Properties

Label 25.34.b.c.24.4
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 24.4
Root \(-48092.0i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.34.b.c.24.9

$q$-expansion

\(f(q)\) \(=\) \(q-96184.0i q^{2} -1.07272e8i q^{3} -6.61431e8 q^{4} -1.03179e13 q^{6} -8.61302e13i q^{7} -7.62595e14i q^{8} -5.94832e15 q^{9} +O(q^{10})\) \(q-96184.0i q^{2} -1.07272e8i q^{3} -6.61431e8 q^{4} -1.03179e13 q^{6} -8.61302e13i q^{7} -7.62595e14i q^{8} -5.94832e15 q^{9} +9.57101e16 q^{11} +7.09533e16i q^{12} -2.44486e18i q^{13} -8.28435e18 q^{14} -7.90311e19 q^{16} -6.06607e19i q^{17} +5.72133e20i q^{18} +1.50149e21 q^{19} -9.23940e21 q^{21} -9.20578e21i q^{22} -1.88217e22i q^{23} -8.18055e22 q^{24} -2.35157e23 q^{26} +4.17567e22i q^{27} +5.69692e22i q^{28} -2.67431e24 q^{29} +2.64569e24 q^{31} +1.05089e24i q^{32} -1.02671e25i q^{33} -5.83459e24 q^{34} +3.93440e24 q^{36} -1.12777e26i q^{37} -1.44419e26i q^{38} -2.62266e26 q^{39} +7.31837e26 q^{41} +8.88682e26i q^{42} -3.88890e26i q^{43} -6.33056e25 q^{44} -1.81034e27 q^{46} +3.28337e27i q^{47} +8.47786e27i q^{48} +3.12582e26 q^{49} -6.50723e27 q^{51} +1.61711e27i q^{52} -3.35567e28i q^{53} +4.01633e27 q^{54} -6.56825e28 q^{56} -1.61068e29i q^{57} +2.57226e29i q^{58} -2.93443e29 q^{59} +7.07686e28 q^{61} -2.54473e29i q^{62} +5.12330e29i q^{63} -5.77794e29 q^{64} -9.87527e29 q^{66} +1.99482e30i q^{67} +4.01229e28i q^{68} -2.01905e30 q^{69} -1.15434e30 q^{71} +4.53616e30i q^{72} +2.83903e30i q^{73} -1.08474e31 q^{74} -9.93131e29 q^{76} -8.24353e30i q^{77} +2.52258e31i q^{78} -6.93520e30 q^{79} -2.85877e31 q^{81} -7.03910e31i q^{82} -4.09391e31i q^{83} +6.11122e30 q^{84} -3.74050e31 q^{86} +2.86880e32i q^{87} -7.29881e31i q^{88} +1.49837e32 q^{89} -2.10577e32 q^{91} +1.24492e31i q^{92} -2.83809e32i q^{93} +3.15807e32 q^{94} +1.12731e32 q^{96} +1.00351e33i q^{97} -3.00654e31i q^{98} -5.69314e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + O(q^{10}) \) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + 301521793111780384q^{11} + 14172220017979782912q^{14} + \)\(12\!\cdots\!32\)\(q^{16} - \)\(34\!\cdots\!00\)\(q^{19} - \)\(21\!\cdots\!56\)\(q^{21} - \)\(42\!\cdots\!00\)\(q^{24} - \)\(72\!\cdots\!76\)\(q^{26} - \)\(26\!\cdots\!00\)\(q^{29} + \)\(21\!\cdots\!04\)\(q^{31} - \)\(17\!\cdots\!48\)\(q^{34} + \)\(33\!\cdots\!92\)\(q^{36} - \)\(11\!\cdots\!12\)\(q^{39} + \)\(89\!\cdots\!64\)\(q^{41} - \)\(14\!\cdots\!28\)\(q^{44} + \)\(69\!\cdots\!44\)\(q^{46} + \)\(14\!\cdots\!16\)\(q^{49} - \)\(10\!\cdots\!76\)\(q^{51} + \)\(36\!\cdots\!00\)\(q^{54} - \)\(78\!\cdots\!00\)\(q^{56} + \)\(35\!\cdots\!00\)\(q^{59} - \)\(56\!\cdots\!16\)\(q^{61} - \)\(38\!\cdots\!44\)\(q^{64} + \)\(10\!\cdots\!28\)\(q^{66} - \)\(63\!\cdots\!72\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} - \)\(40\!\cdots\!68\)\(q^{74} + \)\(58\!\cdots\!00\)\(q^{76} + \)\(44\!\cdots\!00\)\(q^{79} - \)\(26\!\cdots\!48\)\(q^{81} + \)\(47\!\cdots\!52\)\(q^{84} - \)\(25\!\cdots\!16\)\(q^{86} + \)\(67\!\cdots\!00\)\(q^{89} + \)\(38\!\cdots\!64\)\(q^{91} - \)\(29\!\cdots\!28\)\(q^{94} + \)\(63\!\cdots\!44\)\(q^{96} - \)\(33\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 96184.0i − 1.03779i −0.854839 0.518893i \(-0.826344\pi\)
0.854839 0.518893i \(-0.173656\pi\)
\(3\) − 1.07272e8i − 1.43876i −0.694618 0.719379i \(-0.744427\pi\)
0.694618 0.719379i \(-0.255573\pi\)
\(4\) −6.61431e8 −0.0770007
\(5\) 0 0
\(6\) −1.03179e13 −1.49312
\(7\) − 8.61302e13i − 0.979575i −0.871842 0.489788i \(-0.837074\pi\)
0.871842 0.489788i \(-0.162926\pi\)
\(8\) − 7.62595e14i − 0.957876i
\(9\) −5.94832e15 −1.07002
\(10\) 0 0
\(11\) 9.57101e16 0.628027 0.314014 0.949418i \(-0.398326\pi\)
0.314014 + 0.949418i \(0.398326\pi\)
\(12\) 7.09533e16i 0.110785i
\(13\) − 2.44486e18i − 1.01904i −0.860460 0.509518i \(-0.829824\pi\)
0.860460 0.509518i \(-0.170176\pi\)
\(14\) −8.28435e18 −1.01659
\(15\) 0 0
\(16\) −7.90311e19 −1.07107
\(17\) − 6.06607e19i − 0.302343i −0.988508 0.151172i \(-0.951695\pi\)
0.988508 0.151172i \(-0.0483046\pi\)
\(18\) 5.72133e20i 1.11045i
\(19\) 1.50149e21 1.19423 0.597115 0.802156i \(-0.296314\pi\)
0.597115 + 0.802156i \(0.296314\pi\)
\(20\) 0 0
\(21\) −9.23940e21 −1.40937
\(22\) − 9.20578e21i − 0.651758i
\(23\) − 1.88217e22i − 0.639955i −0.947425 0.319977i \(-0.896325\pi\)
0.947425 0.319977i \(-0.103675\pi\)
\(24\) −8.18055e22 −1.37815
\(25\) 0 0
\(26\) −2.35157e23 −1.05754
\(27\) 4.17567e22i 0.100745i
\(28\) 5.69692e22i 0.0754280i
\(29\) −2.67431e24 −1.98447 −0.992235 0.124376i \(-0.960307\pi\)
−0.992235 + 0.124376i \(0.960307\pi\)
\(30\) 0 0
\(31\) 2.64569e24 0.653236 0.326618 0.945156i \(-0.394091\pi\)
