Properties

Label 25.34.b.c.24.11
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 24.11
Root \(66460.8i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.34.b.c.24.2

$q$-expansion

\(f(q)\) \(=\) \(q+132922. i q^{2} +1.17526e8i q^{3} -9.07819e9 q^{4} -1.56217e13 q^{6} -1.34853e13i q^{7} -6.49000e13i q^{8} -8.25322e15 q^{9} +O(q^{10})\) \(q+132922. i q^{2} +1.17526e8i q^{3} -9.07819e9 q^{4} -1.56217e13 q^{6} -1.34853e13i q^{7} -6.49000e13i q^{8} -8.25322e15 q^{9} -7.96470e16 q^{11} -1.06692e18i q^{12} +1.16823e18i q^{13} +1.79249e18 q^{14} -6.93545e19 q^{16} +3.27833e20i q^{17} -1.09703e21i q^{18} -2.21358e21 q^{19} +1.58487e21 q^{21} -1.05868e22i q^{22} -3.93052e22i q^{23} +7.62742e21 q^{24} -1.55283e23 q^{26} -3.16633e23i q^{27} +1.22422e23i q^{28} -1.59428e24 q^{29} +2.07133e24 q^{31} -9.77619e24i q^{32} -9.36057e24i q^{33} -4.35761e25 q^{34} +7.49243e25 q^{36} -4.76387e25i q^{37} -2.94233e26i q^{38} -1.37297e26 q^{39} -6.39281e26 q^{41} +2.10663e26i q^{42} +2.16469e26i q^{43} +7.23051e26 q^{44} +5.22451e27 q^{46} +7.45743e27i q^{47} -8.15093e27i q^{48} +7.54914e27 q^{49} -3.85288e28 q^{51} -1.06055e28i q^{52} -1.70592e28i q^{53} +4.20873e28 q^{54} -8.75196e26 q^{56} -2.60153e29i q^{57} -2.11913e29i q^{58} +5.91397e28 q^{59} +2.18117e29 q^{61} +2.75325e29i q^{62} +1.11297e29i q^{63} +7.03715e29 q^{64} +1.24422e30 q^{66} +2.21631e30i q^{67} -2.97613e30i q^{68} +4.61937e30 q^{69} +2.87498e30 q^{71} +5.35634e29i q^{72} +3.41825e30i q^{73} +6.33220e30 q^{74} +2.00953e31 q^{76} +1.07406e30i q^{77} -1.82498e31i q^{78} -5.87385e30 q^{79} -8.66769e30 q^{81} -8.49743e31i q^{82} +2.11211e31i q^{83} -1.43877e31 q^{84} -2.87734e31 q^{86} -1.87368e32i q^{87} +5.16909e30i q^{88} +2.67672e32 q^{89} +1.57540e31 q^{91} +3.56821e32i q^{92} +2.43435e32i q^{93} -9.91253e32 q^{94} +1.14895e33 q^{96} +1.07066e31i q^{97} +1.00344e33i q^{98} +6.57344e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + O(q^{10}) \) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + 301521793111780384q^{11} + 14172220017979782912q^{14} + \)\(12\!\cdots\!32\)\(q^{16} - \)\(34\!\cdots\!00\)\(q^{19} - \)\(21\!\cdots\!56\)\(q^{21} - \)\(42\!\cdots\!00\)\(q^{24} - \)\(72\!\cdots\!76\)\(q^{26} - \)\(26\!\cdots\!00\)\(q^{29} + \)\(21\!\cdots\!04\)\(q^{31} - \)\(17\!\cdots\!48\)\(q^{34} + \)\(33\!\cdots\!92\)\(q^{36} - \)\(11\!\cdots\!12\)\(q^{39} + \)\(89\!\cdots\!64\)\(q^{41} - \)\(14\!\cdots\!28\)\(q^{44} + \)\(69\!\cdots\!44\)\(q^{46} + \)\(14\!\cdots\!16\)\(q^{49} - \)\(10\!\cdots\!76\)\(q^{51} + \)\(36\!\cdots\!00\)\(q^{54} - \)\(78\!\cdots\!00\)\(q^{56} + \)\(35\!\cdots\!00\)\(q^{59} - \)\(56\!\cdots\!16\)\(q^{61} - \)\(38\!\cdots\!44\)\(q^{64} + \)\(10\!\cdots\!28\)\(q^{66} - \)\(63\!\cdots\!72\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} - \)\(40\!\cdots\!68\)\(q^{74} + \)\(58\!\cdots\!00\)\(q^{76} + \)\(44\!\cdots\!00\)\(q^{79} - \)\(26\!\cdots\!48\)\(q^{81} + \)\(47\!\cdots\!52\)\(q^{84} - \)\(25\!\cdots\!16\)\(q^{86} + \)\(67\!\cdots\!00\)\(q^{89} + \)\(38\!\cdots\!64\)\(q^{91} - \)\(29\!\cdots\!28\)\(q^{94} + \)\(63\!\cdots\!44\)\(q^{96} - \)\(33\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 132922.i 1.43417i 0.696986 + 0.717085i \(0.254524\pi\)
−0.696986 + 0.717085i \(0.745476\pi\)
\(3\) 1.17526e8i 1.57627i 0.615499 + 0.788137i \(0.288954\pi\)
−0.615499 + 0.788137i \(0.711046\pi\)
\(4\) −9.07819e9 −1.05684
\(5\) 0 0
\(6\) −1.56217e13 −2.26064
\(7\) − 1.34853e13i − 0.153371i −0.997055 0.0766854i \(-0.975566\pi\)
0.997055 0.0766854i \(-0.0244337\pi\)
\(8\) − 6.49000e13i − 0.0815192i
\(9\) −8.25322e15 −1.48464
\(10\) 0 0
\(11\) −7.96470e16 −0.522625 −0.261313 0.965254i \(-0.584155\pi\)
−0.261313 + 0.965254i \(0.584155\pi\)
\(12\) − 1.06692e18i − 1.66587i
\(13\) 1.16823e18i 0.486928i 0.969910 + 0.243464i \(0.0782838\pi\)
−0.969910 + 0.243464i \(0.921716\pi\)
\(14\) 1.79249e18 0.219960
\(15\) 0 0
\(16\) −6.93545e19 −0.939928
\(17\) 3.27833e20i 1.63398i 0.576654 + 0.816988i \(0.304358\pi\)
−0.576654 + 0.816988i \(0.695642\pi\)
\(18\) − 1.09703e21i − 2.12923i
\(19\) −2.21358e21 −1.76060 −0.880301 0.474416i \(-0.842659\pi\)
−0.880301 + 0.474416i \(0.842659\pi\)
\(20\) 0 0
\(21\) 1.58487e21 0.241755
\(22\) − 1.05868e22i − 0.749533i
\(23\) − 3.93052e22i − 1.33641i −0.743975 0.668207i \(-0.767062\pi\)
0.743975 0.668207i \(-0.232938\pi\)
\(24\) 7.62742e21 0.128497
\(25\) 0 0
\(26\) −1.55283e23 −0.698337
\(27\) − 3.16633e23i − 0.763930i
\(28\) 1.22422e23i 0.162089i
\(29\) −1.59428e24 −1.18303 −0.591516 0.806293i \(-0.701470\pi\)
−0.591516 + 0.806293i \(0.701470\pi\)
\(30\) 0 0
\(31\) 2.07133e24 0.511425 0.255712 0.966753i \(-0.417690\pi\)
0.255712 + 0.966753i \(0.417690\pi\)
\(32\) − 9.77619e24i − 1.42954i
\(33\) − 9.36057e24i − 0.823801i
\(34\) −4.35761e25 −2.34340
\(35\) 0 0
\(36\) 7.49243e25 1.56903
\(37\) − 4.76387e25i − 0.634792i −0.948293 0.317396i \(-0.897192\pi\)
0.948293 0.317396i \(-0.102808\pi\)
\(38\) − 2.94233e26i − 2.52500i
\(39\) −1.37297e26 −0.767532
\(40\) 0 0
\(41\) −6.39281e26 −1.56588 −0.782940 0.622097i \(-0.786281\pi\)
−0.782940 + 0.622097i \(0.786281\pi\)
\(42\) 2.10663e26i 0.346717i
\(43\) 2.16469e26i 0.241639i 0.992674 + 0.120819i \(0.0385522\pi\)
−0.992674 + 0.120819i \(0.961448\pi\)
\(44\) 7.23051e26 0.552331
\(45\) 0 0
\(46\) 5.22451e27 1.91664
\(47\) 7.45743e27i 1.91856i 0.282462 + 0.959278i \(0.408849\pi\)
