Properties

Label 25.12.b.b.24.2
Level $25$
Weight $12$
Character 25.24
Analytic conductor $19.209$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.2085795140\)
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 24.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.12.b.b.24.1

$q$-expansion

\(f(q)\) \(=\) \(q+24.0000i q^{2} +252.000i q^{3} +1472.00 q^{4} -6048.00 q^{6} +16744.0i q^{7} +84480.0i q^{8} +113643. q^{9} +O(q^{10})\) \(q+24.0000i q^{2} +252.000i q^{3} +1472.00 q^{4} -6048.00 q^{6} +16744.0i q^{7} +84480.0i q^{8} +113643. q^{9} +534612. q^{11} +370944. i q^{12} -577738. i q^{13} -401856. q^{14} +987136. q^{16} +6.90593e6i q^{17} +2.72743e6i q^{18} -1.06614e7 q^{19} -4.21949e6 q^{21} +1.28307e7i q^{22} +1.86433e7i q^{23} -2.12890e7 q^{24} +1.38657e7 q^{26} +7.32791e7i q^{27} +2.46472e7i q^{28} -1.28407e8 q^{29} -5.28432e7 q^{31} +1.96706e8i q^{32} +1.34722e8i q^{33} -1.65742e8 q^{34} +1.67282e8 q^{36} +1.82213e8i q^{37} -2.55874e8i q^{38} +1.45590e8 q^{39} +3.08120e8 q^{41} -1.01268e8i q^{42} -1.71257e7i q^{43} +7.86949e8 q^{44} -4.47439e8 q^{46} -2.68735e9i q^{47} +2.48758e8i q^{48} +1.69697e9 q^{49} -1.74030e9 q^{51} -8.50430e8i q^{52} -1.59606e9i q^{53} -1.75870e9 q^{54} -1.41453e9 q^{56} -2.68668e9i q^{57} -3.08176e9i q^{58} +5.18920e9 q^{59} +6.95648e9 q^{61} -1.26824e9i q^{62} +1.90284e9i q^{63} -2.69930e9 q^{64} -3.23333e9 q^{66} +1.54818e10i q^{67} +1.01655e10i q^{68} -4.69810e9 q^{69} +9.79149e9 q^{71} +9.60056e9i q^{72} +1.46379e9i q^{73} -4.37312e9 q^{74} -1.56936e10 q^{76} +8.95154e9i q^{77} +3.49416e9i q^{78} -3.81168e10 q^{79} +1.66519e9 q^{81} +7.39489e9i q^{82} -2.93351e10i q^{83} -6.21109e9 q^{84} +4.11017e8 q^{86} -3.23585e10i q^{87} +4.51640e10i q^{88} +2.49929e10 q^{89} +9.67365e9 q^{91} +2.74429e10i q^{92} -1.33165e10i q^{93} +6.44964e10 q^{94} -4.95700e10 q^{96} -7.50136e10i q^{97} +4.07272e10i q^{98} +6.07549e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2944q^{4} - 12096q^{6} + 227286q^{9} + O(q^{10}) \) \( 2q + 2944q^{4} - 12096q^{6} + 227286q^{9} + 1069224q^{11} - 803712q^{14} + 1974272q^{16} - 21322840q^{19} - 8438976q^{21} - 42577920q^{24} + 27731424q^{26} - 256813260q^{29} - 105686336q^{31} - 331484832q^{34} + 334564992q^{36} + 291179952q^{39} + 616240884q^{41} + 1573897728q^{44} - 894877056q^{46} + 3393930414q^{49} - 3480590736q^{51} - 3517395840q^{54} - 2829066240q^{56} + 10378407480q^{59} + 13912957324q^{61} - 5398593536q^{64} - 6466666752q^{66} - 9396209088q^{69} + 19582970544q^{71} - 8746239072q^{74} - 31387220480q^{76} - 76233691360q^{79} + 3330376722q^{81} - 12422172672q^{84} + 822033984q^{86} + 49985834220q^{89} + 19347290144q^{91} + 128992727808q^{94} - 99139977216q^{96} + 121509823032q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(0.914574\pi\)
\(3\) 252.000i 0.598734i 0.954138 + 0.299367i \(0.0967754\pi\)
−0.954138 + 0.299367i \(0.903225\pi\)
\(4\) 1472.00 0.718750
\(5\) 0 0
\(6\) −6048.00 −0.317526
\(7\) 16744.0i 0.376548i 0.982117 + 0.188274i \(0.0602893\pi\)
−0.982117 + 0.188274i \(0.939711\pi\)
\(8\) 84480.0i 0.911505i
\(9\) 113643. 0.641518
\(10\) 0 0
\(11\) 534612. 1.00087 0.500436 0.865773i \(-0.333173\pi\)
0.500436 + 0.865773i \(0.333173\pi\)
\(12\) 370944.i 0.430340i
\(13\) − 577738.i − 0.431561i −0.976442 0.215781i \(-0.930770\pi\)
0.976442 0.215781i \(-0.0692296\pi\)
\(14\) −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 \(0.799197\pi\)
\(18\) 2.72743e6i 0.340216i
\(19\) −1.06614e7 −0.987803 −0.493901 0.869518i \(-0.664430\pi\)
−0.493901 + 0.869518i \(0.664430\pi\)
\(20\) 0 0
\(21\) −4.21949e6 −0.225452
\(22\) 1.28307e7i 0.530793i
\(23\) 1.86433e7i 0.603975i 0.953312 + 0.301988i \(0.0976501\pi\)
−0.953312 + 0.301988i \(0.902350\pi\)
\(24\) −2.12890e7 −0.545749
\(25\) 0 0
\(26\) 1.38657e7 0.228870
\(27\) 7.32791e7i 0.982832i
\(28\) 2.46472e7i 0.270644i
\(29\) −1.28407e8 −1.16251 −0.581257 0.813720i \(-0.697439\pi\)
−0.581257 + 0.813720i \(0.697439\pi\)
\(30\) 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
\(33\) 1.34722e8i 0.599256i
\(34\) −1.65742e8 −0.625604
\(35\) 0 0
\(36\) 1.67282e8 0.461091
\(37\) 1.82213e8i 0.431987i 0.976395 + 0.215993i \(0.0692990\pi\)
−0.976395 + 0.215993i \(0.930701\pi\)
\(38\) − 2.55874e8i − 0.523862i
\(39\) 1.45590e8 0.258390
\(40\) 0 0
\(41\) 3.08120e8 0.415345 0.207673 0.978198i \(-0.433411\pi\)
0.207673 + 0.978198i \(0.433411\pi\)
\(42\) − 1.01268e8i − 0.119564i
\(43\) − 1.71257e7i − 0.0177653i −0.999961 0.00888264i \(-0.997173\pi\)
0.999961 0.00888264i \(-0.00282747\pi\)
\(44\) 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 \(-0.326189\pi\)
\(48\) 2.48758e8i 0.140913i
\(49\) 1.69697e9 0.858212
\(50\) 0 0
\(51\) −1.74030e9 −0.706296
\(52\) − 8.50430e8i − 0.310185i
\(53\) − 1.59606e9i − 0.524241i −0.965035 0.262120i \(-0.915578\pi\)
0.965035 0.262120i \(-0.0844217\pi\)
\(54\) −1.75870e9 −0.521225
\(55\) 0 0
\(56\) −1.41453e9 −0.343225