0.326618 + 0.945156i \(0.394091\pi\)
\(32\) 1.05089e24i 0.153667i
\(33\) − 1.02671e25i − 0.903579i
\(34\) −5.83459e24 −0.313768
\(35\) 0 0
\(36\) 3.93440e24 0.0823925
\(37\) − 1.12777e26i − 1.50277i −0.659862 0.751387i \(-0.729385\pi\)
0.659862 0.751387i \(-0.270615\pi\)
\(38\) − 1.44419e26i − 1.23935i
\(39\) −2.62266e26 −1.46614
\(40\) 0 0
\(41\) 7.31837e26 1.79259 0.896294 0.443460i \(-0.146249\pi\)
0.896294 + 0.443460i \(0.146249\pi\)
\(42\) 8.88682e26i 1.46263i
\(43\) − 3.88890e26i − 0.434107i −0.976160 0.217054i \(-0.930355\pi\)
0.976160 0.217054i \(-0.0696446\pi\)
\(44\) −6.33056e25 −0.0483585
\(45\) 0 0
\(46\) −1.81034e27 −0.664136
\(47\) 3.28337e27i 0.844704i 0.906432 + 0.422352i \(0.138795\pi\)
−0.906432 + 0.422352i \(0.861205\pi\)
\(48\) 8.47786e27i 1.54101i
\(49\) 3.12582e26 0.0404323
\(50\) 0 0
\(51\) −6.50723e27 −0.434999
\(52\) 1.61711e27i 0.0784664i
\(53\) − 3.35567e28i − 1.18912i −0.804050 0.594562i \(-0.797325\pi\)
0.804050 0.594562i \(-0.202675\pi\)
\(54\) 4.01633e27 0.104552
\(55\) 0 0
\(56\) −6.56825e28 −0.938312
\(57\) − 1.61068e29i − 1.71821i
\(58\) 2.57226e29i 2.05946i
\(59\) −2.93443e29 −1.77201 −0.886005 0.463676i \(-0.846530\pi\)
−0.886005 + 0.463676i \(0.846530\pi\)
\(60\) 0 0
\(61\) 7.07686e28 0.246546 0.123273 0.992373i \(-0.460661\pi\)
0.123273 + 0.992373i \(0.460661\pi\)
\(62\) − 2.54473e29i − 0.677920i
\(63\) 5.12330e29i 1.04817i
\(64\) −5.77794e29 −0.911598
\(65\) 0 0
\(66\) −9.87527e29 −0.937722
\(67\) 1.99482e30i 1.47798i 0.673715 + 0.738991i \(0.264698\pi\)
−0.673715 + 0.738991i \(0.735302\pi\)
\(68\) 4.01229e28i 0.0232807i
\(69\) −2.01905e30 −0.920739
\(70\) 0 0
\(71\) −1.15434e30 −0.328527 −0.164263 0.986417i \(-0.552525\pi\)
−0.164263 + 0.986417i \(0.552525\pi\)
\(72\) 4.53616e30i 1.02495i
\(73\) 2.83903e30i 0.510908i 0.966821 + 0.255454i \(0.0822250\pi\)
−0.966821 + 0.255454i \(0.917775\pi\)
\(74\) −1.08474e31 −1.55956
\(75\) 0 0
\(76\) −9.93131e29 −0.0919565
\(77\) − 8.24353e30i − 0.615200i
\(78\) 2.52258e31i 1.52154i
\(79\) −6.93520e30 −0.339010 −0.169505 0.985529i \(-0.554217\pi\)
−0.169505 + 0.985529i \(0.554217\pi\)
\(80\) 0 0
\(81\) −2.85877e31 −0.925075
\(82\) − 7.03910e31i − 1.86032i
\(83\) − 4.09391e31i − 0.885828i −0.896564 0.442914i \(-0.853945\pi\)
0.896564 0.442914i \(-0.146055\pi\)
\(84\) 6.11122e30 0.108523
\(85\) 0 0
\(86\) −3.74050e31 −0.450510
\(87\) 2.86880e32i 2.85517i
\(88\) − 7.29881e31i − 0.601572i
\(89\) 1.49837e32 1.02491 0.512454 0.858715i \(-0.328736\pi\)
0.512454 + 0.858715i \(0.328736\pi\)
\(90\) 0 0
\(91\) −2.10577e32 −0.998222
\(92\) 1.24492e31i 0.0492769i
\(93\) − 2.83809e32i − 0.939849i
\(94\) 3.15807e32 0.876623
\(95\) 0 0
\(96\) 1.12731e32 0.221090
\(97\) 1.00351e33i 1.65878i 0.558673 + 0.829388i \(0.311311\pi\)
−0.558673 + 0.829388i \(0.688689\pi\)
\(98\) − 3.00654e31i − 0.0419601i
\(99\) −5.69314e32 −0.672003
\(100\) 0 0
\(101\) −8.22627e31 −0.0698072 −0.0349036 0.999391i \(-0.511112\pi\)
−0.0349036 + 0.999391i \(0.511112\pi\)
\(102\) 6.25891e32i 0.451436i
\(103\) − 1.80950e33i − 1.11108i −0.831491 0.555539i \(-0.812512\pi\)
0.831491 0.555539i \(-0.187488\pi\)
\(104\) −1.86444e33 −0.976110
\(105\) 0 0
\(106\) −3.22762e33 −1.23406
\(107\) − 2.36850e33i − 0.775605i −0.921743 0.387802i \(-0.873234\pi\)
0.921743 0.387802i \(-0.126766\pi\)
\(108\) − 2.76192e31i − 0.00775744i
\(109\) 3.69273e33 0.890862 0.445431 0.895316i \(-0.353051\pi\)
0.445431 + 0.895316i \(0.353051\pi\)
\(110\) 0 0
\(111\) −1.20979e34 −2.16213
\(112\) 6.80697e33i 1.04920i
\(113\) 1.24016e34i 1.65076i 0.564575 + 0.825382i \(0.309040\pi\)
−0.564575 + 0.825382i \(0.690960\pi\)
\(114\) −1.54922e34 −1.78313
\(115\) 0 0
\(116\) 1.76887e33 0.152806
\(117\) 1.45428e34i 1.09039i
\(118\) 2.82246e34i 1.83897i
\(119\) −5.22472e33 −0.296168
\(120\) 0 0
\(121\) −1.40647e34 −0.605582
\(122\) − 6.80681e33i − 0.255862i
\(123\) − 7.85059e34i − 2.57910i
\(124\) −1.74994e33 −0.0502997
\(125\) 0 0
\(126\) 4.92780e34 1.08777
\(127\) − 3.85830e34i − 0.747541i −0.927521 0.373771i \(-0.878065\pi\)
0.927521 0.373771i \(-0.121935\pi\)
\(128\) 6.46016e34i 1.09971i
\(129\) −4.17171e34 −0.624575
\(130\) 0 0
\(131\) 7.28441e34 0.846093 0.423046 0.906108i \(-0.360961\pi\)
0.423046 + 0.906108i \(0.360961\pi\)
\(132\) 6.79095e33i 0.0695762i
\(133\) − 1.29324e35i − 1.16984i
\(134\) 1.91870e35 1.53383
\(135\) 0 0
\(136\) −4.62596e34 −0.289608
\(137\) − 1.31044e35i − 0.726990i −0.931596 0.363495i \(-0.881583\pi\)
0.931596 0.363495i \(-0.118417\pi\)
\(138\) 1.94200e35i 0.955531i
\(139\) 2.86050e35 1.24939 0.624694 0.780870i \(-0.285224\pi\)
0.624694 + 0.780870i \(0.285224\pi\)
\(140\) 0 0