−0.282462 + 0.959278i \(0.591151\pi\)
\(48\) − 8.15093e27i − 1.48159i
\(49\) 7.54914e27 0.976477
\(50\) 0 0
\(51\) −3.85288e28 −2.57560
\(52\) − 1.06055e28i − 0.514605i
\(53\) − 1.70592e28i − 0.604513i −0.953227 0.302257i \(-0.902260\pi\)
0.953227 0.302257i \(-0.0977399\pi\)
\(54\) 4.20873e28 1.09560
\(55\) 0 0
\(56\) −8.75196e26 −0.0125027
\(57\) − 2.60153e29i − 2.77519i
\(58\) − 2.11913e29i − 1.69667i
\(59\) 5.91397e28 0.357126 0.178563 0.983928i \(-0.442855\pi\)
0.178563 + 0.983928i \(0.442855\pi\)
\(60\) 0 0
\(61\) 2.18117e29 0.759884 0.379942 0.925010i \(-0.375944\pi\)
0.379942 + 0.925010i \(0.375944\pi\)
\(62\) 2.75325e29i 0.733470i
\(63\) 1.11297e29i 0.227701i
\(64\) 7.03715e29 1.11027
\(65\) 0 0
\(66\) 1.24422e30 1.18147
\(67\) 2.21631e30i 1.64209i 0.570865 + 0.821044i \(0.306608\pi\)
−0.570865 + 0.821044i \(0.693392\pi\)
\(68\) − 2.97613e30i − 1.72685i
\(69\) 4.61937e30 2.10656
\(70\) 0 0
\(71\) 2.87498e30 0.818223 0.409112 0.912484i \(-0.365839\pi\)
0.409112 + 0.912484i \(0.365839\pi\)
\(72\) 5.35634e29i 0.121027i
\(73\) 3.41825e30i 0.615143i 0.951525 + 0.307571i \(0.0995163\pi\)
−0.951525 + 0.307571i \(0.900484\pi\)
\(74\) 6.33220e30 0.910398
\(75\) 0 0
\(76\) 2.00953e31 1.86068
\(77\) 1.07406e30i 0.0801555i
\(78\) − 1.82498e31i − 1.10077i
\(79\) −5.87385e30 −0.287128 −0.143564 0.989641i \(-0.545856\pi\)
−0.143564 + 0.989641i \(0.545856\pi\)
\(80\) 0 0
\(81\) −8.66769e30 −0.280479
\(82\) − 8.49743e31i − 2.24574i
\(83\) 2.11211e31i 0.457013i 0.973542 + 0.228507i \(0.0733843\pi\)
−0.973542 + 0.228507i \(0.926616\pi\)
\(84\) −1.43877e31 −0.255496
\(85\) 0 0
\(86\) −2.87734e31 −0.346551
\(87\) − 1.87368e32i − 1.86478i
\(88\) 5.16909e30i 0.0426040i
\(89\) 2.67672e32 1.83092 0.915458 0.402413i \(-0.131828\pi\)
0.915458 + 0.402413i \(0.131828\pi\)
\(90\) 0 0
\(91\) 1.57540e31 0.0746806
\(92\) 3.56821e32i 1.41238i
\(93\) 2.43435e32i 0.806146i
\(94\) −9.91253e32 −2.75153
\(95\) 0 0
\(96\) 1.14895e33 2.25334
\(97\) 1.07066e31i 0.0176977i 0.999961 + 0.00884885i \(0.00281671\pi\)
−0.999961 + 0.00884885i \(0.997183\pi\)
\(98\) 1.00344e33i 1.40043i
\(99\) 6.57344e32 0.775911
\(100\) 0 0
\(101\) 1.55093e33 1.31610 0.658051 0.752974i \(-0.271381\pi\)
0.658051 + 0.752974i \(0.271381\pi\)
\(102\) − 5.12131e33i − 3.69384i
\(103\) − 6.99665e32i − 0.429612i −0.976657 0.214806i \(-0.931088\pi\)
0.976657 0.214806i \(-0.0689119\pi\)
\(104\) 7.58184e31 0.0396940
\(105\) 0 0
\(106\) 2.26753e33 0.866974
\(107\) − 6.35208e32i − 0.208010i −0.994577 0.104005i \(-0.966834\pi\)
0.994577 0.104005i \(-0.0331658\pi\)
\(108\) 2.87445e33i 0.807352i
\(109\) −2.09396e33 −0.505163 −0.252581 0.967576i \(-0.581280\pi\)
−0.252581 + 0.967576i \(0.581280\pi\)
\(110\) 0 0
\(111\) 5.59876e33 1.00061
\(112\) 9.35266e32i 0.144158i
\(113\) − 7.28662e33i − 0.969910i −0.874539 0.484955i \(-0.838836\pi\)
0.874539 0.484955i \(-0.161164\pi\)
\(114\) 3.45799e34 3.98009
\(115\) 0 0
\(116\) 1.44731e34 1.25028
\(117\) − 9.64169e33i − 0.722914i
\(118\) 7.86094e33i 0.512179i
\(119\) 4.42093e33 0.250604
\(120\) 0 0
\(121\) −1.68815e34 −0.726863
\(122\) 2.89924e34i 1.08980i
\(123\) − 7.51320e34i − 2.46826i
\(124\) −1.88040e34 −0.540495
\(125\) 0 0
\(126\) −1.47938e34 −0.326562
\(127\) − 4.47086e34i − 0.866222i −0.901341 0.433111i \(-0.857416\pi\)
0.901341 0.433111i \(-0.142584\pi\)
\(128\) 9.56208e33i 0.162775i
\(129\) −2.54407e34 −0.380889
\(130\) 0 0
\(131\) −2.63964e34 −0.306597 −0.153298 0.988180i \(-0.548990\pi\)
−0.153298 + 0.988180i \(0.548990\pi\)
\(132\) 8.49770e34i 0.870626i
\(133\) 2.98508e34i 0.270025i
\(134\) −2.94595e35 −2.35503
\(135\) 0 0
\(136\) 2.12764e34 0.133201
\(137\) 1.03805e35i 0.575874i 0.957649 + 0.287937i \(0.0929694\pi\)
−0.957649 + 0.287937i \(0.907031\pi\)
\(138\) 6.14014e35i 3.02116i
\(139\) 1.77157e34 0.0773774 0.0386887 0.999251i \(-0.487682\pi\)
0.0386887 + 0.999251i \(0.487682\pi\)
\(140\) 0 0
\(141\) −8.76440e35 −3.02417
\(142\) 3.82147e35i 1.17347i
\(143\) − 9.30464e34i − 0.254481i
\(144\) 5.72398e35 1.39546
\(145\) 0 0
\(146\) −4.54358e35 −0.882219
\(147\) 8.87218e35i 1.53920i
\(148\) 4.32473e35i 0.670874i
\(149\) −9.39618e35 −1.30430 −0.652151 0.758089i \(-0.726133\pi\)
−0.652151 + 0.758089i \(0.726133\pi\)
\(150\) 0 0
\(151\) 4.46650e35 0.497563 0.248781 0.968560i \(-0.419970\pi\)
0.248781 + 0.968560i \(0.419970\pi\)
\(152\) 1.43661e35i 0.143523i
\(153\) − 2.70568e36i − 2.42587i
\(154\) −1.42766e35 −0.114956
\(155\) 0 0
\(156\) 1.24641e36 0.811159
\(157\) − 7.30629e35i − 0.427910i −0.976843 0.213955i \(-0.931365\pi\)
0.976843 0.213955i \(-0.0686346\pi\)
\(158\) − 7.80761e35i − 0.411791i
\(159\) 2.00489e36 0.952879
\(160\) 0 0
\(161\) −5.30043e35 −0.204967
\(162\) − 1.15212e36i − 0.402254i
\(163\) 2.77089e35i 0.0874026i 0.999045 + 0.0437013i \(0.0139150\pi\)
−0.999045 + 0.0437013i \(0.986085\pi\)
\(164\) 5.80352e36 1.65489
\(165\) 0 0
\(166\) −2.80745e36 −0.655434
\(167\) 5.22919e36i 1.10564i 0.833302 + 0.552818i \(0.186448\pi\)
−0.833302 + 0.552818i \(0.813552\pi\)
\(168\) − 1.02858e35i − 0.0197077i
\(169\) 4.39136e36 0.762901
\(170\) 0 0
\(171\) 1.82692e37 2.61386
\(172\) − 1.96515e36i − 0.255374i
\(173\) 5.38814e36i 0.636324i 0.948036 + 0.318162i \(0.103066\pi\)
−0.948036 + 0.318162i \(0.896934\pi\)
\(174\) 2.49053e37 2.67441
\(175\) 0 0
\(176\) 5.52388e36 0.491230
