Properties

Label 25.34.b.c.24.7
Level $25$
Weight $34$
Character 25.24
Analytic conductor $172.457$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 20265063301 x^{10} + 151877130254993985540 x^{8} + 529361026408998369955851761920 x^{6} + 862054500541443268723145801151954288640 x^{4} + 544606882049987247786598582768552479276511789056 x^{2} + 39700341273686227371481766034965422500156905479886340096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{47}\cdot 3^{10}\cdot 5^{36}\cdot 7^{2}\cdot 11^{4} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.7
Root \(9129.22i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.34.b.c.24.6

$q$-expansion

\(f(q)\) \(=\) \(q+18258.4i q^{2} -3.67420e7i q^{3} +8.25656e9 q^{4} +6.70852e11 q^{6} +1.51681e13i q^{7} +3.07591e14i q^{8} +4.20908e15 q^{9} +O(q^{10})\) \(q+18258.4i q^{2} -3.67420e7i q^{3} +8.25656e9 q^{4} +6.70852e11 q^{6} +1.51681e13i q^{7} +3.07591e14i q^{8} +4.20908e15 q^{9} -2.61480e17 q^{11} -3.03363e17i q^{12} -3.70714e17i q^{13} -2.76946e17 q^{14} +6.53072e19 q^{16} +3.71730e20i q^{17} +7.68513e19i q^{18} -1.42255e21 q^{19} +5.57308e20 q^{21} -4.77422e21i q^{22} -3.26990e22i q^{23} +1.13015e22 q^{24} +6.76865e21 q^{26} -3.58901e23i q^{27} +1.25237e23i q^{28} -6.41343e23 q^{29} -1.44554e23 q^{31} +3.83459e24i q^{32} +9.60732e24i q^{33} -6.78720e24 q^{34} +3.47526e25 q^{36} -1.34909e26i q^{37} -2.59736e25i q^{38} -1.36208e25 q^{39} +6.37865e26 q^{41} +1.01756e25i q^{42} +3.37044e26i q^{43} -2.15893e27 q^{44} +5.97033e26 q^{46} -3.12828e27i q^{47} -2.39952e27i q^{48} +7.50092e27 q^{49} +1.36581e28 q^{51} -3.06082e27i q^{52} -4.06250e28i q^{53} +6.55298e27 q^{54} -4.66557e27 q^{56} +5.22675e28i q^{57} -1.17099e28i q^{58} +8.77955e28 q^{59} -1.84902e28 q^{61} -2.63933e27i q^{62} +6.38439e28i q^{63} +4.90971e29 q^{64} -1.75415e29 q^{66} -7.69531e29i q^{67} +3.06921e30i q^{68} -1.20143e30 q^{69} +2.72988e30 q^{71} +1.29468e30i q^{72} +4.44452e30i q^{73} +2.46323e30 q^{74} -1.17454e31 q^{76} -3.96617e30i q^{77} -2.48694e29i q^{78} +1.16758e31 q^{79} +1.02118e31 q^{81} +1.16464e31i q^{82} -4.84088e31i q^{83} +4.60145e30 q^{84} -6.15390e30 q^{86} +2.35643e31i q^{87} -8.04290e31i q^{88} -1.18101e32 q^{89} +5.62303e30 q^{91} -2.69981e32i q^{92} +5.31122e30i q^{93} +5.71176e31 q^{94} +1.40891e32 q^{96} +2.71846e31i q^{97} +1.36955e32i q^{98} -1.10059e33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + O(q^{10}) \) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + 301521793111780384q^{11} + 14172220017979782912q^{14} + \)\(12\!\cdots\!32\)\(q^{16} - \)\(34\!\cdots\!00\)\(q^{19} - \)\(21\!\cdots\!56\)\(q^{21} - \)\(42\!\cdots\!00\)\(q^{24} - \)\(72\!\cdots\!76\)\(q^{26} - \)\(26\!\cdots\!00\)\(q^{29} + \)\(21\!\cdots\!04\)\(q^{31} - \)\(17\!\cdots\!48\)\(q^{34} + \)\(33\!\cdots\!92\)\(q^{36} - \)\(11\!\cdots\!12\)\(q^{39} + \)\(89\!\cdots\!64\)\(q^{41} - \)\(14\!\cdots\!28\)\(q^{44} + \)\(69\!\cdots\!44\)\(q^{46} + \)\(14\!\cdots\!16\)\(q^{49} - \)\(10\!\cdots\!76\)\(q^{51} + \)\(36\!\cdots\!00\)\(q^{54} - \)\(78\!\cdots\!00\)\(q^{56} + \)\(35\!\cdots\!00\)\(q^{59} - \)\(56\!\cdots\!16\)\(q^{61} - \)\(38\!\cdots\!44\)\(q^{64} + \)\(10\!\cdots\!28\)\(q^{66} - \)\(63\!\cdots\!72\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} - \)\(40\!\cdots\!68\)\(q^{74} + \)\(58\!\cdots\!00\)\(q^{76} + \)\(44\!\cdots\!00\)\(q^{79} - \)\(26\!\cdots\!48\)\(q^{81} + \)\(47\!\cdots\!52\)\(q^{84} - \)\(25\!\cdots\!16\)\(q^{86} + \)\(67\!\cdots\!00\)\(q^{89} + \)\(38\!\cdots\!64\)\(q^{91} - \)\(29\!\cdots\!28\)\(q^{94} + \)\(63\!\cdots\!44\)\(q^{96} - \)\(33\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18258.4i 0.197001i 0.995137 + 0.0985006i \(0.0314046\pi\)
−0.995137 + 0.0985006i \(0.968595\pi\)
\(3\) − 3.67420e7i − 0.492791i −0.969169 0.246395i \(-0.920754\pi\)
0.969169 0.246395i \(-0.0792462\pi\)
\(4\) 8.25656e9 0.961191
\(5\) 0 0
\(6\) 6.70852e11 0.0970803
\(7\) 1.51681e13i 0.172510i 0.996273 + 0.0862550i \(0.0274900\pi\)
−0.996273 + 0.0862550i \(0.972510\pi\)
\(8\) 3.07591e14i 0.386357i
\(9\) 4.20908e15 0.757157
\(10\) 0 0
\(11\) −2.61480e17 −1.71577 −0.857887 0.513839i \(-0.828223\pi\)
−0.857887 + 0.513839i \(0.828223\pi\)
\(12\) − 3.03363e17i − 0.473666i
\(13\) − 3.70714e17i − 0.154516i −0.997011 0.0772580i \(-0.975383\pi\)
0.997011 0.0772580i \(-0.0246165\pi\)
\(14\) −2.76946e17 −0.0339847
\(15\) 0 0
\(16\) 6.53072e19 0.885078
\(17\) 3.71730e20i 1.85276i 0.376585 + 0.926382i \(0.377098\pi\)
−0.376585 + 0.926382i \(0.622902\pi\)
\(18\) 7.68513e19i 0.149161i
\(19\) −1.42255e21 −1.13145 −0.565723 0.824595i \(-0.691403\pi\)
−0.565723 + 0.824595i \(0.691403\pi\)
\(20\) 0 0
\(21\) 5.57308e20 0.0850113
\(22\) − 4.77422e21i − 0.338009i
\(23\) − 3.26990e22i − 1.11180i −0.831250 0.555898i \(-0.812374\pi\)
0.831250 0.555898i \(-0.187626\pi\)
\(24\) 1.13015e22 0.190393
\(25\) 0 0
\(26\) 6.76865e21 0.0304398
\(27\) − 3.58901e23i − 0.865911i
\(28\) 1.25237e23i 0.165815i
\(29\) −6.41343e23 −0.475909 −0.237954 0.971276i \(-0.576477\pi\)
−0.237954 + 0.971276i \(0.576477\pi\)
\(30\) 0 0
\(31\) −1.44554e23 −0.0356914 −0.0178457 0.999841i \(-0.505681\pi\)
−0.0178457 + 0.999841i \(0.505681\pi\)
\(32\) 3.83459e24i 0.560718i
\(33\) 9.60732e24i 0.845517i
\(34\) −6.78720e24 −0.364997
\(35\) 0 0
\(36\) 3.47526e25 0.727772
\(37\) − 1.34909e26i − 1.79768i −0.438273 0.898842i \(-0.644410\pi\)
0.438273 0.898842i \(-0.355590\pi\)
\(38\) − 2.59736e25i − 0.222896i