\(141\) 3.52215e35 1.21532
\(142\) 1.11029e35i 0.340941i
\(143\) − 2.33998e35i − 0.639982i
\(144\) 4.70102e35 1.14607
\(145\) 0 0
\(146\) 2.73069e35 0.530214
\(147\) − 3.35314e34i − 0.0581723i
\(148\) 7.45944e34i 0.115715i
\(149\) 4.76533e35 0.661485 0.330742 0.943721i \(-0.392701\pi\)
0.330742 + 0.943721i \(0.392701\pi\)
\(150\) 0 0
\(151\) 2.97234e35 0.331115 0.165557 0.986200i \(-0.447058\pi\)
0.165557 + 0.986200i \(0.447058\pi\)
\(152\) − 1.14503e36i − 1.14392i
\(153\) 3.60829e35i 0.323514i
\(154\) −7.92896e35 −0.638446
\(155\) 0 0
\(156\) 1.73471e35 0.112894
\(157\) 2.26342e36i 1.32562i 0.748786 + 0.662812i \(0.230637\pi\)
−0.748786 + 0.662812i \(0.769363\pi\)
\(158\) 6.67056e35i 0.351820i
\(159\) −3.59971e36 −1.71086
\(160\) 0 0
\(161\) −1.62112e36 −0.626884
\(162\) 2.74968e36i 0.960030i
\(163\) − 1.48676e35i − 0.0468970i −0.999725 0.0234485i \(-0.992535\pi\)
0.999725 0.0234485i \(-0.00746458\pi\)
\(164\) −4.84059e35 −0.138031
\(165\) 0 0
\(166\) −3.93768e36 −0.919300
\(167\) 1.71469e36i 0.362546i 0.983433 + 0.181273i \(0.0580217\pi\)
−0.983433 + 0.181273i \(0.941978\pi\)
\(168\) 7.04592e36i 1.35000i
\(169\) −2.21226e35 −0.0384331
\(170\) 0 0
\(171\) −8.93133e36 −1.27785
\(172\) 2.57224e35i 0.0334265i
\(173\) 6.70788e36i 0.792181i 0.918211 + 0.396090i \(0.129633\pi\)
−0.918211 + 0.396090i \(0.870367\pi\)
\(174\) 2.75932e37 2.96306
\(175\) 0 0
\(176\) −7.56408e36 −0.672662
\(177\) 3.14784e37i 2.54949i
\(178\) − 1.44120e37i − 1.06364i
\(179\) 7.92192e36 0.533035 0.266517 0.963830i \(-0.414127\pi\)
0.266517 + 0.963830i \(0.414127\pi\)
\(180\) 0 0
\(181\) 2.84989e37 1.59637 0.798183 0.602416i \(-0.205795\pi\)
0.798183 + 0.602416i \(0.205795\pi\)
\(182\) 2.02541e37i 1.03594i
\(183\) − 7.59152e36i − 0.354720i
\(184\) −1.43533e37 −0.612997
\(185\) 0 0
\(186\) −2.72979e37 −0.975362
\(187\) − 5.80585e36i − 0.189880i
\(188\) − 2.17172e36i − 0.0650428i
\(189\) 3.59651e36 0.0986874
\(190\) 0 0
\(191\) −1.67422e37 −0.386155 −0.193077 0.981184i \(-0.561847\pi\)
−0.193077 + 0.981184i \(0.561847\pi\)
\(192\) 6.19813e37i 1.31157i
\(193\) − 2.80272e37i − 0.544357i −0.962247 0.272179i \(-0.912256\pi\)
0.962247 0.272179i \(-0.0877441\pi\)
\(194\) 9.65216e37 1.72146
\(195\) 0 0
\(196\) −2.06751e35 −0.00311332
\(197\) 5.75950e37i 0.797428i 0.917075 + 0.398714i \(0.130543\pi\)
−0.917075 + 0.398714i \(0.869457\pi\)
\(198\) 5.47589e37i 0.697396i
\(199\) 1.44611e38 1.69483 0.847415 0.530931i \(-0.178158\pi\)
0.847415 + 0.530931i \(0.178158\pi\)
\(200\) 0 0
\(201\) 2.13989e38 2.12646
\(202\) 7.91236e36i 0.0724450i
\(203\) 2.30339e38i 1.94394i
\(204\) 4.30408e36 0.0334952
\(205\) 0 0
\(206\) −1.74045e38 −1.15306
\(207\) 1.11957e38i 0.684766i
\(208\) 1.93220e38i 1.09146i
\(209\) 1.43708e38 0.750008
\(210\) 0 0
\(211\) −4.42459e37 −0.197339 −0.0986695 0.995120i \(-0.531459\pi\)
−0.0986695 + 0.995120i \(0.531459\pi\)
\(212\) 2.21954e37i 0.0915634i
\(213\) 1.23829e38i 0.472670i
\(214\) −2.27812e38 −0.804912
\(215\) 0 0
\(216\) 3.18435e37 0.0965014
\(217\) − 2.27873e38i − 0.639894i
\(218\) − 3.55182e38i − 0.924524i
\(219\) 3.04550e38 0.735073
\(220\) 0 0
\(221\) −1.48307e38 −0.308099
\(222\) 1.16363e39i 2.24383i
\(223\) 4.20819e38i 0.753469i 0.926321 + 0.376734i \(0.122953\pi\)
−0.926321 + 0.376734i \(0.877047\pi\)
\(224\) 9.05132e37 0.150529
\(225\) 0 0
\(226\) 1.19284e39 1.71314
\(227\) − 2.29758e38i − 0.306793i −0.988165 0.153396i \(-0.950979\pi\)
0.988165 0.153396i \(-0.0490211\pi\)
\(228\) 1.06536e38i 0.132303i
\(229\) 1.22734e39 1.41801 0.709007 0.705202i \(-0.249144\pi\)
0.709007 + 0.705202i \(0.249144\pi\)
\(230\) 0 0
\(231\) −8.84304e38 −0.885123
\(232\) 2.03942e39i 1.90088i
\(233\) − 2.11378e39i − 1.83522i −0.397486 0.917608i \(-0.630117\pi\)
0.397486 0.917608i \(-0.369883\pi\)
\(234\) 1.39879e39 1.13159
\(235\) 0 0
\(236\) 1.94093e38 0.136446
\(237\) 7.43956e38i 0.487753i
\(238\) 5.02535e38i 0.307359i
\(239\) 9.09248e38 0.518940 0.259470 0.965751i \(-0.416452\pi\)
0.259470 + 0.965751i \(0.416452\pi\)
\(240\) 0 0
\(241\) −1.71576e39 −0.853443 −0.426722 0.904383i \(-0.640332\pi\)
−0.426722 + 0.904383i \(0.640332\pi\)
\(242\) 1.35280e39i 0.628465i
\(243\) 3.29880e39i 1.43170i
\(244\) −4.68085e37 −0.0189842
\(245\) 0 0
\(246\) −7.55101e39 −2.67655
\(247\) − 3.67093e39i − 1.21696i
\(248\) − 2.01759e39i − 0.625720i
\(249\) −4.39163e39 −1.27449
\(250\) 0 0
\(251\) 3.71395e39 0.944537 0.472269 0.881455i \(-0.343435\pi\)
0.472269 + 0.881455i \(0.343435\pi\)
\(252\) − 3.38871e38i − 0.0807096i
\(253\) − 1.80143e39i − 0.401909i
\(254\) −3.71107e39 −0.775788
\(255\) 0 0
\(256\) 1.25043e39 0.229668
\(257\) 3.64681e39i 0.628082i 0.949409 + 0.314041i \(0.101683\pi\)
