5174i − 0.892707i
\(479\) 13.8432 0.632514 0.316257 0.948674i \(-0.397574\pi\)
0.316257 + 0.948674i \(0.397574\pi\)
\(480\) 0 0
\(481\) 8.68035 0.395790
\(482\) 14.0989i 0.642187i
\(483\) 2.32580i 0.105827i
\(484\) −5.34017 −0.242735
\(485\) 0 0
\(486\) −2.70928 −0.122895
\(487\) − 40.9939i − 1.85761i −0.370570 0.928804i \(-0.620838\pi\)
0.370570 0.928804i \(-0.379162\pi\)
\(488\) − 37.6163i − 1.70281i
\(489\) 6.52359 0.295007
\(490\) 0 0
\(491\) 34.8371 1.57218 0.786088 0.618114i \(-0.212103\pi\)
0.786088 + 0.618114i \(0.212103\pi\)
\(492\) − 50.2967i − 2.26755i
\(493\) − 11.0061i − 0.495692i
\(494\) −61.8720 −2.78375
\(495\) 0 0
\(496\) −64.7624 −2.90792
\(497\) − 5.04718i − 0.226397i
\(498\) − 29.8576i − 1.33795i
\(499\) −15.1506 −0.678235 −0.339117 0.940744i \(-0.610128\pi\)
−0.339117 + 0.940744i \(0.610128\pi\)
\(500\) 0 0
\(501\) 1.97334 0.0881622
\(502\) − 41.5052i − 1.85247i
\(503\) 6.65368i 0.296673i 0.988937 + 0.148337i \(0.0473919\pi\)
−0.988937 + 0.148337i \(0.952608\pi\)
\(504\) 9.75872 0.434688
\(505\) 0 0
\(506\) 5.84324 0.259764
\(507\) 5.83710i 0.259235i
\(508\) 11.9688i 0.531030i
\(509\) −41.3484 −1.83274 −0.916368 0.400337i \(-0.868893\pi\)
−0.916368 + 0.400337i \(0.868893\pi\)
\(510\) 0 0
\(511\) −11.3197 −0.500752
\(512\) − 17.8599i − 0.789303i
\(513\) − 5.26180i − 0.232314i
\(514\) 11.2618 0.496736
\(515\) 0 0
\(516\) 40.5958 1.78713
\(517\) 4.68035i 0.205841i
\(518\) − 5.84324i − 0.256737i
\(519\) −3.75872 −0.164990
\(520\) 0 0
\(521\) 7.67420 0.336213 0.168106 0.985769i \(-0.446235\pi\)
0.168106 + 0.985769i \(0.446235\pi\)
\(522\) − 3.84324i − 0.168214i
\(523\) − 23.2351i − 1.01600i −0.861357 0.508001i \(-0.830385\pi\)
0.861357 0.508001i \(-0.169615\pi\)
\(524\) −46.3545 −2.02501
\(525\) 0 0
\(526\) 50.6824 2.20986
\(527\) − 36.3135i − 1.58184i
\(528\) − 13.8371i − 0.602183i
\(529\) 18.3484 0.797757
\(530\) 0 0
\(531\) 6.15676 0.267180
\(532\) 30.3012i 1.31372i
\(533\) − 40.8781i − 1.77063i
\(534\) −34.7792 −1.50505
\(535\) 0 0
\(536\) 78.5523 3.39294
\(537\) 15.1506i 0.653797i
\(538\) 63.2905i 2.72865i
\(539\) 5.83710 0.251422
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 15.1050i 0.648817i
\(543\) − 4.83710i − 0.207580i
\(544\) 150.439 6.45001
\(545\) 0 0
\(546\) 12.6803 0.542669
\(547\) − 23.0661i − 0.986235i −0.869963 0.493117i \(-0.835857\pi\)
0.869963 0.493117i \(-0.164143\pi\)
\(548\) 82.0288i 3.50409i
\(549\) 4.15676 0.177406