\(177\) 6.95044e36i 0.562928i
\(178\) 3.55794e37i 2.62584i
\(179\) −2.52793e37 −1.70094 −0.850472 0.526020i \(-0.823684\pi\)
−0.850472 + 0.526020i \(0.823684\pi\)
\(180\) 0 0
\(181\) 1.65835e36 0.0928924 0.0464462 0.998921i \(-0.485210\pi\)
0.0464462 + 0.998921i \(0.485210\pi\)
\(182\) 2.09404e36i 0.107105i
\(183\) 2.56343e37i 1.19779i
\(184\) −2.55091e36 −0.108943
\(185\) 0 0
\(186\) −3.23577e37 −1.15615
\(187\) − 2.61109e37i − 0.853957i
\(188\) − 6.77000e37i − 2.02761i
\(189\) −4.26989e36 −0.117165
\(190\) 0 0
\(191\) 6.04480e37 1.39422 0.697110 0.716964i \(-0.254469\pi\)
0.697110 + 0.716964i \(0.254469\pi\)
\(192\) 8.27046e37i 1.75009i
\(193\) − 6.49082e36i − 0.126068i −0.998011 0.0630339i \(-0.979922\pi\)
0.998011 0.0630339i \(-0.0200776\pi\)
\(194\) −1.42313e36 −0.0253815
\(195\) 0 0
\(196\) −6.85326e37 −1.03198
\(197\) 9.24040e37i 1.27938i 0.768635 + 0.639688i \(0.220936\pi\)
−0.768635 + 0.639688i \(0.779064\pi\)
\(198\) 8.73752e37i 1.11279i
\(199\) 1.09203e38 1.27985 0.639925 0.768437i \(-0.278965\pi\)
0.639925 + 0.768437i \(0.278965\pi\)
\(200\) 0 0
\(201\) −2.60473e38 −2.58838
\(202\) 2.06152e38i 1.88751i
\(203\) 2.14993e37i 0.181443i
\(204\) 3.49772e38 2.72200
\(205\) 0 0
\(206\) 9.30006e37 0.616136
\(207\) 3.24395e38i 1.98410i
\(208\) − 8.10223e37i − 0.457677i
\(209\) 1.76305e38 0.920134
\(210\) 0 0
\(211\) −6.11672e37 −0.272808 −0.136404 0.990653i \(-0.543555\pi\)
−0.136404 + 0.990653i \(0.543555\pi\)
\(212\) 1.54866e38i 0.638874i
\(213\) 3.37884e38i 1.28974i
\(214\) 8.44328e37 0.298321
\(215\) 0 0
\(216\) −2.05495e37 −0.0622750
\(217\) − 2.79325e37i − 0.0784377i
\(218\) − 2.78332e38i − 0.724489i
\(219\) −4.01731e38 −0.969634
\(220\) 0 0
\(221\) −3.82986e38 −0.795629
\(222\) 7.44196e38i 1.43504i
\(223\) − 9.45280e38i − 1.69251i −0.532781 0.846253i \(-0.678853\pi\)
0.532781 0.846253i \(-0.321147\pi\)
\(224\) −1.31835e38 −0.219249
\(225\) 0 0
\(226\) 9.68548e38 1.39102
\(227\) 4.90469e38i 0.654914i 0.944866 + 0.327457i \(0.106192\pi\)
−0.944866 + 0.327457i \(0.893808\pi\)
\(228\) 2.36172e39i 2.93294i
\(229\) 4.57241e38 0.528275 0.264137 0.964485i \(-0.414913\pi\)
0.264137 + 0.964485i \(0.414913\pi\)
\(230\) 0 0
\(231\) −1.26230e38 −0.126347
\(232\) 1.03468e38i 0.0964398i
\(233\) 8.69571e38i 0.754975i 0.926015 + 0.377487i \(0.123212\pi\)
−0.926015 + 0.377487i \(0.876788\pi\)
\(234\) 1.28159e39 1.03678
\(235\) 0 0
\(236\) −5.36882e38 −0.377425
\(237\) − 6.90328e38i − 0.452593i
\(238\) 5.87637e38i 0.359409i
\(239\) 1.38618e39 0.791140 0.395570 0.918436i \(-0.370547\pi\)
0.395570 + 0.918436i \(0.370547\pi\)
\(240\) 0 0
\(241\) −2.94732e39 −1.46604 −0.733019 0.680208i \(-0.761890\pi\)
−0.733019 + 0.680208i \(0.761890\pi\)
\(242\) − 2.24392e39i − 1.04244i
\(243\) − 2.77886e39i − 1.20604i
\(244\) −1.98011e39 −0.803076
\(245\) 0 0
\(246\) 9.98665e39 3.53990
\(247\) − 2.58598e39i − 0.857286i
\(248\) − 1.34430e38i − 0.0416910i
\(249\) −2.48228e39 −0.720379
\(250\) 0 0
\(251\) −5.17168e39 −1.31527 −0.657636 0.753336i \(-0.728444\pi\)
−0.657636 + 0.753336i \(0.728444\pi\)
\(252\) − 1.01038e39i − 0.240644i
\(253\) 3.13055e39i 0.698444i
\(254\) 5.94273e39 1.24231
\(255\) 0 0
\(256\) 4.77386e39 0.876820
\(257\) − 4.09624e39i − 0.705488i −0.935720 0.352744i \(-0.885249\pi\)
0.935720 0.352744i \(-0.114751\pi\)
\(258\) − 3.38162e39i − 0.546260i
\(259\) −6.42422e38 −0.0973585
\(260\) 0 0
\(261\) 1.31579e40 1.75638
\(262\) − 3.50865e39i − 0.439712i
\(263\) − 2.59968e39i − 0.305950i −0.988230 0.152975i \(-0.951115\pi\)
0.988230 0.152975i \(-0.0488854\pi\)
\(264\) −6.07501e38 −0.0671556
\(265\) 0 0
\(266\) −3.96781e39 −0.387262
\(267\) 3.14584e40i 2.88603i
\(268\) − 2.01201e40i − 1.73543i
\(269\) −7.20422e39 −0.584351 −0.292176 0.956365i \(-0.594379\pi\)
−0.292176 + 0.956365i \(0.594379\pi\)
\(270\) 0 0
\(271\) −1.89743e40 −1.36199 −0.680993 0.732290i \(-0.738451\pi\)
−0.680993 + 0.732290i \(0.738451\pi\)
\(272\) − 2.27367e40i − 1.53582i
\(273\) 1.85150e39i 0.117717i
\(274\) −1.37979e40 −0.825901
\(275\) 0 0
\(276\) −4.19356e40 −2.22629
\(277\) 3.08340e40i 1.54211i 0.636771 + 0.771053i \(0.280270\pi\)
−0.636771 + 0.771053i \(0.719730\pi\)
\(278\) 2.35480e39i 0.110972i
\(279\) −1.70952e40 −0.759283
\(280\) 0 0
\(281\) −1.56575e40 −0.618112 −0.309056 0.951044i \(-0.600013\pi\)
−0.309056 + 0.951044i \(0.600013\pi\)
\(282\) − 1.16498e41i − 4.33718i
\(283\) − 7.24188e39i − 0.254317i −0.991882 0.127158i \(-0.959414\pi\)
0.991882 0.127158i \(-0.0405857\pi\)
\(284\) −2.60996e40 −0.864732
\(285\) 0 0
\(286\) 1.23679e40 0.364968
\(287\) 8.62090e39i 0.240160i
\(288\) 8.06850e40i 2.12235i
\(289\) −6.72202e40 −1.66988
\(290\) 0 0
\(291\) −1.25830e39 −0.0278964
\(292\) − 3.10315e40i − 0.650108i
\(293\) − 7.67091e40i − 1.51891i −0.650562 0.759453i \(-0.725467\pi\)
0.650562 0.759453i \(-0.274533\pi\)
\(294\) −1.17930e41 −2.20747
\(295\) 0 0
\(296\) −3.09175e39 −0.0517477
\(297\) 2.52188e40i 0.399249i
\(298\) − 1.24895e41i − 1.87059i
\(299\) 4.59177e40 0.650737
\(300\) 0 0
\(301\) 2.91915e39 0.0370604
\(302\) 5.93694e40i 0.713589i
\(303\) 1.82274e41i 2.07454i
\(304\) 1.53522e41 1.65484
\(305\) 0 0
\(306\) 3.59643e41 3.47911
\(307\) − 1.29298e41i − 1.18525i −0.805478 0.592625i \(-0.798091\pi\)
0.805478 0.592625i \(-0.201909\pi\)
\(308\) − 9.75056e39i − 0.0847116i
\(309\) 8.22286e40 0.677186
\(310\) 0 0
