Properties

Label 225.6.b.e.199.1
Level $225$
Weight $6$
Character 225.199
Analytic conductor $36.086$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.6.b.e.199.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{2} +28.0000 q^{4} +192.000i q^{7} -120.000i q^{8} +O(q^{10})\) \(q-2.00000i q^{2} +28.0000 q^{4} +192.000i q^{7} -120.000i q^{8} +148.000 q^{11} -286.000i q^{13} +384.000 q^{14} +656.000 q^{16} +1678.00i q^{17} -1060.00 q^{19} -296.000i q^{22} +2976.00i q^{23} -572.000 q^{26} +5376.00i q^{28} -3410.00 q^{29} -2448.00 q^{31} -5152.00i q^{32} +3356.00 q^{34} +182.000i q^{37} +2120.00i q^{38} +9398.00 q^{41} +1244.00i q^{43} +4144.00 q^{44} +5952.00 q^{46} +12088.0i q^{47} -20057.0 q^{49} -8008.00i q^{52} +23846.0i q^{53} +23040.0 q^{56} +6820.00i q^{58} -20020.0 q^{59} +32302.0 q^{61} +4896.00i q^{62} +10688.0 q^{64} +60972.0i q^{67} +46984.0i q^{68} +32648.0 q^{71} +38774.0i q^{73} +364.000 q^{74} -29680.0 q^{76} +28416.0i q^{77} +33360.0 q^{79} -18796.0i q^{82} +16716.0i q^{83} +2488.00 q^{86} -17760.0i q^{88} +101370. q^{89} +54912.0 q^{91} +83328.0i q^{92} +24176.0 q^{94} -119038. i q^{97} +40114.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 56 q^{4} + O(q^{10}) \) \( 2 q + 56 q^{4} + 296 q^{11} + 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1144 q^{26} - 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 18796 q^{41} + 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 46080 q^{56} - 40040 q^{59} + 64604 q^{61} + 21376 q^{64} + 65296 q^{71} + 728 q^{74} - 59360 q^{76} + 66720 q^{79} + 4976 q^{86} + 202740 q^{89} + 109824 q^{91} + 48352 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.353553i −0.984251 0.176777i \(-0.943433\pi\)
0.984251 0.176777i \(-0.0565670\pi\)
\(3\) 0 0
\(4\) 28.0000 0.875000
\(5\) 0 0
\(6\) 0 0
\(7\) 192.000i 1.48100i 0.672054 + 0.740502i \(0.265412\pi\)
−0.672054 + 0.740502i \(0.734588\pi\)
\(8\) − 120.000i − 0.662913i
\(9\) 0 0
\(10\) 0 0
\(11\) 148.000 0.368791 0.184395 0.982852i \(-0.440967\pi\)
0.184395 + 0.982852i \(0.440967\pi\)
\(12\) 0 0
\(13\) − 286.000i − 0.469362i −0.972072 0.234681i \(-0.924595\pi\)
0.972072 0.234681i \(-0.0754045\pi\)
\(14\) 384.000 0.523614
\(15\) 0 0
\(16\) 656.000 0.640625
\(17\) 1678.00i 1.40822i 0.710092 + 0.704109i \(0.248653\pi\)
−0.710092 + 0.704109i \(0.751347\pi\)
\(18\) 0 0
\(19\) −1060.00 −0.673631 −0.336815 0.941571i \(-0.609350\pi\)
−0.336815 + 0.941571i \(0.609350\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 296.000i − 0.130387i
\(23\) 2976.00i 1.17304i 0.809934 + 0.586521i \(0.199503\pi\)
−0.809934 + 0.586521i \(0.800497\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −572.000 −0.165944
\(27\) 0 0
\(28\) 5376.00i 1.29588i
\(29\) −3410.00 −0.752938 −0.376469 0.926429i \(-0.622862\pi\)
−0.376469 + 0.926429i \(0.622862\pi\)
\(30\) 0 0
\(31\) −2448.00 −0.457517 −0.228758 0.973483i \(-0.573467\pi\)
−0.228758 + 0.973483i \(0.573467\pi\)
\(32\) − 5152.00i − 0.889408i
\(33\) 0 0
\(34\) 3356.00 0.497880
\(35\) 0 0
\(36\) 0 0
\(37\) 182.000i 0.0218558i 0.999940 + 0.0109279i \(0.00347853\pi\)
−0.999940 + 0.0109279i \(0.996521\pi\)
\(38\) 2120.00i 0.238164i
\(39\) 0 0
\(40\) 0 0
\(41\) 9398.00 0.873124 0.436562 0.899674i \(-0.356196\pi\)
0.436562 + 0.899674i \(0.356196\pi\)
\(42\) 0 0
\(43\) 1244.00i 0.102600i 0.998683 + 0.0513002i \(0.0163365\pi\)
−0.998683 + 0.0513002i \(0.983663\pi\)
\(44\) 4144.00 0.322692
\(45\) 0 0
\(46\) 5952.00 0.414733
\(47\) 12088.0i 0.798196i 0.916908 + 0.399098i \(0.130677\pi\)
−0.916908 + 0.399098i \(0.869323\pi\)
\(48\) 0 0
\(49\) −20057.0 −1.19337
\(50\) 0 0
\(51\) 0 0
\(52\) − 8008.00i − 0.410691i
\(53\) 23846.0i 1.16607i 0.812446 + 0.583037i \(0.198136\pi\)
−0.812446 + 0.583037i \(0.801864\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 23040.0 0.981776
\(57\) 0 0
\(58\) 6820.00i 0.266204i
\(59\) −20020.0 −0.748745 −0.374373 0.927278i \(-0.622142\pi\)
−0.374373 + 0.927278i \(0.622142\pi\)
\(60\) 0 0
\(61\) 32302.0 1.11149 0.555744 0.831353i \(-0.312433\pi\)
0.555744 + 0.831353i \(0.312433\pi\)
\(62\) 4896.00i 0.161757i
\(63\) 0 0
\(64\) 10688.0 0.326172
\(65\) 0 0
\(66\) 0 0
\(67\) 60972.0i 1.65937i 0.558231 + 0.829685i \(0.311480\pi\)
−0.558231 + 0.829685i \(0.688520\pi\)
\(68\) 46984.0i 1.23219i
\(69\) 0 0
\(70\) 0 0
\(71\) 32648.0 0.768618 0.384309 0.923204i \(-0.374440\pi\)
0.384309 + 0.923204i \(0.374440\pi\)
\(72\) 0 0
\(73\) 38774.0i 0.851596i 0.904818 + 0.425798i \(0.140007\pi\)
−0.904818 + 0.425798i \(0.859993\pi\)
\(74\) 364.000 0.00772720
\(75\) 0 0
\(76\) −29680.0 −0.589427
\(77\) 28416.0i 0.546180i
