Properties

Label 225.12.b.d.199.2
Level $225$
Weight $12$
Character 225.199
Analytic conductor $172.877$
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: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(172.877215626\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.12.b.d.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+24.0000i q^{2} +1472.00 q^{4} -16744.0i q^{7} +84480.0i q^{8} +O(q^{10})\) \(q+24.0000i q^{2} +1472.00 q^{4} -16744.0i q^{7} +84480.0i q^{8} -534612. q^{11} +577738. i q^{13} +401856. q^{14} +987136. q^{16} +6.90593e6i q^{17} -1.06614e7 q^{19} -1.28307e7i q^{22} +1.86433e7i q^{23} -1.38657e7 q^{26} -2.46472e7i q^{28} +1.28407e8 q^{29} -5.28432e7 q^{31} +1.96706e8i q^{32} -1.65742e8 q^{34} -1.82213e8i q^{37} -2.55874e8i q^{38} -3.08120e8 q^{41} +1.71257e7i q^{43} -7.86949e8 q^{44} -4.47439e8 q^{46} -2.68735e9i q^{47} +1.69697e9 q^{49} +8.50430e8i q^{52} -1.59606e9i q^{53} +1.41453e9 q^{56} +3.08176e9i q^{58} -5.18920e9 q^{59} +6.95648e9 q^{61} -1.26824e9i q^{62} -2.69930e9 q^{64} -1.54818e10i q^{67} +1.01655e10i q^{68} -9.79149e9 q^{71} -1.46379e9i q^{73} +4.37312e9 q^{74} -1.56936e10 q^{76} +8.95154e9i q^{77} -3.81168e10 q^{79} -7.39489e9i q^{82} -2.93351e10i q^{83} -4.11017e8 q^{86} -4.51640e10i q^{88} -2.49929e10 q^{89} +9.67365e9 q^{91} +2.74429e10i q^{92} +6.44964e10 q^{94} +7.50136e10i q^{97} +4.07272e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2944q^{4} + O(q^{10}) \) \( 2q + 2944q^{4} - 1069224q^{11} + 803712q^{14} + 1974272q^{16} - 21322840q^{19} - 27731424q^{26} + 256813260q^{29} - 105686336q^{31} - 331484832q^{34} - 616240884q^{41} - 1573897728q^{44} - 894877056q^{46} + 3393930414q^{49} + 2829066240q^{56} - 10378407480q^{59} + 13912957324q^{61} - 5398593536q^{64} - 19582970544q^{71} + 8746239072q^{74} - 31387220480q^{76} - 76233691360q^{79} - 822033984q^{86} - 49985834220q^{89} + 19347290144q^{91} + 128992727808q^{94} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 24.0000i 0.530330i 0.964203 + 0.265165i \(0.0854264\pi\)
−0.964203 + 0.265165i \(0.914574\pi\)
\(3\) 0 0
\(4\) 1472.00 0.718750
\(5\) 0 0
\(6\) 0 0
\(7\) − 16744.0i − 0.376548i −0.982117 0.188274i \(-0.939711\pi\)
0.982117 0.188274i \(-0.0602893\pi\)
\(8\) 84480.0i 0.911505i
\(9\) 0 0
\(10\) 0 0
\(11\) −534612. −1.00087 −0.500436 0.865773i \(-0.666827\pi\)
−0.500436 + 0.865773i \(0.666827\pi\)
\(12\) 0 0
\(13\) 577738.i 0.431561i 0.976442 + 0.215781i \(0.0692296\pi\)
−0.976442 + 0.215781i \(0.930770\pi\)
\(14\) 401856. 0.199695
\(15\) 0 0
\(16\) 987136. 0.235352
\(17\) 6.90593e6i 1.17965i 0.807531 + 0.589825i \(0.200803\pi\)
−0.807531 + 0.589825i \(0.799197\pi\)
\(18\) 0 0
\(19\) −1.06614e7 −0.987803 −0.493901 0.869518i \(-0.664430\pi\)
−0.493901 + 0.869518i \(0.664430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 1.28307e7i − 0.530793i
\(23\) 1.86433e7i 0.603975i 0.953312 + 0.301988i \(0.0976501\pi\)
−0.953312 + 0.301988i \(0.902350\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.38657e7 −0.228870
\(27\) 0 0
\(28\) − 2.46472e7i − 0.270644i
\(29\) 1.28407e8 1.16251 0.581257 0.813720i \(-0.302561\pi\)
0.581257 + 0.813720i \(0.302561\pi\)
\(30\) 0 0
\(31\) −5.28432e7 −0.331512 −0.165756 0.986167i \(-0.553006\pi\)
−0.165756 + 0.986167i \(0.553006\pi\)
\(32\) 1.96706e8i 1.03632i
\(33\) 0 0
\(34\) −1.65742e8 −0.625604
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.82213e8i − 0.431987i −0.976395 0.215993i \(-0.930701\pi\)
0.976395 0.215993i \(-0.0692990\pi\)
\(38\) − 2.55874e8i − 0.523862i
\(39\) 0 0
\(40\) 0 0
\(41\) −3.08120e8 −0.415345 −0.207673 0.978198i \(-0.566589\pi\)
−0.207673 + 0.978198i \(0.566589\pi\)
\(42\) 0 0
\(43\) 1.71257e7i 0.0177653i 0.999961 + 0.00888264i \(0.00282747\pi\)
−0.999961 + 0.00888264i \(0.997173\pi\)
\(44\) −7.86949e8 −0.719377
\(45\) 0 0
\(46\) −4.47439e8 −0.320306
\(47\) − 2.68735e9i − 1.70917i −0.519310 0.854586i \(-0.673811\pi\)
0.519310 0.854586i \(-0.326189\pi\)
\(48\) 0 0
\(49\) 1.69697e9 0.858212
\(50\) 0 0
\(51\) 0 0
\(52\) 8.50430e8i 0.310185i
\(53\) − 1.59606e9i − 0.524241i −0.965035 0.262120i \(-0.915578\pi\)
0.965035 0.262120i \(-0.0844217\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.41453e9 0.343225
\(57\) 0 0
\(58\) 3.08176e9i 0.616517i
\(59\) −5.18920e9 −0.944963 −0.472481 0.881341i \(-0.656642\pi\)
−0.472481 + 0.881341i \(0.656642\pi\)
\(60\) 0 0
\(61\) 6.95648e9 1.05457 0.527285 0.849689i \(-0.323210\pi\)
0.527285 + 0.849689i \(0.323210\pi\)
\(62\) − 1.26824e9i − 0.175811i
\(63\) 0 0
\(64\) −2.69930e9 −0.314240
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.54818e10i − 1.40091i −0.713696 0.700456i \(-0.752980\pi\)
0.713696 0.700456i \(-0.247020\pi\)
\(68\) 1.01655e10i 0.847874i
\(69\) 0 0
\(70\) 0 0
\(71\) −9.79149e9 −0.644062 −0.322031 0.946729i \(-0.604366\pi\)
−0.322031 + 0.946729i \(0.604366\pi\)