−0.949409 + 0.314041i \(0.898317\pi\)
\(258\) 4.01252e39i 0.648175i
\(259\) −9.71354e39 −1.47208
\(260\) 0 0
\(261\) 1.59076e40 2.12343
\(262\) − 7.00644e39i − 0.878064i
\(263\) − 3.70275e39i − 0.435767i −0.975975 0.217884i \(-0.930085\pi\)
0.975975 0.217884i \(-0.0699153\pi\)
\(264\) −7.82961e39 −0.865517
\(265\) 0 0
\(266\) −1.24389e40 −1.21404
\(267\) − 1.60734e40i − 1.47459i
\(268\) − 1.31944e39i − 0.113806i
\(269\) 1.93097e40 1.56626 0.783129 0.621859i \(-0.213622\pi\)
0.783129 + 0.621859i \(0.213622\pi\)
\(270\) 0 0
\(271\) −5.56561e37 −0.00399502 −0.00199751 0.999998i \(-0.500636\pi\)
−0.00199751 + 0.999998i \(0.500636\pi\)
\(272\) 4.79409e39i 0.323831i
\(273\) 2.25891e40i 1.43620i
\(274\) −1.26044e40 −0.754460
\(275\) 0 0
\(276\) 1.33546e39 0.0708976
\(277\) − 2.59199e40i − 1.29634i −0.761497 0.648168i \(-0.775535\pi\)
0.761497 0.648168i \(-0.224465\pi\)
\(278\) − 2.75134e40i − 1.29660i
\(279\) −1.57374e40 −0.698977
\(280\) 0 0
\(281\) 4.04959e40 1.59866 0.799331 0.600891i \(-0.205187\pi\)
0.799331 + 0.600891i \(0.205187\pi\)
\(282\) − 3.38774e40i − 1.26125i
\(283\) − 4.05121e40i − 1.42269i −0.702846 0.711343i \(-0.748087\pi\)
0.702846 0.711343i \(-0.251913\pi\)
\(284\) 7.63516e38 0.0252968
\(285\) 0 0
\(286\) −2.25069e40 −0.664165
\(287\) − 6.30332e40i − 1.75597i
\(288\) − 6.25102e39i − 0.164428i
\(289\) 3.65748e40 0.908588
\(290\) 0 0
\(291\) 1.07649e41 2.38658
\(292\) − 1.87782e39i − 0.0393403i
\(293\) − 7.90241e38i − 0.0156475i −0.999969 0.00782373i \(-0.997510\pi\)
0.999969 0.00782373i \(-0.00249040\pi\)
\(294\) −3.22519e39 −0.0603704
\(295\) 0 0
\(296\) −8.60035e40 −1.43947
\(297\) 3.99654e39i 0.0632707i
\(298\) − 4.58349e40i − 0.686480i
\(299\) −4.60164e40 −0.652136
\(300\) 0 0
\(301\) −3.34951e40 −0.425241
\(302\) − 2.85891e40i − 0.343627i
\(303\) 8.82452e39i 0.100436i
\(304\) −1.18664e41 −1.27910
\(305\) 0 0
\(306\) 3.47060e40 0.335739
\(307\) 1.53513e41i 1.40722i 0.710585 + 0.703612i \(0.248431\pi\)
−0.710585 + 0.703612i \(0.751569\pi\)
\(308\) 5.45253e39i 0.0473708i
\(309\) −1.94110e41 −1.59857
\(310\) 0 0
\(311\) −2.01085e41 −1.48879 −0.744395 0.667739i \(-0.767262\pi\)
−0.744395 + 0.667739i \(0.767262\pi\)
\(312\) 2.00003e41i 1.40438i
\(313\) 6.89540e40i 0.459280i 0.973276 + 0.229640i \(0.0737549\pi\)
−0.973276 + 0.229640i \(0.926245\pi\)
\(314\) 2.17705e41 1.37571
\(315\) 0 0
\(316\) 4.58716e39 0.0261040
\(317\) − 2.25823e41i − 1.21981i −0.792476 0.609903i \(-0.791208\pi\)
0.792476 0.609903i \(-0.208792\pi\)
\(318\) 3.46234e41i 1.77551i
\(319\) −2.55958e41 −1.24630
\(320\) 0 0
\(321\) −2.54075e41 −1.11591
\(322\) 1.55925e41i 0.650571i
\(323\) − 9.10814e40i − 0.361067i
\(324\) 1.89088e40 0.0712314
\(325\) 0 0
\(326\) −1.43002e40 −0.0486691
\(327\) − 3.96128e41i − 1.28173i
\(328\) − 5.58095e41i − 1.71708i
\(329\) 2.82797e41 0.827451
\(330\) 0 0
\(331\) −1.74008e41 −0.460688 −0.230344 0.973109i \(-0.573985\pi\)
−0.230344 + 0.973109i \(0.573985\pi\)
\(332\) 2.70784e40i 0.0682094i
\(333\) 6.70836e41i 1.60800i
\(334\) 1.64925e41 0.376245
\(335\) 0 0
\(336\) 7.30200e41 1.50954
\(337\) 8.05374e41i 1.58527i 0.609695 + 0.792636i \(0.291292\pi\)
−0.609695 + 0.792636i \(0.708708\pi\)
\(338\) 2.12784e40i 0.0398853i
\(339\) 1.33035e42 2.37505
\(340\) 0 0
\(341\) 2.53219e41 0.410250
\(342\) 8.59052e41i 1.32614i
\(343\) − 6.92795e41i − 1.01918i
\(344\) −2.96565e41 −0.415821
\(345\) 0 0
\(346\) 6.45191e41 0.822115
\(347\) 6.26061e41i 0.760642i 0.924854 + 0.380321i \(0.124187\pi\)
−0.924854 + 0.380321i \(0.875813\pi\)
\(348\) − 1.89751e41i − 0.219850i
\(349\) 9.05089e40 0.100017 0.0500083 0.998749i \(-0.484075\pi\)
0.0500083 + 0.998749i \(0.484075\pi\)
\(350\) 0 0
\(351\) 1.02089e41 0.102663
\(352\) 1.00581e41i 0.0965073i
\(353\) 1.08093e42i 0.989725i 0.868971 + 0.494862i \(0.164782\pi\)
−0.868971 + 0.494862i \(0.835218\pi\)
\(354\) 3.02772e42 2.64583
\(355\) 0 0
\(356\) −9.91071e40 −0.0789186
\(357\) 5.60469e41i 0.426114i
\(358\) − 7.61962e41i − 0.553176i
\(359\) 2.08351e41 0.144456 0.0722282 0.997388i \(-0.476989\pi\)
0.0722282 + 0.997388i \(0.476989\pi\)
\(360\) 0 0
\(361\) 6.73698e41 0.426183
\(362\) − 2.74114e42i − 1.65669i
\(363\) 1.50876e42i 0.871285i
\(364\) 1.39282e41 0.0768638
\(365\) 0 0
\(366\) −7.30183e41 −0.368124
\(367\) − 1.16121e42i − 0.559654i −0.960050 0.279827i \(-0.909723\pi\)
0.960050 0.279827i \(-0.0902771\pi\)
\(368\) 1.48750e42i 0.685437i
\(369\) −4.35320e42 −1.91811
\(370\) 0 0
\(371\) −2.89024e42 −1.16484
\(372\) 1.87720e41i 0.0723690i
\(373\) 1.34955e42i 0.497728i 0.968538 + 0.248864i \(0.0800572\pi\)
−0.968538 + 0.248864i \(0.919943\pi\)