\(311\) −5.39990e40 −0.399797 −0.199898 0.979817i \(-0.564061\pi\)
−0.199898 + 0.979817i \(0.564061\pi\)
\(312\) 8.91061e39i 0.0625686i
\(313\) − 1.08588e41i − 0.723270i −0.932320 0.361635i \(-0.882219\pi\)
0.932320 0.361635i \(-0.117781\pi\)
\(314\) 9.71163e40 0.613696
\(315\) 0 0
\(316\) 5.33240e40 0.303449
\(317\) − 1.38129e41i − 0.746117i −0.927808 0.373059i \(-0.878309\pi\)
0.927808 0.373059i \(-0.121691\pi\)
\(318\) 2.66493e41i 1.36659i
\(319\) 1.26979e41 0.618282
\(320\) 0 0
\(321\) 7.46533e40 0.327881
\(322\) − 7.04541e40i − 0.293957i
\(323\) − 7.25686e41i − 2.87678i
\(324\) 7.86869e40 0.296422
\(325\) 0 0
\(326\) −3.68311e40 −0.125350
\(327\) − 2.46094e41i − 0.796275i
\(328\) 4.14894e40i 0.127649i
\(329\) 1.00566e41 0.294251
\(330\) 0 0
\(331\) −4.29457e41 −1.13699 −0.568496 0.822686i \(-0.692474\pi\)
−0.568496 + 0.822686i \(0.692474\pi\)
\(332\) − 1.91742e41i − 0.482990i
\(333\) 3.93172e41i 0.942439i
\(334\) −6.95072e41 −1.58567
\(335\) 0 0
\(336\) −1.09918e41 −0.227232
\(337\) 6.97107e41i 1.37216i 0.727524 + 0.686082i \(0.240671\pi\)
−0.727524 + 0.686082i \(0.759329\pi\)
\(338\) 5.83706e41i 1.09413i
\(339\) 8.56364e41 1.52885
\(340\) 0 0
\(341\) −1.64975e41 −0.267284
\(342\) 2.42837e42i 3.74872i
\(343\) − 2.06057e41i − 0.303134i
\(344\) 1.40489e40 0.0196982
\(345\) 0 0
\(346\) −7.16200e41 −0.912596
\(347\) 7.56830e41i 0.919522i 0.888043 + 0.459761i \(0.152065\pi\)
−0.888043 + 0.459761i \(0.847935\pi\)
\(348\) 1.70096e42i 1.97078i
\(349\) −4.06051e41 −0.448705 −0.224353 0.974508i \(-0.572027\pi\)
−0.224353 + 0.974508i \(0.572027\pi\)
\(350\) 0 0
\(351\) 3.69901e41 0.371979
\(352\) 7.78644e41i 0.747111i
\(353\) 1.97994e42i 1.81288i 0.422336 + 0.906439i \(0.361210\pi\)
−0.422336 + 0.906439i \(0.638790\pi\)
\(354\) −9.23863e41 −0.807335
\(355\) 0 0
\(356\) −2.42998e42 −1.93499
\(357\) 5.19573e41i 0.395022i
\(358\) − 3.36016e42i − 2.43944i
\(359\) 6.70289e41 0.464733 0.232366 0.972628i \(-0.425353\pi\)
0.232366 + 0.972628i \(0.425353\pi\)
\(360\) 0 0
\(361\) 3.31917e42 2.09972
\(362\) 2.20431e41i 0.133223i
\(363\) − 1.98401e42i − 1.14574i
\(364\) −1.43018e41 −0.0789255
\(365\) 0 0
\(366\) −3.40736e42 −1.71783
\(367\) − 1.36571e42i − 0.658215i −0.944292 0.329108i \(-0.893252\pi\)
0.944292 0.329108i \(-0.106748\pi\)
\(368\) 2.72599e42i 1.25613i
\(369\) 5.27613e42 2.32477
\(370\) 0 0
\(371\) −2.30048e41 −0.0927147
\(372\) − 2.20995e42i − 0.851968i
\(373\) − 3.32617e41i − 0.122673i −0.998117 0.0613364i \(-0.980464\pi\)
0.998117 0.0613364i \(-0.0195363\pi\)
\(374\) 3.47071e42 1.22472
\(375\) 0 0
\(376\) 4.83988e41 0.156399
\(377\) − 1.86249e42i − 0.576051i
\(378\) − 5.67560e41i − 0.168034i
\(379\) −1.32387e42 −0.375230 −0.187615 0.982243i \(-0.560076\pi\)
−0.187615 + 0.982243i \(0.560076\pi\)
\(380\) 0 0
\(381\) 5.25440e42 1.36540
\(382\) 8.03484e42i 1.99955i
\(383\) − 5.38010e42i − 1.28236i −0.767390 0.641181i \(-0.778445\pi\)
0.767390 0.641181i \(-0.221555\pi\)
\(384\) −1.12379e42 −0.256578
\(385\) 0 0
\(386\) 8.62770e41 0.180803
\(387\) − 1.78657e42i − 0.358747i
\(388\) − 9.71963e40i − 0.0187036i
\(389\) 6.93155e42 1.27839 0.639193 0.769046i \(-0.279269\pi\)
0.639193 + 0.769046i \(0.279269\pi\)
\(390\) 0 0
\(391\) 1.28856e43 2.18367
\(392\) − 4.89939e41i − 0.0796017i
\(393\) − 3.10225e42i − 0.483281i
\(394\) −1.22825e43 −1.83484
\(395\) 0 0
\(396\) −5.96750e42 −0.820015
\(397\) − 9.33530e42i − 1.23051i −0.788329 0.615254i \(-0.789053\pi\)
0.788329 0.615254i \(-0.210947\pi\)
\(398\) 1.45154e43i 1.83552i
\(399\) −3.50824e42 −0.425634
\(400\) 0 0
\(401\) 3.40931e42 0.380877 0.190438 0.981699i \(-0.439009\pi\)
0.190438 + 0.981699i \(0.439009\pi\)
\(402\) − 3.46225e43i − 3.71218i
\(403\) 2.41980e42i 0.249027i
\(404\) −1.40796e43 −1.39091
\(405\) 0 0
\(406\) −2.85772e42 −0.260219
\(407\) 3.79428e42i 0.331758i
\(408\) 2.50052e42i 0.209961i
\(409\) −3.12051e42 −0.251646 −0.125823 0.992053i \(-0.540157\pi\)
−0.125823 + 0.992053i \(0.540157\pi\)
\(410\) 0 0
\(411\) −1.21997e43 −0.907736
\(412\) 6.35170e42i 0.454031i
\(413\) − 7.97517e41i − 0.0547727i
\(414\) −4.31190e43 −2.84553
\(415\) 0 0
\(416\) 1.14209e43 0.696081
\(417\) 2.08205e42i 0.121968i
\(418\) 2.34347e43i 1.31963i
\(419\) 3.20466e43 1.73480 0.867402 0.497608i \(-0.165788\pi\)
0.867402 + 0.497608i \(0.165788\pi\)
\(420\) 0 0
\(421\) 5.44061e42 0.272266 0.136133 0.990691i \(-0.456533\pi\)
0.136133 + 0.990691i \(0.456533\pi\)
\(422\) − 8.13043e42i − 0.391253i
\(423\) − 6.15478e43i − 2.84837i
\(424\) −1.10714e42 −0.0492794
\(425\) 0 0
\(426\) −4.49120e43 −1.84971
\(427\) − 2.94137e42i − 0.116544i
\(428\) 5.76654e42i 0.219833i
\(429\) 1.09353e43 0.401132
\(430\) 0 0
\(431\) −3.50771e43 −1.19165 −0.595826 0.803114i \(-0.703175\pi\)
−0.595826 + 0.803114i \(0.703175\pi\)
\(432\) 2.19599e43i 0.718039i
\(433\) − 4.30567e43i − 1.35516i −0.735449 0.677580i \(-0.763029\pi\)
0.735449 0.677580i \(-0.236971\pi\)
\(434\) 3.71284e42 0.112493
\(435\) 0 0
\(436\) 1.90094e43 0.533876
\(437\) 8.70054e43i 2.35289i
\(438\) − 5.33988e43i − 1.39062i
\(439\) 1.81689e43 0.455685 0.227843 0.973698i \(-0.426833\pi\)
0.227843 + 0.973698i \(0.426833\pi\)
\(440\) 0 0
\(441\) −6.23047e43 −1.44972
\(442\) − 5.09071e43i − 1.14107i
\(443\) 2.81575e43i 0.608040i 0.952666 + 0.304020i \(0.0983290\pi\)
−0.952666 + 0.304020i \(0.901671\pi\)
\(444\) −5.08267e43 −1.05748
\(445\) 0 0
\(446\) 1.25648e44 2.42734