\(39\) −1.36208e25 −0.0761440
\(40\) 0 0
\(41\) 6.37865e26 1.56241 0.781205 0.624275i \(-0.214605\pi\)
0.781205 + 0.624275i \(0.214605\pi\)
\(42\) 1.01756e25i 0.0167473i
\(43\) 3.37044e26i 0.376233i 0.982147 + 0.188117i \(0.0602383\pi\)
−0.982147 + 0.188117i \(0.939762\pi\)
\(44\) −2.15893e27 −1.64919
\(45\) 0 0
\(46\) 5.97033e26 0.219025
\(47\) − 3.12828e27i − 0.804807i −0.915462 0.402403i \(-0.868175\pi\)
0.915462 0.402403i \(-0.131825\pi\)
\(48\) − 2.39952e27i − 0.436158i
\(49\) 7.50092e27 0.970240
\(50\) 0 0
\(51\) 1.36581e28 0.913025
\(52\) − 3.06082e27i − 0.148519i
\(53\) − 4.06250e28i − 1.43960i −0.694182 0.719799i \(-0.744234\pi\)
0.694182 0.719799i \(-0.255766\pi\)
\(54\) 6.55298e27 0.170585
\(55\) 0 0
\(56\) −4.66557e27 −0.0666504
\(57\) 5.22675e28i 0.557566i
\(58\) − 1.17099e28i − 0.0937545i
\(59\) 8.77955e28 0.530169 0.265084 0.964225i \(-0.414600\pi\)
0.265084 + 0.964225i \(0.414600\pi\)
\(60\) 0 0
\(61\) −1.84902e28 −0.0644168 −0.0322084 0.999481i \(-0.510254\pi\)
−0.0322084 + 0.999481i \(0.510254\pi\)
\(62\) − 2.63933e27i − 0.00703124i
\(63\) 6.38439e28i 0.130617i
\(64\) 4.90971e29 0.774616
\(65\) 0 0
\(66\) −1.75415e29 −0.166568
\(67\) − 7.69531e29i − 0.570154i −0.958505 0.285077i \(-0.907981\pi\)
0.958505 0.285077i \(-0.0920192\pi\)
\(68\) 3.06921e30i 1.78086i
\(69\) −1.20143e30 −0.547883
\(70\) 0 0
\(71\) 2.72988e30 0.776929 0.388464 0.921464i \(-0.373006\pi\)
0.388464 + 0.921464i \(0.373006\pi\)
\(72\) 1.29468e30i 0.292533i
\(73\) 4.44452e30i 0.799831i 0.916552 + 0.399915i \(0.130960\pi\)
−0.916552 + 0.399915i \(0.869040\pi\)
\(74\) 2.46323e30 0.354146
\(75\) 0 0
\(76\) −1.17454e31 −1.08754
\(77\) − 3.96617e30i − 0.295988i
\(78\) − 2.48694e29i − 0.0150005i
\(79\) 1.16758e31 0.570743 0.285371 0.958417i \(-0.407883\pi\)
0.285371 + 0.958417i \(0.407883\pi\)
\(80\) 0 0
\(81\) 1.02118e31 0.330445
\(82\) 1.16464e31i 0.307796i
\(83\) − 4.84088e31i − 1.04746i −0.851886 0.523728i \(-0.824541\pi\)
0.851886 0.523728i \(-0.175459\pi\)
\(84\) 4.60145e30 0.0817121
\(85\) 0 0
\(86\) −6.15390e30 −0.0741184
\(87\) 2.35643e31i 0.234523i
\(88\) − 8.04290e31i − 0.662901i
\(89\) −1.18101e32 −0.807831 −0.403915 0.914796i \(-0.632351\pi\)
−0.403915 + 0.914796i \(0.632351\pi\)
\(90\) 0 0
\(91\) 5.62303e30 0.0266555
\(92\) − 2.69981e32i − 1.06865i
\(93\) 5.31122e30i 0.0175884i
\(94\) 5.71176e31 0.158548
\(95\) 0 0
\(96\) 1.40891e32 0.276317
\(97\) 2.71846e31i 0.0449355i 0.999748 + 0.0224678i \(0.00715231\pi\)
−0.999748 + 0.0224678i \(0.992848\pi\)
\(98\) 1.36955e32i 0.191138i
\(99\) −1.10059e33 −1.29911
\(100\) 0 0
\(101\) −1.30031e33 −1.10343 −0.551714 0.834033i \(-0.686026\pi\)
−0.551714 + 0.834033i \(0.686026\pi\)
\(102\) 2.49376e32i 0.179867i
\(103\) 1.73233e33i 1.06369i 0.846841 + 0.531846i \(0.178502\pi\)
−0.846841 + 0.531846i \(0.821498\pi\)
\(104\) 1.14028e32 0.0596983
\(105\) 0 0
\(106\) 7.41749e32 0.283603
\(107\) − 3.72062e32i − 0.121838i −0.998143 0.0609191i \(-0.980597\pi\)
0.998143 0.0609191i \(-0.0194032\pi\)
\(108\) − 2.96329e33i − 0.832305i
\(109\) 5.92448e33 1.42927 0.714633 0.699500i \(-0.246594\pi\)
0.714633 + 0.699500i \(0.246594\pi\)
\(110\) 0 0
\(111\) −4.95684e33 −0.885882
\(112\) 9.90588e32i 0.152685i
\(113\) 2.74196e33i 0.364978i 0.983208 + 0.182489i \(0.0584154\pi\)
−0.983208 + 0.182489i \(0.941585\pi\)
\(114\) −9.54322e32 −0.109841
\(115\) 0 0
\(116\) −5.29529e33 −0.457439
\(117\) − 1.56036e33i − 0.116993i
\(118\) 1.60301e33i 0.104444i
\(119\) −5.63844e33 −0.319620
\(120\) 0 0
\(121\) 4.51469e34 1.94388
\(122\) − 3.37602e32i − 0.0126902i
\(123\) − 2.34364e34i − 0.769941i
\(124\) −1.19352e33 −0.0343062
\(125\) 0 0
\(126\) −1.16569e33 −0.0257317
\(127\) − 7.72774e34i − 1.49724i −0.663000 0.748619i \(-0.730717\pi\)
0.663000 0.748619i \(-0.269283\pi\)
\(128\) 4.19033e34i 0.713318i
\(129\) 1.23837e34 0.185404
\(130\) 0 0
\(131\) −1.77088e34 −0.205690 −0.102845 0.994697i \(-0.532795\pi\)
−0.102845 + 0.994697i \(0.532795\pi\)
\(132\) 7.93235e34i 0.812703i
\(133\) − 2.15774e34i − 0.195186i
\(134\) 1.40504e34 0.112321
\(135\) 0 0
\(136\) −1.14341e35 −0.715828
\(137\) 1.26744e35i 0.703133i 0.936163 + 0.351567i \(0.114351\pi\)
−0.936163 + 0.351567i \(0.885649\pi\)
\(138\) − 2.19362e34i − 0.107934i
\(139\) −1.04160e35 −0.454941 −0.227471 0.973785i \(-0.573046\pi\)
−0.227471 + 0.973785i \(0.573046\pi\)
\(140\) 0 0
\(141\) −1.14940e35 −0.396601
\(142\) 4.98434e34i 0.153056i
\(143\) 9.69344e34i 0.265114i
\(144\) 2.74884e35 0.670143
\(145\) 0 0
\(146\) −8.11501e34 −0.157568
\(147\) − 2.75599e35i − 0.478125i
\(148\) − 1.11389e36i − 1.72792i
\(149\) 3.09463e35 0.429572 0.214786 0.976661i \(-0.431095\pi\)
0.214786 + 0.976661i \(0.431095\pi\)
\(150\) 0 0
\(151\) 1.12720e36 1.25569 0.627844 0.778340i \(-0.283938\pi\)
0.627844 + 0.778340i \(0.283938\pi\)
\(152\) − 4.37564e35i − 0.437142i
\(153\) 1.56464e36i 1.40283i
\(154\) 7.24160e34 0.0583100
\(155\) 0 0
\(156\) −1.12461e35 −0.0731889
\(157\) − 1.58631e36i − 0.929061i −0.885557 0.464530i \(-0.846223\pi\)
0.885557 0.464530i \(-0.153777\pi\)
\(158\) 2.13182e35i 0.112437i
\(159\) −1.49264e36 −0.709421
\(160\) 0 0
\(161\) 4.95982e35 0.191796
\(162\) 1.86451e35i 0.0650979i
\(163\) − 3.56599e36i − 1.12483i −0.826857 0.562413i \(-0.809873\pi\)
0.826857 0.562413i \(-0.190127\pi\)
\(164\) 5.26657e36 1.50177
\(165\) 0 0
\(166\) 8.83869e35 0.206350
\(167\) − 6.77798e36i − 1.43311i −0.697532 0.716553i \(-0.745719\pi\)
0.697532 0.716553i \(-0.254281\pi\)