\(78\) 0 0
\(79\) 33360.0 0.601393 0.300696 0.953720i \(-0.402781\pi\)
0.300696 + 0.953720i \(0.402781\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 18796.0i − 0.308696i
\(83\) 16716.0i 0.266340i 0.991093 + 0.133170i \(0.0425157\pi\)
−0.991093 + 0.133170i \(0.957484\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2488.00 0.0362747
\(87\) 0 0
\(88\) − 17760.0i − 0.244476i
\(89\) 101370. 1.35655 0.678273 0.734810i \(-0.262729\pi\)
0.678273 + 0.734810i \(0.262729\pi\)
\(90\) 0 0
\(91\) 54912.0 0.695126
\(92\) 83328.0i 1.02641i
\(93\) 0 0
\(94\) 24176.0 0.282205
\(95\) 0 0
\(96\) 0 0
\(97\) − 119038.i − 1.28457i −0.766468 0.642283i \(-0.777987\pi\)
0.766468 0.642283i \(-0.222013\pi\)
\(98\) 40114.0i 0.421921i
\(99\) 0 0
\(100\) 0 0
\(101\) 89898.0 0.876893 0.438446 0.898757i \(-0.355529\pi\)
0.438446 + 0.898757i \(0.355529\pi\)
\(102\) 0 0
\(103\) 19504.0i 0.181147i 0.995890 + 0.0905734i \(0.0288700\pi\)
−0.995890 + 0.0905734i \(0.971130\pi\)
\(104\) −34320.0 −0.311146
\(105\) 0 0
\(106\) 47692.0 0.412269
\(107\) − 158292.i − 1.33659i −0.743895 0.668297i \(-0.767024\pi\)
0.743895 0.668297i \(-0.232976\pi\)
\(108\) 0 0
\(109\) −36830.0 −0.296917 −0.148459 0.988919i \(-0.547431\pi\)
−0.148459 + 0.988919i \(0.547431\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 125952.i 0.948768i
\(113\) 11186.0i 0.0824098i 0.999151 + 0.0412049i \(0.0131196\pi\)
−0.999151 + 0.0412049i \(0.986880\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −95480.0 −0.658821
\(117\) 0 0
\(118\) 40040.0i 0.264721i
\(119\) −322176. −2.08557
\(120\) 0 0
\(121\) −139147. −0.863993
\(122\) − 64604.0i − 0.392970i
\(123\) 0 0
\(124\) −68544.0 −0.400327
\(125\) 0 0
\(126\) 0 0
\(127\) 70552.0i 0.388150i 0.980987 + 0.194075i \(0.0621706\pi\)
−0.980987 + 0.194075i \(0.937829\pi\)
\(128\) − 186240.i − 1.00473i
\(129\) 0 0
\(130\) 0 0
\(131\) −76452.0 −0.389234 −0.194617 0.980879i \(-0.562346\pi\)
−0.194617 + 0.980879i \(0.562346\pi\)
\(132\) 0 0
\(133\) − 203520.i − 0.997650i
\(134\) 121944. 0.586676
\(135\) 0 0
\(136\) 201360. 0.933525
\(137\) 144918.i 0.659661i 0.944040 + 0.329831i \(0.106992\pi\)
−0.944040 + 0.329831i \(0.893008\pi\)
\(138\) 0 0
\(139\) −112220. −0.492644 −0.246322 0.969188i \(-0.579222\pi\)
−0.246322 + 0.969188i \(0.579222\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 65296.0i − 0.271748i
\(143\) − 42328.0i − 0.173096i
\(144\) 0 0
\(145\) 0 0
\(146\) 77548.0 0.301085
\(147\) 0 0
\(148\) 5096.00i 0.0191238i
\(149\) 403750. 1.48986 0.744932 0.667140i \(-0.232482\pi\)
0.744932 + 0.667140i \(0.232482\pi\)
\(150\) 0 0
\(151\) −446648. −1.59413 −0.797064 0.603895i \(-0.793615\pi\)
−0.797064 + 0.603895i \(0.793615\pi\)
\(152\) 127200.i 0.446558i
\(153\) 0 0
\(154\) 56832.0 0.193104
\(155\) 0 0
\(156\) 0 0
\(157\) − 262258.i − 0.849141i −0.905395 0.424570i \(-0.860425\pi\)
0.905395 0.424570i \(-0.139575\pi\)
\(158\) − 66720.0i − 0.212625i
\(159\) 0 0
\(160\) 0 0
\(161\) −571392. −1.73728
\(162\) 0 0
\(163\) 154564.i 0.455658i 0.973701 + 0.227829i \(0.0731628\pi\)
−0.973701 + 0.227829i \(0.926837\pi\)
\(164\) 263144. 0.763983
\(165\) 0 0
\(166\) 33432.0 0.0941656
\(167\) − 396672.i − 1.10063i −0.834958 0.550314i \(-0.814508\pi\)
0.834958 0.550314i \(-0.185492\pi\)
\(168\) 0 0
\(169\) 289497. 0.779700
\(170\) 0 0
\(171\) 0 0
\(172\) 34832.0i 0.0897754i
\(173\) − 573474.i − 1.45680i −0.685155 0.728398i \(-0.740265\pi\)
0.685155 0.728398i \(-0.259735\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 97088.0 0.236257
\(177\) 0 0
\(178\) − 202740.i − 0.479611i
\(179\) −594460. −1.38672 −0.693362 0.720589i \(-0.743871\pi\)
−0.693362 + 0.720589i \(0.743871\pi\)
\(180\) 0 0
\(181\) −107098. −0.242988 −0.121494 0.992592i \(-0.538769\pi\)
−0.121494 + 0.992592i \(0.538769\pi\)
\(182\) − 109824.i − 0.245764i
\(183\) 0 0
\(184\) 357120. 0.777624
\(185\) 0 0
\(186\) 0 0
\(187\) 248344.i 0.519337i
\(188\) 338464.i 0.698422i
\(189\) 0 0
\(190\) 0 0
\(191\) −469552. −0.931323 −0.465661 0.884963i \(-0.654184\pi\)
−0.465661 + 0.884963i \(0.654184\pi\)
\(192\) 0 0
\(193\) − 52706.0i − 0.101851i −0.998702 0.0509257i \(-0.983783\pi\)
0.998702 0.0509257i \(-0.0162172\pi\)
\(194\) −238076. −0.454163
\(195\) 0 0
\(196\) −561596. −1.04420
\(197\) − 455862.i − 0.836889i −0.908242 0.418444i \(-0.862575\pi\)
0.908242 0.418444i \(-0.137425\pi\)
\(198\) 0 0
\(199\) −865000. −1.54840 −0.774200 0.632940i \(-0.781848\pi\)
−0.774200 + 0.632940i \(0.781848\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 179796.i − 0.310028i
\(203\) − 654720.i − 1.11510i
\(204\) 0 0
\(205\) 0 0
\(206\) 39008.0 0.0640451