\(447\) − 1.10429e44i − 2.05594i
\(448\) − 9.48981e42i − 0.170283i
\(449\) −2.21288e43 −0.382731 −0.191365 0.981519i \(-0.561292\pi\)
−0.191365 + 0.981519i \(0.561292\pi\)
\(450\) 0 0
\(451\) 5.09169e43 0.818368
\(452\) 6.61493e43i 1.02504i
\(453\) 5.24928e43i 0.784295i
\(454\) −6.51938e43 −0.939258
\(455\) 0 0
\(456\) −1.68839e43 −0.226231
\(457\) − 6.39404e43i − 0.826339i −0.910654 0.413169i \(-0.864422\pi\)
0.910654 0.413169i \(-0.135578\pi\)
\(458\) 6.07772e43i 0.757636i
\(459\) 1.03803e44 1.24824
\(460\) 0 0
\(461\) 8.79013e42 0.0983853 0.0491926 0.998789i \(-0.484335\pi\)
0.0491926 + 0.998789i \(0.484335\pi\)
\(462\) − 1.67787e43i − 0.181203i
\(463\) 1.18763e44i 1.23764i 0.785532 + 0.618821i \(0.212389\pi\)
−0.785532 + 0.618821i \(0.787611\pi\)
\(464\) 1.10570e44 1.11196
\(465\) 0 0
\(466\) −1.15585e44 −1.08276
\(467\) 1.78679e44i 1.61564i 0.589431 + 0.807819i \(0.299352\pi\)
−0.589431 + 0.807819i \(0.700648\pi\)
\(468\) 8.75291e43i 0.764005i
\(469\) 2.98876e43 0.251849
\(470\) 0 0
\(471\) 8.58677e43 0.674504
\(472\) − 3.83817e42i − 0.0291126i
\(473\) − 1.72411e43i − 0.126287i
\(474\) 9.17595e43 0.649095
\(475\) 0 0
\(476\) −4.01341e43 −0.264849
\(477\) 1.40793e44i 0.897486i
\(478\) 1.84253e44i 1.13463i
\(479\) −1.06259e44 −0.632165 −0.316082 0.948732i \(-0.602368\pi\)
−0.316082 + 0.948732i \(0.602368\pi\)
\(480\) 0 0
\(481\) 5.56531e43 0.309098
\(482\) − 3.91762e44i − 2.10255i
\(483\) − 6.22936e43i − 0.323084i
\(484\) 1.53254e44 0.768178
\(485\) 0 0
\(486\) 3.69370e44 1.72967
\(487\) 1.85504e44i 0.839700i 0.907593 + 0.419850i \(0.137917\pi\)
−0.907593 + 0.419850i \(0.862083\pi\)
\(488\) − 1.41558e43i − 0.0619451i
\(489\) −3.25651e43 −0.137771
\(490\) 0 0
\(491\) 8.88210e43 0.351294 0.175647 0.984453i \(-0.443798\pi\)
0.175647 + 0.984453i \(0.443798\pi\)
\(492\) 6.82062e44i 2.60855i
\(493\) − 5.22657e44i − 1.93305i
\(494\) 3.43733e44 1.22949
\(495\) 0 0
\(496\) −1.43656e44 −0.480703
\(497\) − 3.87700e43i − 0.125492i
\(498\) − 3.29948e44i − 1.03314i
\(499\) −2.77853e44 −0.841699 −0.420849 0.907131i \(-0.638268\pi\)
−0.420849 + 0.907131i \(0.638268\pi\)
\(500\) 0 0
\(501\) −6.14564e44 −1.74279
\(502\) − 6.87428e44i − 1.88632i
\(503\) − 8.80760e41i − 0.00233876i −0.999999 0.00116938i \(-0.999628\pi\)
0.999999 0.00116938i \(-0.000372226\pi\)
\(504\) 7.22319e42 0.0185620
\(505\) 0 0
\(506\) −4.16117e44 −1.00169
\(507\) 5.16097e44i 1.20254i
\(508\) 4.05873e44i 0.915459i
\(509\) 8.68803e44 1.89705 0.948523 0.316709i \(-0.102578\pi\)
0.948523 + 0.316709i \(0.102578\pi\)
\(510\) 0 0
\(511\) 4.60961e43 0.0943450
\(512\) 7.16687e44i 1.42028i
\(513\) 7.00892e44i 1.34498i
\(514\) 5.44479e44 1.01179
\(515\) 0 0
\(516\) 2.30956e44 0.402539
\(517\) − 5.93962e44i − 1.00269i
\(518\) − 8.53917e43i − 0.139629i
\(519\) −6.33245e44 −1.00302
\(520\) 0 0
\(521\) −2.56437e44 −0.381205 −0.190602 0.981667i \(-0.561044\pi\)
−0.190602 + 0.981667i \(0.561044\pi\)
\(522\) 1.74897e45i 2.51894i
\(523\) 6.37079e43i 0.0889028i 0.999012 + 0.0444514i \(0.0141540\pi\)
−0.999012 + 0.0444514i \(0.985846\pi\)
\(524\) 2.39631e44 0.324024
\(525\) 0 0
\(526\) 3.45554e44 0.438784
\(527\) 6.79052e44i 0.835657i
\(528\) 6.49197e44i 0.774314i
\(529\) −6.79897e44 −0.786004
\(530\) 0 0
\(531\) −4.88093e44 −0.530204
\(532\) − 2.70991e44i − 0.285373i
\(533\) − 7.46830e44i − 0.762471i
\(534\) −4.18149e45 −4.13905
\(535\) 0 0
\(536\) 1.43839e44 0.133862
\(537\) − 2.97097e45i − 2.68116i
\(538\) − 9.57595e44i − 0.838059i
\(539\) −6.01267e44 −0.510332
\(540\) 0 0
\(541\) −1.56706e45 −1.25121 −0.625605 0.780140i \(-0.715148\pi\)
−0.625605 + 0.780140i \(0.715148\pi\)
\(542\) − 2.52209e45i − 1.95332i
\(543\) 1.94899e44i 0.146424i
\(544\) 3.20496e45 2.33583
\(545\) 0 0
\(546\) −2.46104e44 −0.168826
\(547\) − 2.05395e45i − 1.36710i −0.729905 0.683548i \(-0.760436\pi\)
0.729905 0.683548i \(-0.239564\pi\)
\(548\) − 9.42359e44i − 0.608607i
\(549\) −1.80017e45 −1.12816
\(550\) 0 0
\(551\) 3.52906e45 2.08285
\(552\) − 2.99797e44i − 0.171725i
\(553\) 7.92107e43i 0.0440371i
\(554\) −4.09851e45 −2.21164
\(555\) 0 0
\(556\) −1.60827e44 −0.0817756
\(557\) 3.02985e44i 0.149558i 0.997200 + 0.0747792i \(0.0238252\pi\)
−0.997200 + 0.0747792i \(0.976175\pi\)
\(558\) − 2.27231e45i − 1.08894i
\(559\) −2.52887e44 −0.117661
\(560\) 0 0
\(561\) 3.06871e45 1.34607
\(562\) − 2.08122e45i − 0.886478i
\(563\) 2.25447e45i 0.932514i 0.884649 + 0.466257i \(0.154398\pi\)
−0.884649 + 0.466257i \(0.845602\pi\)
\(564\) 7.95649e45 3.19607
\(565\) 0 0
\(566\) 9.62602e44 0.364733
\(567\) 1.16886e44i 0.0430173i
\(568\) − 1.86586e44i − 0.0667009i
\(569\) 3.27232e45 1.13633 0.568163 0.822916i \(-0.307654\pi\)
0.568163 + 0.822916i \(0.307654\pi\)
\(570\) 0 0
\(571\) 3.07381e45 1.00735 0.503675 0.863893i \(-0.331981\pi\)
0.503675 + 0.863893i \(0.331981\pi\)
\(572\) 8.44693e44i 0.268946i
\(573\) 7.10419e45i 2.19768i
\(574\) −1.14590e45 −0.344431
\(575\) 0 0
\(576\) −5.80792e45 −1.64835
\(577\) − 6.81899e43i − 0.0188070i −0.999956 0.00940349i \(-0.997007\pi\)
0.999956 0.00940349i \(-0.00299327\pi\)
\(578\) − 8.93501e45i − 2.39489i
\(579\) 7.62838e44 0.198718
\(580\) 0 0
\(581\) 2.84825e44 0.0700926
\(582\) − 1.67255e44i − 0.0400082i
\(583\) 1.35871e45i 0.315934i
\(584\) 2.21844e44 0.0501460
\(585\) 0 0
\(586\) 1.01963e46 2.17837
\(587\) − 8.96797e45i − 1.86280i −0.364002 0.931398i \(-0.618590\pi\)