\(168\) 1.71423e35i 0.0328447i
\(169\) 5.61870e36 0.976125
\(170\) 0 0
\(171\) −5.98764e36 −0.856683
\(172\) 2.78283e36i 0.361632i
\(173\) 1.72200e36i 0.203364i 0.994817 + 0.101682i \(0.0324224\pi\)
−0.994817 + 0.101682i \(0.967578\pi\)
\(174\) −4.30247e35 −0.0462014
\(175\) 0 0
\(176\) −1.70766e37 −1.51859
\(177\) − 3.22578e36i − 0.261262i
\(178\) − 2.15635e36i − 0.159144i
\(179\) 6.69163e36 0.450253 0.225127 0.974330i \(-0.427720\pi\)
0.225127 + 0.974330i \(0.427720\pi\)
\(180\) 0 0
\(181\) 1.34636e37 0.754163 0.377082 0.926180i \(-0.376928\pi\)
0.377082 + 0.926180i \(0.376928\pi\)
\(182\) 1.02668e35i 0.00525117i
\(183\) 6.79367e35i 0.0317440i
\(184\) 1.00579e37 0.429550
\(185\) 0 0
\(186\) −9.69745e34 −0.00346493
\(187\) − 9.72001e37i − 3.17892i
\(188\) − 2.58289e37i − 0.773573i
\(189\) 5.44386e36 0.149378
\(190\) 0 0
\(191\) −6.65942e36 −0.153598 −0.0767990 0.997047i \(-0.524470\pi\)
−0.0767990 + 0.997047i \(0.524470\pi\)
\(192\) − 1.80393e37i − 0.381723i
\(193\) − 5.05982e37i − 0.982742i −0.870950 0.491371i \(-0.836496\pi\)
0.870950 0.491371i \(-0.163504\pi\)
\(194\) −4.96349e35 −0.00885235
\(195\) 0 0
\(196\) 6.19318e37 0.932586
\(197\) − 2.33016e37i − 0.322621i −0.986904 0.161311i \(-0.948428\pi\)
0.986904 0.161311i \(-0.0515721\pi\)
\(198\) − 2.00951e37i − 0.255926i
\(199\) 1.24079e38 1.45420 0.727100 0.686531i \(-0.240868\pi\)
0.727100 + 0.686531i \(0.240868\pi\)
\(200\) 0 0
\(201\) −2.82742e37 −0.280967
\(202\) − 2.37416e37i − 0.217377i
\(203\) − 9.72798e36i − 0.0820990i
\(204\) 1.12769e38 0.877591
\(205\) 0 0
\(206\) −3.16296e37 −0.209549
\(207\) − 1.37633e38i − 0.841805i
\(208\) − 2.42103e37i − 0.136759i
\(209\) 3.71970e38 1.94130
\(210\) 0 0
\(211\) −2.15498e38 −0.961133 −0.480567 0.876958i \(-0.659569\pi\)
−0.480567 + 0.876958i \(0.659569\pi\)
\(212\) − 3.35423e38i − 1.38373i
\(213\) − 1.00301e38i − 0.382863i
\(214\) 6.79328e36 0.0240023
\(215\) 0 0
\(216\) 1.10395e38 0.334550
\(217\) − 2.19262e36i − 0.00615711i
\(218\) 1.08172e38i 0.281567i
\(219\) 1.63301e38 0.394149
\(220\) 0 0
\(221\) 1.37805e38 0.286282
\(222\) − 9.05042e37i − 0.174520i
\(223\) − 1.94520e37i − 0.0348285i −0.999848 0.0174142i \(-0.994457\pi\)
0.999848 0.0174142i \(-0.00554340\pi\)
\(224\) −5.81636e37 −0.0967294
\(225\) 0 0
\(226\) −5.00639e37 −0.0719011
\(227\) − 1.37345e39i − 1.83395i −0.398947 0.916974i \(-0.630624\pi\)
0.398947 0.916974i \(-0.369376\pi\)
\(228\) 4.31550e38i 0.535927i
\(229\) 3.69693e38 0.427126 0.213563 0.976929i \(-0.431493\pi\)
0.213563 + 0.976929i \(0.431493\pi\)
\(230\) 0 0
\(231\) −1.45725e38 −0.145860
\(232\) − 1.97271e38i − 0.183871i
\(233\) − 2.02428e39i − 1.75751i −0.477275 0.878754i \(-0.658375\pi\)
0.477275 0.878754i \(-0.341625\pi\)
\(234\) 2.84898e37 0.0230477
\(235\) 0 0
\(236\) 7.24889e38 0.509593
\(237\) − 4.28994e38i − 0.281257i
\(238\) − 1.02949e38i − 0.0629655i
\(239\) 6.31944e38 0.360673 0.180336 0.983605i \(-0.442281\pi\)
0.180336 + 0.983605i \(0.442281\pi\)
\(240\) 0 0
\(241\) −1.82829e39 −0.909420 −0.454710 0.890640i \(-0.650257\pi\)
−0.454710 + 0.890640i \(0.650257\pi\)
\(242\) 8.24311e38i 0.382946i
\(243\) − 2.37036e39i − 1.02875i
\(244\) −1.52665e38 −0.0619168
\(245\) 0 0
\(246\) 4.27913e38 0.151679
\(247\) 5.27360e38i 0.174826i
\(248\) − 4.44636e37i − 0.0137896i
\(249\) −1.77864e39 −0.516177
\(250\) 0 0
\(251\) −2.17903e39 −0.554174 −0.277087 0.960845i \(-0.589369\pi\)
−0.277087 + 0.960845i \(0.589369\pi\)
\(252\) 5.27131e38i 0.125548i
\(253\) 8.55015e39i 1.90759i
\(254\) 1.41096e39 0.294958
\(255\) 0 0
\(256\) 3.45232e39 0.634091
\(257\) 1.48926e39i 0.256492i 0.991742 + 0.128246i \(0.0409348\pi\)
−0.991742 + 0.128246i \(0.959065\pi\)
\(258\) 2.26107e38i 0.0365249i
\(259\) 2.04632e39 0.310118
\(260\) 0 0
\(261\) −2.69947e39 −0.360338
\(262\) − 3.23335e38i − 0.0405211i
\(263\) 1.17319e40i 1.38069i 0.723479 + 0.690346i \(0.242542\pi\)
−0.723479 + 0.690346i \(0.757458\pi\)
\(264\) −2.95512e39 −0.326671
\(265\) 0 0
\(266\) 3.93970e38 0.0384518
\(267\) 4.33929e39i 0.398091i
\(268\) − 6.35369e39i − 0.548027i
\(269\) −2.30936e40 −1.87318 −0.936590 0.350428i \(-0.886036\pi\)
−0.936590 + 0.350428i \(0.886036\pi\)
\(270\) 0 0
\(271\) 2.16176e40 1.55172 0.775861 0.630903i \(-0.217316\pi\)
0.775861 + 0.630903i \(0.217316\pi\)
\(272\) 2.42766e40i 1.63984i
\(273\) − 2.06602e38i − 0.0131356i
\(274\) −2.31414e39 −0.138518
\(275\) 0 0
\(276\) −9.91967e39 −0.526620
\(277\) 1.86721e40i 0.933847i 0.884298 + 0.466924i \(0.154638\pi\)
−0.884298 + 0.466924i \(0.845362\pi\)
\(278\) − 1.90179e39i − 0.0896239i
\(279\) −6.08441e38 −0.0270240
\(280\) 0 0
\(281\) −6.80173e39 −0.268513 −0.134256 0.990947i \(-0.542865\pi\)
−0.134256 + 0.990947i \(0.542865\pi\)
\(282\) − 2.09862e39i − 0.0781309i
\(283\) − 3.60211e40i − 1.26497i −0.774573 0.632485i \(-0.782035\pi\)
0.774573 0.632485i \(-0.217965\pi\)
\(284\) 2.25395e40 0.746777
\(285\) 0 0
\(286\) −1.76987e39 −0.0522278
\(287\) 9.67521e39i 0.269531i
\(288\) 1.61401e40i 0.424552i
\(289\) −9.79285e40 −2.43274
\(290\) 0 0
\(291\) 9.98819e38 0.0221438
\(292\) 3.66965e40i 0.768790i
\(293\) 6.02024e40i 1.19206i 0.802962 + 0.596030i \(0.203256\pi\)
−0.802962 + 0.596030i \(0.796744\pi\)
\(294\) 5.03201e39 0.0941912
\(295\) 0 0
\(296\) 4.14968e40 0.694547
\(297\) 9.38457e40i 1.48571i
\(298\) 5.65032e39i 0.0846261i
\(299\) −1.21220e40 −0.171790
\(300\) 0 0
\(301\) −5.11233e39 −0.0649040
\(302\) 2.05809e40i 0.247372i
\(303\) 4.77760e40i 0.543759i