\(72\) 0 0
\(73\) − 1.46379e9i − 0.0826425i −0.999146 0.0413212i \(-0.986843\pi\)
0.999146 0.0413212i \(-0.0131567\pi\)
\(74\) 4.37312e9 0.229096
\(75\) 0 0
\(76\) −1.56936e10 −0.709983
\(77\) 8.95154e9i 0.376876i
\(78\) 0 0
\(79\) −3.81168e10 −1.39370 −0.696848 0.717219i \(-0.745415\pi\)
−0.696848 + 0.717219i \(0.745415\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 7.39489e9i − 0.220270i
\(83\) − 2.93351e10i − 0.817444i −0.912659 0.408722i \(-0.865975\pi\)
0.912659 0.408722i \(-0.134025\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.11017e8 −0.00942146
\(87\) 0 0
\(88\) − 4.51640e10i − 0.912300i
\(89\) −2.49929e10 −0.474430 −0.237215 0.971457i \(-0.576235\pi\)
−0.237215 + 0.971457i \(0.576235\pi\)
\(90\) 0 0
\(91\) 9.67365e9 0.162503
\(92\) 2.74429e10i 0.434107i
\(93\) 0 0
\(94\) 6.44964e10 0.906425
\(95\) 0 0
\(96\) 0 0
\(97\) 7.50136e10i 0.886942i 0.896289 + 0.443471i \(0.146253\pi\)
−0.896289 + 0.443471i \(0.853747\pi\)
\(98\) 4.07272e10i 0.455136i
\(99\) 0 0
\(100\) 0 0
\(101\) −8.17430e10 −0.773896 −0.386948 0.922101i \(-0.626471\pi\)
−0.386948 + 0.922101i \(0.626471\pi\)
\(102\) 0 0
\(103\) 2.25755e11i 1.91881i 0.282025 + 0.959407i \(0.408994\pi\)
−0.282025 + 0.959407i \(0.591006\pi\)
\(104\) −4.88073e10 −0.393370
\(105\) 0 0
\(106\) 3.83053e10 0.278021
\(107\) − 9.02413e10i − 0.622006i −0.950409 0.311003i \(-0.899335\pi\)
0.950409 0.311003i \(-0.100665\pi\)
\(108\) 0 0
\(109\) −7.34827e10 −0.457445 −0.228723 0.973492i \(-0.573455\pi\)
−0.228723 + 0.973492i \(0.573455\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.65286e10i − 0.0886211i
\(113\) − 8.51469e10i − 0.434748i −0.976088 0.217374i \(-0.930251\pi\)
0.976088 0.217374i \(-0.0697491\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.89015e11 0.835557
\(117\) 0 0
\(118\) − 1.24541e11i − 0.501142i
\(119\) 1.15633e11 0.444195
\(120\) 0 0
\(121\) 4.98320e8 0.00174658
\(122\) 1.66955e11i 0.559270i
\(123\) 0 0
\(124\) −7.77851e10 −0.238274
\(125\) 0 0
\(126\) 0 0
\(127\) − 2.62717e11i − 0.705615i −0.935696 0.352808i \(-0.885227\pi\)
0.935696 0.352808i \(-0.114773\pi\)
\(128\) 3.38071e11i 0.869668i
\(129\) 0 0
\(130\) 0 0
\(131\) −6.31529e11 −1.43021 −0.715107 0.699015i \(-0.753622\pi\)
−0.715107 + 0.699015i \(0.753622\pi\)
\(132\) 0 0
\(133\) 1.78515e11i 0.371955i
\(134\) 3.71564e11 0.742946
\(135\) 0 0
\(136\) −5.83413e11 −1.07526
\(137\) 2.97199e11i 0.526119i 0.964780 + 0.263059i \(0.0847315\pi\)
−0.964780 + 0.263059i \(0.915268\pi\)
\(138\) 0 0
\(139\) −5.96794e11 −0.975535 −0.487767 0.872974i \(-0.662189\pi\)
−0.487767 + 0.872974i \(0.662189\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 2.34996e11i − 0.341565i
\(143\) − 3.08866e11i − 0.431938i
\(144\) 0 0
\(145\) 0 0
\(146\) 3.51310e10 0.0438278
\(147\) 0 0
\(148\) − 2.68218e11i − 0.310491i
\(149\) −1.11543e12 −1.24428 −0.622142 0.782905i \(-0.713737\pi\)
−0.622142 + 0.782905i \(0.713737\pi\)
\(150\) 0 0
\(151\) −8.24447e11 −0.854653 −0.427326 0.904097i \(-0.640544\pi\)
−0.427326 + 0.904097i \(0.640544\pi\)
\(152\) − 9.00677e11i − 0.900387i
\(153\) 0 0
\(154\) −2.14837e11 −0.199869
\(155\) 0 0
\(156\) 0 0
\(157\) 1.31512e12i 1.10031i 0.835062 + 0.550156i \(0.185432\pi\)
−0.835062 + 0.550156i \(0.814568\pi\)
\(158\) − 9.14804e11i − 0.739119i
\(159\) 0 0
\(160\) 0 0
\(161\) 3.12163e11 0.227425
\(162\) 0 0
\(163\) 3.57833e11i 0.243584i 0.992556 + 0.121792i \(0.0388640\pi\)
−0.992556 + 0.121792i \(0.961136\pi\)
\(164\) −4.53553e11 −0.298529
\(165\) 0 0
\(166\) 7.04042e11 0.433515
\(167\) − 2.75483e12i − 1.64117i −0.571521 0.820587i \(-0.693646\pi\)
0.571521 0.820587i \(-0.306354\pi\)
\(168\) 0 0
\(169\) 1.45838e12 0.813755
\(170\) 0 0
\(171\) 0 0
\(172\) 2.52090e10i 0.0127688i
\(173\) − 9.50387e11i − 0.466280i −0.972443 0.233140i \(-0.925100\pi\)
0.972443 0.233140i \(-0.0749001\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.27735e11 −0.235557
\(177\) 0 0
\(178\) − 5.99830e11i − 0.251604i
\(179\) 1.68138e12 0.683873 0.341936 0.939723i \(-0.388917\pi\)
0.341936 + 0.939723i \(0.388917\pi\)
\(180\) 0 0
\(181\) −9.96774e11 −0.381386 −0.190693 0.981650i \(-0.561073\pi\)
−0.190693 + 0.981650i \(0.561073\pi\)
\(182\) 2.32167e11i 0.0861804i
\(183\) 0 0
\(184\) −1.57498e12 −0.550526
\(185\) 0 0
\(186\) 0 0
\(187\) − 3.69200e12i − 1.18068i
\(188\) − 3.95578e12i − 1.22847i
\(189\) 0 0
\(190\) 0 0
\(191\) −2.76240e12 −0.786328 −0.393164 0.919468i \(-0.628619\pi\)
−0.393164 + 0.919468i \(0.628619\pi\)
\(192\) 0 0
\(193\) − 5.44239e12i − 1.46293i −0.681878 0.731466i \(-0.738836\pi\)
0.681878 0.731466i \(-0.261164\pi\)
\(194\) −1.80033e12 −0.470372
\(195\) 0 0
\(196\) 2.49793e12 0.616840
\(197\) 2.87609e12i 0.690619i 0.938489 + 0.345309i \(0.112226\pi\)
−0.938489 + 0.345309i \(0.887774\pi\)
\(198\) 0 0
\(199\) −7.28391e11 −0.165452 −0.0827262 0.996572i \(-0.526363\pi\)