\(57\) − 2.68668e9i − 0.591431i
\(58\) − 3.08176e9i − 0.616517i
\(59\) 5.18920e9 0.944963 0.472481 0.881341i \(-0.343358\pi\)
0.472481 + 0.881341i \(0.343358\pi\)
\(60\) 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
\(63\) 1.90284e9i 0.241562i
\(64\) −2.69930e9 −0.314240
\(65\) 0 0
\(66\) −3.23333e9 −0.317804
\(67\) 1.54818e10i 1.40091i 0.713696 + 0.700456i \(0.247020\pi\)
−0.713696 + 0.700456i \(0.752980\pi\)
\(68\) 1.01655e10i 0.847874i
\(69\) −4.69810e9 −0.361620
\(70\) 0 0
\(71\) 9.79149e9 0.644062 0.322031 0.946729i \(-0.395634\pi\)
0.322031 + 0.946729i \(0.395634\pi\)
\(72\) 9.60056e9i 0.584747i
\(73\) 1.46379e9i 0.0826425i 0.999146 + 0.0413212i \(0.0131567\pi\)
−0.999146 + 0.0413212i \(0.986843\pi\)
\(74\) −4.37312e9 −0.229096
\(75\) 0 0
\(76\) −1.56936e10 −0.709983
\(77\) 8.95154e9i 0.376876i
\(78\) 3.49416e9i 0.137032i
\(79\) −3.81168e10 −1.39370 −0.696848 0.717219i \(-0.745415\pi\)
−0.696848 + 0.717219i \(0.745415\pi\)
\(80\) 0 0
\(81\) 1.66519e9 0.0530635
\(82\) 7.39489e9i 0.220270i
\(83\) − 2.93351e10i − 0.817444i −0.912659 0.408722i \(-0.865975\pi\)
0.912659 0.408722i \(-0.134025\pi\)
\(84\) −6.21109e9 −0.162043
\(85\) 0 0
\(86\) 4.11017e8 0.00942146
\(87\) − 3.23585e10i − 0.696037i
\(88\) 4.51640e10i 0.912300i
\(89\) 2.49929e10 0.474430 0.237215 0.971457i \(-0.423765\pi\)
0.237215 + 0.971457i \(0.423765\pi\)
\(90\) 0 0
\(91\) 9.67365e9 0.162503
\(92\) 2.74429e10i 0.434107i
\(93\) − 1.33165e10i − 0.198488i
\(94\) 6.44964e10 0.906425
\(95\) 0 0
\(96\) −4.95700e10 −0.620479
\(97\) − 7.50136e10i − 0.886942i −0.896289 0.443471i \(-0.853747\pi\)
0.896289 0.443471i \(-0.146253\pi\)
\(98\) 4.07272e10i 0.455136i
\(99\) 6.07549e10 0.642078
\(100\) 0 0
\(101\) 8.17430e10 0.773896 0.386948 0.922101i \(-0.373529\pi\)
0.386948 + 0.922101i \(0.373529\pi\)
\(102\) − 4.17671e10i − 0.374570i
\(103\) − 2.25755e11i − 1.91881i −0.282025 0.959407i \(-0.591006\pi\)
0.282025 0.959407i \(-0.408994\pi\)
\(104\) 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 \(-0.100665\pi\)
\(108\) 1.07867e11i 0.706411i
\(109\) −7.34827e10 −0.457445 −0.228723 0.973492i \(-0.573455\pi\)
−0.228723 + 0.973492i \(0.573455\pi\)
\(110\) 0 0
\(111\) −4.59178e10 −0.258645
\(112\) 1.65286e10i 0.0886211i
\(113\) − 8.51469e10i − 0.434748i −0.976088 0.217374i \(-0.930251\pi\)
0.976088 0.217374i \(-0.0697491\pi\)
\(114\) 6.44803e10 0.313654
\(115\) 0 0
\(116\) −1.89015e11 −0.835557
\(117\) − 6.56559e10i − 0.276854i
\(118\) 1.24541e11i 0.501142i
\(119\) −1.15633e11 −0.444195
\(120\) 0 0
\(121\) 4.98320e8 0.00174658
\(122\) 1.66955e11i 0.559270i
\(123\) 7.76464e10i 0.248681i
\(124\) −7.77851e10 −0.238274
\(125\) 0 0
\(126\) −4.56681e10 −0.128108
\(127\) 2.62717e11i 0.705615i 0.935696 + 0.352808i \(0.114773\pi\)
−0.935696 + 0.352808i \(0.885227\pi\)
\(128\) 3.38071e11i 0.869668i
\(129\) 4.31568e9 0.0106367
\(130\) 0 0
\(131\) 6.31529e11 1.43021 0.715107 0.699015i \(-0.246378\pi\)
0.715107 + 0.699015i \(0.246378\pi\)
\(132\) 1.98311e11i 0.430715i
\(133\) − 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 \(0.915268\pi\)
\(138\) − 1.12755e11i − 0.191778i
\(139\) −5.96794e11 −0.975535 −0.487767 0.872974i \(-0.662189\pi\)
−0.487767 + 0.872974i \(0.662189\pi\)
\(140\) 0 0
\(141\) 6.77212e11 1.02334
\(142\) 2.34996e11i 0.341565i
\(143\) − 3.08866e11i − 0.431938i
\(144\) 1.12181e11 0.150982
\(145\) 0 0
\(146\) −3.51310e10 −0.0438278
\(147\) 4.27635e11i 0.513840i
\(148\) 2.68218e11i 0.310491i
\(149\) 1.11543e12 1.24428 0.622142 0.782905i \(-0.286263\pi\)
0.622142 + 0.782905i \(0.286263\pi\)
\(150\) 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
\(153\) 7.84811e11i 0.756767i
\(154\) −2.14837e11 −0.199869
\(155\) 0 0
\(156\) 2.14308e11 0.185718
\(157\) − 1.31512e12i − 1.10031i −0.835062 0.550156i \(-0.814568\pi\)
0.835062 0.550156i \(-0.185432\pi\)
\(158\) − 9.14804e11i − 0.739119i
\(159\) 4.02206e11 0.313881
\(160\) 0 0
\(161\) −3.12163e11 −0.227425
\(162\) 3.99645e10i 0.0281412i
\(163\) − 3.57833e11i − 0.243584i −0.992556 0.121792i \(-0.961136\pi\)
0.992556 0.121792i \(-0.0388640\pi\)
\(164\) 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 \(-0.306354\pi\)
\(168\) − 3.56462e11i − 0.205500i
\(169\) 1.45838e12 0.813755
\(170\) 0 0
\(171\) −1.21160e12 −0.633693
\(172\) − 2.52090e10i − 0.0127688i
\(173\) − 9.50387e11i − 0.466280i −0.972443 0.233140i \(-0.925100\pi\)
0.972443 0.233140i \(-0.0749001\pi\)
\(174\) 7.76603e11 0.369129
\(175\) 0 0
\(176\) 5.27735e11 0.235557
\(177\) 1.30768e12i 0.565781i
\(178\) 5.99830e11i 0.251604i
\(179\) −1.68138e12 −0.683873 −0.341936 0.939723i \(-0.611083\pi\)
−0.341936 + 0.939723i \(0.611083\pi\)
\(180\) 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
\(183\) 1.75303e12i 0.631406i
\(184\) −1.57498e12 −0.550526
\(185\) 0 0
\(186\) 3.19595e11 0.105264
\(187\) 3.69200e12i 1.18068i
\(188\) − 3.95578e12i − 1.22847i
\(189\) −1.22698e12 −0.370083
\(190\) 0 0