\(304\) −9.29029e40 −1.00142
\(305\) 0 0
\(306\) −2.85679e40 −0.276360
\(307\) − 2.05983e41i − 1.88820i −0.329664 0.944098i \(-0.606936\pi\)
0.329664 0.944098i \(-0.393064\pi\)
\(308\) − 3.27469e40i − 0.284501i
\(309\) 6.36493e40 0.524178
\(310\) 0 0
\(311\) −1.19843e41 −0.887294 −0.443647 0.896202i \(-0.646316\pi\)
−0.443647 + 0.896202i \(0.646316\pi\)
\(312\) − 4.18962e39i − 0.0294188i
\(313\) 2.13282e41i 1.42060i 0.703897 + 0.710302i \(0.251442\pi\)
−0.703897 + 0.710302i \(0.748558\pi\)
\(314\) 2.89636e40 0.183026
\(315\) 0 0
\(316\) 9.64022e40 0.548593
\(317\) 2.31538e41i 1.25068i 0.780354 + 0.625338i \(0.215039\pi\)
−0.780354 + 0.625338i \(0.784961\pi\)
\(318\) − 2.72534e40i − 0.139757i
\(319\) 1.67699e41 0.816552
\(320\) 0 0
\(321\) −1.36703e40 −0.0600407
\(322\) 9.05586e39i 0.0377840i
\(323\) − 5.28805e41i − 2.09630i
\(324\) 8.43142e40 0.317620
\(325\) 0 0
\(326\) 6.51094e40 0.221592
\(327\) − 2.17677e41i − 0.704329i
\(328\) 1.96201e41i 0.603647i
\(329\) 4.74502e40 0.138837
\(330\) 0 0
\(331\) −3.27790e40 −0.0867830 −0.0433915 0.999058i \(-0.513816\pi\)
−0.0433915 + 0.999058i \(0.513816\pi\)
\(332\) − 3.99690e41i − 1.00680i
\(333\) − 5.67844e41i − 1.36113i
\(334\) 1.23755e41 0.282324
\(335\) 0 0
\(336\) 3.63962e40 0.0752416
\(337\) − 5.05149e41i − 0.994321i −0.867659 0.497160i \(-0.834376\pi\)
0.867659 0.497160i \(-0.165624\pi\)
\(338\) 1.02589e41i 0.192298i
\(339\) 1.00745e41 0.179858
\(340\) 0 0
\(341\) 3.77981e40 0.0612383
\(342\) − 1.09325e41i − 0.168767i
\(343\) 2.31040e41i 0.339886i
\(344\) −1.03672e41 −0.145360
\(345\) 0 0
\(346\) −3.14411e40 −0.0400629
\(347\) − 5.54748e40i − 0.0673999i −0.999432 0.0337000i \(-0.989271\pi\)
0.999432 0.0337000i \(-0.0107291\pi\)
\(348\) 1.94560e41i 0.225422i
\(349\) 4.36140e41 0.481955 0.240978 0.970531i \(-0.422532\pi\)
0.240978 + 0.970531i \(0.422532\pi\)
\(350\) 0 0
\(351\) −1.33050e41 −0.133797
\(352\) − 1.00267e42i − 0.962065i
\(353\) 2.19124e41i 0.200635i 0.994955 + 0.100318i \(0.0319859\pi\)
−0.994955 + 0.100318i \(0.968014\pi\)
\(354\) 5.88978e40 0.0514689
\(355\) 0 0
\(356\) −9.75113e41 −0.776479
\(357\) 2.07168e41i 0.157506i
\(358\) 1.22179e41i 0.0887004i
\(359\) 1.83249e42 1.27053 0.635264 0.772295i \(-0.280891\pi\)
0.635264 + 0.772295i \(0.280891\pi\)
\(360\) 0 0
\(361\) 4.42885e41 0.280170
\(362\) 2.45825e41i 0.148571i
\(363\) − 1.65879e42i − 0.957925i
\(364\) 4.64269e40 0.0256211
\(365\) 0 0
\(366\) −1.24042e40 −0.00625360
\(367\) − 2.46022e42i − 1.18572i −0.805304 0.592862i \(-0.797998\pi\)
0.805304 0.592862i \(-0.202002\pi\)
\(368\) − 2.13548e42i − 0.984026i
\(369\) 2.68483e42 1.18299
\(370\) 0 0
\(371\) 6.16205e41 0.248345
\(372\) 4.38524e40i 0.0169058i
\(373\) 9.17264e41i 0.338297i 0.985591 + 0.169149i \(0.0541018\pi\)
−0.985591 + 0.169149i \(0.945898\pi\)
\(374\) 1.77472e42 0.626251
\(375\) 0 0
\(376\) 9.62231e41 0.310942
\(377\) 2.37755e41i 0.0735355i
\(378\) 9.93964e40i 0.0294277i
\(379\) −5.49346e42 −1.55704 −0.778518 0.627622i \(-0.784028\pi\)
−0.778518 + 0.627622i \(0.784028\pi\)
\(380\) 0 0
\(381\) −2.83933e42 −0.737825
\(382\) − 1.21591e41i − 0.0302590i
\(383\) 2.70725e42i 0.645281i 0.946522 + 0.322640i \(0.104570\pi\)
−0.946522 + 0.322640i \(0.895430\pi\)
\(384\) 1.53961e42 0.351517
\(385\) 0 0
\(386\) 9.23844e41 0.193601
\(387\) 1.41865e42i 0.284868i
\(388\) 2.24452e41i 0.0431916i
\(389\) −5.33079e42 −0.983157 −0.491579 0.870833i \(-0.663580\pi\)
−0.491579 + 0.870833i \(0.663580\pi\)
\(390\) 0 0
\(391\) 1.21552e43 2.05990
\(392\) 2.30721e42i 0.374859i
\(393\) 6.50657e41i 0.101362i
\(394\) 4.25451e41 0.0635568
\(395\) 0 0
\(396\) −9.08712e42 −1.24869
\(397\) − 8.08784e42i − 1.06608i −0.846091 0.533039i \(-0.821050\pi\)
0.846091 0.533039i \(-0.178950\pi\)
\(398\) 2.26550e42i 0.286479i
\(399\) −7.92799e41 −0.0961857
\(400\) 0 0
\(401\) −1.16113e43 −1.29717 −0.648587 0.761140i \(-0.724640\pi\)
−0.648587 + 0.761140i \(0.724640\pi\)
\(402\) − 5.16242e41i − 0.0553507i
\(403\) 5.35882e40i 0.00551488i
\(404\) −1.07361e43 −1.06061
\(405\) 0 0
\(406\) 1.77618e41 0.0161736
\(407\) 3.52761e43i 3.08442i
\(408\) 4.20111e42i 0.352753i
\(409\) −4.40488e42 −0.355222 −0.177611 0.984101i \(-0.556837\pi\)
−0.177611 + 0.984101i \(0.556837\pi\)
\(410\) 0 0
\(411\) 4.65683e42 0.346498
\(412\) 1.43031e43i 1.02241i
\(413\) 1.33169e42i 0.0914594i
\(414\) 2.51296e42 0.165836
\(415\) 0 0
\(416\) 1.42154e42 0.0866399
\(417\) 3.82704e42i 0.224191i
\(418\) 6.79158e42i 0.382439i
\(419\) −2.34926e43 −1.27174 −0.635871 0.771796i \(-0.719359\pi\)
−0.635871 + 0.771796i \(0.719359\pi\)
\(420\) 0 0
\(421\) −4.10149e42 −0.205252 −0.102626 0.994720i \(-0.532724\pi\)
−0.102626 + 0.994720i \(0.532724\pi\)
\(422\) − 3.93466e42i − 0.189344i
\(423\) − 1.31672e43i − 0.609365i
\(424\) 1.24959e43 0.556199
\(425\) 0 0
\(426\) 1.83135e42 0.0754245
\(427\) − 2.80461e41i − 0.0111125i
\(428\) − 3.07196e42i − 0.117110i
\(429\) 3.56157e42 0.130646
\(430\) 0 0
\(431\) 1.92923e43 0.655404 0.327702 0.944781i \(-0.393726\pi\)
0.327702 + 0.944781i \(0.393726\pi\)
\(432\) − 2.34389e43i − 0.766398i
\(433\) 1.20050e43i 0.377843i 0.981992 + 0.188922i \(0.0604992\pi\)
−0.981992 + 0.188922i \(0.939501\pi\)
\(434\) 4.00338e40 0.00121296
\(435\) 0 0
\(436\) 4.89159e43 1.37380
\(437\) 4.65160e43i 1.25794i
\(438\) 2.98162e42i 0.0776478i
\(439\) −3.55254e43 −0.890993 −0.445496 0.895284i \(-0.646973\pi\)
−0.445496 + 0.895284i \(0.646973\pi\)
\(440\) 0 0
\(441\) 3.15720e43 0.734625