−0.0827262 + 0.996572i \(0.526363\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 1.96183e12i − 0.410421i
\(203\) − 2.15004e12i − 0.437742i
\(204\) 0 0
\(205\) 0 0
\(206\) −5.41812e12 −1.01760
\(207\) 0 0
\(208\) 5.70306e11i 0.101569i
\(209\) 5.69972e12 0.988665
\(210\) 0 0
\(211\) −6.79317e12 −1.11820 −0.559099 0.829101i \(-0.688853\pi\)
−0.559099 + 0.829101i \(0.688853\pi\)
\(212\) − 2.34939e12i − 0.376798i
\(213\) 0 0
\(214\) 2.16579e12 0.329868
\(215\) 0 0
\(216\) 0 0
\(217\) 8.84806e11i 0.124830i
\(218\) − 1.76358e12i − 0.242597i
\(219\) 0 0
\(220\) 0 0
\(221\) −3.98982e12 −0.509092
\(222\) 0 0
\(223\) − 7.33486e12i − 0.890667i −0.895365 0.445333i \(-0.853085\pi\)
0.895365 0.445333i \(-0.146915\pi\)
\(224\) 3.29365e12 0.390223
\(225\) 0 0
\(226\) 2.04352e12 0.230560
\(227\) 1.35984e12i 0.149743i 0.997193 + 0.0748713i \(0.0238546\pi\)
−0.997193 + 0.0748713i \(0.976145\pi\)
\(228\) 0 0
\(229\) 1.18244e13 1.24075 0.620375 0.784305i \(-0.286980\pi\)
0.620375 + 0.784305i \(0.286980\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.08478e13i 1.05964i
\(233\) − 1.75634e13i − 1.67552i −0.546038 0.837761i \(-0.683865\pi\)
0.546038 0.837761i \(-0.316135\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.63851e12 −0.679192
\(237\) 0 0
\(238\) 2.77519e12i 0.235570i
\(239\) −7.13958e12 −0.592221 −0.296111 0.955154i \(-0.595690\pi\)
−0.296111 + 0.955154i \(0.595690\pi\)
\(240\) 0 0
\(241\) −2.31307e11 −0.0183271 −0.00916357 0.999958i \(-0.502917\pi\)
−0.00916357 + 0.999958i \(0.502917\pi\)
\(242\) 1.19597e10i 0 0.000926264i
\(243\) 0 0
\(244\) 1.02399e13 0.757972
\(245\) 0 0
\(246\) 0 0
\(247\) − 6.15951e12i − 0.426297i
\(248\) − 4.46419e12i − 0.302175i
\(249\) 0 0
\(250\) 0 0
\(251\) −1.29831e13 −0.822567 −0.411284 0.911507i \(-0.634919\pi\)
−0.411284 + 0.911507i \(0.634919\pi\)
\(252\) 0 0
\(253\) − 9.96692e12i − 0.604502i
\(254\) 6.30521e12 0.374209
\(255\) 0 0
\(256\) −1.36419e13 −0.775451
\(257\) − 2.39612e13i − 1.33314i −0.745442 0.666571i \(-0.767761\pi\)
0.745442 0.666571i \(-0.232239\pi\)
\(258\) 0 0
\(259\) −3.05098e12 −0.162664
\(260\) 0 0
\(261\) 0 0
\(262\) − 1.51567e13i − 0.758485i
\(263\) − 2.42737e13i − 1.18954i −0.803895 0.594771i \(-0.797243\pi\)
0.803895 0.594771i \(-0.202757\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.28436e12 −0.197259
\(267\) 0 0
\(268\) − 2.27892e13i − 1.00691i
\(269\) 2.58377e13 1.11845 0.559225 0.829016i \(-0.311099\pi\)
0.559225 + 0.829016i \(0.311099\pi\)
\(270\) 0 0
\(271\) −3.76793e12 −0.156593 −0.0782964 0.996930i \(-0.524948\pi\)
−0.0782964 + 0.996930i \(0.524948\pi\)
\(272\) 6.81710e12i 0.277633i
\(273\) 0 0
\(274\) −7.13277e12 −0.279017
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.64189e13i − 0.604931i −0.953160 0.302466i \(-0.902190\pi\)
0.953160 0.302466i \(-0.0978098\pi\)
\(278\) − 1.43230e13i − 0.517355i
\(279\) 0 0
\(280\) 0 0
\(281\) −2.10357e13 −0.716263 −0.358132 0.933671i \(-0.616586\pi\)
−0.358132 + 0.933671i \(0.616586\pi\)
\(282\) 0 0
\(283\) − 1.67132e13i − 0.547310i −0.961828 0.273655i \(-0.911767\pi\)
0.961828 0.273655i \(-0.0882327\pi\)
\(284\) −1.44131e13 −0.462920
\(285\) 0 0
\(286\) 7.41278e12 0.229070
\(287\) 5.15917e12i 0.156397i
\(288\) 0 0
\(289\) −1.34200e13 −0.391575
\(290\) 0 0
\(291\) 0 0
\(292\) − 2.15470e12i − 0.0593993i
\(293\) − 2.39269e13i − 0.647312i −0.946175 0.323656i \(-0.895088\pi\)
0.946175 0.323656i \(-0.104912\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.53934e13 0.393758
\(297\) 0 0
\(298\) − 2.67704e13i − 0.659881i
\(299\) −1.07709e13 −0.260652
\(300\) 0 0
\(301\) 2.86753e11 0.00668947
\(302\) − 1.97867e13i − 0.453248i
\(303\) 0 0
\(304\) −1.05243e13 −0.232481
\(305\) 0 0
\(306\) 0 0
\(307\) 1.53111e13i 0.320439i 0.987081 + 0.160219i \(0.0512202\pi\)
−0.987081 + 0.160219i \(0.948780\pi\)
\(308\) 1.31767e13i 0.270880i
\(309\) 0 0
\(310\) 0 0
\(311\) −4.98752e13 −0.972080 −0.486040 0.873936i \(-0.661559\pi\)
−0.486040 + 0.873936i \(0.661559\pi\)
\(312\) 0 0
\(313\) 9.94808e13i 1.87174i 0.352345 + 0.935870i \(0.385384\pi\)
−0.352345 + 0.935870i \(0.614616\pi\)
\(314\) −3.15628e13 −0.583529
\(315\) 0 0
\(316\) −5.61080e13 −1.00172
\(317\) − 8.33692e13i − 1.46278i −0.681958 0.731392i \(-0.738871\pi\)
0.681958 0.731392i \(-0.261129\pi\)
\(318\) 0 0
\(319\) −6.86477e13 −1.16353
\(320\) 0 0
\(321\) 0 0
\(322\) 7.49191e12i 0.120611i
\(323\) − 7.36271e13i − 1.16526i
\(324\) 0 0
\(325\) 0 0
\(326\) −8.58799e12 −0.129180
\(327\) 0 0
\(328\) − 2.60300e13i − 0.378589i
\(329\) −4.49970e13 −0.643585
\(330\) 0 0
\(331\) −6.35840e13 −0.879618 −0.439809 0.898091i \(-0.644954\pi\)
−0.439809 + 0.898091i \(0.644954\pi\)
\(332\) − 4.31813e13i − 0.587538i
\(333\) 0 0
\(334\) 6.61160e13 0.870364
\(335\) 0 0
\(336\) 0 0
\(337\) 1.21001e14i 1.51644i 0.651997 + 0.758221i \(0.273931\pi\)
−0.651997 + 0.758221i \(0.726069\pi\)