\(550\) 0 0
\(551\) 7.46412 0.317982
\(552\) 19.5174i 0.830718i
\(553\) 8.73367i 0.371393i
\(554\) −70.4801 −2.99441
\(555\) 0 0
\(556\) 45.8264 1.94347
\(557\) 10.5958i 0.448960i 0.974479 + 0.224480i \(0.0720683\pi\)
−0.974479 + 0.224480i \(0.927932\pi\)
\(558\) − 12.6803i − 0.536802i
\(559\) 32.9939 1.39549
\(560\) 0 0
\(561\) 7.75872 0.327574
\(562\) 25.5174i 1.07639i
\(563\) 36.2122i 1.52616i 0.646303 + 0.763080i \(0.276314\pi\)
−0.646303 + 0.763080i \(0.723686\pi\)
\(564\) −24.9939 −1.05243
\(565\) 0 0
\(566\) −38.5835 −1.62179
\(567\) 1.07838i 0.0452876i
\(568\) − 42.3545i − 1.77716i
\(569\) 27.5753 1.15602 0.578008 0.816031i \(-0.303830\pi\)
0.578008 + 0.816031i \(0.303830\pi\)
\(570\) 0 0
\(571\) −27.9299 −1.16883 −0.584414 0.811456i \(-0.698676\pi\)
−0.584414 + 0.811456i \(0.698676\pi\)
\(572\) − 23.1773i − 0.969091i
\(573\) − 2.52359i − 0.105425i
\(574\) −27.5174 −1.14856
\(575\) 0 0
\(576\) 24.8576 1.03573
\(577\) 41.4017i 1.72358i 0.507268 + 0.861788i \(0.330655\pi\)
−0.507268 + 0.861788i \(0.669345\pi\)
\(578\) 117.035i 4.86800i
\(579\) −0.0266620 −0.00110803
\(580\) 0 0
\(581\) −11.8843 −0.493043
\(582\) − 39.7731i − 1.64865i
\(583\) − 0.156755i − 0.00649215i
\(584\) −94.9914 −3.93077
\(585\) 0 0
\(586\) 42.6947 1.76370
\(587\) 8.48255i 0.350112i 0.984558 + 0.175056i \(0.0560107\pi\)
−0.984558 + 0.175056i \(0.943989\pi\)
\(588\) 31.1711i 1.28548i
\(589\) 24.6270 1.01474
\(590\) 0 0
\(591\) 21.1194 0.868737
\(592\) − 27.6742i − 1.13740i
\(593\) − 7.56093i − 0.310490i −0.987876 0.155245i \(-0.950383\pi\)
0.987876 0.155245i \(-0.0496167\pi\)
\(594\) 2.70928 0.111163
\(595\) 0 0
\(596\) −96.6512 −3.95899
\(597\) 10.5236i 0.430702i
\(598\) 25.3607i 1.03708i
\(599\) −5.67420 −0.231842 −0.115921 0.993258i \(-0.536982\pi\)
−0.115921 + 0.993258i \(0.536982\pi\)
\(600\) 0 0
\(601\) −1.31965 −0.0538298 −0.0269149 0.999638i \(-0.508568\pi\)
−0.0269149 + 0.999638i \(0.508568\pi\)
\(602\) − 22.2101i − 0.905215i
\(603\) 8.68035i 0.353491i
\(604\) −122.482 −4.98373
\(605\) 0 0
\(606\) 42.1978 1.71417
\(607\) − 2.24128i − 0.0909706i −0.998965 0.0454853i \(-0.985517\pi\)
0.998965 0.0454853i \(-0.0144834\pi\)
\(608\) 102.024i 4.13763i
\(609\) −1.52973 −0.0619879
\(610\) 0 0
\(611\) −20.3135 −0.821797
\(612\) 41.4329i 1.67483i
\(613\) − 42.8638i − 1.73125i −0.500692 0.865626i \(-0.666921\pi\)
0.500692 0.865626i \(-0.333079\pi\)
\(614\) −51.2639 −2.06884