\(207\) 0 0
\(208\) − 187616.i − 0.300685i
\(209\) −156880. −0.248429
\(210\) 0 0
\(211\) 1.10565e6 1.70967 0.854835 0.518900i \(-0.173658\pi\)
0.854835 + 0.518900i \(0.173658\pi\)
\(212\) 667688.i 1.02031i
\(213\) 0 0
\(214\) −316584. −0.472557
\(215\) 0 0
\(216\) 0 0
\(217\) − 470016.i − 0.677584i
\(218\) 73660.0i 0.104976i
\(219\) 0 0
\(220\) 0 0
\(221\) 479908. 0.660963
\(222\) 0 0
\(223\) − 1.12158e6i − 1.51031i −0.655545 0.755156i \(-0.727561\pi\)
0.655545 0.755156i \(-0.272439\pi\)
\(224\) 989184. 1.31722
\(225\) 0 0
\(226\) 22372.0 0.0291363
\(227\) 23348.0i 0.0300736i 0.999887 + 0.0150368i \(0.00478654\pi\)
−0.999887 + 0.0150368i \(0.995213\pi\)
\(228\) 0 0
\(229\) 596010. 0.751043 0.375522 0.926814i \(-0.377464\pi\)
0.375522 + 0.926814i \(0.377464\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 409200.i 0.499132i
\(233\) − 485334.i − 0.585667i −0.956163 0.292834i \(-0.905402\pi\)
0.956163 0.292834i \(-0.0945982\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −560560. −0.655152
\(237\) 0 0
\(238\) 644352.i 0.737362i
\(239\) −48880.0 −0.0553524 −0.0276762 0.999617i \(-0.508811\pi\)
−0.0276762 + 0.999617i \(0.508811\pi\)
\(240\) 0 0
\(241\) −110798. −0.122882 −0.0614411 0.998111i \(-0.519570\pi\)
−0.0614411 + 0.998111i \(0.519570\pi\)
\(242\) 278294.i 0.305468i
\(243\) 0 0
\(244\) 904456. 0.972552
\(245\) 0 0
\(246\) 0 0
\(247\) 303160.i 0.316176i
\(248\) 293760.i 0.303294i
\(249\) 0 0
\(250\) 0 0
\(251\) 1.64375e6 1.64684 0.823419 0.567434i \(-0.192064\pi\)
0.823419 + 0.567434i \(0.192064\pi\)
\(252\) 0 0
\(253\) 440448.i 0.432607i
\(254\) 141104. 0.137232
\(255\) 0 0
\(256\) −30464.0 −0.0290527
\(257\) − 1.30624e6i − 1.23365i −0.787102 0.616823i \(-0.788419\pi\)
0.787102 0.616823i \(-0.211581\pi\)
\(258\) 0 0
\(259\) −34944.0 −0.0323685
\(260\) 0 0
\(261\) 0 0
\(262\) 152904.i 0.137615i
\(263\) 2.12834e6i 1.89736i 0.316231 + 0.948682i \(0.397583\pi\)
−0.316231 + 0.948682i \(0.602417\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −407040. −0.352722
\(267\) 0 0
\(268\) 1.70722e6i 1.45195i
\(269\) −1.44109e6 −1.21426 −0.607128 0.794604i \(-0.707679\pi\)
−0.607128 + 0.794604i \(0.707679\pi\)
\(270\) 0 0
\(271\) −93248.0 −0.0771288 −0.0385644 0.999256i \(-0.512278\pi\)
−0.0385644 + 0.999256i \(0.512278\pi\)
\(272\) 1.10077e6i 0.902139i
\(273\) 0 0
\(274\) 289836. 0.233225
\(275\) 0 0
\(276\) 0 0
\(277\) − 110298.i − 0.0863711i −0.999067 0.0431855i \(-0.986249\pi\)
0.999067 0.0431855i \(-0.0137507\pi\)
\(278\) 224440.i 0.174176i
\(279\) 0 0
\(280\) 0 0
\(281\) 192198. 0.145205 0.0726027 0.997361i \(-0.476869\pi\)
0.0726027 + 0.997361i \(0.476869\pi\)
\(282\) 0 0
\(283\) 331884.i 0.246332i 0.992386 + 0.123166i \(0.0393047\pi\)
−0.992386 + 0.123166i \(0.960695\pi\)
\(284\) 914144. 0.672541
\(285\) 0 0
\(286\) −84656.0 −0.0611988
\(287\) 1.80442e6i 1.29310i
\(288\) 0 0
\(289\) −1.39583e6 −0.983076
\(290\) 0 0
\(291\) 0 0
\(292\) 1.08567e6i 0.745146i
\(293\) 2.19481e6i 1.49358i 0.665063 + 0.746788i \(0.268405\pi\)
−0.665063 + 0.746788i \(0.731595\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 21840.0 0.0144885
\(297\) 0 0
\(298\) − 807500.i − 0.526747i
\(299\) 851136. 0.550581
\(300\) 0 0
\(301\) −238848. −0.151952
\(302\) 893296.i 0.563609i
\(303\) 0 0
\(304\) −695360. −0.431545
\(305\) 0 0
\(306\) 0 0
\(307\) − 2.37751e6i − 1.43971i −0.694123 0.719857i \(-0.744207\pi\)
0.694123 0.719857i \(-0.255793\pi\)
\(308\) 795648.i 0.477908i
\(309\) 0 0
\(310\) 0 0
\(311\) 2.37305e6 1.39125 0.695626 0.718405i \(-0.255127\pi\)
0.695626 + 0.718405i \(0.255127\pi\)
\(312\) 0 0
\(313\) 1.42941e6i 0.824702i 0.911025 + 0.412351i \(0.135292\pi\)
−0.911025 + 0.412351i \(0.864708\pi\)
\(314\) −524516. −0.300217
\(315\) 0 0
\(316\) 934080. 0.526219
\(317\) − 2.12462e6i − 1.18750i −0.804650 0.593750i \(-0.797647\pi\)
0.804650 0.593750i \(-0.202353\pi\)
\(318\) 0 0
\(319\) −504680. −0.277677
\(320\) 0 0
\(321\) 0 0
\(322\) 1.14278e6i 0.614221i
\(323\) − 1.77868e6i − 0.948618i
\(324\) 0 0
\(325\) 0 0
\(326\) 309128. 0.161100
\(327\) 0 0
\(328\) − 1.12776e6i − 0.578805i
\(329\) −2.32090e6 −1.18213
\(330\) 0 0
\(331\) 3.09985e6 1.55515 0.777573 0.628793i \(-0.216451\pi\)
0.777573 + 0.628793i \(0.216451\pi\)
\(332\) 468048.i 0.233048i
\(333\) 0 0
\(334\) −793344. −0.389131
\(335\) 0 0
\(336\) 0 0
\(337\) 2.40008e6i 1.15120i 0.817731 + 0.575601i \(0.195232\pi\)
−0.817731 + 0.575601i \(0.804768\pi\)
\(338\) − 578994.i − 0.275665i
\(339\) 0 0
\(340\) 0 0
\(341\) −362304. −0.168728
\(342\) 0 0
\(343\) − 624000.i − 0.286384i
\(344\) 149280. 0.0680151
\(345\) 0 0