\(374\) −5.58430e41 −0.197055
\(375\) 0 0
\(376\) 2.50388e42 0.809122
\(377\) 6.53832e42i 2.02225i
\(378\) − 3.45927e41i − 0.102416i
\(379\) −3.19198e42 −0.904717 −0.452358 0.891836i \(-0.649417\pi\)
−0.452358 + 0.891836i \(0.649417\pi\)
\(380\) 0 0
\(381\) −4.13890e42 −1.07553
\(382\) 1.61033e42i 0.400746i
\(383\) 3.06206e42i 0.729849i 0.931037 + 0.364925i \(0.118905\pi\)
−0.931037 + 0.364925i \(0.881095\pi\)
\(384\) 6.92997e42 1.58222
\(385\) 0 0
\(386\) −2.69577e42 −0.564926
\(387\) 2.31324e42i 0.464504i
\(388\) − 6.63752e41i − 0.127727i
\(389\) −5.44019e42 −1.00333 −0.501667 0.865061i \(-0.667280\pi\)
−0.501667 + 0.865061i \(0.667280\pi\)
\(390\) 0 0
\(391\) −1.14174e42 −0.193486
\(392\) − 2.38374e41i − 0.0387292i
\(393\) − 7.81416e42i − 1.21732i
\(394\) 5.53971e42 0.827560
\(395\) 0 0
\(396\) 3.76562e41 0.0517447
\(397\) 4.34919e42i 0.573277i 0.958039 + 0.286639i \(0.0925379\pi\)
−0.958039 + 0.286639i \(0.907462\pi\)
\(398\) − 1.39093e43i − 1.75887i
\(399\) −1.38729e43 −1.68311
\(400\) 0 0
\(401\) −1.13262e42 −0.126532 −0.0632660 0.997997i \(-0.520152\pi\)
−0.0632660 + 0.997997i \(0.520152\pi\)
\(402\) − 2.05823e43i − 2.20681i
\(403\) − 6.46834e42i − 0.665671i
\(404\) 5.44111e40 0.00537520
\(405\) 0 0
\(406\) 2.21549e43 2.01739
\(407\) − 1.07939e43i − 0.943783i
\(408\) 4.96238e42i 0.416675i
\(409\) −6.57876e42 −0.530530 −0.265265 0.964176i \(-0.585459\pi\)
−0.265265 + 0.964176i \(0.585459\pi\)
\(410\) 0 0
\(411\) −1.40574e43 −1.04596
\(412\) 1.19686e42i 0.0855538i
\(413\) 2.52743e43i 1.73582i
\(414\) 1.07685e43 0.710640
\(415\) 0 0
\(416\) 2.56928e42 0.156593
\(417\) − 3.06853e43i − 1.79757i
\(418\) − 1.38224e43i − 0.778349i
\(419\) 2.13895e43 1.15789 0.578947 0.815365i \(-0.303464\pi\)
0.578947 + 0.815365i \(0.303464\pi\)
\(420\) 0 0
\(421\) −5.34625e42 −0.267544 −0.133772 0.991012i \(-0.542709\pi\)
−0.133772 + 0.991012i \(0.542709\pi\)
\(422\) 4.25575e42i 0.204796i
\(423\) − 1.95305e43i − 0.903852i
\(424\) −2.55902e43 −1.13903
\(425\) 0 0
\(426\) 1.19104e43 0.490531
\(427\) − 6.09531e42i − 0.241511i
\(428\) 1.56660e42i 0.0597221i
\(429\) −2.51016e43 −0.920779
\(430\) 0 0
\(431\) 2.33735e43 0.794053 0.397026 0.917807i \(-0.370042\pi\)
0.397026 + 0.917807i \(0.370042\pi\)
\(432\) − 3.30008e42i − 0.107905i
\(433\) 1.61313e43i 0.507714i 0.967242 + 0.253857i \(0.0816993\pi\)
−0.967242 + 0.253857i \(0.918301\pi\)
\(434\) −2.19178e43 −0.664074
\(435\) 0 0
\(436\) −2.44249e42 −0.0685970
\(437\) − 2.82605e43i − 0.764252i
\(438\) − 2.92928e43i − 0.762849i
\(439\) −4.60088e43 −1.15392 −0.576961 0.816772i \(-0.695762\pi\)
−0.576961 + 0.816772i \(0.695762\pi\)
\(440\) 0 0
\(441\) −1.85934e42 −0.0432635
\(442\) 1.42648e43i 0.319741i
\(443\) 5.92450e43i 1.27935i 0.768644 + 0.639676i \(0.220932\pi\)
−0.768644 + 0.639676i \(0.779068\pi\)
\(444\) 8.00193e42 0.166485
\(445\) 0 0
\(446\) 4.04761e43 0.781940
\(447\) − 5.11189e43i − 0.951716i
\(448\) 4.97655e43i 0.892978i
\(449\) 1.64413e43 0.284361 0.142181 0.989841i \(-0.454589\pi\)
0.142181 + 0.989841i \(0.454589\pi\)
\(450\) 0 0
\(451\) 7.00442e43 1.12579
\(452\) − 8.20283e42i − 0.127110i
\(453\) − 3.18850e43i − 0.476394i
\(454\) −2.20991e43 −0.318385
\(455\) 0 0
\(456\) −1.22830e44 −1.64583
\(457\) 2.69494e42i 0.0348283i 0.999848 + 0.0174141i \(0.00554338\pi\)
−0.999848 + 0.0174141i \(0.994457\pi\)
\(458\) − 1.18051e44i − 1.47160i
\(459\) 2.53299e42 0.0304596
\(460\) 0 0
\(461\) −8.91542e43 −0.997876 −0.498938 0.866638i \(-0.666276\pi\)
−0.498938 + 0.866638i \(0.666276\pi\)
\(462\) 8.50559e43i 0.918569i
\(463\) 7.81223e43i 0.814121i 0.913401 + 0.407060i \(0.133446\pi\)
−0.913401 + 0.407060i \(0.866554\pi\)
\(464\) 2.11354e44 2.12551
\(465\) 0 0
\(466\) −2.03312e44 −1.90456
\(467\) 1.34336e44i 1.21469i 0.794439 + 0.607344i \(0.207765\pi\)
−0.794439 + 0.607344i \(0.792235\pi\)
\(468\) − 9.61907e42i − 0.0839608i
\(469\) 1.71814e44 1.44780
\(470\) 0 0
\(471\) 2.42802e44 1.90725
\(472\) 2.23779e44i 1.69737i
\(473\) − 3.72207e43i − 0.272631i
\(474\) 7.15567e43 0.506183
\(475\) 0 0
\(476\) 3.45579e42 0.0228052
\(477\) 1.99606e44i 1.27239i
\(478\) − 8.74551e43i − 0.538549i
\(479\) −6.08543e43 −0.362039 −0.181020 0.983479i \(-0.557940\pi\)
−0.181020 + 0.983479i \(0.557940\pi\)
\(480\) 0 0
\(481\) −2.75725e44 −1.53138
\(482\) 1.65028e44i 0.885692i
\(483\) 1.73901e44i 0.901933i
\(484\) 9.30285e42 0.0466302
\(485\) 0 0
\(486\) 3.17292e44 1.48580
\(487\) − 2.29550e44i − 1.03908i −0.854446 0.519541i \(-0.826103\pi\)
0.854446 0.519541i \(-0.173897\pi\)
\(488\) − 5.39678e43i − 0.236161i
\(489\) −1.59488e43 −0.0674734
\(490\) 0 0
\(491\) −5.01787e43 −0.198461 −0.0992305 0.995064i \(-0.531638\pi\)