\(615\) 0 0
\(616\) −9.75872 −0.393190
\(617\) 11.3607i 0.457364i 0.973501 + 0.228682i \(0.0734416\pi\)
−0.973501 + 0.228682i \(0.926558\pi\)
\(618\) 18.5236i 0.745128i
\(619\) 45.1917 1.81641 0.908203 0.418530i \(-0.137455\pi\)
0.908203 + 0.418530i \(0.137455\pi\)
\(620\) 0 0
\(621\) −2.15676 −0.0865476
\(622\) 56.5646i 2.26803i
\(623\) 13.8432i 0.554618i
\(624\) 60.0554 2.40414
\(625\) 0 0
\(626\) −17.1050 −0.683655
\(627\) 5.26180i 0.210136i
\(628\) 58.7091i 2.34275i
\(629\) 15.5174 0.618721
\(630\) 0 0
\(631\) −9.78992 −0.389731 −0.194865 0.980830i \(-0.562427\pi\)
−0.194865 + 0.980830i \(0.562427\pi\)
\(632\) 73.2905i 2.91534i
\(633\) − 9.57531i − 0.380584i
\(634\) 84.9647 3.37438
\(635\) 0 0
\(636\) 0.837101 0.0331932
\(637\) 25.3340i 1.00377i
\(638\) 3.84324i 0.152156i
\(639\) 4.68035 0.185152
\(640\) 0 0
\(641\) 0.210079 0.00829764 0.00414882 0.999991i \(-0.498679\pi\)
0.00414882 + 0.999991i \(0.498679\pi\)
\(642\) − 17.1773i − 0.677933i
\(643\) − 14.5236i − 0.572754i −0.958117 0.286377i \(-0.907549\pi\)
0.958117 0.286377i \(-0.0924511\pi\)
\(644\) 12.4202 0.489423
\(645\) 0 0
\(646\) −110.606 −4.35172
\(647\) 15.4641i 0.607957i 0.952679 + 0.303979i \(0.0983152\pi\)
−0.952679 + 0.303979i \(0.901685\pi\)
\(648\) 9.04945i 0.355496i
\(649\) −6.15676 −0.241674
\(650\) 0 0
\(651\) −5.04718 −0.197815
\(652\) − 34.8371i − 1.36433i
\(653\) 17.8310i 0.697779i 0.937164 + 0.348890i \(0.113441\pi\)
−0.937164 + 0.348890i \(0.886559\pi\)
\(654\) 6.26794 0.245096
\(655\) 0 0
\(656\) −130.325 −5.08835
\(657\) − 10.4969i − 0.409524i
\(658\) 13.6742i 0.533076i
\(659\) 32.3135 1.25876 0.629378 0.777099i \(-0.283310\pi\)
0.629378 + 0.777099i \(0.283310\pi\)
\(660\) 0 0
\(661\) −5.68649 −0.221179 −0.110589 0.993866i \(-0.535274\pi\)
−0.110589 + 0.993866i \(0.535274\pi\)
\(662\) − 52.0288i − 2.02215i
\(663\) 33.6742i 1.30780i
\(664\) −99.7296 −3.87026
\(665\) 0 0
\(666\) 5.41855 0.209965
\(667\) − 3.05947i − 0.118463i
\(668\) − 10.5380i − 0.407726i
\(669\) 2.15676 0.0833850
\(670\) 0 0
\(671\) −4.15676 −0.160470
\(672\) − 20.9093i − 0.806595i
\(673\) 21.0205i 0.810281i 0.914254 + 0.405141i \(0.132777\pi\)
−0.914254 + 0.405141i \(0.867223\pi\)
\(674\) 36.5835 1.40915
\(675\) 0 0
\(676\) 31.1711 1.19889
\(677\) 36.7526i 1.41252i 0.707954 + 0.706258i \(0.249618\pi\)
−0.707954 + 0.706258i \(0.750382\pi\)
\(678\) − 16.2557i − 0.624295i
\(679\) −15.8310 −0.607536
\(680\) 0 0