\(346\) −1.14695e6 −0.515055
\(347\) − 1.77741e6i − 0.792436i −0.918156 0.396218i \(-0.870322\pi\)
0.918156 0.396218i \(-0.129678\pi\)
\(348\) 0 0
\(349\) 2.14805e6 0.944019 0.472010 0.881593i \(-0.343529\pi\)
0.472010 + 0.881593i \(0.343529\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 762496.i − 0.328005i
\(353\) − 661854.i − 0.282700i −0.989960 0.141350i \(-0.954856\pi\)
0.989960 0.141350i \(-0.0451443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.83836e6 1.18698
\(357\) 0 0
\(358\) 1.18892e6i 0.490281i
\(359\) −259320. −0.106194 −0.0530970 0.998589i \(-0.516909\pi\)
−0.0530970 + 0.998589i \(0.516909\pi\)
\(360\) 0 0
\(361\) −1.35250e6 −0.546222
\(362\) 214196.i 0.0859093i
\(363\) 0 0
\(364\) 1.53754e6 0.608236
\(365\) 0 0
\(366\) 0 0
\(367\) − 1.49993e6i − 0.581307i −0.956828 0.290653i \(-0.906127\pi\)
0.956828 0.290653i \(-0.0938726\pi\)
\(368\) 1.95226e6i 0.751480i
\(369\) 0 0
\(370\) 0 0
\(371\) −4.57843e6 −1.72696
\(372\) 0 0
\(373\) 2.23807e6i 0.832918i 0.909154 + 0.416459i \(0.136729\pi\)
−0.909154 + 0.416459i \(0.863271\pi\)
\(374\) 496688. 0.183614
\(375\) 0 0
\(376\) 1.45056e6 0.529135
\(377\) 975260.i 0.353400i
\(378\) 0 0
\(379\) −3.15934e6 −1.12979 −0.564896 0.825162i \(-0.691084\pi\)
−0.564896 + 0.825162i \(0.691084\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 939104.i 0.329272i
\(383\) 342216.i 0.119207i 0.998222 + 0.0596037i \(0.0189837\pi\)
−0.998222 + 0.0596037i \(0.981016\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −105412. −0.0360099
\(387\) 0 0
\(388\) − 3.33306e6i − 1.12399i
\(389\) 88470.0 0.0296430 0.0148215 0.999890i \(-0.495282\pi\)
0.0148215 + 0.999890i \(0.495282\pi\)
\(390\) 0 0
\(391\) −4.99373e6 −1.65190
\(392\) 2.40684e6i 0.791101i
\(393\) 0 0
\(394\) −911724. −0.295885
\(395\) 0 0
\(396\) 0 0
\(397\) − 5.45674e6i − 1.73763i −0.495138 0.868814i \(-0.664883\pi\)
0.495138 0.868814i \(-0.335117\pi\)
\(398\) 1.73000e6i 0.547442i
\(399\) 0 0
\(400\) 0 0
\(401\) −4.04680e6 −1.25676 −0.628378 0.777908i \(-0.716281\pi\)
−0.628378 + 0.777908i \(0.716281\pi\)
\(402\) 0 0
\(403\) 700128.i 0.214741i
\(404\) 2.51714e6 0.767281
\(405\) 0 0
\(406\) −1.30944e6 −0.394249
\(407\) 26936.0i 0.00806022i
\(408\) 0 0
\(409\) 2.71207e6 0.801664 0.400832 0.916151i \(-0.368721\pi\)
0.400832 + 0.916151i \(0.368721\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 546112.i 0.158503i
\(413\) − 3.84384e6i − 1.10889i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.47347e6 −0.417454
\(417\) 0 0
\(418\) 313760.i 0.0878328i
\(419\) 3.71746e6 1.03445 0.517227 0.855848i \(-0.326964\pi\)
0.517227 + 0.855848i \(0.326964\pi\)
\(420\) 0 0
\(421\) 3.55250e6 0.976853 0.488426 0.872605i \(-0.337571\pi\)
0.488426 + 0.872605i \(0.337571\pi\)
\(422\) − 2.21130e6i − 0.604460i
\(423\) 0 0
\(424\) 2.86152e6 0.773005
\(425\) 0 0
\(426\) 0 0
\(427\) 6.20198e6i 1.64612i
\(428\) − 4.43218e6i − 1.16952i
\(429\) 0 0
\(430\) 0 0
\(431\) 4.06205e6 1.05330 0.526650 0.850082i \(-0.323448\pi\)
0.526650 + 0.850082i \(0.323448\pi\)
\(432\) 0 0
\(433\) − 7.26287e6i − 1.86161i −0.365518 0.930804i \(-0.619108\pi\)
0.365518 0.930804i \(-0.380892\pi\)
\(434\) −940032. −0.239562
\(435\) 0 0
\(436\) −1.03124e6 −0.259803
\(437\) − 3.15456e6i − 0.790197i
\(438\) 0 0
\(439\) 5.41028e6 1.33986 0.669928 0.742426i \(-0.266325\pi\)
0.669928 + 0.742426i \(0.266325\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 959816.i − 0.233686i
\(443\) − 6.51524e6i − 1.57733i −0.614826 0.788663i \(-0.710774\pi\)
0.614826 0.788663i \(-0.289226\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2.24315e6 −0.533976
\(447\) 0 0
\(448\) 2.05210e6i 0.483062i
\(449\) −509950. −0.119375 −0.0596873 0.998217i \(-0.519010\pi\)
−0.0596873 + 0.998217i \(0.519010\pi\)
\(450\) 0 0
\(451\) 1.39090e6 0.322000
\(452\) 313208.i 0.0721085i
\(453\) 0 0
\(454\) 46696.0 0.0106326
\(455\) 0 0
\(456\) 0 0
\(457\) 1.22084e6i 0.273444i 0.990609 + 0.136722i \(0.0436568\pi\)
−0.990609 + 0.136722i \(0.956343\pi\)
\(458\) − 1.19202e6i − 0.265534i
\(459\) 0 0
\(460\) 0 0
\(461\) 4.07210e6 0.892413 0.446207 0.894930i \(-0.352775\pi\)
0.446207 + 0.894930i \(0.352775\pi\)
\(462\) 0 0
\(463\) − 2.02294e6i − 0.438561i −0.975662 0.219280i \(-0.929629\pi\)
0.975662 0.219280i \(-0.0703709\pi\)
\(464\) −2.23696e6 −0.482351
\(465\) 0 0
\(466\) −970668. −0.207065
\(467\) − 3.25097e6i − 0.689797i −0.938640 0.344898i \(-0.887913\pi\)
0.938640 0.344898i \(-0.112087\pi\)
\(468\) 0 0
\(469\) −1.17066e7 −2.45753
\(470\) 0 0
\(471\) 0 0
\(472\) 2.40240e6i 0.496353i
\(473\) 184112.i 0.0378381i
\(474\) 0 0
\(475\) 0 0
\(476\) −9.02093e6 −1.82488