\(338\) 3.50011e13i 0.431559i
\(339\) 0 0
\(340\) 0 0
\(341\) 2.82506e13 0.331802
\(342\) 0 0
\(343\) − 6.15223e13i − 0.699705i
\(344\) −1.44678e12 −0.0161931
\(345\) 0 0
\(346\) 2.28093e13 0.247283
\(347\) 1.55662e14i 1.66100i 0.557020 + 0.830499i \(0.311945\pi\)
−0.557020 + 0.830499i \(0.688055\pi\)
\(348\) 0 0
\(349\) 2.56430e13 0.265112 0.132556 0.991176i \(-0.457682\pi\)
0.132556 + 0.991176i \(0.457682\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 1.05162e14i − 1.03722i
\(353\) 2.49098e13i 0.241885i 0.992659 + 0.120943i \(0.0385917\pi\)
−0.992659 + 0.120943i \(0.961408\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.67896e13 −0.340996
\(357\) 0 0
\(358\) 4.03532e13i 0.362678i
\(359\) 1.57584e14 1.39474 0.697370 0.716712i \(-0.254354\pi\)
0.697370 + 0.716712i \(0.254354\pi\)
\(360\) 0 0
\(361\) −2.82438e12 −0.0242457
\(362\) − 2.39226e13i − 0.202260i
\(363\) 0 0
\(364\) 1.42396e13 0.116799
\(365\) 0 0
\(366\) 0 0
\(367\) − 1.77901e14i − 1.39481i −0.716676 0.697406i \(-0.754338\pi\)
0.716676 0.697406i \(-0.245662\pi\)
\(368\) 1.84034e13i 0.142146i
\(369\) 0 0
\(370\) 0 0
\(371\) −2.67244e13 −0.197402
\(372\) 0 0
\(373\) 5.51617e13i 0.395585i 0.980244 + 0.197792i \(0.0633772\pi\)
−0.980244 + 0.197792i \(0.936623\pi\)
\(374\) 8.86079e13 0.626150
\(375\) 0 0
\(376\) 2.27027e14 1.55792
\(377\) 7.41854e13i 0.501696i
\(378\) 0 0
\(379\) −1.46463e14 −0.962083 −0.481042 0.876698i \(-0.659741\pi\)
−0.481042 + 0.876698i \(0.659741\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 6.62977e13i − 0.417013i
\(383\) 2.31450e14i 1.43504i 0.696539 + 0.717519i \(0.254722\pi\)
−0.696539 + 0.717519i \(0.745278\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.30617e14 0.775837
\(387\) 0 0
\(388\) 1.10420e14i 0.637490i
\(389\) −1.49872e14 −0.853093 −0.426547 0.904466i \(-0.640270\pi\)
−0.426547 + 0.904466i \(0.640270\pi\)
\(390\) 0 0
\(391\) −1.28749e14 −0.712480
\(392\) 1.43360e14i 0.782264i
\(393\) 0 0
\(394\) −6.90262e13 −0.366256
\(395\) 0 0
\(396\) 0 0
\(397\) 2.08111e14i 1.05912i 0.848271 + 0.529562i \(0.177644\pi\)
−0.848271 + 0.529562i \(0.822356\pi\)
\(398\) − 1.74814e13i − 0.0877443i
\(399\) 0 0
\(400\) 0 0
\(401\) 1.33408e14 0.642521 0.321261 0.946991i \(-0.395893\pi\)
0.321261 + 0.946991i \(0.395893\pi\)
\(402\) 0 0
\(403\) − 3.05295e13i − 0.143068i
\(404\) −1.20326e14 −0.556238
\(405\) 0 0
\(406\) 5.16010e13 0.232148
\(407\) 9.74134e13i 0.432364i
\(408\) 0 0
\(409\) 2.06168e14 0.890722 0.445361 0.895351i \(-0.353075\pi\)
0.445361 + 0.895351i \(0.353075\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 3.32312e14i 1.37915i
\(413\) 8.68880e13i 0.355824i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.13645e14 −0.447235
\(417\) 0 0
\(418\) 1.36793e14i 0.524319i
\(419\) 7.34035e13 0.277677 0.138838 0.990315i \(-0.455663\pi\)
0.138838 + 0.990315i \(0.455663\pi\)
\(420\) 0 0
\(421\) 1.71112e14 0.630563 0.315282 0.948998i \(-0.397901\pi\)
0.315282 + 0.948998i \(0.397901\pi\)
\(422\) − 1.63036e14i − 0.593014i
\(423\) 0 0
\(424\) 1.34835e14 0.477848
\(425\) 0 0
\(426\) 0 0
\(427\) − 1.16479e14i − 0.397096i
\(428\) − 1.32835e14i − 0.447067i
\(429\) 0 0
\(430\) 0 0
\(431\) 7.17758e13 0.232463 0.116231 0.993222i \(-0.462919\pi\)
0.116231 + 0.993222i \(0.462919\pi\)
\(432\) 0 0
\(433\) − 9.98812e13i − 0.315356i −0.987491 0.157678i \(-0.949599\pi\)
0.987491 0.157678i \(-0.0504007\pi\)
\(434\) −2.12353e13 −0.0662012
\(435\) 0 0
\(436\) −1.08166e14 −0.328789
\(437\) − 1.98764e14i − 0.596608i
\(438\) 0 0
\(439\) 2.90312e13 0.0849788 0.0424894 0.999097i \(-0.486471\pi\)
0.0424894 + 0.999097i \(0.486471\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 9.57557e13i − 0.269987i
\(443\) 3.28370e14i 0.914414i 0.889360 + 0.457207i \(0.151150\pi\)
−0.889360 + 0.457207i \(0.848850\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.76037e14 0.472347
\(447\) 0 0
\(448\) 4.51970e13i 0.118326i
\(449\) −6.12368e14 −1.58364 −0.791822 0.610752i \(-0.790867\pi\)
−0.791822 + 0.610752i \(0.790867\pi\)
\(450\) 0 0
\(451\) 1.64725e14 0.415708
\(452\) − 1.25336e14i − 0.312475i
\(453\) 0 0
\(454\) −3.26361e13 −0.0794130
\(455\) 0 0
\(456\) 0 0
\(457\) 3.03483e14i 0.712189i 0.934450 + 0.356095i \(0.115892\pi\)
−0.934450 + 0.356095i \(0.884108\pi\)
\(458\) 2.83786e14i 0.658007i
\(459\) 0 0
\(460\) 0 0
\(461\) 7.29308e14 1.63138 0.815691 0.578487i \(-0.196357\pi\)
0.815691 + 0.578487i \(0.196357\pi\)
\(462\) 0 0
\(463\) − 1.22188e14i − 0.266891i −0.991056 0.133445i \(-0.957396\pi\)
0.991056 0.133445i \(-0.0426041\pi\)
\(464\) 1.26755e14 0.273600
\(465\) 0 0
\(466\) 4.21520e14 0.888579
\(467\) 6.17381e14i 1.28621i 0.765780 + 0.643103i \(0.222353\pi\)
−0.765780 + 0.643103i \(0.777647\pi\)
\(468\) 0 0
\(469\) −2.59228e14 −0.527510
\(470\) 0 0
\(471\) 0 0
\(472\) − 4.38384e14i − 0.861338i
\(473\) − 9.15561e12i − 0.0177808i