\(191\) 2.76240e12 0.786328 0.393164 0.919468i \(-0.371381\pi\)
0.393164 + 0.919468i \(0.371381\pi\)
\(192\) − 6.80223e11i − 0.188146i
\(193\) 5.44239e12i 1.46293i 0.681878 + 0.731466i \(0.261164\pi\)
−0.681878 + 0.731466i \(0.738836\pi\)
\(194\) 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 \(0.887774\pi\)
\(198\) 1.45812e12i 0.340513i
\(199\) −7.28391e11 −0.165452 −0.0827262 0.996572i \(-0.526363\pi\)
−0.0827262 + 0.996572i \(0.526363\pi\)
\(200\) 0 0
\(201\) −3.90142e12 −0.838773
\(202\) 1.96183e12i 0.410421i
\(203\) − 2.15004e12i − 0.437742i
\(204\) −2.56171e12 −0.507651
\(205\) 0 0
\(206\) 5.41812e12 1.01760
\(207\) 2.11868e12i 0.387461i
\(208\) − 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
\(213\) 2.46745e12i 0.385622i
\(214\) 2.16579e12 0.329868
\(215\) 0 0
\(216\) −6.19062e12 −0.895856
\(217\) − 8.84806e11i − 0.124830i
\(218\) − 1.76358e12i − 0.242597i
\(219\) −3.68875e11 −0.0494808
\(220\) 0 0
\(221\) 3.98982e12 0.509092
\(222\) − 1.10203e12i − 0.137167i
\(223\) 7.33486e12i 0.890667i 0.895365 + 0.445333i \(0.146915\pi\)
−0.895365 + 0.445333i \(0.853085\pi\)
\(224\) −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 \(0.976145\pi\)
\(228\) − 3.95479e12i − 0.425091i
\(229\) 1.18244e13 1.24075 0.620375 0.784305i \(-0.286980\pi\)
0.620375 + 0.784305i \(0.286980\pi\)
\(230\) 0 0
\(231\) −2.25579e12 −0.225649
\(232\) − 1.08478e13i − 1.05964i
\(233\) − 1.75634e13i − 1.67552i −0.546038 0.837761i \(-0.683865\pi\)
0.546038 0.837761i \(-0.316135\pi\)
\(234\) 1.57574e12 0.146824
\(235\) 0 0
\(236\) 7.63851e12 0.679192
\(237\) − 9.60545e12i − 0.834452i
\(238\) − 2.77519e12i − 0.235570i
\(239\) 7.13958e12 0.592221 0.296111 0.955154i \(-0.404310\pi\)
0.296111 + 0.955154i \(0.404310\pi\)
\(240\) 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
\(243\) 1.34008e13i 1.01460i
\(244\) 1.02399e13 0.757972
\(245\) 0 0
\(246\) −1.86351e12 −0.131883
\(247\) 6.15951e12i 0.426297i
\(248\) − 4.46419e12i − 0.302175i
\(249\) 7.39245e12 0.489431
\(250\) 0 0
\(251\) 1.29831e13 0.822567 0.411284 0.911507i \(-0.365081\pi\)
0.411284 + 0.911507i \(0.365081\pi\)
\(252\) 2.80098e12i 0.173623i
\(253\) 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 \(-0.232239\pi\)
\(258\) 1.03576e11i 0.00564095i
\(259\) −3.05098e12 −0.162664
\(260\) 0 0
\(261\) −1.45925e13 −0.745774
\(262\) 1.51567e13i 0.758485i
\(263\) − 2.42737e13i − 1.18954i −0.803895 0.594771i \(-0.797243\pi\)
0.803895 0.594771i \(-0.202757\pi\)
\(264\) −1.13813e13 −0.546225
\(265\) 0 0
\(266\) 4.28436e12 0.197259
\(267\) 6.29822e12i 0.284057i
\(268\) 2.27892e13i 1.00691i
\(269\) −2.58377e13 −1.11845 −0.559225 0.829016i \(-0.688901\pi\)
−0.559225 + 0.829016i \(0.688901\pi\)
\(270\) 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
\(273\) 2.43776e12i 0.0972963i
\(274\) −7.13277e12 −0.279017
\(275\) 0 0
\(276\) −6.91561e12 −0.259915
\(277\) 1.64189e13i 0.604931i 0.953160 + 0.302466i \(0.0978098\pi\)
−0.953160 + 0.302466i \(0.902190\pi\)
\(278\) − 1.43230e13i − 0.517355i
\(279\) −6.00526e12 −0.212671
\(280\) 0 0
\(281\) 2.10357e13 0.716263 0.358132 0.933671i \(-0.383414\pi\)
0.358132 + 0.933671i \(0.383414\pi\)
\(282\) 1.62531e13i 0.542707i
\(283\) 1.67132e13i 0.547310i 0.961828 + 0.273655i \(0.0882327\pi\)
−0.961828 + 0.273655i \(0.911767\pi\)
\(284\) 1.44131e13 0.462920
\(285\) 0 0
\(286\) 7.41278e12 0.229070
\(287\) 5.15917e12i 0.156397i
\(288\) 2.23543e13i 0.664817i
\(289\) −1.34200e13 −0.391575
\(290\) 0 0
\(291\) 1.89034e13 0.531042
\(292\) 2.15470e12i 0.0593993i
\(293\) − 2.39269e13i − 0.647312i −0.946175 0.323656i \(-0.895088\pi\)
0.946175 0.323656i \(-0.104912\pi\)
\(294\) −1.02632e13 −0.272505
\(295\) 0 0
\(296\) −1.53934e13 −0.393758
\(297\) 3.91759e13i 0.983690i
\(298\) 2.67704e13i 0.659881i
\(299\) 1.07709e13 0.260652
\(300\) 0 0
\(301\) 2.86753e11 0.00668947
\(302\) − 1.97867e13i − 0.453248i
\(303\) 2.05992e13i 0.463358i
\(304\) −1.05243e13 −0.232481
\(305\) 0 0
\(306\) −1.88355e13 −0.401336
\(307\) − 1.53111e13i − 0.320439i −0.987081 0.160219i \(-0.948780\pi\)
0.987081 0.160219i \(-0.0512202\pi\)
\(308\) 1.31767e13i 0.270880i
\(309\) 5.68903e13 1.14886
\(310\) 0 0
\(311\) 4.98752e13 0.972080 0.486040 0.873936i \(-0.338441\pi\)
0.486040 + 0.873936i \(0.338441\pi\)
\(312\) 1.22994e13i 0.235524i
\(313\) − 9.94808e13i − 1.87174i −0.352345 0.935870i \(-0.614616\pi\)
0.352345 0.935870i \(-0.385384\pi\)
\(314\) 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 \(-0.261129\pi\)
\(318\) 9.65294e12i 0.166460i
\(319\) −6.86477e13 −1.16353
\(320\) 0 0
\(321\) 2.27408e13 0.372416
\(322\) − 7.49191e12i − 0.120611i
\(323\) − 7.36271e13i − 1.16526i
\(324\) 2.45116e12 0.0381394
\(325\) 0 0
\(326\) 8.58799e12 0.129180
\(327\) − 1.85176e13i − 0.273888i
\(328\) 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
\(333\) 2.07073e13i 0.277127i
\(334\) 6.61160e13 0.870364
\(335\) 0 0
\(336\) −4.16521e12 −0.0530604