0.364002 0.931398i \(-0.381410\pi\)
\(588\) − 8.05433e45i − 1.62669i
\(589\) −4.58506e45 −0.900416
\(590\) 0 0
\(591\) −1.08598e46 −2.01665
\(592\) 3.30395e45i 0.596659i
\(593\) 3.75910e45i 0.660209i 0.943944 + 0.330104i \(0.107084\pi\)
−0.943944 + 0.330104i \(0.892916\pi\)
\(594\) −3.35213e45 −0.572590
\(595\) 0 0
\(596\) 8.53004e45 1.37844
\(597\) 1.28342e46i 2.01740i
\(598\) 6.10345e45i 0.933268i
\(599\) −8.80446e45 −1.30966 −0.654832 0.755774i \(-0.727261\pi\)
−0.654832 + 0.755774i \(0.727261\pi\)
\(600\) 0 0
\(601\) −4.53048e45 −0.637846 −0.318923 0.947781i \(-0.603321\pi\)
−0.318923 + 0.947781i \(0.603321\pi\)
\(602\) 3.88018e44i 0.0531508i
\(603\) − 1.82917e46i − 2.43791i
\(604\) −4.05477e45 −0.525844
\(605\) 0 0
\(606\) −2.42281e46 −2.97524
\(607\) 1.02463e46i 1.22448i 0.790671 + 0.612242i \(0.209732\pi\)
−0.790671 + 0.612242i \(0.790268\pi\)
\(608\) 2.16404e46i 2.51684i
\(609\) −2.52672e45 −0.286003
\(610\) 0 0
\(611\) −8.71203e45 −0.934199
\(612\) 2.45627e46i 2.56376i
\(613\) 4.85425e45i 0.493201i 0.969117 + 0.246601i \(0.0793136\pi\)
−0.969117 + 0.246601i \(0.920686\pi\)
\(614\) 1.71865e46 1.69985
\(615\) 0 0
\(616\) 6.97068e43 0.00653421
\(617\) − 1.62668e46i − 1.48455i −0.670093 0.742277i \(-0.733746\pi\)
0.670093 0.742277i \(-0.266254\pi\)
\(618\) 1.09300e46i 0.971200i
\(619\) 2.50157e45 0.216430 0.108215 0.994128i \(-0.465486\pi\)
0.108215 + 0.994128i \(0.465486\pi\)
\(620\) 0 0
\(621\) −1.24453e46 −1.02093
\(622\) − 7.17763e45i − 0.573376i
\(623\) − 3.60964e45i − 0.280809i
\(624\) 9.52220e45 0.721425
\(625\) 0 0
\(626\) 1.44337e46 1.03729
\(627\) 2.07204e46i 1.45038i
\(628\) 6.63279e45i 0.452233i
\(629\) 1.56175e46 1.03723
\(630\) 0 0
\(631\) 6.94339e45 0.437610 0.218805 0.975769i \(-0.429784\pi\)
0.218805 + 0.975769i \(0.429784\pi\)
\(632\) 3.81213e44i 0.0234065i
\(633\) − 7.18871e45i − 0.430021i
\(634\) 1.83603e46 1.07006
\(635\) 0 0
\(636\) −1.82008e46 −1.00704
\(637\) 8.81916e45i 0.475474i
\(638\) 1.68783e46i 0.886721i
\(639\) −2.37278e46 −1.21477
\(640\) 0 0
\(641\) 4.34529e45 0.211282 0.105641 0.994404i \(-0.466311\pi\)
0.105641 + 0.994404i \(0.466311\pi\)
\(642\) 9.92303e45i 0.470236i
\(643\) − 2.56764e46i − 1.18591i −0.805235 0.592956i \(-0.797961\pi\)
0.805235 0.592956i \(-0.202039\pi\)
\(644\) 4.81183e45 0.216618
\(645\) 0 0
\(646\) 9.64592e46 4.12579
\(647\) − 1.55928e46i − 0.650136i −0.945691 0.325068i \(-0.894613\pi\)
0.945691 0.325068i \(-0.105387\pi\)
\(648\) 5.62533e44i 0.0228644i
\(649\) −4.71030e45 −0.186643
\(650\) 0 0
\(651\) 3.28279e45 0.123639
\(652\) − 2.51547e45i − 0.0923706i
\(653\) − 1.54838e45i − 0.0554384i −0.999616 0.0277192i \(-0.991176\pi\)
0.999616 0.0277192i \(-0.00882442\pi\)
\(654\) 3.27112e46 1.14199
\(655\) 0 0
\(656\) 4.43370e46 1.47182
\(657\) − 2.82115e46i − 0.913267i
\(658\) 1.33673e46i 0.422005i
\(659\) 1.05260e46 0.324080 0.162040 0.986784i \(-0.448193\pi\)
0.162040 + 0.986784i \(0.448193\pi\)
\(660\) 0 0
\(661\) −1.48865e46 −0.435981 −0.217990 0.975951i \(-0.569950\pi\)
−0.217990 + 0.975951i \(0.569950\pi\)
\(662\) − 5.70840e46i − 1.63064i
\(663\) − 4.50107e46i − 1.25413i
\(664\) 1.37076e45 0.0372554
\(665\) 0 0
\(666\) −5.22611e46 −1.35162
\(667\) 6.26634e46i 1.58102i
\(668\) − 4.74716e46i − 1.16848i
\(669\) 1.11095e47 2.66786
\(670\) 0 0
\(671\) −1.73724e46 −0.397134
\(672\) − 1.54940e46i − 0.345597i
\(673\) 2.78172e46i 0.605432i 0.953081 + 0.302716i \(0.0978933\pi\)
−0.953081 + 0.302716i \(0.902107\pi\)
\(674\) −9.26605e46 −1.96791
\(675\) 0 0
\(676\) −3.98656e46 −0.806265
\(677\) − 9.09948e45i − 0.179599i −0.995960 0.0897995i \(-0.971377\pi\)
0.995960 0.0897995i \(-0.0286226\pi\)
\(678\) 1.13829e47i 2.19262i
\(679\) 1.44381e44 0.00271431
\(680\) 0 0
\(681\) −5.76426e46 −1.03233
\(682\) − 2.19288e46i − 0.383330i
\(683\) − 2.07397e46i − 0.353883i −0.984221 0.176942i \(-0.943380\pi\)
0.984221 0.176942i \(-0.0566204\pi\)
\(684\) −1.65851e47 −2.76244
\(685\) 0 0
\(686\) 2.73894e46 0.434746
\(687\) 5.37376e46i 0.832707i
\(688\) − 1.50131e46i − 0.227123i
\(689\) 1.99291e46 0.294354
\(690\) 0 0
\(691\) 7.80876e46 1.09950 0.549749 0.835330i \(-0.314723\pi\)
0.549749 + 0.835330i \(0.314723\pi\)
\(692\) − 4.89146e46i − 0.672493i
\(693\) − 8.86448e45i − 0.119002i
\(694\) −1.00599e47 −1.31875
\(695\) 0 0
\(696\) −1.21602e46 −0.152016
\(697\) − 2.09578e47i − 2.55861i
\(698\) − 5.39729e46i − 0.643519i
\(699\) −1.02197e47 −1.19005
\(700\) 0 0
\(701\) −7.28838e46 −0.809624 −0.404812 0.914400i \(-0.632663\pi\)
−0.404812 + 0.914400i \(0.632663\pi\)
\(702\) 4.91678e46i 0.533480i
\(703\) 1.05452e47i 1.11761i
\(704\) −5.60488e46 −0.580253
\(705\) 0 0
\(706\) −2.63176e47 −2.59997
\(707\) − 2.09147e46i − 0.201852i
\(708\) − 6.30974e46i − 0.594926i
\(709\) −4.67334e46 −0.430491 −0.215246 0.976560i \(-0.569055\pi\)
−0.215246 + 0.976560i \(0.569055\pi\)
\(710\) 0 0
\(711\) 4.84782e46 0.426283
\(712\) − 1.73719e46i − 0.149255i
\(713\) − 8.14142e46i − 0.683476i
\(714\) −6.90624e46 −0.566528
\(715\) 0 0
\(716\) 2.29490e47 1.79763
\(717\) 1.62911e47i 1.24705i
\(718\) 8.90958e46i 0.666506i
\(719\) 1.27931e47 0.935299 0.467649 0.883914i \(-0.345101\pi\)
0.467649 + 0.883914i \(0.345101\pi\)
\(720\) 0 0
\(721\) −9.43520e45 −0.0658899
\(722\) 4.41189e47i 3.01135i
\(723\) − 3.46385e47i − 2.31088i
\(724\) −1.50548e46 −0.0981725
\(725\) 0 0
\(726\) 2.63718e47 1.64318