\(477\) 0 0
\(478\) 97760.0i 0.0195700i
\(479\) −3.27936e6 −0.653056 −0.326528 0.945188i \(-0.605879\pi\)
−0.326528 + 0.945188i \(0.605879\pi\)
\(480\) 0 0
\(481\) 52052.0 0.0102583
\(482\) 221596.i 0.0434455i
\(483\) 0 0
\(484\) −3.89612e6 −0.755994
\(485\) 0 0
\(486\) 0 0
\(487\) − 8.53197e6i − 1.63015i −0.579357 0.815074i \(-0.696696\pi\)
0.579357 0.815074i \(-0.303304\pi\)
\(488\) − 3.87624e6i − 0.736819i
\(489\) 0 0
\(490\) 0 0
\(491\) −1.51265e6 −0.283162 −0.141581 0.989927i \(-0.545219\pi\)
−0.141581 + 0.989927i \(0.545219\pi\)
\(492\) 0 0
\(493\) − 5.72198e6i − 1.06030i
\(494\) 606320. 0.111785
\(495\) 0 0
\(496\) −1.60589e6 −0.293097
\(497\) 6.26842e6i 1.13833i
\(498\) 0 0
\(499\) 6.49190e6 1.16713 0.583567 0.812065i \(-0.301657\pi\)
0.583567 + 0.812065i \(0.301657\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 3.28750e6i − 0.582245i
\(503\) 8.61770e6i 1.51870i 0.650684 + 0.759349i \(0.274482\pi\)
−0.650684 + 0.759349i \(0.725518\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 880896. 0.152950
\(507\) 0 0
\(508\) 1.97546e6i 0.339632i
\(509\) 2.67323e6 0.457343 0.228671 0.973504i \(-0.426562\pi\)
0.228671 + 0.973504i \(0.426562\pi\)
\(510\) 0 0
\(511\) −7.44461e6 −1.26122
\(512\) − 5.89875e6i − 0.994455i
\(513\) 0 0
\(514\) −2.61248e6 −0.436160
\(515\) 0 0
\(516\) 0 0
\(517\) 1.78902e6i 0.294367i
\(518\) 69888.0i 0.0114440i
\(519\) 0 0
\(520\) 0 0
\(521\) −6.18500e6 −0.998264 −0.499132 0.866526i \(-0.666348\pi\)
−0.499132 + 0.866526i \(0.666348\pi\)
\(522\) 0 0
\(523\) 6.89452e6i 1.10217i 0.834448 + 0.551087i \(0.185787\pi\)
−0.834448 + 0.551087i \(0.814213\pi\)
\(524\) −2.14066e6 −0.340580
\(525\) 0 0
\(526\) 4.25667e6 0.670820
\(527\) − 4.10774e6i − 0.644283i
\(528\) 0 0
\(529\) −2.42023e6 −0.376026
\(530\) 0 0
\(531\) 0 0
\(532\) − 5.69856e6i − 0.872943i
\(533\) − 2.68783e6i − 0.409811i
\(534\) 0 0
\(535\) 0 0
\(536\) 7.31664e6 1.10002
\(537\) 0 0
\(538\) 2.88218e6i 0.429304i
\(539\) −2.96844e6 −0.440104
\(540\) 0 0
\(541\) 155502. 0.0228425 0.0114212 0.999935i \(-0.496364\pi\)
0.0114212 + 0.999935i \(0.496364\pi\)
\(542\) 186496.i 0.0272691i
\(543\) 0 0
\(544\) 8.64506e6 1.25248
\(545\) 0 0
\(546\) 0 0
\(547\) 1.26544e7i 1.80831i 0.427201 + 0.904157i \(0.359500\pi\)
−0.427201 + 0.904157i \(0.640500\pi\)
\(548\) 4.05770e6i 0.577204i
\(549\) 0 0
\(550\) 0 0
\(551\) 3.61460e6 0.507202
\(552\) 0 0
\(553\) 6.40512e6i 0.890665i
\(554\) −220596. −0.0305368
\(555\) 0 0
\(556\) −3.14216e6 −0.431064
\(557\) 7.07786e6i 0.966638i 0.875444 + 0.483319i \(0.160569\pi\)
−0.875444 + 0.483319i \(0.839431\pi\)
\(558\) 0 0
\(559\) 355784. 0.0481567
\(560\) 0 0
\(561\) 0 0
\(562\) − 384396.i − 0.0513379i
\(563\) 846636.i 0.112571i 0.998415 + 0.0562854i \(0.0179257\pi\)
−0.998415 + 0.0562854i \(0.982074\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 663768. 0.0870914
\(567\) 0 0
\(568\) − 3.91776e6i − 0.509527i
\(569\) 4.96041e6 0.642299 0.321149 0.947029i \(-0.395931\pi\)
0.321149 + 0.947029i \(0.395931\pi\)
\(570\) 0 0
\(571\) 8.96505e6 1.15070 0.575351 0.817907i \(-0.304866\pi\)
0.575351 + 0.817907i \(0.304866\pi\)
\(572\) − 1.18518e6i − 0.151459i
\(573\) 0 0
\(574\) 3.60883e6 0.457180
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.86080e6i − 0.357724i −0.983874 0.178862i \(-0.942758\pi\)
0.983874 0.178862i \(-0.0572415\pi\)
\(578\) 2.79165e6i 0.347570i
\(579\) 0 0
\(580\) 0 0
\(581\) −3.20947e6 −0.394451
\(582\) 0 0
\(583\) 3.52921e6i 0.430037i
\(584\) 4.65288e6 0.564534
\(585\) 0 0
\(586\) 4.38961e6 0.528059
\(587\) 6.74027e6i 0.807387i 0.914894 + 0.403694i \(0.132274\pi\)
−0.914894 + 0.403694i \(0.867726\pi\)
\(588\) 0 0
\(589\) 2.59488e6 0.308197
\(590\) 0 0
\(591\) 0 0
\(592\) 119392.i 0.0140014i
\(593\) − 1.78609e6i − 0.208578i −0.994547 0.104289i \(-0.966743\pi\)
0.994547 0.104289i \(-0.0332566\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.13050e7 1.30363
\(597\) 0 0
\(598\) − 1.70227e6i − 0.194660i
\(599\) 4.94620e6 0.563254 0.281627 0.959524i \(-0.409126\pi\)
0.281627 + 0.959524i \(0.409126\pi\)
\(600\) 0 0
\(601\) −4.58100e6 −0.517337 −0.258669 0.965966i \(-0.583284\pi\)
−0.258669 + 0.965966i \(0.583284\pi\)
\(602\) 477696.i 0.0537230i
\(603\) 0 0
\(604\) −1.25061e7 −1.39486
\(605\) 0 0
\(606\) 0 0
\(607\) 7.07999e6i 0.779940i 0.920828 + 0.389970i \(0.127515\pi\)
−0.920828 + 0.389970i \(0.872485\pi\)
\(608\) 5.46112e6i 0.599132i
\(609\) 0 0
\(610\) 0 0
\(611\) 3.45717e6 0.374643
\(612\) 0 0
\(613\) − 5.09609e6i − 0.547754i −0.961765 0.273877i \(-0.911694\pi\)
0.961765 0.273877i \(-0.0883061\pi\)
\(614\) −4.75502e6 −0.509016