\(337\) − 1.21001e14i − 1.51644i −0.651997 0.758221i \(-0.726069\pi\)
0.651997 0.758221i \(-0.273931\pi\)
\(338\) 3.50011e13i 0.431559i
\(339\) 2.14570e13 0.260298
\(340\) 0 0
\(341\) −2.82506e13 −0.331802
\(342\) − 2.90783e13i − 0.336067i
\(343\) 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 \(0.688055\pi\)
\(348\) − 4.76317e13i − 0.500276i
\(349\) 2.56430e13 0.265112 0.132556 0.991176i \(-0.457682\pi\)
0.132556 + 0.991176i \(0.457682\pi\)
\(350\) 0 0
\(351\) 4.23361e13 0.424152
\(352\) 1.05162e14i 1.03722i
\(353\) 2.49098e13i 0.241885i 0.992659 + 0.120943i \(0.0385917\pi\)
−0.992659 + 0.120943i \(0.961408\pi\)
\(354\) −3.13843e13 −0.300051
\(355\) 0 0
\(356\) 3.67896e13 0.340996
\(357\) − 2.91395e13i − 0.265954i
\(358\) − 4.03532e13i − 0.362678i
\(359\) −1.57584e14 −1.39474 −0.697370 0.716712i \(-0.745646\pi\)
−0.697370 + 0.716712i \(0.745646\pi\)
\(360\) 0 0
\(361\) −2.82438e12 −0.0242457
\(362\) − 2.39226e13i − 0.202260i
\(363\) 1.25577e11i 0.00104574i
\(364\) 1.42396e13 0.116799
\(365\) 0 0
\(366\) −4.20728e13 −0.334854
\(367\) 1.77901e14i 1.39481i 0.716676 + 0.697406i \(0.245662\pi\)
−0.716676 + 0.697406i \(0.754338\pi\)
\(368\) 1.84034e13i 0.142146i
\(369\) 3.50157e13 0.266452
\(370\) 0 0
\(371\) 2.67244e13 0.197402
\(372\) − 1.96019e13i − 0.142663i
\(373\) − 5.51617e13i − 0.395585i −0.980244 0.197792i \(-0.936623\pi\)
0.980244 0.197792i \(-0.0633772\pi\)
\(374\) −8.86079e13 −0.626150
\(375\) 0 0
\(376\) 2.27027e14 1.55792
\(377\) 7.41854e13i 0.501696i
\(378\) − 2.94476e13i − 0.196266i
\(379\) −1.46463e14 −0.962083 −0.481042 0.876698i \(-0.659741\pi\)
−0.481042 + 0.876698i \(0.659741\pi\)
\(380\) 0 0
\(381\) −6.62047e13 −0.422476
\(382\) 6.62977e13i 0.417013i
\(383\) 2.31450e14i 1.43504i 0.696539 + 0.717519i \(0.254722\pi\)
−0.696539 + 0.717519i \(0.745278\pi\)
\(384\) −8.51940e13 −0.520700
\(385\) 0 0
\(386\) −1.30617e14 −0.775837
\(387\) − 1.94622e12i − 0.0113967i
\(388\) − 1.10420e14i − 0.637490i
\(389\) 1.49872e14 0.853093 0.426547 0.904466i \(-0.359730\pi\)
0.426547 + 0.904466i \(0.359730\pi\)
\(390\) 0 0
\(391\) −1.28749e14 −0.712480
\(392\) 1.43360e14i 0.782264i
\(393\) 1.59145e14i 0.856317i
\(394\) −6.90262e13 −0.366256
\(395\) 0 0
\(396\) 8.94312e13 0.461494
\(397\) − 2.08111e14i − 1.05912i −0.848271 0.529562i \(-0.822356\pi\)
0.848271 0.529562i \(-0.177644\pi\)
\(398\) − 1.74814e13i − 0.0877443i
\(399\) 4.49857e13 0.222702
\(400\) 0 0
\(401\) −1.33408e14 −0.642521 −0.321261 0.946991i \(-0.604107\pi\)
−0.321261 + 0.946991i \(0.604107\pi\)
\(402\) − 9.36341e13i − 0.444827i
\(403\) 3.05295e13i 0.143068i
\(404\) 1.20326e14 0.556238
\(405\) 0 0
\(406\) 5.16010e13 0.232148
\(407\) 9.74134e13i 0.432364i
\(408\) − 1.47020e14i − 0.643793i
\(409\) 2.06168e14 0.890722 0.445361 0.895351i \(-0.353075\pi\)
0.445361 + 0.895351i \(0.353075\pi\)
\(410\) 0 0
\(411\) −7.48941e13 −0.315005
\(412\) − 3.32312e14i − 1.37915i
\(413\) 8.68880e13i 0.355824i
\(414\) −5.08483e13 −0.205482
\(415\) 0 0
\(416\) 1.13645e14 0.447235
\(417\) − 1.50392e14i − 0.584085i
\(418\) − 1.36793e14i − 0.524319i
\(419\) −7.34035e13 −0.277677 −0.138838 0.990315i \(-0.544337\pi\)
−0.138838 + 0.990315i \(0.544337\pi\)
\(420\) 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
\(423\) − 3.05398e14i − 1.09646i
\(424\) 1.34835e14 0.477848
\(425\) 0 0
\(426\) −5.92189e13 −0.204507
\(427\) 1.16479e14i 0.397096i
\(428\) − 1.32835e14i − 0.447067i
\(429\) 7.78341e13 0.258616
\(430\) 0 0
\(431\) −7.17758e13 −0.232463 −0.116231 0.993222i \(-0.537081\pi\)
−0.116231 + 0.993222i \(0.537081\pi\)
\(432\) 7.23364e13i 0.231311i
\(433\) 9.98812e13i 0.315356i 0.987491 + 0.157678i \(0.0504007\pi\)
−0.987491 + 0.157678i \(0.949599\pi\)
\(434\) 2.12353e13 0.0662012
\(435\) 0 0
\(436\) −1.08166e14 −0.328789
\(437\) − 1.98764e14i − 0.596608i
\(438\) − 8.85301e12i − 0.0262412i
\(439\) 2.90312e13 0.0849788 0.0424894 0.999097i \(-0.486471\pi\)
0.0424894 + 0.999097i \(0.486471\pi\)
\(440\) 0 0
\(441\) 1.92848e14 0.550558
\(442\) 9.57557e13i 0.269987i
\(443\) 3.28370e14i 0.914414i 0.889360 + 0.457207i \(0.151150\pi\)
−0.889360 + 0.457207i \(0.848850\pi\)
\(444\) −6.75909e13 −0.185901
\(445\) 0 0
\(446\) −1.76037e14 −0.472347
\(447\) 2.81089e14i 0.744994i
\(448\) − 4.51970e13i − 0.118326i
\(449\) 6.12368e14 1.58364 0.791822 0.610752i \(-0.209133\pi\)
0.791822 + 0.610752i \(0.209133\pi\)
\(450\) 0 0
\(451\) 1.64725e14 0.415708
\(452\) − 1.25336e14i − 0.312475i
\(453\) − 2.07761e14i − 0.511709i
\(454\) −3.26361e13 −0.0794130
\(455\) 0 0
\(456\) 2.26971e14 0.539092
\(457\) − 3.03483e14i − 0.712189i −0.934450 0.356095i \(-0.884108\pi\)
0.934450 0.356095i \(-0.115892\pi\)
\(458\) 2.83786e14i 0.658007i
\(459\) −5.06060e14 −1.15940
\(460\) 0 0
\(461\) −7.29308e14 −1.63138 −0.815691 0.578487i \(-0.803643\pi\)
−0.815691 + 0.578487i \(0.803643\pi\)
\(462\) − 5.41389e13i − 0.119668i
\(463\) 1.22188e14i 0.266891i 0.991056 + 0.133445i \(0.0426041\pi\)
−0.991056 + 0.133445i \(0.957396\pi\)