−0.0992305 + 0.995064i \(0.531638\pi\)
\(492\) 5.19262e43i 0.198592i
\(493\) 1.62226e44i 0.599992i
\(494\) −3.53085e44 −1.26295
\(495\) 0 0
\(496\) −2.09092e44 −0.699663
\(497\) 9.94235e43i 0.321817i
\(498\) 4.22405e44i 1.32265i
\(499\) 1.28404e44 0.388974 0.194487 0.980905i \(-0.437696\pi\)
0.194487 + 0.980905i \(0.437696\pi\)
\(500\) 0 0
\(501\) 1.83939e44 0.521615
\(502\) − 3.57222e44i − 0.980228i
\(503\) − 3.85407e44i − 1.02341i −0.859163 0.511703i \(-0.829015\pi\)
0.859163 0.511703i \(-0.170985\pi\)
\(504\) 3.90700e44 1.00401
\(505\) 0 0
\(506\) −1.73268e44 −0.417096
\(507\) 2.37314e43i 0.0552959i
\(508\) 2.55200e43i 0.0575612i
\(509\) −5.40769e44 −1.18078 −0.590389 0.807119i \(-0.701026\pi\)
−0.590389 + 0.807119i \(0.701026\pi\)
\(510\) 0 0
\(511\) 2.44526e44 0.500473
\(512\) 4.34652e44i 0.861365i
\(513\) 6.26972e43i 0.120313i
\(514\) 3.50765e44 0.651815
\(515\) 0 0
\(516\) 2.75930e43 0.0480927
\(517\) 3.14251e44i 0.530497i
\(518\) 9.34287e44i 1.52770i
\(519\) 7.19570e44 1.13976
\(520\) 0 0
\(521\) −9.57656e42 −0.0142360 −0.00711799 0.999975i \(-0.502266\pi\)
−0.00711799 + 0.999975i \(0.502266\pi\)
\(522\) − 1.53006e45i − 2.20366i
\(523\) 7.88861e43i 0.110084i 0.998484 + 0.0550418i \(0.0175292\pi\)
−0.998484 + 0.0550418i \(0.982471\pi\)
\(524\) −4.81813e43 −0.0651497
\(525\) 0 0
\(526\) −3.56145e44 −0.452233
\(527\) − 1.60489e44i − 0.197502i
\(528\) 8.11417e44i 0.967798i
\(529\) 5.10749e44 0.590458
\(530\) 0 0
\(531\) 1.74549e45 1.89609
\(532\) 8.55386e43i 0.0900783i
\(533\) − 1.78924e45i − 1.82671i
\(534\) −1.54601e45 −1.53031
\(535\) 0 0
\(536\) 1.52124e45 1.41572
\(537\) − 8.49804e44i − 0.766907i
\(538\) − 1.85729e45i − 1.62544i
\(539\) 2.99173e43 0.0253926
\(540\) 0 0
\(541\) −1.96609e45 −1.56981 −0.784907 0.619613i \(-0.787289\pi\)
−0.784907 + 0.619613i \(0.787289\pi\)
\(542\) 5.35323e42i 0.00414598i
\(543\) − 3.05715e45i − 2.29678i
\(544\) 6.37476e43 0.0464603
\(545\) 0 0
\(546\) 2.17271e45 1.49047
\(547\) − 4.44935e44i − 0.296146i −0.988976 0.148073i \(-0.952693\pi\)
0.988976 0.148073i \(-0.0473070\pi\)
\(548\) 8.66767e43i 0.0559787i
\(549\) −4.20954e44 −0.263810
\(550\) 0 0
\(551\) −4.01544e45 −2.36991
\(552\) 1.53972e45i 0.881954i
\(553\) 5.97331e44i 0.332086i
\(554\) −2.49308e45 −1.34532
\(555\) 0 0
\(556\) −1.89202e44 −0.0962037
\(557\) − 1.02978e45i − 0.508318i −0.967162 0.254159i \(-0.918201\pi\)
0.967162 0.254159i \(-0.0817986\pi\)
\(558\) 1.51368e45i 0.725389i
\(559\) −9.50782e44 −0.442370
\(560\) 0 0
\(561\) −6.22807e44 −0.273191
\(562\) − 3.89506e45i − 1.65907i
\(563\) 3.82628e45i 1.58266i 0.611388 + 0.791331i \(0.290612\pi\)
−0.611388 + 0.791331i \(0.709388\pi\)
\(564\) −2.32966e44 −0.0935808
\(565\) 0 0
\(566\) −3.89662e45 −1.47644
\(567\) 2.46227e45i 0.906180i
\(568\) 8.80294e44i 0.314688i
\(569\) 1.77526e45 0.616468 0.308234 0.951311i \(-0.400262\pi\)
0.308234 + 0.951311i \(0.400262\pi\)
\(570\) 0 0
\(571\) 1.00861e45 0.330544 0.165272 0.986248i \(-0.447150\pi\)
0.165272 + 0.986248i \(0.447150\pi\)
\(572\) 1.54774e44i 0.0492791i
\(573\) 1.79597e45i 0.555583i
\(574\) −6.06279e45 −1.82233
\(575\) 0 0
\(576\) 3.43690e45 0.975430
\(577\) 3.07790e45i 0.848895i 0.905452 + 0.424448i \(0.139532\pi\)
−0.905452 + 0.424448i \(0.860468\pi\)
\(578\) − 3.51791e45i − 0.942921i
\(579\) −3.00654e45 −0.783198
\(580\) 0 0
\(581\) −3.52609e45 −0.867735
\(582\) − 1.03541e46i − 2.47676i
\(583\) − 3.21171e45i − 0.746803i
\(584\) 2.16503e45 0.489387
\(585\) 0 0
\(586\) −7.60085e43 −0.0162387
\(587\) 3.80587e45i 0.790544i 0.918564 + 0.395272i \(0.129350\pi\)
−0.918564 + 0.395272i \(0.870650\pi\)
\(588\) 2.21787e43i 0.00447931i
\(589\) 3.97247e45 0.780114
\(590\) 0 0
\(591\) 6.17835e45 1.14731
\(592\) 8.91292e45i 1.60958i
\(593\) 1.41796e45i 0.249036i 0.992217 + 0.124518i \(0.0397384\pi\)
−0.992217 + 0.124518i \(0.960262\pi\)
\(594\) 3.84403e44 0.0656615
\(595\) 0 0
\(596\) −3.15194e44 −0.0509348
\(597\) − 1.55128e46i − 2.43845i
\(598\) 4.42605e45i 0.676778i
\(599\) −7.86874e45 −1.17048 −0.585239 0.810861i \(-0.698999\pi\)
−0.585239 + 0.810861i \(0.698999\pi\)
\(600\) 0 0
\(601\) 6.55936e45 0.923491 0.461746 0.887012i \(-0.347223\pi\)
0.461746 + 0.887012i \(0.347223\pi\)
\(602\) 3.22170e45i 0.441309i
\(603\) − 1.18658e46i − 1.58147i
\(604\) −1.96600e44 −0.0254961
\(605\) 0 0
\(606\) 8.48778e44 0.104231
\(607\) 1.16821e46i 1.39607i 0.716062 + 0.698037i \(0.245943\pi\)
−0.716062 + 0.698037i \(0.754057\pi\)
\(608\) 1.57790e45i 0.183514i
\(609\) 2.47090e46 2.79685
\(610\) 0 0
\(611\) 8.02738e45 0.860783
\(612\) − 2.38664e44i − 0.0249108i
\(613\) 2.09504e45i 0.212860i 0.994320 + 0.106430i \(0.0339420\pi\)