\(474\) 0 0
\(475\) 0 0
\(476\) 1.70212e14 0.319265
\(477\) 0 0
\(478\) − 1.71350e14i − 0.314073i
\(479\) 1.05084e15 1.90410 0.952052 0.305938i \(-0.0989700\pi\)
0.952052 + 0.305938i \(0.0989700\pi\)
\(480\) 0 0
\(481\) 1.05272e14 0.186429
\(482\) − 5.55137e12i − 0.00971944i
\(483\) 0 0
\(484\) 7.33527e11 0.00125536
\(485\) 0 0
\(486\) 0 0
\(487\) − 2.19910e14i − 0.363777i −0.983319 0.181889i \(-0.941779\pi\)
0.983319 0.181889i \(-0.0582210\pi\)
\(488\) 5.87683e14i 0.961246i
\(489\) 0 0
\(490\) 0 0
\(491\) 4.83863e14 0.765199 0.382599 0.923914i \(-0.375029\pi\)
0.382599 + 0.923914i \(0.375029\pi\)
\(492\) 0 0
\(493\) 8.86768e14i 1.37136i
\(494\) 1.47828e14 0.226078
\(495\) 0 0
\(496\) −5.21634e13 −0.0780219
\(497\) 1.63949e14i 0.242520i
\(498\) 0 0
\(499\) 1.08878e14 0.157538 0.0787691 0.996893i \(-0.474901\pi\)
0.0787691 + 0.996893i \(0.474901\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 3.11593e14i − 0.436232i
\(503\) 5.06588e14i 0.701506i 0.936468 + 0.350753i \(0.114074\pi\)
−0.936468 + 0.350753i \(0.885926\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.39206e14 0.320586
\(507\) 0 0
\(508\) − 3.86720e14i − 0.507161i
\(509\) 8.57534e13 0.111251 0.0556254 0.998452i \(-0.482285\pi\)
0.0556254 + 0.998452i \(0.482285\pi\)
\(510\) 0 0
\(511\) −2.45097e13 −0.0311188
\(512\) 3.64965e14i 0.458423i
\(513\) 0 0
\(514\) 5.75069e14 0.707005
\(515\) 0 0
\(516\) 0 0
\(517\) 1.43669e15i 1.71066i
\(518\) − 7.32235e13i − 0.0862654i
\(519\) 0 0
\(520\) 0 0
\(521\) −9.27575e14 −1.05862 −0.529312 0.848428i \(-0.677550\pi\)
−0.529312 + 0.848428i \(0.677550\pi\)
\(522\) 0 0
\(523\) 2.18187e13i 0.0243820i 0.999926 + 0.0121910i \(0.00388061\pi\)
−0.999926 + 0.0121910i \(0.996119\pi\)
\(524\) −9.29610e14 −1.02797
\(525\) 0 0
\(526\) 5.82569e14 0.630850
\(527\) − 3.64931e14i − 0.391069i
\(528\) 0 0
\(529\) 6.05238e14 0.635214
\(530\) 0 0
\(531\) 0 0
\(532\) 2.62774e14i 0.267343i
\(533\) − 1.78013e14i − 0.179247i
\(534\) 0 0
\(535\) 0 0
\(536\) 1.30790e15 1.27694
\(537\) 0 0
\(538\) 6.20105e14i 0.593147i
\(539\) −9.07218e14 −0.858961
\(540\) 0 0
\(541\) −1.69527e15 −1.57273 −0.786363 0.617765i \(-0.788038\pi\)
−0.786363 + 0.617765i \(0.788038\pi\)
\(542\) − 9.04304e13i − 0.0830459i
\(543\) 0 0
\(544\) −1.35844e15 −1.22249
\(545\) 0 0
\(546\) 0 0
\(547\) 7.52145e14i 0.656706i 0.944555 + 0.328353i \(0.106494\pi\)
−0.944555 + 0.328353i \(0.893506\pi\)
\(548\) 4.37477e14i 0.378148i
\(549\) 0 0
\(550\) 0 0
\(551\) −1.36900e15 −1.14834
\(552\) 0 0
\(553\) 6.38228e14i 0.524793i
\(554\) 3.94054e14 0.320813
\(555\) 0 0
\(556\) −8.78480e14 −0.701166
\(557\) − 1.87489e14i − 0.148174i −0.997252 0.0740870i \(-0.976396\pi\)
0.997252 0.0740870i \(-0.0236043\pi\)
\(558\) 0 0
\(559\) −9.89417e12 −0.00766681
\(560\) 0 0
\(561\) 0 0
\(562\) − 5.04857e14i − 0.379856i
\(563\) 2.44971e14i 0.182524i 0.995827 + 0.0912618i \(0.0290900\pi\)
−0.995827 + 0.0912618i \(0.970910\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4.01116e14 0.290255
\(567\) 0 0
\(568\) − 8.27185e14i − 0.587066i
\(569\) 1.35243e15 0.950596 0.475298 0.879825i \(-0.342340\pi\)
0.475298 + 0.879825i \(0.342340\pi\)
\(570\) 0 0
\(571\) 1.43223e15 0.987447 0.493723 0.869619i \(-0.335636\pi\)
0.493723 + 0.869619i \(0.335636\pi\)
\(572\) − 4.54650e14i − 0.310455i
\(573\) 0 0
\(574\) −1.23820e14 −0.0829422
\(575\) 0 0
\(576\) 0 0
\(577\) − 8.77659e14i − 0.571293i −0.958335 0.285647i \(-0.907792\pi\)
0.958335 0.285647i \(-0.0922083\pi\)
\(578\) − 3.22081e14i − 0.207664i
\(579\) 0 0
\(580\) 0 0
\(581\) −4.91187e14 −0.307807
\(582\) 0 0
\(583\) 8.53271e14i 0.524698i
\(584\) 1.23661e14 0.0753290
\(585\) 0 0
\(586\) 5.74245e14 0.343289
\(587\) 2.43425e15i 1.44164i 0.693124 + 0.720818i \(0.256234\pi\)
−0.693124 + 0.720818i \(0.743766\pi\)
\(588\) 0 0
\(589\) 5.63383e14 0.327469
\(590\) 0 0
\(591\) 0 0
\(592\) − 1.79869e14i − 0.101669i
\(593\) − 3.03318e14i − 0.169863i −0.996387 0.0849313i \(-0.972933\pi\)
0.996387 0.0849313i \(-0.0270671\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.64192e15 −0.894329
\(597\) 0 0
\(598\) − 2.58502e14i − 0.138232i
\(599\) −1.70198e15 −0.901795 −0.450898 0.892576i \(-0.648896\pi\)
−0.450898 + 0.892576i \(0.648896\pi\)
\(600\) 0 0
\(601\) 2.33922e15 1.21692 0.608458 0.793586i \(-0.291788\pi\)
0.608458 + 0.793586i \(0.291788\pi\)
\(602\) 6.88207e12i 0.00354763i
\(603\) 0 0
\(604\) −1.21359e15 −0.614282
\(605\) 0 0
\(606\) 0 0
\(607\) − 2.49607e15i − 1.22947i −0.788732 0.614737i \(-0.789262\pi\)
0.788732 0.614737i \(-0.210738\pi\)
\(608\) − 2.09717e15i − 1.02368i
\(609\) 0 0
\(610\) 0 0
\(611\) 1.55258e15 0.737612
\(612\) 0 0
\(613\) − 2.47301e15i − 1.15397i −0.816756 0.576983i \(-0.804230\pi\)
0.816756 0.576983i \(-0.195770\pi\)
\(614\) −3.67466e14 −0.169938
\(615\) 0 0
\(616\) −7.56226e14 −0.343525