\(464\) −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 \(0.777647\pi\)
\(468\) − 9.66455e13i − 0.198989i
\(469\) −2.59228e14 −0.527510
\(470\) 0 0
\(471\) 3.31409e14 0.658794
\(472\) 4.38384e14i 0.861338i
\(473\) − 9.15561e12i − 0.0177808i
\(474\) 2.30531e14 0.442535
\(475\) 0 0
\(476\) −1.70212e14 −0.319265
\(477\) − 1.81381e14i − 0.336310i
\(478\) 1.71350e14i 0.314073i
\(479\) −1.05084e15 −1.90410 −0.952052 0.305938i \(-0.901030\pi\)
−0.952052 + 0.305938i \(0.901030\pi\)
\(480\) 0 0
\(481\) 1.05272e14 0.186429
\(482\) − 5.55137e12i − 0.00971944i
\(483\) − 7.86651e13i − 0.136167i
\(484\) 7.33527e11 0.00125536
\(485\) 0 0
\(486\) −3.21619e14 −0.538074
\(487\) 2.19910e14i 0.363777i 0.983319 + 0.181889i \(0.0582210\pi\)
−0.983319 + 0.181889i \(0.941779\pi\)
\(488\) 5.87683e14i 0.961246i
\(489\) 9.01739e13 0.145842
\(490\) 0 0
\(491\) −4.83863e14 −0.765199 −0.382599 0.923914i \(-0.624971\pi\)
−0.382599 + 0.923914i \(0.624971\pi\)
\(492\) 1.14295e14i 0.178740i
\(493\) − 8.86768e14i − 1.37136i
\(494\) −1.47828e14 −0.226078
\(495\) 0 0
\(496\) −5.21634e13 −0.0780219
\(497\) 1.63949e14i 0.242520i
\(498\) 1.77419e14i 0.259560i
\(499\) 1.08878e14 0.157538 0.0787691 0.996893i \(-0.474901\pi\)
0.0787691 + 0.996893i \(0.474901\pi\)
\(500\) 0 0
\(501\) 6.94218e14 0.982626
\(502\) 3.11593e14i 0.436232i
\(503\) 5.06588e14i 0.701506i 0.936468 + 0.350753i \(0.114074\pi\)
−0.936468 + 0.350753i \(0.885926\pi\)
\(504\) −1.60752e14 −0.220185
\(505\) 0 0
\(506\) −2.39206e14 −0.320586
\(507\) 3.67512e14i 0.487222i
\(508\) 3.86720e14i 0.507161i
\(509\) −8.57534e13 −0.111251 −0.0556254 0.998452i \(-0.517715\pi\)
−0.0556254 + 0.998452i \(0.517715\pi\)
\(510\) 0 0
\(511\) −2.45097e13 −0.0311188
\(512\) 3.64965e14i 0.458423i
\(513\) − 7.81259e14i − 0.970844i
\(514\) 5.75069e14 0.707005
\(515\) 0 0
\(516\) 6.35268e12 0.00764511
\(517\) − 1.43669e15i − 1.71066i
\(518\) − 7.32235e13i − 0.0862654i
\(519\) 2.39498e14 0.279178
\(520\) 0 0
\(521\) 9.27575e14 1.05862 0.529312 0.848428i \(-0.322450\pi\)
0.529312 + 0.848428i \(0.322450\pi\)
\(522\) − 3.50220e14i − 0.395506i
\(523\) − 2.18187e13i − 0.0243820i −0.999926 0.0121910i \(-0.996119\pi\)
0.999926 0.0121910i \(-0.00388061\pi\)
\(524\) 9.29610e14 1.02797
\(525\) 0 0
\(526\) 5.82569e14 0.630850
\(527\) − 3.64931e14i − 0.391069i
\(528\) 1.32989e14i 0.141036i
\(529\) 6.05238e14 0.635214
\(530\) 0 0
\(531\) 5.89717e14 0.606211
\(532\) − 2.62774e14i − 0.267343i
\(533\) − 1.78013e14i − 0.179247i
\(534\) −1.51157e14 −0.150644
\(535\) 0 0
\(536\) −1.30790e15 −1.27694
\(537\) − 4.23709e14i − 0.409458i
\(538\) − 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
\(543\) − 2.51187e14i − 0.228349i
\(544\) −1.35844e15 −1.22249
\(545\) 0 0
\(546\) −5.85062e13 −0.0515991
\(547\) − 7.52145e14i − 0.656706i −0.944555 0.328353i \(-0.893506\pi\)
0.944555 0.328353i \(-0.106494\pi\)
\(548\) 4.37477e14i 0.378148i
\(549\) 7.90555e14 0.676526
\(550\) 0 0
\(551\) 1.36900e15 1.14834
\(552\) − 3.96896e14i − 0.329619i
\(553\) − 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 \(-0.0236043\pi\)
\(558\) − 1.44126e14i − 0.112786i
\(559\) −9.89417e12 −0.00766681
\(560\) 0 0
\(561\) −9.30383e14 −0.706913
\(562\) 5.04857e14i 0.379856i
\(563\) 2.44971e14i 0.182524i 0.995827 + 0.0912618i \(0.0290900\pi\)
−0.995827 + 0.0912618i \(0.970910\pi\)
\(564\) 9.96856e14 0.735525
\(565\) 0 0
\(566\) −4.01116e14 −0.290255
\(567\) 2.78819e13i 0.0199809i
\(568\) 8.27185e14i 0.587066i
\(569\) −1.35243e15 −0.950596 −0.475298 0.879825i \(-0.657660\pi\)
−0.475298 + 0.879825i \(0.657660\pi\)
\(570\) 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
\(573\) 6.96126e14i 0.470801i
\(574\) −1.23820e14 −0.0829422
\(575\) 0 0
\(576\) −3.06756e14 −0.201590
\(577\) 8.77659e14i 0.571293i 0.958335 + 0.285647i \(0.0922083\pi\)
−0.958335 + 0.285647i \(0.907792\pi\)
\(578\) − 3.22081e14i − 0.207664i
\(579\) −1.37148e15 −0.875907
\(580\) 0 0
\(581\) 4.91187e14 0.307807
\(582\) 4.53682e14i 0.281628i
\(583\) − 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 \(0.743766\pi\)
\(588\) 6.29479e14i 0.369323i
\(589\) 5.63383e14 0.327469
\(590\) 0 0
\(591\) −7.24775e14 −0.413497
\(592\) 1.79869e14i 0.101669i
\(593\) − 3.03318e14i − 0.169863i −0.996387 0.0849313i \(-0.972933\pi\)
0.996387 0.0849313i \(-0.0270671\pi\)
\(594\) −9.40221e14 −0.521680
\(595\) 0 0
\(596\) 1.64192e15 0.894329
\(597\) − 1.83555e14i − 0.0990619i
\(598\) 2.58502e14i 0.138232i
\(599\) 1.70198e15 0.901795 0.450898 0.892576i \(-0.351104\pi\)
0.450898 + 0.892576i \(0.351104\pi\)
\(600\) 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
\(603\) 1.75940e15i 0.898710i
\(604\) −1.21359e15 −0.614282
\(605\) 0 0
\(606\) −4.94381e14 −0.245733
\(607\) 2.49607e15i 1.22947i 0.788732 + 0.614737i \(0.210738\pi\)
−0.788732 + 0.614737i \(0.789262\pi\)
\(608\) − 2.09717e15i − 1.02368i
\(609\) 5.41810e14 0.262091
\(610\) 0 0
\(611\) −1.55258e15 −0.737612