\(681\) 9.65983 0.370165
\(682\) 12.6803i 0.485556i
\(683\) 17.3074i 0.662248i 0.943587 + 0.331124i \(0.107428\pi\)
−0.943587 + 0.331124i \(0.892572\pi\)
\(684\) −28.0989 −1.07439
\(685\) 0 0
\(686\) 37.5052 1.43195
\(687\) − 3.36069i − 0.128218i
\(688\) − 105.189i − 4.01030i
\(689\) 0.680346 0.0259191
\(690\) 0 0
\(691\) 17.6742 0.672358 0.336179 0.941798i \(-0.390865\pi\)
0.336179 + 0.941798i \(0.390865\pi\)
\(692\) 20.0722i 0.763032i
\(693\) − 1.07838i − 0.0409642i
\(694\) 17.1773 0.652040
\(695\) 0 0
\(696\) −12.8371 −0.486589
\(697\) − 73.0759i − 2.76795i
\(698\) 43.7731i 1.65684i
\(699\) 2.39803 0.0907019
\(700\) 0 0
\(701\) −17.1050 −0.646048 −0.323024 0.946391i \(-0.604700\pi\)
−0.323024 + 0.946391i \(0.604700\pi\)
\(702\) 11.7587i 0.443804i
\(703\) 10.5236i 0.396905i
\(704\) −24.8576 −0.936857
\(705\) 0 0
\(706\) 35.7731 1.34634
\(707\) − 16.7961i − 0.631681i
\(708\) − 32.8781i − 1.23564i
\(709\) −25.1506 −0.944551 −0.472276 0.881451i \(-0.656567\pi\)
−0.472276 + 0.881451i \(0.656567\pi\)
\(710\) 0 0
\(711\) −8.09890 −0.303732
\(712\) 116.169i 4.35361i
\(713\) − 10.0944i − 0.378037i
\(714\) 22.6681 0.848331
\(715\) 0 0
\(716\) 80.9069 3.02363
\(717\) − 7.20394i − 0.269036i
\(718\) 8.99386i 0.335648i
\(719\) 1.78992 0.0667528 0.0333764 0.999443i \(-0.489374\pi\)
0.0333764 + 0.999443i \(0.489374\pi\)
\(720\) 0 0
\(721\) 7.37298 0.274584
\(722\) − 23.5341i − 0.875848i
\(723\) 5.20394i 0.193536i
\(724\) −25.8310 −0.960000
\(725\) 0 0
\(726\) −2.70928 −0.100551
\(727\) 25.9877i 0.963831i 0.876218 + 0.481915i \(0.160059\pi\)
−0.876218 + 0.481915i \(0.839941\pi\)
\(728\) − 42.3545i − 1.56976i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 58.9816 2.18151
\(732\) − 22.1978i − 0.820454i
\(733\) 41.0205i 1.51513i 0.652761 + 0.757564i \(0.273610\pi\)
−0.652761 + 0.757564i \(0.726390\pi\)
\(734\) −97.9253 −3.61449
\(735\) 0 0
\(736\) 41.8187 1.54146
\(737\) − 8.68035i − 0.319745i
\(738\) − 25.5174i − 0.939310i
\(739\) 47.6163 1.75160 0.875798 0.482678i \(-0.160336\pi\)
0.875798 + 0.482678i \(0.160336\pi\)
\(740\) 0 0
\(741\) −22.8371 −0.838942
\(742\) − 0.457980i − 0.0168130i
\(743\) − 0.550252i − 0.0201868i −0.999949 0.0100934i \(-0.996787\pi\)
0.999949 0.0100934i \(-0.00321288\pi\)
\(744\) −42.3545 −1.55279
\(745\) 0 0
\(746\) 7.61425 0.278778
\(747\) − 11.0205i − 0.403220i
\(748\) − 41.4329i − 1.51494i
\(749\) −6.83710 −0.249822
\(750\) 0 0