\(727\) − 1.68652e47i − 1.02724i −0.858017 0.513621i \(-0.828304\pi\)
0.858017 0.513621i \(-0.171696\pi\)
\(728\) − 1.02243e45i − 0.00608790i
\(729\) 2.78403e47 1.62057
\(730\) 0 0
\(731\) −7.09659e46 −0.394832
\(732\) − 2.32713e47i − 1.26587i
\(733\) − 3.19951e47i − 1.70164i −0.525456 0.850821i \(-0.676105\pi\)
0.525456 0.850821i \(-0.323895\pi\)
\(734\) 1.81532e47 0.943992
\(735\) 0 0
\(736\) −3.84255e47 −1.91045
\(737\) − 1.76523e47i − 0.858196i
\(738\) 7.01311e47i 3.33412i
\(739\) 1.33356e47 0.619980 0.309990 0.950740i \(-0.399674\pi\)
0.309990 + 0.950740i \(0.399674\pi\)
\(740\) 0 0
\(741\) 3.03919e47 1.35132
\(742\) − 3.05783e46i − 0.132969i
\(743\) − 3.22840e46i − 0.137300i −0.997641 0.0686502i \(-0.978131\pi\)
0.997641 0.0686502i \(-0.0218692\pi\)
\(744\) 1.57989e46 0.0657164
\(745\) 0 0
\(746\) 4.42119e46 0.175934
\(747\) − 1.74317e47i − 0.678502i
\(748\) 2.37040e47i 0.902497i
\(749\) −8.56597e45 −0.0319027
\(750\) 0 0
\(751\) 2.39672e46 0.0854196 0.0427098 0.999088i \(-0.486401\pi\)
0.0427098 + 0.999088i \(0.486401\pi\)
\(752\) − 5.17206e47i − 1.80331i
\(753\) − 6.07806e47i − 2.07323i
\(754\) 2.47565e47 0.826154
\(755\) 0 0
\(756\) 3.87628e46 0.123824
\(757\) − 6.92584e46i − 0.216466i −0.994126 0.108233i \(-0.965481\pi\)
0.994126 0.108233i \(-0.0345193\pi\)
\(758\) − 1.75970e47i − 0.538143i
\(759\) −3.67919e47 −1.10094
\(760\) 0 0
\(761\) −4.97287e47 −1.42482 −0.712411 0.701763i \(-0.752397\pi\)
−0.712411 + 0.701763i \(0.752397\pi\)
\(762\) 6.98423e47i 1.95822i
\(763\) 2.82377e46i 0.0774773i
\(764\) −5.48759e47 −1.47347
\(765\) 0 0
\(766\) 7.15131e47 1.83912
\(767\) 6.90891e46i 0.173895i
\(768\) 5.61051e47i 1.38211i
\(769\) −2.89778e47 −0.698683 −0.349342 0.936995i \(-0.613595\pi\)
−0.349342 + 0.936995i \(0.613595\pi\)
\(770\) 0 0
\(771\) 4.81414e47 1.11204
\(772\) 5.89249e46i 0.133234i
\(773\) 5.10656e47i 1.13023i 0.825012 + 0.565115i \(0.191168\pi\)
−0.825012 + 0.565115i \(0.808832\pi\)
\(774\) 2.37473e47 0.514504
\(775\) 0 0
\(776\) 6.94857e44 0.00144270
\(777\) − 7.55010e46i − 0.153464i
\(778\) 9.21352e47i 1.83342i
\(779\) 1.41510e48 2.75689
\(780\) 0 0
\(781\) −2.28983e47 −0.427624
\(782\) 1.71277e48i 3.13175i
\(783\) 5.04799e47i 0.903753i
\(784\) −5.23567e47 −0.917819
\(785\) 0 0
\(786\) 4.12356e47 0.693107
\(787\) 3.90010e47i 0.641938i 0.947090 + 0.320969i \(0.104009\pi\)
−0.947090 + 0.320969i \(0.895991\pi\)
\(788\) − 8.38862e47i − 1.35210i
\(789\) 3.05529e47 0.482261
\(790\) 0 0
\(791\) −9.82622e46 −0.148756
\(792\) − 4.26617e46i − 0.0632517i
\(793\) 2.54812e47i 0.370009i
\(794\) 1.24086e48 1.76476
\(795\) 0 0
\(796\) −9.91367e47 −1.35260
\(797\) 6.38986e47i 0.853943i 0.904265 + 0.426972i \(0.140420\pi\)
−0.904265 + 0.426972i \(0.859580\pi\)
\(798\) − 4.66320e47i − 0.610431i
\(799\) −2.44479e48 −3.13488
\(800\) 0 0
\(801\) −2.20916e48 −2.71826
\(802\) 4.53171e47i 0.546242i
\(803\) − 2.72253e47i − 0.321489i
\(804\) 2.36463e48 2.73551
\(805\) 0 0
\(806\) −3.21644e47 −0.357147
\(807\) − 8.46680e47i − 0.921098i
\(808\) − 1.00655e47i − 0.107288i
\(809\) 3.00339e47 0.313661 0.156831 0.987626i \(-0.449872\pi\)
0.156831 + 0.987626i \(0.449872\pi\)
\(810\) 0 0
\(811\) −1.57205e48 −1.57624 −0.788121 0.615521i \(-0.788946\pi\)
−0.788121 + 0.615521i \(0.788946\pi\)
\(812\) − 1.95175e47i − 0.191756i
\(813\) − 2.22997e48i − 2.14686i
\(814\) −5.04341e47 −0.475797
\(815\) 0 0
\(816\) 2.67215e48 2.42088
\(817\) − 4.79172e47i − 0.425430i
\(818\) − 4.14782e47i − 0.360903i
\(819\) −1.30021e47 −0.110874
\(820\) 0 0
\(821\) 2.23751e48 1.83275 0.916374 0.400324i \(-0.131102\pi\)
0.916374 + 0.400324i \(0.131102\pi\)
\(822\) − 1.62160e48i − 1.30185i
\(823\) 5.02472e47i 0.395380i 0.980265 + 0.197690i \(0.0633439\pi\)
−0.980265 + 0.197690i \(0.936656\pi\)
\(824\) −4.54083e46 −0.0350216
\(825\) 0 0
\(826\) 1.06007e47 0.0785533
\(827\) 1.52395e48i 1.10695i 0.832865 + 0.553475i \(0.186699\pi\)
−0.832865 + 0.553475i \(0.813301\pi\)
\(828\) − 2.94492e48i − 2.09688i
\(829\) −7.14872e47 −0.498975 −0.249487 0.968378i \(-0.580262\pi\)
−0.249487 + 0.968378i \(0.580262\pi\)
\(830\) 0 0
\(831\) −3.62379e48 −2.43078
\(832\) 8.22104e47i 0.540620i
\(833\) 2.47486e48i 1.59554i
\(834\) −2.76749e47 −0.174923
\(835\) 0 0
\(836\) −1.60053e48 −0.972436
\(837\) − 6.55851e47i − 0.390693i
\(838\) 4.25968e48i 2.48800i
\(839\) −2.41800e48 −1.38479 −0.692394 0.721519i \(-0.743444\pi\)
−0.692394 + 0.721519i \(0.743444\pi\)
\(840\) 0 0
\(841\) 7.25637e47 0.399563
\(842\) 7.23174e47i 0.390475i
\(843\) − 1.84015e48i − 0.974315i
\(844\) 5.55287e47 0.288315
\(845\) 0 0
\(846\) 8.18103e48 4.08505
\(847\) 2.27652e47i 0.111480i
\(848\) 1.18313e48i 0.568199i
\(849\) 8.51107e47 0.400873
\(850\) 0 0
\(851\) −1.87245e48 −0.848345
\(852\) − 3.06737e48i − 1.36306i
\(853\) − 5.58595e47i − 0.243466i −0.992563 0.121733i \(-0.961155\pi\)
0.992563 0.121733i \(-0.0388452\pi\)
\(854\) 3.90972e47 0.167144
\(855\) 0 0
\(856\) −4.12250e46 −0.0169568
\(857\) − 7.99378e47i − 0.322529i −0.986911 0.161264i \(-0.948443\pi\)
0.986911 0.161264i \(-0.0515572\pi\)
\(858\) 1.45354e48i 0.575290i
\(859\) −3.21299e48 −1.24745 −0.623723 0.781646i \(-0.714381\pi\)
−0.623723 + 0.781646i \(0.714381\pi\)
\(860\) 0 0
\(861\) −1.01318e48 −0.378559
\(862\) − 4.66251e48i − 1.70903i
\(863\) − 2.15382e48i − 0.774519i −0.921971 0.387259i \(-0.873422\pi\)