\(612\) 1.15524e15i 0.543926i
\(613\) 2.47301e15i 1.15397i 0.816756 + 0.576983i \(0.195770\pi\)
−0.816756 + 0.576983i \(0.804230\pi\)
\(614\) 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 \(-0.00174388\pi\)
\(618\) 1.36537e15i 0.609274i
\(619\) −4.22545e15 −1.86885 −0.934425 0.356160i \(-0.884086\pi\)
−0.934425 + 0.356160i \(0.884086\pi\)
\(620\) 0 0
\(621\) −1.36616e15 −0.593606
\(622\) 1.19700e15i 0.515523i
\(623\) 4.18481e14i 0.178645i
\(624\) 1.43717e14 0.0608126
\(625\) 0 0
\(626\) 2.38754e15 0.992640
\(627\) − 1.43633e15i − 0.591947i
\(628\) − 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
\(633\) − 1.71188e15i − 0.669503i
\(634\) 2.00086e15 0.775758
\(635\) 0 0
\(636\) 5.92047e14 0.225602
\(637\) − 9.80401e14i − 0.370371i
\(638\) − 1.64755e15i − 0.617055i
\(639\) 1.11273e15 0.413177
\(640\) 0 0
\(641\) 1.00830e15 0.368018 0.184009 0.982925i \(-0.441092\pi\)
0.184009 + 0.982925i \(0.441092\pi\)
\(642\) 5.45779e14i 0.197503i
\(643\) 3.03982e14i 0.109066i 0.998512 + 0.0545328i \(0.0173670\pi\)
−0.998512 + 0.0545328i \(0.982633\pi\)
\(644\) −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 \(-0.203125\pi\)
\(648\) 1.40675e14i 0.0483676i
\(649\) 2.77421e15 0.945788
\(650\) 0 0
\(651\) 2.22971e14 0.0747400
\(652\) − 5.26730e14i − 0.175076i
\(653\) − 1.18539e15i − 0.390695i −0.980734 0.195347i \(-0.937417\pi\)
0.980734 0.195347i \(-0.0625834\pi\)
\(654\) 4.44423e14 0.145251
\(655\) 0 0
\(656\) 3.04157e14 0.0977522
\(657\) 1.66350e14i 0.0530167i
\(658\) 1.07993e15i 0.341312i
\(659\) 2.26510e15 0.709934 0.354967 0.934879i \(-0.384492\pi\)
0.354967 + 0.934879i \(0.384492\pi\)
\(660\) 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
\(663\) 1.00543e15i 0.304810i
\(664\) 2.47823e15 0.745104
\(665\) 0 0
\(666\) −4.96974e14 −0.146969
\(667\) − 2.39392e15i − 0.702130i
\(668\) − 4.05512e15i − 1.17959i
\(669\) −1.84839e15 −0.533272
\(670\) 0 0
\(671\) 3.71902e15 1.05549
\(672\) − 8.30000e14i − 0.233640i
\(673\) 4.74120e15i 1.32375i 0.749615 + 0.661874i \(0.230239\pi\)
−0.749615 + 0.661874i \(0.769761\pi\)
\(674\) 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 \(0.938846\pi\)
\(678\) 5.14968e14i 0.138044i
\(679\) 1.25603e15 0.333976
\(680\) 0 0
\(681\) −3.42680e14 −0.0896559
\(682\) − 6.78014e14i − 0.175964i
\(683\) − 3.03116e15i − 0.780359i −0.920739 0.390180i \(-0.872413\pi\)
0.920739 0.390180i \(-0.127587\pi\)
\(684\) −1.78347e15 −0.455467
\(685\) 0 0
\(686\) −1.47654e15 −0.371075
\(687\) 2.97975e15i 0.742879i
\(688\) − 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
\(693\) 1.01728e15i 0.241773i
\(694\) −3.73588e15 −0.880878
\(695\) 0 0
\(696\) 2.73364e15 0.634441
\(697\) 2.12786e15i 0.489962i
\(698\) 6.15433e14i 0.140597i
\(699\) 4.42597e15 1.00319
\(700\) 0 0
\(701\) 5.72747e15 1.27795 0.638974 0.769228i \(-0.279359\pi\)
0.638974 + 0.769228i \(0.279359\pi\)
\(702\) 1.01607e15i 0.224941i
\(703\) − 1.94265e15i − 0.426718i
\(704\) −1.44308e15 −0.314514
\(705\) 0 0
\(706\) −5.97836e14 −0.128279
\(707\) 1.36870e15i 0.291409i
\(708\) 1.92490e15i 0.406655i
\(709\) −6.98326e14 −0.146388 −0.0731938 0.997318i \(-0.523319\pi\)
−0.0731938 + 0.997318i \(0.523319\pi\)
\(710\) 0 0
\(711\) −4.33171e15 −0.894081
\(712\) 2.11140e15i 0.432445i
\(713\) − 9.85170e14i − 0.200225i
\(714\) 6.99348e14 0.141044
\(715\) 0 0
\(716\) −2.47500e15 −0.491534
\(717\) 1.79917e15i 0.354583i
\(718\) − 3.78202e15i − 0.739672i
\(719\) −9.70979e15 −1.88452 −0.942260 0.334882i \(-0.891304\pi\)
−0.942260 + 0.334882i \(0.891304\pi\)
\(720\) 0 0
\(721\) 3.78004e15 0.722525
\(722\) − 6.77852e13i − 0.0128582i
\(723\) − 5.82893e13i − 0.0109731i
\(724\) −1.46725e15 −0.274121
\(725\) 0 0
\(726\) −3.01384e12 −0.000554586 0
\(727\) − 2.46469e15i − 0.450114i −0.974346 0.225057i \(-0.927743\pi\)
0.974346 0.225057i \(-0.0722568\pi\)
\(728\) 8.17230e14i 0.148123i
\(729\) −3.08202e15 −0.554413
\(730\) 0 0
\(731\) 1.18269e14 0.0209568
\(732\) 2.58046e15i 0.453823i
\(733\) 7.91285e15i 1.38121i 0.723230 + 0.690607i \(0.242657\pi\)
−0.723230 + 0.690607i \(0.757343\pi\)
\(734\) −4.26963e15 −0.739711
\(735\) 0 0
\(736\) −3.66725e15 −0.625911
\(737\) 8.27677e15i 1.40213i
\(738\) 8.40378e14i 0.141307i
\(739\) 8.40694e15 1.40312 0.701558 0.712613i \(-0.252488\pi\)
0.701558 + 0.712613i \(0.252488\pi\)
\(740\) 0 0
\(741\) −1.55220e15 −0.255239
\(742\) 6.41385e14i 0.104688i
\(743\) 1.36287e15i 0.220809i 0.993887 + 0.110404i \(0.0352146\pi\)
−0.993887 + 0.110404i \(0.964785\pi\)
\(744\) 1.12498e15 0.180922
\(745\) 0 0
\(746\) 1.32388e15 0.209790
\(747\) − 3.33373e15i − 0.524405i
\(748\) 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
\(753\) 3.27173e15i 0.492499i
\(754\) −1.78045e15 −0.266065
\(755\) 0 0
\(756\) −1.80612e15 −0.265997
\(757\) 6.67049e14i 0.0975282i 0.998810 + 0.0487641i \(0.0155283\pi\)
−0.998810 + 0.0487641i \(0.984472\pi\)
\(758\) − 3.51511e15i − 0.510222i
\(759\) −2.51166e15 −0.361936