\(751\) 41.5585 1.51649 0.758245 0.651969i \(-0.226057\pi\)
0.758245 + 0.651969i \(0.226057\pi\)
\(752\) 64.7624i 2.36164i
\(753\) − 15.3197i − 0.558279i
\(754\) −16.6803 −0.607462
\(755\) 0 0
\(756\) 5.75872 0.209443
\(757\) 1.31965i 0.0479636i 0.999712 + 0.0239818i \(0.00763438\pi\)
−0.999712 + 0.0239818i \(0.992366\pi\)
\(758\) − 54.1855i − 1.96811i
\(759\) 2.15676 0.0782853
\(760\) 0 0
\(761\) −2.21461 −0.0802797 −0.0401399 0.999194i \(-0.512780\pi\)
−0.0401399 + 0.999194i \(0.512780\pi\)
\(762\) 6.07223i 0.219974i
\(763\) − 2.49484i − 0.0903192i
\(764\) −13.4764 −0.487559
\(765\) 0 0
\(766\) −90.9192 −3.28504
\(767\) − 26.7214i − 0.964853i
\(768\) − 27.6803i − 0.998828i
\(769\) 14.3668 0.518081 0.259041 0.965866i \(-0.416594\pi\)
0.259041 + 0.965866i \(0.416594\pi\)
\(770\) 0 0
\(771\) 4.15676 0.149702
\(772\) 0.142380i 0.00512436i
\(773\) − 40.1568i − 1.44434i −0.691717 0.722169i \(-0.743145\pi\)
0.691717 0.722169i \(-0.256855\pi\)
\(774\) 20.5958 0.740302
\(775\) 0 0
\(776\) −132.849 −4.76900
\(777\) − 2.15676i − 0.0773732i
\(778\) 34.7792i 1.24690i
\(779\) 49.5585 1.77562
\(780\) 0 0
\(781\) −4.68035 −0.167476
\(782\) 45.3361i 1.62122i
\(783\) − 1.41855i − 0.0506949i
\(784\) 80.7686 2.88459
\(785\) 0 0
\(786\) −23.5174 −0.838840
\(787\) 49.5897i 1.76768i 0.467788 + 0.883841i \(0.345051\pi\)
−0.467788 + 0.883841i \(0.654949\pi\)
\(788\) − 112.781i − 4.01767i
\(789\) 18.7070 0.665987
\(790\) 0 0
\(791\) −6.47027 −0.230056
\(792\) − 9.04945i − 0.321558i
\(793\) − 18.0410i − 0.640656i
\(794\) −14.4124 −0.511477
\(795\) 0 0
\(796\) 56.1978 1.99188
\(797\) − 46.7091i − 1.65452i −0.561818 0.827261i \(-0.689898\pi\)
0.561818 0.827261i \(-0.310102\pi\)
\(798\) 15.3730i 0.544198i
\(799\) −36.3135 −1.28468
\(800\) 0 0
\(801\) −12.8371 −0.453577
\(802\) − 5.41855i − 0.191336i
\(803\) 10.4969i 0.370429i
\(804\) 46.3545 1.63480
\(805\) 0 0
\(806\) −55.0349 −1.93852
\(807\) 23.3607i 0.822335i
\(808\) − 140.948i − 4.95853i
\(809\) 18.5814 0.653289 0.326644 0.945147i \(-0.394082\pi\)
0.326644 + 0.945147i \(0.394082\pi\)
\(810\) 0 0
\(811\) 27.3028 0.958732 0.479366 0.877615i \(-0.340867\pi\)
0.479366 + 0.877615i \(0.340867\pi\)
\(812\) 8.16904i 0.286677i
\(813\) 5.57531i 0.195535i
\(814\) −5.41855 −0.189920
\(815\) 0 0
\(816\) 107.358 3.75829
\(817\) 40.0000i 1.39942i
\(818\) 70.9770i 2.48165i
\(819\) 4.68035 0.163545
\(820\) 0 0
\(821\) −31.2085 −1.08918 −0.544592 0.838701i \(-0.683315\pi\)