\(615\) 0 0
\(616\) 3.40992e6 0.362070
\(617\) 1.30003e7i 1.37480i 0.726279 + 0.687400i \(0.241248\pi\)
−0.726279 + 0.687400i \(0.758752\pi\)
\(618\) 0 0
\(619\) −4.84406e6 −0.508139 −0.254070 0.967186i \(-0.581769\pi\)
−0.254070 + 0.967186i \(0.581769\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 4.74610e6i − 0.491882i
\(623\) 1.94630e7i 2.00905i
\(624\) 0 0
\(625\) 0 0
\(626\) 2.85883e6 0.291576
\(627\) 0 0
\(628\) − 7.34322e6i − 0.742998i
\(629\) −305396. −0.0307777
\(630\) 0 0
\(631\) 6.22775e6 0.622670 0.311335 0.950300i \(-0.399224\pi\)
0.311335 + 0.950300i \(0.399224\pi\)
\(632\) − 4.00320e6i − 0.398671i
\(633\) 0 0
\(634\) −4.24924e6 −0.419845
\(635\) 0 0
\(636\) 0 0
\(637\) 5.73630e6i 0.560123i
\(638\) 1.00936e6i 0.0981735i
\(639\) 0 0
\(640\) 0 0
\(641\) −1.53280e6 −0.147347 −0.0736734 0.997282i \(-0.523472\pi\)
−0.0736734 + 0.997282i \(0.523472\pi\)
\(642\) 0 0
\(643\) 1.74382e7i 1.66332i 0.555287 + 0.831659i \(0.312609\pi\)
−0.555287 + 0.831659i \(0.687391\pi\)
\(644\) −1.59990e7 −1.52012
\(645\) 0 0
\(646\) −3.55736e6 −0.335387
\(647\) 4.25469e6i 0.399583i 0.979838 + 0.199792i \(0.0640265\pi\)
−0.979838 + 0.199792i \(0.935974\pi\)
\(648\) 0 0
\(649\) −2.96296e6 −0.276130
\(650\) 0 0
\(651\) 0 0
\(652\) 4.32779e6i 0.398701i
\(653\) 3.01085e6i 0.276316i 0.990410 + 0.138158i \(0.0441181\pi\)
−0.990410 + 0.138158i \(0.955882\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.16509e6 0.559345
\(657\) 0 0
\(658\) 4.64179e6i 0.417947i
\(659\) −8.11462e6 −0.727871 −0.363936 0.931424i \(-0.618567\pi\)
−0.363936 + 0.931424i \(0.618567\pi\)
\(660\) 0 0
\(661\) 2.47370e6 0.220213 0.110107 0.993920i \(-0.464881\pi\)
0.110107 + 0.993920i \(0.464881\pi\)
\(662\) − 6.19970e6i − 0.549827i
\(663\) 0 0
\(664\) 2.00592e6 0.176560
\(665\) 0 0
\(666\) 0 0
\(667\) − 1.01482e7i − 0.883228i
\(668\) − 1.11068e7i − 0.963049i
\(669\) 0 0
\(670\) 0 0
\(671\) 4.78070e6 0.409907
\(672\) 0 0
\(673\) − 5.77063e6i − 0.491117i −0.969382 0.245559i \(-0.921029\pi\)
0.969382 0.245559i \(-0.0789714\pi\)
\(674\) 4.80016e6 0.407011
\(675\) 0 0
\(676\) 8.10592e6 0.682237
\(677\) − 1.67197e7i − 1.40203i −0.713147 0.701014i \(-0.752731\pi\)
0.713147 0.701014i \(-0.247269\pi\)
\(678\) 0 0
\(679\) 2.28553e7 1.90245
\(680\) 0 0
\(681\) 0 0
\(682\) 724608.i 0.0596544i
\(683\) 7.14532e6i 0.586097i 0.956098 + 0.293049i \(0.0946698\pi\)
−0.956098 + 0.293049i \(0.905330\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.24800e6 −0.101252
\(687\) 0 0
\(688\) 816064.i 0.0657284i
\(689\) 6.81996e6 0.547310
\(690\) 0 0
\(691\) −8.78395e6 −0.699833 −0.349917 0.936781i \(-0.613790\pi\)
−0.349917 + 0.936781i \(0.613790\pi\)
\(692\) − 1.60573e7i − 1.27470i
\(693\) 0 0
\(694\) −3.55482e6 −0.280169
\(695\) 0 0
\(696\) 0 0
\(697\) 1.57698e7i 1.22955i
\(698\) − 4.29610e6i − 0.333761i
\(699\) 0 0
\(700\) 0 0
\(701\) 1.60141e7 1.23086 0.615428 0.788193i \(-0.288983\pi\)
0.615428 + 0.788193i \(0.288983\pi\)
\(702\) 0 0
\(703\) − 192920.i − 0.0147228i
\(704\) 1.58182e6 0.120289
\(705\) 0 0
\(706\) −1.32371e6 −0.0999495
\(707\) 1.72604e7i 1.29868i
\(708\) 0 0
\(709\) 1.91354e7 1.42962 0.714811 0.699318i \(-0.246513\pi\)
0.714811 + 0.699318i \(0.246513\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 1.21644e7i − 0.899271i
\(713\) − 7.28525e6i − 0.536686i
\(714\) 0 0
\(715\) 0 0
\(716\) −1.66449e7 −1.21338
\(717\) 0 0
\(718\) 518640.i 0.0375452i
\(719\) 1.02934e7 0.742566 0.371283 0.928520i \(-0.378918\pi\)
0.371283 + 0.928520i \(0.378918\pi\)
\(720\) 0 0
\(721\) −3.74477e6 −0.268279
\(722\) 2.70500e6i 0.193119i
\(723\) 0 0
\(724\) −2.99874e6 −0.212615
\(725\) 0 0
\(726\) 0 0
\(727\) − 1.93264e7i − 1.35618i −0.734981 0.678088i \(-0.762809\pi\)
0.734981 0.678088i \(-0.237191\pi\)
\(728\) − 6.58944e6i − 0.460808i
\(729\) 0 0
\(730\) 0 0
\(731\) −2.08743e6 −0.144484
\(732\) 0 0
\(733\) − 5.26197e6i − 0.361733i −0.983508 0.180866i \(-0.942110\pi\)
0.983508 0.180866i \(-0.0578902\pi\)
\(734\) −2.99986e6 −0.205523
\(735\) 0 0
\(736\) 1.53324e7 1.04331
\(737\) 9.02386e6i 0.611961i
\(738\) 0 0
\(739\) −2.82944e7 −1.90585 −0.952927 0.303199i \(-0.901945\pi\)
−0.952927 + 0.303199i \(0.901945\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 9.15686e6i 0.610572i
\(743\) 2.09863e7i 1.39464i 0.716759 + 0.697321i \(0.245625\pi\)
−0.716759 + 0.697321i \(0.754375\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.47615e6 0.294481
\(747\) 0 0
\(748\) 6.95363e6i 0.454420i
\(749\) 3.03921e7 1.97950
\(750\) 0 0
\(751\) −1.89668e7 −1.22714 −0.613572 0.789639i \(-0.710268\pi\)
−0.613572 + 0.789639i \(0.710268\pi\)