\(760\) 0 0
\(761\) −7.74408e15 −1.09990 −0.549951 0.835197i \(-0.685354\pi\)
−0.549951 + 0.835197i \(0.685354\pi\)
\(762\) − 1.58891e15i − 0.224052i
\(763\) − 1.23039e15i − 0.172250i
\(764\) 4.06626e15 0.565173
\(765\) 0 0
\(766\) −5.55479e15 −0.761043
\(767\) − 2.99800e15i − 0.407809i
\(768\) − 3.43775e15i − 0.464288i
\(769\) −2.52411e15 −0.338465 −0.169232 0.985576i \(-0.554129\pi\)
−0.169232 + 0.985576i \(0.554129\pi\)
\(770\) 0 0
\(771\) 6.03822e15 0.798197
\(772\) 8.01119e15i 1.05148i
\(773\) − 1.11453e16i − 1.45246i −0.687453 0.726229i \(-0.741271\pi\)
0.687453 0.726229i \(-0.258729\pi\)
\(774\) 4.67092e13 0.00604404
\(775\) 0 0
\(776\) 6.33715e15 0.808452
\(777\) − 7.68847e14i − 0.0973922i
\(778\) 3.59692e15i 0.452421i
\(779\) −3.28500e15 −0.410279
\(780\) 0 0
\(781\) 5.23465e15 0.644624
\(782\) − 3.08998e15i − 0.377849i
\(783\) − 9.40952e15i − 1.14256i
\(784\) 1.67514e15 0.201981
\(785\) 0 0
\(786\) −3.81949e15 −0.454131
\(787\) − 1.32271e16i − 1.56172i −0.624705 0.780861i \(-0.714781\pi\)
0.624705 0.780861i \(-0.285219\pi\)
\(788\) 4.23361e15i 0.496382i
\(789\) 6.11698e15 0.712219
\(790\) 0 0
\(791\) 1.42570e15 0.163703
\(792\) 5.13257e15i 0.585257i
\(793\) − 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 \(-0.0404730\pi\)
\(798\) 1.07966e15i 0.118106i
\(799\) 1.85587e16 2.01623
\(800\) 0 0
\(801\) 2.84027e15 0.304355
\(802\) − 3.20179e15i − 0.340748i
\(803\) 7.82560e14i 0.0827146i
\(804\) −5.74289e15 −0.602868
\(805\) 0 0
\(806\) −7.32708e14 −0.0758732
\(807\) − 6.51110e15i − 0.669653i
\(808\) 6.90565e15i 0.705410i
\(809\) −5.60472e15 −0.568639 −0.284320 0.958730i \(-0.591768\pi\)
−0.284320 + 0.958730i \(0.591768\pi\)
\(810\) 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
\(813\) − 9.49519e14i − 0.0937574i
\(814\) −2.33792e15 −0.229296
\(815\) 0 0
\(816\) −1.71791e15 −0.166228
\(817\) 1.82584e14i 0.0175486i
\(818\) 4.94802e15i 0.472377i
\(819\) 1.09934e15 0.104249
\(820\) 0 0
\(821\) 2.79111e14 0.0261150 0.0130575 0.999915i \(-0.495844\pi\)
0.0130575 + 0.999915i \(0.495844\pi\)
\(822\) − 1.79746e15i − 0.167057i
\(823\) − 1.35265e16i − 1.24878i −0.781112 0.624391i \(-0.785347\pi\)
0.781112 0.624391i \(-0.214653\pi\)
\(824\) 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 \(-0.00389930\pi\)
\(828\) 3.11869e15i 0.278488i
\(829\) −1.80459e16 −1.60077 −0.800385 0.599486i \(-0.795372\pi\)
−0.800385 + 0.599486i \(0.795372\pi\)
\(830\) 0 0
\(831\) −4.13757e15 −0.362193
\(832\) 1.55949e15i 0.135614i
\(833\) 1.17191e16i 1.01239i
\(834\) 3.60941e15 0.309758
\(835\) 0 0
\(836\) −8.38999e15 −0.710603
\(837\) − 3.87230e15i − 0.325821i
\(838\) − 1.76168e15i − 0.147260i
\(839\) 7.96183e15 0.661184 0.330592 0.943774i \(-0.392752\pi\)
0.330592 + 0.943774i \(0.392752\pi\)
\(840\) 0 0
\(841\) 4.28775e15 0.351440
\(842\) 4.10669e15i 0.334407i
\(843\) 5.30100e15i 0.428851i
\(844\) −9.99954e15 −0.803705
\(845\) 0 0
\(846\) 7.32956e15 0.581488
\(847\) 8.34387e12i 0 0.000657671i
\(848\) − 1.57552e15i − 0.123381i
\(849\) −4.21172e15 −0.327693
\(850\) 0 0
\(851\) −3.39705e15 −0.260909
\(852\) 3.63209e15i 0.277165i
\(853\) − 1.49826e16i − 1.13598i −0.823037 0.567988i \(-0.807722\pi\)
0.823037 0.567988i \(-0.192278\pi\)
\(854\) −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 \(0.692696\pi\)
\(858\) 1.86802e15i 0.137152i
\(859\) −5.44237e15 −0.397032 −0.198516 0.980098i \(-0.563612\pi\)
−0.198516 + 0.980098i \(0.563612\pi\)
\(860\) 0 0
\(861\) −1.30011e15 −0.0936403
\(862\) − 1.72262e15i − 0.123282i
\(863\) 1.08110e16i 0.768787i 0.923169 + 0.384393i \(0.125589\pi\)
−0.923169 + 0.384393i \(0.874411\pi\)
\(864\) −1.44145e16 −1.01853
\(865\) 0 0
\(866\) −2.39715e15 −0.167243
\(867\) − 3.38185e15i − 0.234449i
\(868\) − 1.30243e15i − 0.0897217i
\(869\) −2.03777e16 −1.39491
\(870\) 0 0
\(871\) 8.94444e15 0.604579
\(872\) − 6.20782e15i − 0.416964i
\(873\) − 8.52477e15i − 0.568989i
\(874\) 4.77033e15 0.316399
\(875\) 0 0
\(876\) −5.42985e14 −0.0355644
\(877\) 2.81024e16i 1.82914i 0.404431 + 0.914568i \(0.367470\pi\)
−0.404431 + 0.914568i \(0.632530\pi\)
\(878\) 6.96749e14i 0.0450668i
\(879\) 6.02957e15 0.387568
\(880\) 0 0
\(881\) 4.22209e15 0.268016 0.134008 0.990980i \(-0.457215\pi\)
0.134008 + 0.990980i \(0.457215\pi\)
\(882\) 4.62836e15i 0.291978i
\(883\) 5.16092e14i 0.0323551i 0.999869 + 0.0161776i \(0.00514970\pi\)
−0.999869 + 0.0161776i \(0.994850\pi\)
\(884\) 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 \(-0.0559504\pi\)
\(888\) − 3.87913e15i − 0.235756i
\(889\) −4.39894e15 −0.265698
\(890\) 0 0
\(891\) 8.90230e14 0.0531098
\(892\) 1.07969e16i 0.640167i
\(893\) 2.86510e16i 1.68832i
\(894\) −6.74614e15 −0.395093
\(895\) 0 0
\(896\) −5.66067e15 −0.327472
\(897\) 2.71427e15i 0.156061i
\(898\) 1.46968e16i 0.839854i
\(899\) 6.78541e15 0.385388
\(900\) 0 0
\(901\) 1.10223e16 0.618421
\(902\) 3.95340e15i 0.220462i
\(903\) 7.22617e13i 0.00400521i
\(904\) 7.19321e15 0.396275