\(442\) 2.51611e42i 0.0563978i
\(443\) 4.50074e43i 0.971901i 0.873986 + 0.485951i \(0.161526\pi\)
−0.873986 + 0.485951i \(0.838474\pi\)
\(444\) −4.09265e43 −0.851501
\(445\) 0 0
\(446\) 3.55163e41 0.00686124
\(447\) − 1.13703e43i − 0.211689i
\(448\) 7.44711e42i 0.133629i
\(449\) 4.48522e43 0.775745 0.387872 0.921713i \(-0.373210\pi\)
0.387872 + 0.921713i \(0.373210\pi\)
\(450\) 0 0
\(451\) −1.66789e44 −2.68074
\(452\) 2.26392e43i 0.350814i
\(453\) − 4.14156e43i − 0.618791i
\(454\) 2.50771e43 0.361290
\(455\) 0 0
\(456\) −1.60770e43 −0.215419
\(457\) − 5.99465e43i − 0.774723i −0.921928 0.387362i \(-0.873386\pi\)
0.921928 0.387362i \(-0.126614\pi\)
\(458\) 6.75002e42i 0.0841442i
\(459\) 1.33414e44 1.60433
\(460\) 0 0
\(461\) 3.64117e43 0.407546 0.203773 0.979018i \(-0.434680\pi\)
0.203773 + 0.979018i \(0.434680\pi\)
\(462\) − 2.66071e42i − 0.0287346i
\(463\) − 4.93497e43i − 0.514278i −0.966374 0.257139i \(-0.917220\pi\)
0.966374 0.257139i \(-0.0827798\pi\)
\(464\) −4.18844e43 −0.421216
\(465\) 0 0
\(466\) 3.69601e43 0.346231
\(467\) − 6.19636e43i − 0.560284i −0.959959 0.280142i \(-0.909619\pi\)
0.959959 0.280142i \(-0.0903815\pi\)
\(468\) − 1.28833e43i − 0.112452i
\(469\) 1.16723e43 0.0983572
\(470\) 0 0
\(471\) −5.82843e43 −0.457832
\(472\) 2.70051e43i 0.204834i
\(473\) − 8.81305e43i − 0.645531i
\(474\) 7.83275e42 0.0554079
\(475\) 0 0
\(476\) −4.65542e43 −0.307216
\(477\) − 1.70994e44i − 1.09000i
\(478\) 1.15383e43i 0.0710530i
\(479\) −1.65648e44 −0.985483 −0.492741 0.870176i \(-0.664005\pi\)
−0.492741 + 0.870176i \(0.664005\pi\)
\(480\) 0 0
\(481\) −5.00127e43 −0.277771
\(482\) − 3.33818e43i − 0.179157i
\(483\) − 1.82234e43i − 0.0945152i
\(484\) 3.72758e44 1.86844
\(485\) 0 0
\(486\) 4.32790e43 0.202665
\(487\) − 3.41088e44i − 1.54397i −0.635642 0.771984i \(-0.719264\pi\)
0.635642 0.771984i \(-0.280736\pi\)
\(488\) − 5.68741e42i − 0.0248879i
\(489\) −1.31022e44 −0.554303
\(490\) 0 0
\(491\) 4.35160e44 1.72109 0.860547 0.509371i \(-0.170122\pi\)
0.860547 + 0.509371i \(0.170122\pi\)
\(492\) − 1.93505e44i − 0.740060i
\(493\) − 2.38406e44i − 0.881747i
\(494\) −9.62876e42 −0.0344410
\(495\) 0 0
\(496\) −9.44044e42 −0.0315896
\(497\) 4.14072e43i 0.134028i
\(498\) − 3.24751e43i − 0.101687i
\(499\) 6.19626e44 1.87703 0.938514 0.345242i \(-0.112203\pi\)
0.938514 + 0.345242i \(0.112203\pi\)
\(500\) 0 0
\(501\) −2.49037e44 −0.706222
\(502\) − 3.97856e43i − 0.109173i
\(503\) 5.28443e44i 1.40322i 0.712560 + 0.701611i \(0.247536\pi\)
−0.712560 + 0.701611i \(0.752464\pi\)
\(504\) −1.96378e43 −0.0504648
\(505\) 0 0
\(506\) −1.56112e44 −0.375797
\(507\) − 2.06443e44i − 0.481025i
\(508\) − 6.38046e44i − 1.43913i
\(509\) 4.72823e43 0.103242 0.0516209 0.998667i \(-0.483561\pi\)
0.0516209 + 0.998667i \(0.483561\pi\)
\(510\) 0 0
\(511\) −6.74151e43 −0.137979
\(512\) 4.22980e44i 0.838235i
\(513\) 5.10556e44i 0.979731i
\(514\) −2.71916e43 −0.0505293
\(515\) 0 0
\(516\) 1.02247e44 0.178209
\(517\) 8.17985e44i 1.38087i
\(518\) 3.73626e43i 0.0610937i
\(519\) 6.32699e43 0.100216
\(520\) 0 0
\(521\) −4.38738e44 −0.652203 −0.326102 0.945335i \(-0.605735\pi\)
−0.326102 + 0.945335i \(0.605735\pi\)
\(522\) − 4.92881e43i − 0.0709869i
\(523\) − 1.40016e44i − 0.195389i −0.995216 0.0976944i \(-0.968853\pi\)
0.995216 0.0976944i \(-0.0311468\pi\)
\(524\) −1.46214e44 −0.197707
\(525\) 0 0
\(526\) −2.14205e44 −0.271998
\(527\) − 5.37351e43i − 0.0661277i
\(528\) 6.27428e44i 0.748349i
\(529\) −2.04220e44 −0.236091
\(530\) 0 0
\(531\) 3.69539e44 0.401421
\(532\) − 1.78156e44i − 0.187611i
\(533\) − 2.36465e44i − 0.241417i
\(534\) −7.92286e43 −0.0784244
\(535\) 0 0
\(536\) 2.36701e44 0.220283
\(537\) − 2.45864e44i − 0.221881i
\(538\) − 4.21653e44i − 0.369018i
\(539\) −1.96134e45 −1.66471
\(540\) 0 0
\(541\) 1.56960e45 1.25324 0.626622 0.779324i \(-0.284437\pi\)
0.626622 + 0.779324i \(0.284437\pi\)
\(542\) 3.94704e44i 0.305691i
\(543\) − 4.94681e44i − 0.371645i
\(544\) −1.42543e45 −1.03888
\(545\) 0 0
\(546\) 3.77222e42 0.00258773
\(547\) 2.11683e45i 1.40895i 0.709730 + 0.704474i \(0.248817\pi\)
−0.709730 + 0.704474i \(0.751183\pi\)
\(548\) 1.04647e45i 0.675845i
\(549\) −7.78267e43 −0.0487736
\(550\) 0 0
\(551\) 9.12345e44 0.538465
\(552\) − 3.69548e44i − 0.211678i
\(553\) 1.77100e44i 0.0984588i
\(554\) −3.40922e44 −0.183969
\(555\) 0 0
\(556\) −8.60001e44 −0.437285
\(557\) 1.56733e45i 0.773661i 0.922151 + 0.386830i \(0.126430\pi\)
−0.922151 + 0.386830i \(0.873570\pi\)
\(558\) − 1.11092e43i − 0.00532375i
\(559\) 1.24947e44 0.0581341
\(560\) 0 0
\(561\) −3.57133e45 −1.56654
\(562\) − 1.24189e44i − 0.0528974i
\(563\) 1.29413e44i 0.0535291i 0.999642 + 0.0267646i \(0.00852044\pi\)
−0.999642 + 0.0267646i \(0.991480\pi\)
\(564\) −9.49006e44 −0.381209
\(565\) 0 0
\(566\) 6.57688e44 0.249200
\(567\) 1.54894e44i 0.0570050i
\(568\) 8.39687e44i 0.300172i
\(569\) 2.93499e45 1.01919 0.509593 0.860415i \(-0.329796\pi\)
0.509593 + 0.860415i \(0.329796\pi\)
\(570\) 0 0
\(571\) −9.20277e44 −0.301594 −0.150797 0.988565i \(-0.548184\pi\)
−0.150797 + 0.988565i \(0.548184\pi\)
\(572\) 8.00345e44i 0.254825i
\(573\) 2.44681e44i 0.0756917i
\(574\) −1.76654e44 −0.0530979
\(575\) 0 0
\(576\) 2.06654e45 0.586506
\(577\) 6.78370e45i 1.87097i 0.353371 + 0.935483i \(0.385035\pi\)
−0.353371 + 0.935483i \(0.614965\pi\)
\(578\) − 1.78802e45i − 0.479251i
\(579\) −1.85908e45 −0.484286
\(580\) 0 0
\(581\) 7.34270e44 0.180697
\(582\) 1.82369e43i 0.00436236i
\(583\) 1.06226e46i 2.47003i