−0.544592 + 0.838701i \(0.683315\pi\)
\(822\) 41.6163i 1.45154i
\(823\) 50.1855i 1.74936i 0.484704 + 0.874678i \(0.338927\pi\)
−0.484704 + 0.874678i \(0.661073\pi\)
\(824\) 61.8720 2.15541
\(825\) 0 0
\(826\) −17.9877 −0.625873
\(827\) 27.3874i 0.952352i 0.879350 + 0.476176i \(0.157977\pi\)
−0.879350 + 0.476176i \(0.842023\pi\)
\(828\) 11.5174i 0.400259i
\(829\) 26.1978 0.909887 0.454943 0.890520i \(-0.349659\pi\)
0.454943 + 0.890520i \(0.349659\pi\)
\(830\) 0 0
\(831\) −26.0144 −0.902429
\(832\) − 107.886i − 3.74029i
\(833\) 45.2885i 1.56915i
\(834\) 23.2495 0.805065
\(835\) 0 0
\(836\) 28.0989 0.971821
\(837\) − 4.68035i − 0.161776i
\(838\) − 7.68649i − 0.265525i
\(839\) −7.20394 −0.248708 −0.124354 0.992238i \(-0.539686\pi\)
−0.124354 + 0.992238i \(0.539686\pi\)
\(840\) 0 0
\(841\) −26.9877 −0.930611
\(842\) − 31.0928i − 1.07153i
\(843\) 9.41855i 0.324392i
\(844\) −51.1338 −1.76010
\(845\) 0 0
\(846\) −12.6803 −0.435959
\(847\) 1.07838i 0.0370535i
\(848\) − 2.16904i − 0.0744852i
\(849\) −14.2413 −0.488759
\(850\) 0 0
\(851\) 4.31351 0.147865
\(852\) − 24.9939i − 0.856275i
\(853\) − 39.8043i − 1.36287i −0.731877 0.681437i \(-0.761356\pi\)
0.731877 0.681437i \(-0.238644\pi\)
\(854\) −12.1445 −0.415575
\(855\) 0 0
\(856\) −57.3751 −1.96104
\(857\) − 36.9504i − 1.26220i −0.775701 0.631100i \(-0.782604\pi\)
0.775701 0.631100i \(-0.217396\pi\)
\(858\) − 11.7587i − 0.401436i
\(859\) −57.5052 −1.96205 −0.981025 0.193879i \(-0.937893\pi\)
−0.981025 + 0.193879i \(0.937893\pi\)
\(860\) 0 0
\(861\) −10.1568 −0.346141
\(862\) − 63.8597i − 2.17507i
\(863\) 1.89657i 0.0645599i 0.999479 + 0.0322800i \(0.0102768\pi\)
−0.999479 + 0.0322800i \(0.989723\pi\)
\(864\) 19.3896 0.659648
\(865\) 0 0
\(866\) 40.6225 1.38041
\(867\) 43.1978i 1.46707i
\(868\) 26.9528i 0.914838i
\(869\) 8.09890 0.274736
\(870\) 0 0
\(871\) 37.6742 1.27654
\(872\) − 20.9360i − 0.708982i
\(873\) − 14.6803i − 0.496854i
\(874\) −30.7460 −1.04000
\(875\) 0 0
\(876\) −56.0554 −1.89394
\(877\) 32.5380i 1.09873i 0.835583 + 0.549365i \(0.185130\pi\)
−0.835583 + 0.549365i \(0.814870\pi\)
\(878\) 12.9483i 0.436983i
\(879\) 15.7587 0.531529
\(880\) 0 0
\(881\) 18.1978 0.613099 0.306550 0.951855i \(-0.400825\pi\)
0.306550 + 0.951855i \(0.400825\pi\)
\(882\) 15.8143i 0.532496i
\(883\) − 36.3956i − 1.22481i −0.790545 0.612405i \(-0.790202\pi\)
0.790545 0.612405i \(-0.209798\pi\)
\(884\) 179.826 6.04821
\(885\) 0 0