\(905\) 0 0
\(906\) 4.98626e15 0.271375
\(907\) 8.43778e13i 0.00456445i 0.999997 + 0.00228222i \(0.000726455\pi\)
−0.999997 + 0.00228222i \(0.999274\pi\)
\(908\) 2.00168e15i 0.107628i
\(909\) 9.28952e15 0.496468
\(910\) 0 0
\(911\) −1.10091e16 −0.581298 −0.290649 0.956830i \(-0.593871\pi\)
−0.290649 + 0.956830i \(0.593871\pi\)
\(912\) − 2.65212e15i − 0.139194i
\(913\) − 1.56829e16i − 0.818158i
\(914\) 7.28359e15 0.377695
\(915\) 0 0
\(916\) 1.74055e16 0.891789
\(917\) 1.05743e16i 0.538544i
\(918\) − 1.21455e16i − 0.614864i
\(919\) 4.86351e15 0.244746 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(920\) 0 0
\(921\) 3.85840e15 0.191857
\(922\) − 1.75034e16i − 0.865171i
\(923\) − 5.65691e15i − 0.277952i
\(924\) −3.32052e15 −0.162185
\(925\) 0 0
\(926\) −2.93252e15 −0.141540
\(927\) − 2.56555e16i − 1.23095i
\(928\) − 2.52584e16i − 1.20474i
\(929\) −3.57534e15 −0.169524 −0.0847620 0.996401i \(-0.527013\pi\)
−0.0847620 + 0.996401i \(0.527013\pi\)
\(930\) 0 0
\(931\) −1.80921e16 −0.847744
\(932\) − 2.58533e16i − 1.20428i
\(933\) 1.25685e16i 0.582017i
\(934\) −1.48171e16 −0.682113
\(935\) 0 0
\(936\) 5.54661e15 0.252354
\(937\) − 3.86373e16i − 1.74759i −0.486295 0.873795i \(-0.661652\pi\)
0.486295 0.873795i \(-0.338348\pi\)
\(938\) − 6.22147e15i − 0.279754i
\(939\) 2.50692e16 1.12067
\(940\) 0 0
\(941\) −3.48997e16 −1.54198 −0.770991 0.636846i \(-0.780239\pi\)
−0.770991 + 0.636846i \(0.780239\pi\)
\(942\) 7.95383e15i 0.349378i
\(943\) 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 \(0.791875\pi\)
\(948\) − 1.41392e16i − 0.599763i
\(949\) 8.45688e14 0.0356653
\(950\) 0 0
\(951\) 2.10091e16 0.875817
\(952\) − 9.76867e15i − 0.404886i
\(953\) 4.00334e16i 1.64973i 0.565332 + 0.824863i \(0.308748\pi\)
−0.565332 + 0.824863i \(0.691252\pi\)
\(954\) 4.35313e15 0.178355
\(955\) 0 0
\(956\) 1.05095e16 0.425659
\(957\) − 1.72992e16i − 0.696644i
\(958\) − 2.52201e16i − 1.00980i
\(959\) −4.97630e15 −0.198109
\(960\) 0 0
\(961\) −2.26161e16 −0.890100
\(962\) 2.52652e15i 0.0988688i
\(963\) − 1.02553e16i − 0.399028i
\(964\) −3.40484e14 −0.0131726
\(965\) 0 0
\(966\) 1.88796e15 0.0722136
\(967\) − 1.84953e16i − 0.703422i −0.936109 0.351711i \(-0.885600\pi\)
0.936109 0.351711i \(-0.114400\pi\)
\(968\) 4.20981e13i 0.00159202i
\(969\) 1.85540e16 0.697682
\(970\) 0 0
\(971\) −2.14877e16 −0.798884 −0.399442 0.916759i \(-0.630796\pi\)
−0.399442 + 0.916759i \(0.630796\pi\)
\(972\) 1.97260e16i 0.729246i
\(973\) − 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 \(0.949806\pi\)
\(978\) 2.16417e15i 0.0773443i
\(979\) 1.33615e16 0.474844
\(980\) 0 0
\(981\) −8.35079e15 −0.293459
\(982\) − 1.16127e16i − 0.405808i
\(983\) 1.18924e16i 0.413263i 0.978419 + 0.206631i \(0.0662501\pi\)
−0.978419 + 0.206631i \(0.933750\pi\)
\(984\) −6.55956e15 −0.226674
\(985\) 0 0
\(986\) 2.12824e16 0.727274
\(987\) 1.13392e16i 0.385336i
\(988\) 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
\(993\) − 1.60232e16i − 0.526657i
\(994\) −3.93477e15 −0.128616
\(995\) 0 0
\(996\) 1.08817e16 0.351779
\(997\) 2.14004e16i 0.688016i 0.938967 + 0.344008i \(0.111785\pi\)
−0.938967 + 0.344008i \(0.888215\pi\)
\(998\) 2.61307e15i 0.0835473i
\(999\) −1.33524e16 −0.424571
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 25.12.b.b.24.2 2
3.2 odd 2 225.12.b.d.199.1 2
5.2 odd 4 1.12.a.a.1.1 1
5.3 odd 4 25.12.a.b.1.1 1
5.4 even 2 inner 25.12.b.b.24.1 2
15.2 even 4 9.12.a.b.1.1 1
15.8 even 4 225.12.a.b.1.1 1
15.14 odd 2 225.12.b.d.199.2 2
20.7 even 4 16.12.a.a.1.1 1
35.2 odd 12 49.12.c.b.18.1 2
35.12 even 12 49.12.c.c.18.1 2
35.17 even 12 49.12.c.c.30.1 2
35.27 even 4 49.12.a.a.1.1 1
35.32 odd 12 49.12.c.b.30.1 2
40.27 even 4 64.12.a.f.1.1 1
40.37 odd 4 64.12.a.b.1.1 1
45.2 even 12 81.12.c.b.28.1 2
45.7 odd 12 81.12.c.d.28.1 2
45.22 odd 12 81.12.c.d.55.1 2
45.32 even 12 81.12.c.b.55.1 2
55.32 even 4 121.12.a.b.1.1 1
60.47 odd 4 144.12.a.d.1.1 1
65.12 odd 4 169.12.a.a.1.1 1
80.27 even 4 256.12.b.c.129.2 2
80.37 odd 4 256.12.b.e.129.1 2
80.67 even 4 256.12.b.c.129.1 2
80.77 odd 4 256.12.b.e.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.12.a.a.1.1 1 5.2 odd 4
9.12.a.b.1.1 1 15.2 even 4
16.12.a.a.1.1 1 20.7 even 4
25.12.a.b.1.1 1 5.3 odd 4
25.12.b.b.24.1 2 5.4 even 2 inner
25.12.b.b.24.2 2 1.1 even 1 trivial
49.12.a.a.1.1 1 35.27 even 4
49.12.c.b.18.1 2 35.2 odd 12
49.12.c.b.30.1 2 35.32 odd 12
49.12.c.c.18.1 2 35.12 even 12
49.12.c.c.30.1 2 35.17 even 12
64.12.a.b.1.1 1 40.37 odd 4
64.12.a.f.1.1 1 40.27 even 4
81.12.c.b.28.1 2 45.2 even 12
81.12.c.b.55.1 2 45.32 even 12
81.12.c.d.28.1 2 45.7 odd 12
81.12.c.d.55.1 2 45.22 odd 12
121.12.a.b.1.1 1 55.32 even 4
144.12.a.d.1.1 1 60.47 odd 4
169.12.a.a.1.1 1 65.12 odd 4
225.12.a.b.1.1 1 15.8 even 4
225.12.b.d.199.1 2 3.2 odd 2
225.12.b.d.199.2 2 15.14 odd 2
256.12.b.c.129.1 2 80.67 even 4
256.12.b.c.129.2 2 80.27 even 4
256.12.b.e.129.1 2 80.37 odd 4
256.12.b.e.129.2 2 80.77 odd 4