0.921971 0.387259i \(-0.126578\pi\)
\(864\) −3.09546e48 −1.09206
\(865\) 0 0
\(866\) 5.72316e48 1.94353
\(867\) − 7.90010e48i − 2.63219i
\(868\) 2.53577e47i 0.0828962i
\(869\) 4.67835e47 0.150060
\(870\) 0 0
\(871\) −2.58917e48 −0.799578
\(872\) 1.35898e47i 0.0411805i
\(873\) − 8.83637e46i − 0.0262748i
\(874\) −1.15649e49 −3.37445
\(875\) 0 0
\(876\) 3.64700e48 1.02475
\(877\) 4.36089e48i 1.20249i 0.799064 + 0.601245i \(0.205329\pi\)
−0.799064 + 0.601245i \(0.794671\pi\)
\(878\) 2.41504e48i 0.653530i
\(879\) 9.01528e48 2.39421
\(880\) 0 0
\(881\) −4.71885e48 −1.20707 −0.603537 0.797335i \(-0.706242\pi\)
−0.603537 + 0.797335i \(0.706242\pi\)
\(882\) − 8.28163e48i − 2.07914i
\(883\) 2.47214e48i 0.609146i 0.952489 + 0.304573i \(0.0985138\pi\)
−0.952489 + 0.304573i \(0.901486\pi\)
\(884\) 3.47682e48 0.840853
\(885\) 0 0
\(886\) −3.74274e48 −0.872033
\(887\) − 6.65983e48i − 1.52308i −0.648116 0.761542i \(-0.724443\pi\)
0.648116 0.761542i \(-0.275557\pi\)
\(888\) − 3.63360e47i − 0.0815686i
\(889\) −6.02908e47 −0.132853
\(890\) 0 0
\(891\) 6.90355e47 0.146585
\(892\) 8.58144e48i 1.78871i
\(893\) − 1.65076e49i − 3.37781i
\(894\) 1.46784e49 2.94856
\(895\) 0 0
\(896\) 1.28948e47 0.0249650
\(897\) 5.39651e48i 1.02574i
\(898\) − 2.94139e48i − 0.548901i
\(899\) −3.30227e48 −0.605032
\(900\) 0 0
\(901\) 5.59256e48 0.987760
\(902\) 6.76795e48i 1.17368i
\(903\) 3.43075e47i 0.0584173i
\(904\) −4.72902e47 −0.0790663
\(905\) 0 0
\(906\) −6.97743e48 −1.12481
\(907\) 4.30056e48i 0.680777i 0.940285 + 0.340388i \(0.110559\pi\)
−0.940285 + 0.340388i \(0.889441\pi\)
\(908\) − 4.45257e48i − 0.692140i
\(909\) −1.28002e49 −1.95394
\(910\) 0 0
\(911\) −2.43257e48 −0.358106 −0.179053 0.983839i \(-0.557303\pi\)
−0.179053 + 0.983839i \(0.557303\pi\)
\(912\) 1.80427e49i 2.60848i
\(913\) − 1.68224e48i − 0.238847i
\(914\) 8.49906e48 1.18511
\(915\) 0 0
\(916\) −4.15093e48 −0.558303
\(917\) 3.55963e47i 0.0470230i
\(918\) 1.37976e49i 1.79019i
\(919\) −1.59035e48 −0.202669 −0.101334 0.994852i \(-0.532311\pi\)
−0.101334 + 0.994852i \(0.532311\pi\)
\(920\) 0 0
\(921\) 1.51959e49 1.86828
\(922\) 1.16840e48i 0.141101i
\(923\) 3.35865e48i 0.398416i
\(924\) 1.14594e48 0.133529
\(925\) 0 0
\(926\) −1.57862e49 −1.77499
\(927\) 5.77449e48i 0.637820i
\(928\) 1.55859e49i 1.69119i
\(929\) 1.48456e49 1.58248 0.791239 0.611507i \(-0.209436\pi\)
0.791239 + 0.611507i \(0.209436\pi\)
\(930\) 0 0
\(931\) −1.67106e49 −1.71919
\(932\) − 7.89414e48i − 0.797888i
\(933\) − 6.34627e48i − 0.630190i
\(934\) −2.37502e49 −2.31710
\(935\) 0 0
\(936\) −6.25746e47 −0.0589314
\(937\) − 1.00831e49i − 0.933020i −0.884516 0.466510i \(-0.845511\pi\)
0.884516 0.466510i \(-0.154489\pi\)
\(938\) 3.97271e48i 0.361193i
\(939\) 1.27619e49 1.14007
\(940\) 0 0
\(941\) 9.04832e48 0.780440 0.390220 0.920722i \(-0.372399\pi\)
0.390220 + 0.920722i \(0.372399\pi\)
\(942\) 1.14137e49i 0.967353i
\(943\) 2.51271e49i 2.09266i
\(944\) −4.10161e48 −0.335673
\(945\) 0 0
\(946\) 2.29172e48 0.181116
\(947\) − 9.36347e48i − 0.727214i −0.931552 0.363607i \(-0.881545\pi\)
0.931552 0.363607i \(-0.118455\pi\)
\(948\) 6.26693e48i 0.478319i
\(949\) −3.99331e48 −0.299530
\(950\) 0 0
\(951\) 1.62337e49 1.17609
\(952\) − 2.86919e47i − 0.0204291i
\(953\) − 1.92193e49i − 1.34494i −0.740123 0.672472i \(-0.765233\pi\)
0.740123 0.672472i \(-0.234767\pi\)
\(954\) −1.87144e49 −1.28715
\(955\) 0 0
\(956\) −1.25840e49 −0.836109
\(957\) 1.49233e49i 0.974582i
\(958\) − 1.41241e49i − 0.906631i
\(959\) 1.39984e48 0.0883223
\(960\) 0 0
\(961\) −1.21131e49 −0.738444
\(962\) 7.39750e48i 0.443298i
\(963\) 5.24251e48i 0.308820i
\(964\) 2.67563e49 1.54937
\(965\) 0 0
\(966\) 8.28017e48 0.463358
\(967\) − 1.11182e49i − 0.611642i −0.952089 0.305821i \(-0.901069\pi\)
0.952089 0.305821i \(-0.0989309\pi\)
\(968\) 1.09561e48i 0.0592533i
\(969\) 8.52867e49 4.53460
\(970\) 0 0
\(971\) 1.55536e49 0.799308 0.399654 0.916666i \(-0.369130\pi\)
0.399654 + 0.916666i \(0.369130\pi\)
\(972\) 2.52270e49i 1.27459i
\(973\) − 2.38902e47i − 0.0118674i
\(974\) −2.46574e49 −1.20427
\(975\) 0 0
\(976\) −1.51274e49 −0.714236
\(977\) − 2.43707e49i − 1.13138i −0.824618 0.565690i \(-0.808610\pi\)
0.824618 0.565690i \(-0.191390\pi\)
\(978\) − 4.32860e48i − 0.197586i
\(979\) −2.13193e49 −0.956883
\(980\) 0 0
\(981\) 1.72819e49 0.749986
\(982\) 1.18062e49i 0.503815i
\(983\) − 1.27767e48i − 0.0536148i −0.999641 0.0268074i \(-0.991466\pi\)
0.999641 0.0268074i \(-0.00853409\pi\)
\(984\) −4.87607e48 −0.201210
\(985\) 0 0
\(986\) 6.94723e49 2.77231
\(987\) 1.18191e49i 0.463820i
\(988\) 2.34760e49i 0.906015i
\(989\) 8.50838e48 0.322930
\(990\) 0 0
\(991\) −8.47131e48 −0.310982 −0.155491 0.987837i \(-0.549696\pi\)
−0.155491 + 0.987837i \(0.549696\pi\)
\(992\) − 2.02497e49i − 0.731100i
\(993\) − 5.04722e49i − 1.79221i
\(994\) 5.15336e48 0.179976
\(995\) 0 0
\(996\) 2.25346e49 0.761326
\(997\) − 4.23201e49i − 1.40629i −0.711045 0.703147i \(-0.751778\pi\)
0.711045 0.703147i \(-0.248222\pi\)
\(998\) − 3.69327e49i − 1.20714i
\(999\) −1.50840e49 −0.484936
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.24.11 12
5.2 odd 4 25.34.a.c.1.2 6
5.3 odd 4 5.34.a.b.1.5 6
5.4 even 2 inner 25.34.b.c.24.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.a.b.1.5 6 5.3 odd 4
25.34.a.c.1.2 6 5.2 odd 4
25.34.b.c.24.2 12 5.4 even 2 inner
25.34.b.c.24.11 12 1.1 even 1 trivial