\(617\) − 2.43368e13i − 0.0109571i −0.999985 0.00547854i \(-0.998256\pi\)
0.999985 0.00547854i \(-0.00174388\pi\)
\(618\) 0 0
\(619\) −4.22545e15 −1.86885 −0.934425 0.356160i \(-0.884086\pi\)
−0.934425 + 0.356160i \(0.884086\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 1.19700e15i − 0.515523i
\(623\) 4.18481e14i 0.178645i
\(624\) 0 0
\(625\) 0 0
\(626\) −2.38754e15 −0.992640
\(627\) 0 0
\(628\) 1.93585e15i 0.790850i
\(629\) 1.25835e15 0.509594
\(630\) 0 0
\(631\) −4.26326e15 −1.69660 −0.848302 0.529513i \(-0.822375\pi\)
−0.848302 + 0.529513i \(0.822375\pi\)
\(632\) − 3.22011e15i − 1.27036i
\(633\) 0 0
\(634\) 2.00086e15 0.775758
\(635\) 0 0
\(636\) 0 0
\(637\) 9.80401e14i 0.370371i
\(638\) − 1.64755e15i − 0.617055i
\(639\) 0 0
\(640\) 0 0
\(641\) −1.00830e15 −0.368018 −0.184009 0.982925i \(-0.558908\pi\)
−0.184009 + 0.982925i \(0.558908\pi\)
\(642\) 0 0
\(643\) − 3.03982e14i − 0.109066i −0.998512 0.0545328i \(-0.982633\pi\)
0.998512 0.0545328i \(-0.0173670\pi\)
\(644\) 4.59504e14 0.163462
\(645\) 0 0
\(646\) 1.76705e15 0.617974
\(647\) − 3.43583e15i − 1.19140i −0.803207 0.595700i \(-0.796875\pi\)
0.803207 0.595700i \(-0.203125\pi\)
\(648\) 0 0
\(649\) 2.77421e15 0.945788
\(650\) 0 0
\(651\) 0 0
\(652\) 5.26730e14i 0.175076i
\(653\) − 1.18539e15i − 0.390695i −0.980734 0.195347i \(-0.937417\pi\)
0.980734 0.195347i \(-0.0625834\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.04157e14 −0.0977522
\(657\) 0 0
\(658\) − 1.07993e15i − 0.341312i
\(659\) −2.26510e15 −0.709934 −0.354967 0.934879i \(-0.615508\pi\)
−0.354967 + 0.934879i \(0.615508\pi\)
\(660\) 0 0
\(661\) −5.33012e15 −1.64297 −0.821484 0.570232i \(-0.806853\pi\)
−0.821484 + 0.570232i \(0.806853\pi\)
\(662\) − 1.52602e15i − 0.466488i
\(663\) 0 0
\(664\) 2.47823e15 0.745104
\(665\) 0 0
\(666\) 0 0
\(667\) 2.39392e15i 0.702130i
\(668\) − 4.05512e15i − 1.17959i
\(669\) 0 0
\(670\) 0 0
\(671\) −3.71902e15 −1.05549
\(672\) 0 0
\(673\) − 4.74120e15i − 1.32375i −0.749615 0.661874i \(-0.769761\pi\)
0.749615 0.661874i \(-0.230239\pi\)
\(674\) −2.90403e15 −0.804215
\(675\) 0 0
\(676\) 2.14673e15 0.584886
\(677\) 1.41307e15i 0.381880i 0.981602 + 0.190940i \(0.0611535\pi\)
−0.981602 + 0.190940i \(0.938846\pi\)
\(678\) 0 0
\(679\) 1.25603e15 0.333976
\(680\) 0 0
\(681\) 0 0
\(682\) 6.78014e14i 0.175964i
\(683\) − 3.03116e15i − 0.780359i −0.920739 0.390180i \(-0.872413\pi\)
0.920739 0.390180i \(-0.127587\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.47654e15 0.371075
\(687\) 0 0
\(688\) 1.69054e13i 0.00418109i
\(689\) 9.22102e14 0.226242
\(690\) 0 0
\(691\) −2.74731e15 −0.663405 −0.331703 0.943384i \(-0.607623\pi\)
−0.331703 + 0.943384i \(0.607623\pi\)
\(692\) − 1.39897e15i − 0.335139i
\(693\) 0 0
\(694\) −3.73588e15 −0.880878
\(695\) 0 0
\(696\) 0 0
\(697\) − 2.12786e15i − 0.489962i
\(698\) 6.15433e14i 0.140597i
\(699\) 0 0
\(700\) 0 0
\(701\) −5.72747e15 −1.27795 −0.638974 0.769228i \(-0.720641\pi\)
−0.638974 + 0.769228i \(0.720641\pi\)
\(702\) 0 0
\(703\) 1.94265e15i 0.426718i
\(704\) 1.44308e15 0.314514
\(705\) 0 0
\(706\) −5.97836e14 −0.128279
\(707\) 1.36870e15i 0.291409i
\(708\) 0 0
\(709\) −6.98326e14 −0.146388 −0.0731938 0.997318i \(-0.523319\pi\)
−0.0731938 + 0.997318i \(0.523319\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 2.11140e15i − 0.432445i
\(713\) − 9.85170e14i − 0.200225i
\(714\) 0 0
\(715\) 0 0
\(716\) 2.47500e15 0.491534
\(717\) 0 0
\(718\) 3.78202e15i 0.739672i
\(719\) 9.70979e15 1.88452 0.942260 0.334882i \(-0.108696\pi\)
0.942260 + 0.334882i \(0.108696\pi\)
\(720\) 0 0
\(721\) 3.78004e15 0.722525
\(722\) − 6.77852e13i − 0.0128582i
\(723\) 0 0
\(724\) −1.46725e15 −0.274121
\(725\) 0 0
\(726\) 0 0
\(727\) 2.46469e15i 0.450114i 0.974346 + 0.225057i \(0.0722568\pi\)
−0.974346 + 0.225057i \(0.927743\pi\)
\(728\) 8.17230e14i 0.148123i
\(729\) 0 0
\(730\) 0 0
\(731\) −1.18269e14 −0.0209568
\(732\) 0 0
\(733\) − 7.91285e15i − 1.38121i −0.723230 0.690607i \(-0.757343\pi\)
0.723230 0.690607i \(-0.242657\pi\)
\(734\) 4.26963e15 0.739711
\(735\) 0 0
\(736\) −3.66725e15 −0.625911
\(737\) 8.27677e15i 1.40213i
\(738\) 0 0
\(739\) 8.40694e15 1.40312 0.701558 0.712613i \(-0.252488\pi\)
0.701558 + 0.712613i \(0.252488\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 6.41385e14i − 0.104688i
\(743\) 1.36287e15i 0.220809i 0.993887 + 0.110404i \(0.0352146\pi\)
−0.993887 + 0.110404i \(0.964785\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.32388e15 −0.209790
\(747\) 0 0
\(748\) − 5.43462e15i − 0.848614i
\(749\) −1.51100e15 −0.234215
\(750\) 0 0
\(751\) 6.81722e15 1.04133 0.520664 0.853762i \(-0.325684\pi\)
0.520664 + 0.853762i \(0.325684\pi\)
\(752\) − 2.65278e15i − 0.402256i
\(753\) 0 0
\(754\) −1.78045e15 −0.266065
\(755\) 0 0
\(756\) 0 0
\(757\) − 6.67049e14i − 0.0975282i −0.998810 0.0487641i \(-0.984472\pi\)
0.998810 0.0487641i \(-0.0155283\pi\)