\(752\) 7.92973e6i 0.511345i
\(753\) 0 0
\(754\) 1.95052e6 0.124946
\(755\) 0 0
\(756\) 0 0
\(757\) − 1.08257e7i − 0.686617i −0.939223 0.343309i \(-0.888452\pi\)
0.939223 0.343309i \(-0.111548\pi\)
\(758\) 6.31868e6i 0.399442i
\(759\) 0 0
\(760\) 0 0
\(761\) −1.90534e7 −1.19264 −0.596322 0.802745i \(-0.703372\pi\)
−0.596322 + 0.802745i \(0.703372\pi\)
\(762\) 0 0
\(763\) − 7.07136e6i − 0.439736i
\(764\) −1.31475e7 −0.814908
\(765\) 0 0
\(766\) 684432. 0.0421462
\(767\) 5.72572e6i 0.351432i
\(768\) 0 0
\(769\) 1.57826e7 0.962415 0.481208 0.876607i \(-0.340198\pi\)
0.481208 + 0.876607i \(0.340198\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 1.47577e6i − 0.0891199i
\(773\) − 2.44049e7i − 1.46902i −0.678598 0.734510i \(-0.737412\pi\)
0.678598 0.734510i \(-0.262588\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.42846e7 −0.851555
\(777\) 0 0
\(778\) − 176940.i − 0.0104804i
\(779\) −9.96188e6 −0.588163
\(780\) 0 0
\(781\) 4.83190e6 0.283459
\(782\) 9.98746e6i 0.584034i
\(783\) 0 0
\(784\) −1.31574e7 −0.764504
\(785\) 0 0
\(786\) 0 0
\(787\) 3.37607e7i 1.94301i 0.237019 + 0.971505i \(0.423830\pi\)
−0.237019 + 0.971505i \(0.576170\pi\)
\(788\) − 1.27641e7i − 0.732278i
\(789\) 0 0
\(790\) 0 0
\(791\) −2.14771e6 −0.122049
\(792\) 0 0
\(793\) − 9.23837e6i − 0.521690i
\(794\) −1.09135e7 −0.614344
\(795\) 0 0
\(796\) −2.42200e7 −1.35485
\(797\) − 2.19885e7i − 1.22617i −0.790019 0.613083i \(-0.789929\pi\)
0.790019 0.613083i \(-0.210071\pi\)
\(798\) 0 0
\(799\) −2.02837e7 −1.12403
\(800\) 0 0
\(801\) 0 0
\(802\) 8.09360e6i 0.444330i
\(803\) 5.73855e6i 0.314061i
\(804\) 0 0
\(805\) 0 0
\(806\) 1.40026e6 0.0759224
\(807\) 0 0
\(808\) − 1.07878e7i − 0.581303i
\(809\) −2.93597e7 −1.57717 −0.788587 0.614923i \(-0.789187\pi\)
−0.788587 + 0.614923i \(0.789187\pi\)
\(810\) 0 0
\(811\) 3.17703e7 1.69617 0.848083 0.529863i \(-0.177757\pi\)
0.848083 + 0.529863i \(0.177757\pi\)
\(812\) − 1.83322e7i − 0.975716i
\(813\) 0 0
\(814\) 53872.0 0.00284972
\(815\) 0 0
\(816\) 0 0
\(817\) − 1.31864e6i − 0.0691148i
\(818\) − 5.42414e6i − 0.283431i
\(819\) 0 0
\(820\) 0 0
\(821\) 2.71430e6 0.140540 0.0702699 0.997528i \(-0.477614\pi\)
0.0702699 + 0.997528i \(0.477614\pi\)
\(822\) 0 0
\(823\) 1.25866e7i 0.647753i 0.946099 + 0.323877i \(0.104986\pi\)
−0.946099 + 0.323877i \(0.895014\pi\)
\(824\) 2.34048e6 0.120084
\(825\) 0 0
\(826\) −7.68768e6 −0.392053
\(827\) 8.72355e6i 0.443537i 0.975099 + 0.221768i \(0.0711828\pi\)
−0.975099 + 0.221768i \(0.928817\pi\)
\(828\) 0 0
\(829\) 1.06178e7 0.536597 0.268299 0.963336i \(-0.413539\pi\)
0.268299 + 0.963336i \(0.413539\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 3.05677e6i − 0.153093i
\(833\) − 3.36556e7i − 1.68053i
\(834\) 0 0
\(835\) 0 0
\(836\) −4.39264e6 −0.217375
\(837\) 0 0
\(838\) − 7.43492e6i − 0.365735i
\(839\) 1.67765e7 0.822805 0.411403 0.911454i \(-0.365039\pi\)
0.411403 + 0.911454i \(0.365039\pi\)
\(840\) 0 0
\(841\) −8.88305e6 −0.433084
\(842\) − 7.10500e6i − 0.345370i
\(843\) 0 0
\(844\) 3.09583e7 1.49596
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.67162e7i − 1.27958i
\(848\) 1.56430e7i 0.747016i
\(849\) 0 0
\(850\) 0 0
\(851\) −541632. −0.0256378
\(852\) 0 0
\(853\) 2.20186e7i 1.03613i 0.855340 + 0.518067i \(0.173348\pi\)
−0.855340 + 0.518067i \(0.826652\pi\)
\(854\) 1.24040e7 0.581991
\(855\) 0 0
\(856\) −1.89950e7 −0.886045
\(857\) − 3.16676e7i − 1.47287i −0.676510 0.736434i \(-0.736508\pi\)
0.676510 0.736434i \(-0.263492\pi\)
\(858\) 0 0
\(859\) −1.58064e7 −0.730886 −0.365443 0.930834i \(-0.619082\pi\)
−0.365443 + 0.930834i \(0.619082\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 8.12410e6i − 0.372398i
\(863\) − 1.44287e7i − 0.659476i −0.944072 0.329738i \(-0.893040\pi\)
0.944072 0.329738i \(-0.106960\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.45257e7 −0.658178
\(867\) 0 0
\(868\) − 1.31604e7i − 0.592886i
\(869\) 4.93728e6 0.221788
\(870\) 0 0
\(871\) 1.74380e7 0.778845
\(872\) 4.41960e6i 0.196830i
\(873\) 0 0
\(874\) −6.30912e6 −0.279377
\(875\) 0 0
\(876\) 0 0
\(877\) 247902.i 0.0108838i 0.999985 + 0.00544191i \(0.00173222\pi\)
−0.999985 + 0.00544191i \(0.998268\pi\)
\(878\) − 1.08206e7i − 0.473711i
\(879\) 0 0
\(880\) 0 0
\(881\) −4.10268e7 −1.78085 −0.890426 0.455128i \(-0.849594\pi\)
−0.890426 + 0.455128i \(0.849594\pi\)
\(882\) 0 0
\(883\) − 4.18015e7i − 1.80422i −0.431503 0.902112i \(-0.642016\pi\)
0.431503 0.902112i \(-0.357984\pi\)
\(884\) 1.34374e7 0.578343
\(885\) 0 0
\(886\) −1.30305e7 −0.557669
\(887\) 2.10476e7i 0.898241i 0.893471 + 0.449120i \(0.148263\pi\)
−0.893471 + 0.449120i \(0.851737\pi\)