\(584\) −1.36709e45 −0.309020
\(585\) 0 0
\(586\) −1.09920e45 −0.234837
\(587\) − 1.92291e45i − 0.399421i −0.979855 0.199710i \(-0.936000\pi\)
0.979855 0.199710i \(-0.0640001\pi\)
\(588\) − 2.27550e45i − 0.459570i
\(589\) 2.05636e44 0.0403828
\(590\) 0 0
\(591\) −8.56148e44 −0.158985
\(592\) − 8.81055e45i − 1.59109i
\(593\) − 3.84061e45i − 0.674524i −0.941411 0.337262i \(-0.890499\pi\)
0.941411 0.337262i \(-0.109501\pi\)
\(594\) −1.71348e45 −0.292686
\(595\) 0 0
\(596\) 2.55510e45 0.412900
\(597\) − 4.55893e45i − 0.716616i
\(598\) − 2.21328e44i − 0.0338429i
\(599\) 1.48464e45 0.220840 0.110420 0.993885i \(-0.464780\pi\)
0.110420 + 0.993885i \(0.464780\pi\)
\(600\) 0 0
\(601\) 5.01345e45 0.705843 0.352921 0.935653i \(-0.385188\pi\)
0.352921 + 0.935653i \(0.385188\pi\)
\(602\) − 9.33431e43i − 0.0127862i
\(603\) − 3.23902e45i − 0.431696i
\(604\) 9.30680e45 1.20695
\(605\) 0 0
\(606\) −8.72316e44 −0.107121
\(607\) − 3.29284e45i − 0.393511i −0.980453 0.196756i \(-0.936959\pi\)
0.980453 0.196756i \(-0.0630406\pi\)
\(608\) − 5.45491e45i − 0.634422i
\(609\) −3.57426e44 −0.0404576
\(610\) 0 0
\(611\) −1.15970e45 −0.124355
\(612\) 1.29186e46i 1.34839i
\(613\) − 1.00399e46i − 1.02008i −0.860152 0.510038i \(-0.829631\pi\)
0.860152 0.510038i \(-0.170369\pi\)
\(614\) 3.76092e45 0.371977
\(615\) 0 0
\(616\) 1.21996e45 0.114357
\(617\) − 1.43698e46i − 1.31144i −0.755006 0.655718i \(-0.772366\pi\)
0.755006 0.655718i \(-0.227634\pi\)
\(618\) 1.16214e45i 0.103264i
\(619\) −1.61224e46 −1.39487 −0.697435 0.716648i \(-0.745675\pi\)
−0.697435 + 0.716648i \(0.745675\pi\)
\(620\) 0 0
\(621\) −1.17357e46 −0.962716
\(622\) − 2.18815e45i − 0.174798i
\(623\) − 1.79138e45i − 0.139359i
\(624\) −8.89535e44 −0.0673934
\(625\) 0 0
\(626\) −3.89420e45 −0.279861
\(627\) − 1.36669e46i − 0.956657i
\(628\) − 1.30975e46i − 0.893004i
\(629\) 5.01498e46 3.33068
\(630\) 0 0
\(631\) 2.47683e46 1.56103 0.780516 0.625136i \(-0.214957\pi\)
0.780516 + 0.625136i \(0.214957\pi\)
\(632\) 3.59138e45i 0.220510i
\(633\) 7.91785e45i 0.473638i
\(634\) −4.22752e45 −0.246384
\(635\) 0 0
\(636\) −1.23241e46 −0.681889
\(637\) − 2.78069e45i − 0.149918i
\(638\) 3.06192e45i 0.160862i
\(639\) 1.14903e46 0.588257
\(640\) 0 0
\(641\) −3.57575e46 −1.73864 −0.869321 0.494249i \(-0.835443\pi\)
−0.869321 + 0.494249i \(0.835443\pi\)
\(642\) − 2.49599e44i − 0.0118281i
\(643\) 1.88292e46i 0.869663i 0.900512 + 0.434831i \(0.143192\pi\)
−0.900512 + 0.434831i \(0.856808\pi\)
\(644\) 4.09511e45 0.184352
\(645\) 0 0
\(646\) 9.65515e45 0.412974
\(647\) − 2.33501e46i − 0.973573i −0.873521 0.486787i \(-0.838169\pi\)
0.873521 0.486787i \(-0.161831\pi\)
\(648\) 3.14105e45i 0.127669i
\(649\) −2.29568e46 −0.909649
\(650\) 0 0
\(651\) −8.05612e43 −0.00303417
\(652\) − 2.94428e46i − 1.08117i
\(653\) 2.35876e46i 0.844531i 0.906472 + 0.422266i \(0.138765\pi\)
−0.906472 + 0.422266i \(0.861235\pi\)
\(654\) 3.97445e45 0.138754
\(655\) 0 0
\(656\) 4.16572e46 1.38285
\(657\) 1.87074e46i 0.605598i
\(658\) 8.66366e44i 0.0273511i
\(659\) −2.73242e46 −0.841276 −0.420638 0.907228i \(-0.638194\pi\)
−0.420638 + 0.907228i \(0.638194\pi\)
\(660\) 0 0
\(661\) −2.72690e46 −0.798629 −0.399315 0.916814i \(-0.630752\pi\)
−0.399315 + 0.916814i \(0.630752\pi\)
\(662\) − 5.98494e44i − 0.0170963i
\(663\) − 5.06325e45i − 0.141077i
\(664\) 1.48901e46 0.404692
\(665\) 0 0
\(666\) 1.03679e46 0.268144
\(667\) 2.09713e46i 0.529114i
\(668\) − 5.59628e46i − 1.37749i
\(669\) −7.14706e44 −0.0171631
\(670\) 0 0
\(671\) 4.83482e45 0.110525
\(672\) 2.13705e45i 0.0476674i
\(673\) − 1.12643e46i − 0.245163i −0.992458 0.122581i \(-0.960883\pi\)
0.992458 0.122581i \(-0.0391173\pi\)
\(674\) 9.22324e45 0.195882
\(675\) 0 0
\(676\) 4.63912e46 0.938242
\(677\) 4.69606e46i 0.926874i 0.886130 + 0.463437i \(0.153384\pi\)
−0.886130 + 0.463437i \(0.846616\pi\)
\(678\) 1.83945e45i 0.0354322i
\(679\) −4.12340e44 −0.00775183
\(680\) 0 0
\(681\) −5.04634e46 −0.903753
\(682\) 6.90134e44i 0.0120640i
\(683\) − 1.23577e46i − 0.210860i −0.994427 0.105430i \(-0.966378\pi\)
0.994427 0.105430i \(-0.0336220\pi\)
\(684\) −4.94373e46 −0.823435
\(685\) 0 0
\(686\) −4.21842e45 −0.0669579
\(687\) − 1.35833e46i − 0.210484i
\(688\) 2.20114e46i 0.332996i
\(689\) −1.50602e46 −0.222441
\(690\) 0 0
\(691\) −3.45406e46 −0.486342 −0.243171 0.969983i \(-0.578188\pi\)
−0.243171 + 0.969983i \(0.578188\pi\)
\(692\) 1.42178e46i 0.195471i
\(693\) − 1.66939e46i − 0.224109i
\(694\) 1.01288e45 0.0132779
\(695\) 0 0
\(696\) −7.24815e45 −0.0906097
\(697\) 2.37113e47i 2.89478i
\(698\) 7.96324e45i 0.0949457i
\(699\) −7.43761e46 −0.866084
\(700\) 0 0
\(701\) 4.58600e46 0.509433 0.254716 0.967016i \(-0.418018\pi\)
0.254716 + 0.967016i \(0.418018\pi\)
\(702\) − 2.42928e45i − 0.0263582i
\(703\) 1.91915e47i 2.03398i
\(704\) −1.28379e47 −1.32907
\(705\) 0 0
\(706\) −4.00086e45 −0.0395254
\(707\) − 1.97233e46i − 0.190352i
\(708\) − 2.66339e46i − 0.251123i
\(709\) −1.01825e46 −0.0937978 −0.0468989 0.998900i \(-0.514934\pi\)
−0.0468989 + 0.998900i \(0.514934\pi\)
\(710\) 0 0
\(711\) 4.91445e46 0.432142
\(712\) − 3.63269e46i − 0.312111i
\(713\) 4.72678e45i 0.0396815i
\(714\) −3.78256e45 −0.0310288
\(715\) 0 0
\(716\) 5.52498e46 0.432779
\(717\) − 2.32189e46i − 0.177736i
\(718\) 3.34585e46i 0.250295i
\(719\) 2.37751e46 0.173818 0.0869092 0.996216i \(-0.472301\pi\)
0.0869092 + 0.996216i \(0.472301\pi\)
\(720\) 0 0
\(721\) −2.62762e46 −0.183498
\(722\) 8.08638e45i 0.0551938i