\(758\) − 3.51511e15i − 0.510222i
\(759\) 0 0
\(760\) 0 0
\(761\) 7.74408e15 1.09990 0.549951 0.835197i \(-0.314646\pi\)
0.549951 + 0.835197i \(0.314646\pi\)
\(762\) 0 0
\(763\) 1.23039e15i 0.172250i
\(764\) −4.06626e15 −0.565173
\(765\) 0 0
\(766\) −5.55479e15 −0.761043
\(767\) − 2.99800e15i − 0.407809i
\(768\) 0 0
\(769\) −2.52411e15 −0.338465 −0.169232 0.985576i \(-0.554129\pi\)
−0.169232 + 0.985576i \(0.554129\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 8.01119e15i − 1.05148i
\(773\) − 1.11453e16i − 1.45246i −0.687453 0.726229i \(-0.741271\pi\)
0.687453 0.726229i \(-0.258729\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −6.33715e15 −0.808452
\(777\) 0 0
\(778\) − 3.59692e15i − 0.452421i
\(779\) 3.28500e15 0.410279
\(780\) 0 0
\(781\) 5.23465e15 0.644624
\(782\) − 3.08998e15i − 0.377849i
\(783\) 0 0
\(784\) 1.67514e15 0.201981
\(785\) 0 0
\(786\) 0 0
\(787\) 1.32271e16i 1.56172i 0.624705 + 0.780861i \(0.285219\pi\)
−0.624705 + 0.780861i \(0.714781\pi\)
\(788\) 4.23361e15i 0.496382i
\(789\) 0 0
\(790\) 0 0
\(791\) −1.42570e15 −0.163703
\(792\) 0 0
\(793\) 4.01902e15i 0.455112i
\(794\) −4.99466e15 −0.561685
\(795\) 0 0
\(796\) −1.07219e15 −0.118919
\(797\) − 2.30248e15i − 0.253615i −0.991927 0.126807i \(-0.959527\pi\)
0.991927 0.126807i \(-0.0404730\pi\)
\(798\) 0 0
\(799\) 1.85587e16 2.01623
\(800\) 0 0
\(801\) 0 0
\(802\) 3.20179e15i 0.340748i
\(803\) 7.82560e14i 0.0827146i
\(804\) 0 0
\(805\) 0 0
\(806\) 7.32708e14 0.0758732
\(807\) 0 0
\(808\) − 6.90565e15i − 0.705410i
\(809\) 5.60472e15 0.568639 0.284320 0.958730i \(-0.408232\pi\)
0.284320 + 0.958730i \(0.408232\pi\)
\(810\) 0 0
\(811\) −5.08516e15 −0.508968 −0.254484 0.967077i \(-0.581906\pi\)
−0.254484 + 0.967077i \(0.581906\pi\)
\(812\) − 3.16486e15i − 0.314627i
\(813\) 0 0
\(814\) −2.33792e15 −0.229296
\(815\) 0 0
\(816\) 0 0
\(817\) − 1.82584e14i − 0.0175486i
\(818\) 4.94802e15i 0.472377i
\(819\) 0 0
\(820\) 0 0
\(821\) −2.79111e14 −0.0261150 −0.0130575 0.999915i \(-0.504156\pi\)
−0.0130575 + 0.999915i \(0.504156\pi\)
\(822\) 0 0
\(823\) 1.35265e16i 1.24878i 0.781112 + 0.624391i \(0.214653\pi\)
−0.781112 + 0.624391i \(0.785347\pi\)
\(824\) −1.90718e16 −1.74901
\(825\) 0 0
\(826\) −2.08531e15 −0.188704
\(827\) − 2.72544e14i − 0.0244994i −0.999925 0.0122497i \(-0.996101\pi\)
0.999925 0.0122497i \(-0.00389930\pi\)
\(828\) 0 0
\(829\) −1.80459e16 −1.60077 −0.800385 0.599486i \(-0.795372\pi\)
−0.800385 + 0.599486i \(0.795372\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 1.55949e15i − 0.135614i
\(833\) 1.17191e16i 1.01239i
\(834\) 0 0
\(835\) 0 0
\(836\) 8.38999e15 0.710603
\(837\) 0 0
\(838\) 1.76168e15i 0.147260i
\(839\) −7.96183e15 −0.661184 −0.330592 0.943774i \(-0.607248\pi\)
−0.330592 + 0.943774i \(0.607248\pi\)
\(840\) 0 0
\(841\) 4.28775e15 0.351440
\(842\) 4.10669e15i 0.334407i
\(843\) 0 0
\(844\) −9.99954e15 −0.803705
\(845\) 0 0
\(846\) 0 0
\(847\) − 8.34387e12i 0 0.000657671i
\(848\) − 1.57552e15i − 0.123381i
\(849\) 0 0
\(850\) 0 0
\(851\) 3.39705e15 0.260909
\(852\) 0 0
\(853\) 1.49826e16i 1.13598i 0.823037 + 0.567988i \(0.192278\pi\)
−0.823037 + 0.567988i \(0.807722\pi\)
\(854\) 2.79550e15 0.210592
\(855\) 0 0
\(856\) 7.62358e15 0.566961
\(857\) 2.22561e16i 1.64458i 0.569068 + 0.822290i \(0.307304\pi\)
−0.569068 + 0.822290i \(0.692696\pi\)
\(858\) 0 0
\(859\) −5.44237e15 −0.397032 −0.198516 0.980098i \(-0.563612\pi\)
−0.198516 + 0.980098i \(0.563612\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.72262e15i 0.123282i
\(863\) 1.08110e16i 0.768787i 0.923169 + 0.384393i \(0.125589\pi\)
−0.923169 + 0.384393i \(0.874411\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.39715e15 0.167243
\(867\) 0 0
\(868\) 1.30243e15i 0.0897217i
\(869\) 2.03777e16 1.39491
\(870\) 0 0
\(871\) 8.94444e15 0.604579
\(872\) − 6.20782e15i − 0.416964i
\(873\) 0 0
\(874\) 4.77033e15 0.316399
\(875\) 0 0
\(876\) 0 0
\(877\) − 2.81024e16i − 1.82914i −0.404431 0.914568i \(-0.632530\pi\)
0.404431 0.914568i \(-0.367470\pi\)
\(878\) 6.96749e14i 0.0450668i
\(879\) 0 0
\(880\) 0 0
\(881\) −4.22209e15 −0.268016 −0.134008 0.990980i \(-0.542785\pi\)
−0.134008 + 0.990980i \(0.542785\pi\)
\(882\) 0 0
\(883\) − 5.16092e14i − 0.0323551i −0.999869 0.0161776i \(-0.994850\pi\)
0.999869 0.0161776i \(-0.00514970\pi\)
\(884\) −5.87302e15 −0.365910
\(885\) 0 0
\(886\) −7.88088e15 −0.484941
\(887\) − 5.71906e15i − 0.349740i −0.984592 0.174870i \(-0.944050\pi\)
0.984592 0.174870i \(-0.0559504\pi\)
\(888\) 0 0
\(889\) −4.39894e15 −0.265698
\(890\) 0 0
\(891\) 0 0
\(892\) − 1.07969e16i − 0.640167i
\(893\) 2.86510e16i 1.68832i
\(894\) 0 0
\(895\) 0 0
\(896\) 5.66067e15 0.327472
\(897\) 0 0
\(898\) − 1.46968e16i − 0.839854i
\(899\) −6.78541e15 −0.385388
\(900\) 0 0
\(901\) 1.10223e16 0.618421
\(902\) 3.95340e15i 0.220462i
\(903\) 0 0
\(904\) 7.19321e15 0.396275