−0.994320 + 0.106430i \(0.966058\pi\)
\(614\) 1.47655e46 1.46040
\(615\) 0 0
\(616\) −6.28648e45 −0.589285
\(617\) 1.98589e45i 0.181238i 0.995886 + 0.0906192i \(0.0288846\pi\)
−0.995886 + 0.0906192i \(0.971115\pi\)
\(618\) 1.86702e46i 1.65898i
\(619\) 8.02615e45 0.694403 0.347201 0.937791i \(-0.387132\pi\)
0.347201 + 0.937791i \(0.387132\pi\)
\(620\) 0 0
\(621\) 7.85931e44 0.0644723
\(622\) 1.93412e46i 1.54505i
\(623\) − 1.29055e46i − 1.00397i
\(624\) 2.07272e46 1.57035
\(625\) 0 0
\(626\) 6.63227e45 0.476635
\(627\) − 1.54159e46i − 1.07908i
\(628\) − 1.49709e45i − 0.102074i
\(629\) −6.84116e45 −0.454354
\(630\) 0 0
\(631\) −2.32504e46 −1.46536 −0.732682 0.680571i \(-0.761732\pi\)
−0.732682 + 0.680571i \(0.761732\pi\)
\(632\) 5.28875e45i 0.324729i
\(633\) 4.74637e45i 0.283923i
\(634\) −2.17206e46 −1.26590
\(635\) 0 0
\(636\) 2.38096e45 0.131738
\(637\) − 7.64220e44i − 0.0412020i
\(638\) 2.46191e46i 1.29339i
\(639\) 6.86638e45 0.351531
\(640\) 0 0
\(641\) 1.10320e46 0.536412 0.268206 0.963362i \(-0.413569\pi\)
0.268206 + 0.963362i \(0.413569\pi\)
\(642\) 2.44379e46i 1.15807i
\(643\) − 1.90982e46i − 0.882085i −0.897486 0.441042i \(-0.854609\pi\)
0.897486 0.441042i \(-0.145391\pi\)
\(644\) 1.07226e45 0.0482705
\(645\) 0 0
\(646\) −8.76058e45 −0.374711
\(647\) 2.77860e46i 1.15853i 0.815141 + 0.579263i \(0.196659\pi\)
−0.815141 + 0.579263i \(0.803341\pi\)
\(648\) 2.18009e46i 0.886107i
\(649\) −2.80855e46 −1.11287
\(650\) 0 0
\(651\) −2.44445e46 −0.920652
\(652\) 9.83389e43i 0.00361110i
\(653\) 1.10232e46i 0.394676i 0.980335 + 0.197338i \(0.0632297\pi\)
−0.980335 + 0.197338i \(0.936770\pi\)
\(654\) −3.81012e46 −1.33017
\(655\) 0 0
\(656\) −5.78379e46 −1.91999
\(657\) − 1.68875e46i − 0.546683i
\(658\) − 2.72006e46i − 0.858718i
\(659\) −2.82127e46 −0.868632 −0.434316 0.900761i \(-0.643010\pi\)
−0.434316 + 0.900761i \(0.643010\pi\)
\(660\) 0 0
\(661\) 3.86133e44 0.0113087 0.00565435 0.999984i \(-0.498200\pi\)
0.00565435 + 0.999984i \(0.498200\pi\)
\(662\) 1.67368e46i 0.478096i
\(663\) 1.59093e46i 0.443279i
\(664\) −3.12199e46 −0.848514
\(665\) 0 0
\(666\) 6.45237e46 1.66876
\(667\) 5.03350e46i 1.26997i
\(668\) − 1.13415e45i − 0.0279163i
\(669\) 4.51423e46 1.08406
\(670\) 0 0
\(671\) 6.77327e45 0.154838
\(672\) − 9.70957e45i − 0.216574i
\(673\) 3.62752e46i 0.789517i 0.918785 + 0.394759i \(0.129172\pi\)
−0.918785 + 0.394759i \(0.870828\pi\)
\(674\) 7.74641e46 1.64517
\(675\) 0 0
\(676\) 1.46326e44 0.00295937
\(677\) − 6.71996e46i − 1.32634i −0.748471 0.663168i \(-0.769212\pi\)
0.748471 0.663168i \(-0.230788\pi\)
\(678\) − 1.27959e47i − 2.46479i
\(679\) 8.64325e46 1.62490
\(680\) 0 0
\(681\) −2.46468e46 −0.441400
\(682\) − 2.43556e46i − 0.425752i
\(683\) − 5.58915e45i − 0.0953684i −0.998862 0.0476842i \(-0.984816\pi\)
0.998862 0.0476842i \(-0.0151841\pi\)
\(684\) 5.90746e45 0.0983955
\(685\) 0 0
\(686\) −6.66358e46 −1.05769
\(687\) − 1.31660e47i − 2.04018i
\(688\) 3.07344e46i 0.464960i
\(689\) −8.20415e46 −1.21176
\(690\) 0 0
\(691\) 5.89510e46 0.830048 0.415024 0.909810i \(-0.363773\pi\)
0.415024 + 0.909810i \(0.363773\pi\)
\(692\) − 4.43680e45i − 0.0609985i
\(693\) 4.90352e46i 0.658278i
\(694\) 6.02171e46 0.789384
\(695\) 0 0
\(696\) 2.18773e47 2.73490
\(697\) − 4.43938e46i − 0.541977i
\(698\) − 8.70551e45i − 0.103796i
\(699\) −2.26750e47 −2.64043
\(700\) 0 0
\(701\) −9.67830e46 −1.07511 −0.537554 0.843229i \(-0.680651\pi\)
−0.537554 + 0.843229i \(0.680651\pi\)
\(702\) − 9.81937e45i − 0.106542i
\(703\) − 1.69334e47i − 1.79466i
\(704\) −5.53007e46 −0.572508
\(705\) 0 0
\(706\) 1.03968e47 1.02712
\(707\) 7.08530e45i 0.0683814i
\(708\) − 2.08208e46i − 0.196313i
\(709\) 1.89011e46 0.174110 0.0870552 0.996203i \(-0.472254\pi\)
0.0870552 + 0.996203i \(0.472254\pi\)
\(710\) 0 0
\(711\) 4.12528e46 0.362748
\(712\) − 1.14265e47i − 0.981735i
\(713\) − 4.97963e46i − 0.418042i
\(714\) 5.39081e46 0.442215
\(715\) 0 0
\(716\) −5.23980e45 −0.0410440
\(717\) − 9.75372e46i − 0.746628i
\(718\) − 2.00400e46i − 0.149915i
\(719\) −1.61675e47 −1.18199 −0.590997 0.806674i \(-0.701265\pi\)
−0.590997 + 0.806674i \(0.701265\pi\)
\(720\) 0 0
\(721\) −1.55853e47 −1.08838
\(722\) − 6.47989e46i − 0.442287i
\(723\) 1.84053e47i 1.22790i
\(724\) −1.88501e46 −0.122921
\(725\) 0 0
\(726\) 1.45118e47 0.904208
\(727\) 6.77908e44i 0.00412908i 0.999998 + 0.00206454i \(0.000657164\pi\)
−0.999998 + 0.00206454i \(0.999343\pi\)
\(728\) 1.60585e47i 0.956173i
\(729\) 1.94950e47 1.13480
\(730\) 0 0
\(731\) −2.35903e46 −0.131249
\(732\) 5.02127e45i 0.0273137i
\(733\) − 1.03172e46i − 0.0548714i −0.999624 0.0274357i \(-0.991266\pi\)
0.999624 0.0274357i \(-0.00873415\pi\)