\(888\) 0 0
\(889\) −1.35460e7 −0.574852
\(890\) 0 0
\(891\) 0 0
\(892\) − 3.14041e7i − 1.32152i
\(893\) − 1.28133e7i − 0.537690i
\(894\) 0 0
\(895\) 0 0
\(896\) 3.57581e7 1.48800
\(897\) 0 0
\(898\) 1.01990e6i 0.0422053i
\(899\) 8.34768e6 0.344482
\(900\) 0 0
\(901\) −4.00136e7 −1.64208
\(902\) − 2.78181e6i − 0.113844i
\(903\) 0 0
\(904\) 1.34232e6 0.0546305
\(905\) 0 0
\(906\) 0 0
\(907\) 7.48309e6i 0.302039i 0.988531 + 0.151019i \(0.0482556\pi\)
−0.988531 + 0.151019i \(0.951744\pi\)
\(908\) 653744.i 0.0263144i
\(909\) 0 0
\(910\) 0 0
\(911\) 6.63165e6 0.264744 0.132372 0.991200i \(-0.457741\pi\)
0.132372 + 0.991200i \(0.457741\pi\)
\(912\) 0 0
\(913\) 2.47397e6i 0.0982239i
\(914\) 2.44168e6 0.0966772
\(915\) 0 0
\(916\) 1.66883e7 0.657163
\(917\) − 1.46788e7i − 0.576457i
\(918\) 0 0
\(919\) 1.68976e7 0.659990 0.329995 0.943983i \(-0.392953\pi\)
0.329995 + 0.943983i \(0.392953\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 8.14420e6i − 0.315516i
\(923\) − 9.33733e6i − 0.360760i
\(924\) 0 0
\(925\) 0 0
\(926\) −4.04587e6 −0.155055
\(927\) 0 0
\(928\) 1.75683e7i 0.669669i
\(929\) −1.28653e7 −0.489081 −0.244541 0.969639i \(-0.578637\pi\)
−0.244541 + 0.969639i \(0.578637\pi\)
\(930\) 0 0
\(931\) 2.12604e7 0.803892
\(932\) − 1.35894e7i − 0.512459i
\(933\) 0 0
\(934\) −6.50194e6 −0.243880
\(935\) 0 0
\(936\) 0 0
\(937\) 1.06887e7i 0.397718i 0.980028 + 0.198859i \(0.0637236\pi\)
−0.980028 + 0.198859i \(0.936276\pi\)
\(938\) 2.34132e7i 0.868870i
\(939\) 0 0
\(940\) 0 0
\(941\) −2.82455e7 −1.03986 −0.519930 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(942\) 0 0
\(943\) 2.79684e7i 1.02421i
\(944\) −1.31331e7 −0.479665
\(945\) 0 0
\(946\) 368224. 0.0133778
\(947\) 1.70892e7i 0.619222i 0.950863 + 0.309611i \(0.100199\pi\)
−0.950863 + 0.309611i \(0.899801\pi\)
\(948\) 0 0
\(949\) 1.10894e7 0.399706
\(950\) 0 0
\(951\) 0 0
\(952\) 3.86611e7i 1.38255i
\(953\) 2.22259e7i 0.792735i 0.918092 + 0.396367i \(0.129729\pi\)
−0.918092 + 0.396367i \(0.870271\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.36864e6 −0.0484333
\(957\) 0 0
\(958\) 6.55872e6i 0.230890i
\(959\) −2.78243e7 −0.976961
\(960\) 0 0
\(961\) −2.26364e7 −0.790678
\(962\) − 104104.i − 0.00362685i
\(963\) 0 0
\(964\) −3.10234e6 −0.107522
\(965\) 0 0
\(966\) 0 0
\(967\) 2.41551e7i 0.830696i 0.909663 + 0.415348i \(0.136340\pi\)
−0.909663 + 0.415348i \(0.863660\pi\)
\(968\) 1.66976e7i 0.572752i
\(969\) 0 0
\(970\) 0 0
\(971\) 5.48313e7 1.86630 0.933149 0.359491i \(-0.117050\pi\)
0.933149 + 0.359491i \(0.117050\pi\)
\(972\) 0 0
\(973\) − 2.15462e7i − 0.729608i
\(974\) −1.70639e7 −0.576344
\(975\) 0 0
\(976\) 2.11901e7 0.712047
\(977\) 1.56612e7i 0.524915i 0.964944 + 0.262457i \(0.0845329\pi\)
−0.964944 + 0.262457i \(0.915467\pi\)
\(978\) 0 0
\(979\) 1.50028e7 0.500281
\(980\) 0 0
\(981\) 0 0
\(982\) 3.02530e6i 0.100113i
\(983\) − 1.63420e7i − 0.539412i −0.962943 0.269706i \(-0.913073\pi\)
0.962943 0.269706i \(-0.0869266\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.14440e7 −0.374873
\(987\) 0 0
\(988\) 8.48848e6i 0.276654i
\(989\) −3.70214e6 −0.120355
\(990\) 0 0
\(991\) 1.37576e7 0.444997 0.222498 0.974933i \(-0.428579\pi\)
0.222498 + 0.974933i \(0.428579\pi\)
\(992\) 1.26121e7i 0.406919i
\(993\) 0 0
\(994\) 1.25368e7 0.402459
\(995\) 0 0
\(996\) 0 0
\(997\) − 1.29097e7i − 0.411320i −0.978624 0.205660i \(-0.934066\pi\)
0.978624 0.205660i \(-0.0659341\pi\)
\(998\) − 1.29838e7i − 0.412644i
\(999\) 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 225.6.b.e.199.1 2
3.2 odd 2 25.6.b.a.24.2 2
5.2 odd 4 225.6.a.f.1.1 1
5.3 odd 4 45.6.a.b.1.1 1
5.4 even 2 inner 225.6.b.e.199.2 2
12.11 even 2 400.6.c.j.49.1 2
15.2 even 4 25.6.a.a.1.1 1
15.8 even 4 5.6.a.a.1.1 1
15.14 odd 2 25.6.b.a.24.1 2
20.3 even 4 720.6.a.a.1.1 1
60.23 odd 4 80.6.a.e.1.1 1
60.47 odd 4 400.6.a.g.1.1 1
60.59 even 2 400.6.c.j.49.2 2
105.83 odd 4 245.6.a.b.1.1 1
120.53 even 4 320.6.a.j.1.1 1
120.83 odd 4 320.6.a.g.1.1 1
165.98 odd 4 605.6.a.a.1.1 1
195.38 even 4 845.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 1 15.8 even 4
25.6.a.a.1.1 1 15.2 even 4
25.6.b.a.24.1 2 15.14 odd 2
25.6.b.a.24.2 2 3.2 odd 2
45.6.a.b.1.1 1 5.3 odd 4
80.6.a.e.1.1 1 60.23 odd 4
225.6.a.f.1.1 1 5.2 odd 4
225.6.b.e.199.1 2 1.1 even 1 trivial
225.6.b.e.199.2 2 5.4 even 2 inner
245.6.a.b.1.1 1 105.83 odd 4
320.6.a.g.1.1 1 120.83 odd 4
320.6.a.j.1.1 1 120.53 even 4
400.6.a.g.1.1 1 60.47 odd 4
400.6.c.j.49.1 2 12.11 even 2
400.6.c.j.49.2 2 60.59 even 2
605.6.a.a.1.1 1 165.98 odd 4
720.6.a.a.1.1 1 20.3 even 4
845.6.a.b.1.1 1 195.38 even 4