\(905\) 0 0
\(906\) 0 0
\(907\) − 8.43778e13i − 0.00456445i −0.999997 0.00228222i \(-0.999274\pi\)
0.999997 0.00228222i \(-0.000726455\pi\)
\(908\) 2.00168e15i 0.107628i
\(909\) 0 0
\(910\) 0 0
\(911\) 1.10091e16 0.581298 0.290649 0.956830i \(-0.406129\pi\)
0.290649 + 0.956830i \(0.406129\pi\)
\(912\) 0 0
\(913\) 1.56829e16i 0.818158i
\(914\) −7.28359e15 −0.377695
\(915\) 0 0
\(916\) 1.74055e16 0.891789
\(917\) 1.05743e16i 0.538544i
\(918\) 0 0
\(919\) 4.86351e15 0.244746 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.75034e16i 0.865171i
\(923\) − 5.65691e15i − 0.277952i
\(924\) 0 0
\(925\) 0 0
\(926\) 2.93252e15 0.141540
\(927\) 0 0
\(928\) 2.52584e16i 1.20474i
\(929\) 3.57534e15 0.169524 0.0847620 0.996401i \(-0.472987\pi\)
0.0847620 + 0.996401i \(0.472987\pi\)
\(930\) 0 0
\(931\) −1.80921e16 −0.847744
\(932\) − 2.58533e16i − 1.20428i
\(933\) 0 0
\(934\) −1.48171e16 −0.682113
\(935\) 0 0
\(936\) 0 0
\(937\) 3.86373e16i 1.74759i 0.486295 + 0.873795i \(0.338348\pi\)
−0.486295 + 0.873795i \(0.661652\pi\)
\(938\) − 6.22147e15i − 0.279754i
\(939\) 0 0
\(940\) 0 0
\(941\) 3.48997e16 1.54198 0.770991 0.636846i \(-0.219761\pi\)
0.770991 + 0.636846i \(0.219761\pi\)
\(942\) 0 0
\(943\) − 5.74437e15i − 0.250858i
\(944\) −5.12245e15 −0.222398
\(945\) 0 0
\(946\) 2.19735e14 0.00942969
\(947\) 2.85123e16i 1.21649i 0.793751 + 0.608243i \(0.208125\pi\)
−0.793751 + 0.608243i \(0.791875\pi\)
\(948\) 0 0
\(949\) 8.45688e14 0.0356653
\(950\) 0 0
\(951\) 0 0
\(952\) 9.76867e15i 0.404886i
\(953\) 4.00334e16i 1.64973i 0.565332 + 0.824863i \(0.308748\pi\)
−0.565332 + 0.824863i \(0.691252\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.05095e16 −0.425659
\(957\) 0 0
\(958\) 2.52201e16i 1.00980i
\(959\) 4.97630e15 0.198109
\(960\) 0 0
\(961\) −2.26161e16 −0.890100
\(962\) 2.52652e15i 0.0988688i
\(963\) 0 0
\(964\) −3.40484e14 −0.0131726
\(965\) 0 0
\(966\) 0 0
\(967\) 1.84953e16i 0.703422i 0.936109 + 0.351711i \(0.114400\pi\)
−0.936109 + 0.351711i \(0.885600\pi\)
\(968\) 4.20981e13i 0.00159202i
\(969\) 0 0
\(970\) 0 0
\(971\) 2.14877e16 0.798884 0.399442 0.916759i \(-0.369204\pi\)
0.399442 + 0.916759i \(0.369204\pi\)
\(972\) 0 0
\(973\) 9.99271e15i 0.367335i
\(974\) 5.27784e15 0.192922
\(975\) 0 0
\(976\) 6.86699e15 0.248195
\(977\) 8.73880e15i 0.314074i 0.987593 + 0.157037i \(0.0501942\pi\)
−0.987593 + 0.157037i \(0.949806\pi\)
\(978\) 0 0
\(979\) 1.33615e16 0.474844
\(980\) 0 0
\(981\) 0 0
\(982\) 1.16127e16i 0.405808i
\(983\) 1.18924e16i 0.413263i 0.978419 + 0.206631i \(0.0662501\pi\)
−0.978419 + 0.206631i \(0.933750\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.12824e16 −0.727274
\(987\) 0 0
\(988\) − 9.06679e15i − 0.306401i
\(989\) −3.19279e14 −0.0107298
\(990\) 0 0
\(991\) 2.34409e16 0.779056 0.389528 0.921015i \(-0.372638\pi\)
0.389528 + 0.921015i \(0.372638\pi\)
\(992\) − 1.03946e16i − 0.343552i
\(993\) 0 0
\(994\) −3.93477e15 −0.128616
\(995\) 0 0
\(996\) 0 0
\(997\) − 2.14004e16i − 0.688016i −0.938967 0.344008i \(-0.888215\pi\)
0.938967 0.344008i \(-0.111785\pi\)
\(998\) 2.61307e15i 0.0835473i
\(999\) 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 225.12.b.d.199.2 2
3.2 odd 2 25.12.b.b.24.1 2
5.2 odd 4 225.12.a.b.1.1 1
5.3 odd 4 9.12.a.b.1.1 1
5.4 even 2 inner 225.12.b.d.199.1 2
15.2 even 4 25.12.a.b.1.1 1
15.8 even 4 1.12.a.a.1.1 1
15.14 odd 2 25.12.b.b.24.2 2
20.3 even 4 144.12.a.d.1.1 1
45.13 odd 12 81.12.c.b.55.1 2
45.23 even 12 81.12.c.d.55.1 2
45.38 even 12 81.12.c.d.28.1 2
45.43 odd 12 81.12.c.b.28.1 2
60.23 odd 4 16.12.a.a.1.1 1
105.23 even 12 49.12.c.b.18.1 2
105.38 odd 12 49.12.c.c.30.1 2
105.53 even 12 49.12.c.b.30.1 2
105.68 odd 12 49.12.c.c.18.1 2
105.83 odd 4 49.12.a.a.1.1 1
120.53 even 4 64.12.a.b.1.1 1
120.83 odd 4 64.12.a.f.1.1 1
165.98 odd 4 121.12.a.b.1.1 1
195.38 even 4 169.12.a.a.1.1 1
240.53 even 4 256.12.b.e.129.1 2
240.83 odd 4 256.12.b.c.129.1 2
240.173 even 4 256.12.b.e.129.2 2
240.203 odd 4 256.12.b.c.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.12.a.a.1.1 1 15.8 even 4
9.12.a.b.1.1 1 5.3 odd 4
16.12.a.a.1.1 1 60.23 odd 4
25.12.a.b.1.1 1 15.2 even 4
25.12.b.b.24.1 2 3.2 odd 2
25.12.b.b.24.2 2 15.14 odd 2
49.12.a.a.1.1 1 105.83 odd 4
49.12.c.b.18.1 2 105.23 even 12
49.12.c.b.30.1 2 105.53 even 12
49.12.c.c.18.1 2 105.68 odd 12
49.12.c.c.30.1 2 105.38 odd 12
64.12.a.b.1.1 1 120.53 even 4
64.12.a.f.1.1 1 120.83 odd 4
81.12.c.b.28.1 2 45.43 odd 12
81.12.c.b.55.1 2 45.13 odd 12
81.12.c.d.28.1 2 45.38 even 12
81.12.c.d.55.1 2 45.23 even 12
121.12.a.b.1.1 1 165.98 odd 4
144.12.a.d.1.1 1 20.3 even 4
169.12.a.a.1.1 1 195.38 even 4
225.12.a.b.1.1 1 5.2 odd 4
225.12.b.d.199.1 2 5.4 even 2 inner
225.12.b.d.199.2 2 1.1 even 1 trivial
256.12.b.c.129.1 2 240.83 odd 4
256.12.b.c.129.2 2 240.203 odd 4
256.12.b.e.129.1 2 240.53 even 4
256.12.b.e.129.2 2 240.173 even 4