\(723\) 6.71752e46i 0.448154i
\(724\) 1.11163e47 0.724895
\(725\) 0 0
\(726\) 3.02869e46 0.188712
\(727\) 7.60235e46i 0.463053i 0.972829 + 0.231526i \(0.0743719\pi\)
−0.972829 + 0.231526i \(0.925628\pi\)
\(728\) 1.72959e45i 0.0102985i
\(729\) −3.03238e46 −0.176514
\(730\) 0 0
\(731\) −1.25289e47 −0.697072
\(732\) 5.60924e45i 0.0305120i
\(733\) − 9.46431e46i − 0.503354i −0.967811 0.251677i \(-0.919018\pi\)
0.967811 0.251677i \(-0.0809820\pi\)
\(734\) 4.49197e46 0.233589
\(735\) 0 0
\(736\) 1.25387e47 0.623404
\(737\) 2.01217e47i 0.978255i
\(738\) 4.90207e46i 0.233050i
\(739\) −1.36554e46 −0.0634851 −0.0317425 0.999496i \(-0.510106\pi\)
−0.0317425 + 0.999496i \(0.510106\pi\)
\(740\) 0 0
\(741\) 1.93763e46 0.0861529
\(742\) 1.12509e46i 0.0489243i
\(743\) 2.14591e47i 0.912632i 0.889818 + 0.456316i \(0.150831\pi\)
−0.889818 + 0.456316i \(0.849169\pi\)
\(744\) −1.63368e45 −0.00679538
\(745\) 0 0
\(746\) −1.67478e46 −0.0666449
\(747\) − 2.03757e47i − 0.793089i
\(748\) − 8.02539e47i − 3.05555i
\(749\) 5.64349e45 0.0210183
\(750\) 0 0
\(751\) −3.00124e47 −1.06965 −0.534825 0.844963i \(-0.679622\pi\)
−0.534825 + 0.844963i \(0.679622\pi\)
\(752\) − 2.04300e47i − 0.712317i
\(753\) 8.00619e46i 0.273092i
\(754\) −4.34103e45 −0.0144866
\(755\) 0 0
\(756\) 4.49476e46 0.143581
\(757\) − 3.75997e47i − 1.17517i −0.809161 0.587587i \(-0.800078\pi\)
0.809161 0.587587i \(-0.199922\pi\)
\(758\) − 1.00302e47i − 0.306738i
\(759\) 3.14150e47 0.940043
\(760\) 0 0
\(761\) 2.05125e47 0.587723 0.293861 0.955848i \(-0.405060\pi\)
0.293861 + 0.955848i \(0.405060\pi\)
\(762\) − 5.18417e46i − 0.145352i
\(763\) 8.98632e46i 0.246563i
\(764\) −5.49839e46 −0.147637
\(765\) 0 0
\(766\) −4.94302e46 −0.127121
\(767\) − 3.25470e46i − 0.0819195i
\(768\) − 1.26845e47i − 0.312474i
\(769\) 3.91739e47 0.944522 0.472261 0.881459i \(-0.343438\pi\)
0.472261 + 0.881459i \(0.343438\pi\)
\(770\) 0 0
\(771\) 5.47185e46 0.126397
\(772\) − 4.17767e47i − 0.944602i
\(773\) 6.18683e47i 1.36933i 0.728860 + 0.684663i \(0.240051\pi\)
−0.728860 + 0.684663i \(0.759949\pi\)
\(774\) −2.59023e46 −0.0561193
\(775\) 0 0
\(776\) −8.36174e45 −0.0173611
\(777\) − 7.51860e46i − 0.152823i
\(778\) − 9.73319e46i − 0.193683i
\(779\) −9.07396e47 −1.76778
\(780\) 0 0
\(781\) −7.13811e47 −1.33303
\(782\) 2.21935e47i 0.405802i
\(783\) 2.30179e47i 0.412095i
\(784\) 4.89864e47 0.858738
\(785\) 0 0
\(786\) −1.18800e46 −0.0199684
\(787\) − 2.50651e45i − 0.00412559i −0.999998 0.00206280i \(-0.999343\pi\)
0.999998 0.00206280i \(-0.000656609\pi\)
\(788\) − 1.92391e47i − 0.310101i
\(789\) 4.31052e47 0.680392
\(790\) 0 0
\(791\) −4.15904e46 −0.0629624
\(792\) − 3.38532e47i − 0.501920i
\(793\) 6.85457e45i 0.00995342i
\(794\) 1.47671e47 0.210019
\(795\) 0 0
\(796\) 1.02447e48 1.39776
\(797\) − 7.37335e47i − 0.985378i −0.870206 0.492689i \(-0.836014\pi\)
0.870206 0.492689i \(-0.163986\pi\)
\(798\) − 1.44753e46i − 0.0189487i
\(799\) 1.16288e48 1.49112
\(800\) 0 0
\(801\) −4.97099e47 −0.611655
\(802\) − 2.12004e47i − 0.255545i
\(803\) − 1.16216e48i − 1.37233i
\(804\) −2.33447e47 −0.270062
\(805\) 0 0
\(806\) −9.78438e44 −0.00108644
\(807\) 8.48506e47i 0.923085i
\(808\) − 3.99963e47i − 0.426317i
\(809\) 9.92771e47 1.03681 0.518404 0.855136i \(-0.326526\pi\)
0.518404 + 0.855136i \(0.326526\pi\)
\(810\) 0 0
\(811\) 1.07185e47 0.107470 0.0537351 0.998555i \(-0.482887\pi\)
0.0537351 + 0.998555i \(0.482887\pi\)
\(812\) − 8.03197e46i − 0.0789128i
\(813\) − 7.94274e47i − 0.764674i
\(814\) −6.44087e47 −0.607634
\(815\) 0 0
\(816\) 8.91973e47 0.808098
\(817\) − 4.79463e47i − 0.425688i
\(818\) − 8.04262e46i − 0.0699791i
\(819\) 2.36678e46 0.0201824
\(820\) 0 0
\(821\) −1.20939e48 −0.990610 −0.495305 0.868719i \(-0.664944\pi\)
−0.495305 + 0.868719i \(0.664944\pi\)
\(822\) 8.50264e46i 0.0682604i
\(823\) 2.45743e48i 1.93367i 0.255396 + 0.966837i \(0.417794\pi\)
−0.255396 + 0.966837i \(0.582206\pi\)
\(824\) −5.32848e47 −0.410965
\(825\) 0 0
\(826\) −2.43146e46 −0.0180176
\(827\) 1.80057e48i 1.30789i 0.756544 + 0.653943i \(0.226886\pi\)
−0.756544 + 0.653943i \(0.773114\pi\)
\(828\) − 1.13637e48i − 0.809135i
\(829\) −6.57735e47 −0.459094 −0.229547 0.973298i \(-0.573724\pi\)
−0.229547 + 0.973298i \(0.573724\pi\)
\(830\) 0 0
\(831\) 6.86049e47 0.460191
\(832\) − 1.82010e47i − 0.119691i
\(833\) 2.78832e48i 1.79763i
\(834\) −6.98757e46 −0.0441658
\(835\) 0 0
\(836\) 3.07119e48 1.86596
\(837\) 5.18807e46i 0.0309055i
\(838\) − 4.28937e47i − 0.250534i
\(839\) −6.26140e47 −0.358591 −0.179295 0.983795i \(-0.557382\pi\)
−0.179295 + 0.983795i \(0.557382\pi\)
\(840\) 0 0
\(841\) −1.40475e48 −0.773511
\(842\) − 7.48868e46i − 0.0404348i
\(843\) 2.49910e47i 0.132321i
\(844\) −1.77928e48 −0.923832
\(845\) 0 0
\(846\) 2.40413e47 0.120046
\(847\) 6.84793e47i 0.335338i
\(848\) − 2.65311e48i − 1.27416i
\(849\) −1.32349e48 −0.623365
\(850\) 0 0
\(851\) −4.41140e48 −1.99866
\(852\) − 8.28145e47i − 0.368005i
\(853\) 3.16331e48i 1.37874i 0.724410 + 0.689370i \(0.242112\pi\)
−0.724410 + 0.689370i \(0.757888\pi\)
\(854\) 5.12079e45 0.00218918
\(855\) 0 0
\(856\) 1.14443e47 0.0470730
\(857\) 3.13666e48i 1.26556i 0.774330 + 0.632782i \(0.218087\pi\)
−0.774330 + 0.632782i \(0.781913\pi\)
\(858\) 6.50286e46i 0.0257374i
\(859\) −2.20270e48 −0.855202 −0.427601 0.903968i \(-0.640641\pi\)
−0.427601 + 0.903968i \(0.640641\pi\)
\(860\) 0 0
\(861\) 3.55487e47 0.132822
\(862\) 3.52247e47i 0.129115i