\(886\) −54.7214 −1.83840
\(887\) 27.8699i 0.935780i 0.883787 + 0.467890i \(0.154986\pi\)
−0.883787 + 0.467890i \(0.845014\pi\)
\(888\) − 18.0989i − 0.607359i
\(889\) 2.41694 0.0810616
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) − 11.5174i − 0.385633i
\(893\) − 24.6270i − 0.824112i
\(894\) −49.0349 −1.63997
\(895\) 0 0
\(896\) −30.8059 −1.02915
\(897\) 9.36069i 0.312544i
\(898\) − 58.4412i − 1.95021i
\(899\) 6.63931 0.221433
\(900\) 0 0
\(901\) 1.21622 0.0405182
\(902\) 25.5174i 0.849638i
\(903\) − 8.19779i − 0.272805i
\(904\) −54.2967 −1.80588
\(905\) 0 0
\(906\) −62.1399 −2.06446
\(907\) − 27.9376i − 0.927653i −0.885926 0.463826i \(-0.846476\pi\)
0.885926 0.463826i \(-0.153524\pi\)
\(908\) − 51.5851i − 1.71191i
\(909\) 15.5753 0.516600
\(910\) 0 0
\(911\) −11.8843 −0.393744 −0.196872 0.980429i \(-0.563078\pi\)
−0.196872 + 0.980429i \(0.563078\pi\)
\(912\) 72.8080i 2.41091i
\(913\) 11.0205i 0.364726i
\(914\) 76.3234 2.52455
\(915\) 0 0
\(916\) −17.9467 −0.592975
\(917\) 9.36069i 0.309117i
\(918\) 21.0205i 0.693781i
\(919\) 45.6041 1.50434 0.752170 0.658970i \(-0.229007\pi\)
0.752170 + 0.658970i \(0.229007\pi\)
\(920\) 0 0
\(921\) −18.9216 −0.623489
\(922\) 3.98771i 0.131328i
\(923\) − 20.3135i − 0.668627i
\(924\) −5.75872 −0.189448
\(925\) 0 0
\(926\) 62.8659 2.06590
\(927\) 6.83710i 0.224560i
\(928\) 27.5052i 0.902901i
\(929\) 25.1506 0.825165 0.412582 0.910920i \(-0.364627\pi\)
0.412582 + 0.910920i \(0.364627\pi\)
\(930\) 0 0
\(931\) −30.7136 −1.00660
\(932\) − 12.8059i − 0.419471i
\(933\) 20.8781i 0.683520i
\(934\) 38.3545 1.25500
\(935\) 0 0
\(936\) 39.2762 1.28378
\(937\) 5.33403i 0.174255i 0.996197 + 0.0871276i \(0.0277688\pi\)
−0.996197 + 0.0871276i \(0.972231\pi\)
\(938\) − 25.3607i − 0.828056i
\(939\) −6.31351 −0.206034
\(940\) 0 0
\(941\) 56.8203 1.85229 0.926144 0.377170i \(-0.123103\pi\)
0.926144 + 0.377170i \(0.123103\pi\)
\(942\) 29.7854i 0.970460i
\(943\) − 20.3135i − 0.661499i
\(944\) −85.1917 −2.77275
\(945\) 0 0
\(946\) −20.5958 −0.669628
\(947\) − 20.9939i − 0.682209i −0.940025 0.341104i \(-0.889199\pi\)
0.940025 0.341104i \(-0.110801\pi\)
\(948\) 43.2495i 1.40468i
\(949\) −45.5585 −1.47889
\(950\) 0 0
\(951\) 31.3607 1.01694
\(952\) − 75.7152i − 2.45395i
\(953\) 25.2351i 0.817446i 0.912658 + 0.408723i \(0.134026\pi\)
−0.912658 + 0.408723i \(0.865974\pi\)
\(954\) 0.424694 0.0137500
\(955\) 0 0
\(956\) −38.4703 −1.24422
\(957\) 1.41855i 0.0458552i
\(958\) − 37.5052i − 1.21174i
\(959\) 16.5646 0.534900
\(960\) 0 0
\(961\) −9.09436 −0.293367
\(962\) − 23.5174i − 0.758233i
\(963\) − 6.34017i − 0.204309i
\(964\) 27.7899 0.895053
\(965\) 0 0
\(966\) 6.30122 0.202739
\(967\) 13.1317i 0.422287i 0.977455 + 0.211144i \(0.0677187\pi\)
−0.977455 + 0.211144i \(0.932281\pi\)
\(968\) 9.04945i 0.290860i
\(969\) −40.8248 −1.31148
\(970\) 0 0
\(971\) 8.94053 0.286915 0.143458 0.989656i \(-0.454178\pi\)
0.143458 + 0.989656i \(0.454178\pi\)
\(972\) 5.34017i 0.171286i
\(973\) − 9.25404i − 0.296671i
\(974\) −111.064 −3.55871
\(975\) 0 0
\(976\) −57.5174 −1.84109
\(977\) 50.3956i 1.61230i 0.591713 + 0.806149i \(0.298452\pi\)
−0.591713 + 0.806149i \(0.701548\pi\)
\(978\) − 17.6742i − 0.565159i
\(979\) 12.8371 0.410276
\(980\) 0 0
\(981\) 2.31351 0.0738647
\(982\) − 94.3833i − 3.01189i
\(983\) 32.1978i 1.02695i 0.858105 + 0.513475i \(0.171642\pi\)
−0.858105 + 0.513475i \(0.828358\pi\)
\(984\) −85.2327 −2.71712
\(985\) 0 0
\(986\) −29.8187 −0.949620
\(987\) 5.04718i 0.160654i
\(988\) 121.954i 3.87988i
\(989\) 16.3956 0.521349
\(990\) 0 0
\(991\) −46.7747 −1.48585 −0.742924 0.669376i \(-0.766562\pi\)
−0.742924 + 0.669376i \(0.766562\pi\)
\(992\) 90.7501i 2.88132i
\(993\) − 19.2039i − 0.609419i
\(994\) −13.6742 −0.433719
\(995\) 0 0
\(996\) −58.8515 −1.86478
\(997\) 38.2122i 1.21019i 0.796153 + 0.605096i \(0.206865\pi\)
−0.796153 + 0.605096i \(0.793135\pi\)
\(998\) 41.0472i 1.29933i
\(999\) 2.00000 0.0632772
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 825.2.c.g.199.1 6
3.2 odd 2 2475.2.c.r.199.6 6
5.2 odd 4 825.2.a.k.1.3 3
5.3 odd 4 165.2.a.c.1.1 3
5.4 even 2 inner 825.2.c.g.199.6 6
15.2 even 4 2475.2.a.bb.1.1 3
15.8 even 4 495.2.a.e.1.3 3
15.14 odd 2 2475.2.c.r.199.1 6
20.3 even 4 2640.2.a.be.1.2 3
35.13 even 4 8085.2.a.bk.1.1 3
55.32 even 4 9075.2.a.cf.1.1 3
55.43 even 4 1815.2.a.m.1.3 3
60.23 odd 4 7920.2.a.cj.1.2 3
165.98 odd 4 5445.2.a.z.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.1 3 5.3 odd 4
495.2.a.e.1.3 3 15.8 even 4
825.2.a.k.1.3 3 5.2 odd 4
825.2.c.g.199.1 6 1.1 even 1 trivial
825.2.c.g.199.6 6 5.4 even 2 inner
1815.2.a.m.1.3 3 55.43 even 4
2475.2.a.bb.1.1 3 15.2 even 4
2475.2.c.r.199.1 6 15.14 odd 2
2475.2.c.r.199.6 6 3.2 odd 2
2640.2.a.be.1.2 3 20.3 even 4
5445.2.a.z.1.1 3 165.98 odd 4
7920.2.a.cj.1.2 3 60.23 odd 4
8085.2.a.bk.1.1 3 35.13 even 4
9075.2.a.cf.1.1 3 55.32 even 4