\(863\) 5.00806e47i 0.180091i 0.995938 + 0.0900453i \(0.0287012\pi\)
−0.995938 + 0.0900453i \(0.971299\pi\)
\(864\) 1.37624e48 0.485532
\(865\) 0 0
\(866\) −2.19192e47 −0.0744356
\(867\) 3.59809e48i 1.19883i
\(868\) − 1.81035e46i − 0.00591816i
\(869\) −3.05300e48 −0.979266
\(870\) 0 0
\(871\) −2.85276e47 −0.0880979
\(872\) 1.82232e48i 0.552207i
\(873\) 1.14422e47i 0.0340233i
\(874\) −8.49310e47 −0.247815
\(875\) 0 0
\(876\) 1.34830e48 0.378852
\(877\) 1.77120e48i 0.488400i 0.969725 + 0.244200i \(0.0785253\pi\)
−0.969725 + 0.244200i \(0.921475\pi\)
\(878\) − 6.48638e47i − 0.175527i
\(879\) 2.21196e48 0.587436
\(880\) 0 0
\(881\) 1.59935e48 0.409111 0.204555 0.978855i \(-0.434425\pi\)
0.204555 + 0.978855i \(0.434425\pi\)
\(882\) 5.76455e47i 0.144722i
\(883\) − 2.38704e48i − 0.588177i −0.955778 0.294089i \(-0.904984\pi\)
0.955778 0.294089i \(-0.0950161\pi\)
\(884\) 1.13780e48 0.275171
\(885\) 0 0
\(886\) −8.21764e47 −0.191466
\(887\) − 5.13361e48i − 1.17404i −0.809572 0.587020i \(-0.800301\pi\)
0.809572 0.587020i \(-0.199699\pi\)
\(888\) − 1.52468e48i − 0.342266i
\(889\) 1.17215e48 0.258289
\(890\) 0 0
\(891\) −2.67018e48 −0.566968
\(892\) − 1.60607e47i − 0.0334768i
\(893\) 4.45015e48i 0.910595i
\(894\) 2.07604e47 0.0417030
\(895\) 0 0
\(896\) −6.35594e47 −0.123054
\(897\) 4.45386e47i 0.0846566i
\(898\) 8.18931e47i 0.152823i
\(899\) 9.27089e46 0.0169858
\(900\) 0 0
\(901\) 1.51015e49 2.66724
\(902\) − 3.04531e48i − 0.528109i
\(903\) 1.87837e47i 0.0319841i
\(904\) −8.43402e47 −0.141012
\(905\) 0 0
\(906\) 7.56185e47 0.121903
\(907\) − 5.18589e48i − 0.820924i −0.911878 0.410462i \(-0.865368\pi\)
0.911878 0.410462i \(-0.134632\pi\)
\(908\) − 1.13400e49i − 1.76277i
\(909\) −5.47311e48 −0.835469
\(910\) 0 0
\(911\) −6.63747e48 −0.977123 −0.488562 0.872529i \(-0.662478\pi\)
−0.488562 + 0.872529i \(0.662478\pi\)
\(912\) 3.41344e48i 0.493489i
\(913\) 1.26580e49i 1.79720i
\(914\) 1.09453e48 0.152621
\(915\) 0 0
\(916\) 3.05239e48 0.410549
\(917\) − 2.68609e47i − 0.0354835i
\(918\) 2.43594e48i 0.316054i
\(919\) 7.10984e48 0.906054 0.453027 0.891497i \(-0.350344\pi\)
0.453027 + 0.891497i \(0.350344\pi\)
\(920\) 0 0
\(921\) −7.56822e48 −0.930486
\(922\) 6.64821e47i 0.0802869i
\(923\) − 1.01200e48i − 0.120048i
\(924\) −1.20319e48 −0.140199
\(925\) 0 0
\(926\) 9.01048e47 0.101313
\(927\) 7.29151e48i 0.805382i
\(928\) − 2.45929e48i − 0.266851i
\(929\) 1.17316e49 1.25054 0.625272 0.780407i \(-0.284988\pi\)
0.625272 + 0.780407i \(0.284988\pi\)
\(930\) 0 0
\(931\) −1.06705e49 −1.09777
\(932\) − 1.67136e49i − 1.68930i
\(933\) 4.40329e48i 0.437250i
\(934\) 1.13136e48 0.110376
\(935\) 0 0
\(936\) 4.79954e47 0.0452010
\(937\) − 7.60807e48i − 0.703998i −0.936000 0.351999i \(-0.885502\pi\)
0.936000 0.351999i \(-0.114498\pi\)
\(938\) 2.13119e47i 0.0193765i
\(939\) 7.83643e48 0.700061
\(940\) 0 0
\(941\) −5.90432e48 −0.509262 −0.254631 0.967038i \(-0.581954\pi\)
−0.254631 + 0.967038i \(0.581954\pi\)
\(942\) − 1.06418e48i − 0.0901935i
\(943\) − 2.08575e49i − 1.73708i
\(944\) 5.73368e48 0.469241
\(945\) 0 0
\(946\) 1.60912e48 0.127170
\(947\) 7.50195e47i 0.0582639i 0.999576 + 0.0291319i \(0.00927430\pi\)
−0.999576 + 0.0291319i \(0.990726\pi\)
\(948\) − 3.54201e48i − 0.270341i
\(949\) 1.64765e48 0.123587
\(950\) 0 0
\(951\) 8.50718e48 0.616321
\(952\) − 1.73433e48i − 0.123487i
\(953\) − 1.06031e49i − 0.741993i −0.928634 0.370997i \(-0.879016\pi\)
0.928634 0.370997i \(-0.120984\pi\)
\(954\) 3.12208e48 0.214732
\(955\) 0 0
\(956\) 5.21769e48 0.346675
\(957\) − 6.16159e48i − 0.402389i
\(958\) − 3.02447e48i − 0.194141i
\(959\) −1.92247e48 −0.121297
\(960\) 0 0
\(961\) −1.63826e49 −0.998726
\(962\) − 9.13154e47i − 0.0547212i
\(963\) − 1.56604e48i − 0.0922507i
\(964\) −1.50954e49 −0.874126
\(965\) 0 0
\(966\) 3.32731e47 0.0186196
\(967\) 3.31961e48i 0.182621i 0.995822 + 0.0913104i \(0.0291056\pi\)
−0.995822 + 0.0913104i \(0.970894\pi\)
\(968\) 1.38868e49i 0.751030i
\(969\) −1.94294e49 −1.03304
\(970\) 0 0
\(971\) 1.67736e49 0.862003 0.431001 0.902351i \(-0.358160\pi\)
0.431001 + 0.902351i \(0.358160\pi\)
\(972\) − 1.95710e49i − 0.988826i
\(973\) − 1.57991e48i − 0.0784819i
\(974\) 6.22773e48 0.304164
\(975\) 0 0
\(976\) −1.20754e48 −0.0570139
\(977\) − 2.50785e49i − 1.16424i −0.813104 0.582118i \(-0.802224\pi\)
0.813104 0.582118i \(-0.197776\pi\)
\(978\) − 2.39225e48i − 0.109198i
\(979\) 3.08812e49 1.38605
\(980\) 0 0
\(981\) 2.49366e49 1.08218
\(982\) 7.94535e48i 0.339058i
\(983\) 1.24304e49i 0.521616i 0.965391 + 0.260808i \(0.0839890\pi\)
−0.965391 + 0.260808i \(0.916011\pi\)
\(984\) 7.20883e48 0.297472
\(985\) 0 0
\(986\) 4.35293e48 0.173705
\(987\) − 1.74342e48i − 0.0684177i
\(988\) 4.35418e48i 0.168042i
\(989\) 1.10210e49 0.418295
\(990\) 0 0
\(991\) −2.48644e49 −0.912774 −0.456387 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(992\) − 5.54307e47i − 0.0200128i
\(993\) 1.20437e48i 0.0427658i
\(994\) −7.56031e47 −0.0264037
\(995\) 0 0
\(996\) −1.46854e49 −0.496144
\(997\) 2.01041e49i 0.668058i 0.942563 + 0.334029i \(0.108408\pi\)
−0.942563 + 0.334029i \(0.891592\pi\)
\(998\) 1.13134e49i 0.369777i
\(999\) −4.84191e49 −1.55663
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 25.34.b.c.24.7 12
5.2 odd 4 5.34.a.b.1.3 6
5.3 odd 4 25.34.a.c.1.4 6
5.4 even 2 inner 25.34.b.c.24.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.a.b.1.3 6 5.2 odd 4
25.34.a.c.1.4 6 5.3 odd 4
25.34.b.c.24.6 12 5.4 even 2 inner
25.34.b.c.24.7 12 1.1 even 1 trivial