\(734\) −1.11690e47 −0.580802
\(735\) 0 0
\(736\) 1.97795e46 0.0983401
\(737\) 1.90924e47i 0.928213i
\(738\) 4.18708e47i 1.99059i
\(739\) −3.22918e46 −0.150127 −0.0750636 0.997179i \(-0.523916\pi\)
−0.0750636 + 0.997179i \(0.523916\pi\)
\(740\) 0 0
\(741\) −3.93790e47 −1.75091
\(742\) 2.77995e47i 1.20885i
\(743\) − 3.20296e47i − 1.36218i −0.732198 0.681092i \(-0.761506\pi\)
0.732198 0.681092i \(-0.238494\pi\)
\(744\) −2.16432e47 −0.900259
\(745\) 0 0
\(746\) 1.29805e47 0.516535
\(747\) 2.43519e47i 0.947856i
\(748\) 3.84017e45i 0.0146209i
\(749\) −2.03999e47 −0.759763
\(750\) 0 0
\(751\) −5.04227e47 −1.79708 −0.898540 0.438891i \(-0.855371\pi\)
−0.898540 + 0.438891i \(0.855371\pi\)
\(752\) − 2.59488e47i − 0.904739i
\(753\) − 3.98404e47i − 1.35896i
\(754\) 6.28882e47 2.09866
\(755\) 0 0
\(756\) −2.37884e45 −0.00759900
\(757\) − 5.03192e47i − 1.57272i −0.617767 0.786361i \(-0.711962\pi\)
0.617767 0.786361i \(-0.288038\pi\)
\(758\) 3.07017e47i 0.938903i
\(759\) −1.93243e47 −0.578249
\(760\) 0 0
\(761\) −3.34360e47 −0.958004 −0.479002 0.877814i \(-0.659001\pi\)
−0.479002 + 0.877814i \(0.659001\pi\)
\(762\) 3.98096e47i 1.11617i
\(763\) − 3.18055e47i − 0.872666i
\(764\) 1.10738e46 0.0297342
\(765\) 0 0
\(766\) 2.94521e47 0.757428
\(767\) 7.17429e47i 1.80574i
\(768\) − 1.34137e47i − 0.330436i
\(769\) 5.27752e47 1.27246 0.636231 0.771498i \(-0.280492\pi\)
0.636231 + 0.771498i \(0.280492\pi\)
\(770\) 0 0
\(771\) 3.91202e47 0.903658
\(772\) 1.85380e46i 0.0419159i
\(773\) − 1.62889e47i − 0.360520i −0.983619 0.180260i \(-0.942306\pi\)
0.983619 0.180260i \(-0.0576939\pi\)
\(774\) 2.22497e47 0.482056
\(775\) 0 0
\(776\) 7.65272e47 1.58890
\(777\) 1.04200e48i 2.11797i
\(778\) 5.23259e47i 1.04125i
\(779\) 1.09884e48 2.14076
\(780\) 0 0
\(781\) −1.10482e47 −0.206324
\(782\) 1.09817e47i 0.200797i
\(783\) − 1.11670e47i − 0.199926i
\(784\) −2.47037e46 −0.0433059
\(785\) 0 0
\(786\) −7.51598e47 −1.26332
\(787\) 1.50460e46i 0.0247649i 0.999923 + 0.0123825i \(0.00394156\pi\)
−0.999923 + 0.0123825i \(0.996058\pi\)
\(788\) − 3.80951e46i − 0.0614025i
\(789\) −3.97203e47 −0.626963
\(790\) 0 0
\(791\) 1.06816e48 1.61705
\(792\) 4.34156e47i 0.643696i
\(793\) − 1.73020e47i − 0.251239i
\(794\) 4.18322e47 0.594939
\(795\) 0 0
\(796\) −9.56503e46 −0.130503
\(797\) − 5.86479e47i − 0.783773i −0.920014 0.391887i \(-0.871823\pi\)
0.920014 0.391887i \(-0.128177\pi\)
\(798\) 1.33435e48i 1.74671i
\(799\) 1.99171e47 0.255391
\(800\) 0 0
\(801\) −8.91280e47 −1.09667
\(802\) 1.08940e47i 0.131313i
\(803\) 2.71724e47i 0.320864i
\(804\) −1.41539e47 −0.163739
\(805\) 0 0
\(806\) −6.22151e47 −0.690824
\(807\) − 2.07140e48i − 2.25347i
\(808\) 6.27331e46i 0.0668667i
\(809\) −1.28166e48 −1.33852 −0.669258 0.743030i \(-0.733388\pi\)
−0.669258 + 0.743030i \(0.733388\pi\)
\(810\) 0 0
\(811\) −2.05018e47 −0.205564 −0.102782 0.994704i \(-0.532774\pi\)
−0.102782 + 0.994704i \(0.532774\pi\)
\(812\) − 1.52353e47i − 0.149685i
\(813\) 5.97037e45i 0.00574787i
\(814\) −1.03820e48 −0.979445
\(815\) 0 0
\(816\) 5.14274e47 0.465915
\(817\) − 5.83913e47i − 0.518423i
\(818\) 6.32772e47i 0.550576i
\(819\) 1.25258e48 1.06812
\(820\) 0 0
\(821\) 8.74853e47 0.716594 0.358297 0.933608i \(-0.383358\pi\)
0.358297 + 0.933608i \(0.383358\pi\)
\(822\) 1.35210e48i 1.08549i
\(823\) 2.06347e47i 0.162368i 0.996699 + 0.0811841i \(0.0258702\pi\)
−0.996699 + 0.0811841i \(0.974130\pi\)
\(824\) −1.37992e48 −1.06427
\(825\) 0 0
\(826\) 2.43099e48 1.80141
\(827\) − 2.59525e48i − 1.88512i −0.334038 0.942559i \(-0.608412\pi\)
0.334038 0.942559i \(-0.391588\pi\)
\(828\) − 7.40521e46i − 0.0527274i
\(829\) 2.60174e48 1.81599 0.907996 0.418978i \(-0.137612\pi\)
0.907996 + 0.418978i \(0.137612\pi\)
\(830\) 0 0
\(831\) −2.78050e48 −1.86511
\(832\) 1.41263e48i 0.928950i
\(833\) − 1.89615e46i − 0.0122244i
\(834\) −2.95143e48 −1.86549
\(835\) 0 0
\(836\) −9.50527e46 −0.0577512
\(837\) 1.10475e47i 0.0658104i
\(838\) − 2.05733e48i − 1.20165i
\(839\) 2.01476e48 1.15386 0.576928 0.816795i \(-0.304251\pi\)
0.576928 + 0.816795i \(0.304251\pi\)
\(840\) 0 0
\(841\) 5.33585e48 2.93812
\(842\) 5.14224e47i 0.277653i
\(843\) − 4.34409e48i − 2.30009i
\(844\) 2.92656e46 0.0151952
\(845\) 0 0
\(846\) −1.87852e48 −0.938006
\(847\) 1.21140e48i 0.593213i
\(848\) 2.65202e48i 1.27364i
\(849\) −4.34583e48 −2.04690
\(850\) 0 0
\(851\) −2.12266e48 −0.961707
\(852\) − 8.19042e46i − 0.0363959i
\(853\) − 1.85939e48i − 0.810422i −0.914223 0.405211i \(-0.867198\pi\)
0.914223 0.405211i \(-0.132802\pi\)
\(854\) −5.86272e47 −0.250636
\(855\) 0 0
\(856\) −1.80621e48 −0.742933
\(857\) 1.28606e48i 0.518891i 0.965758 + 0.259446i \(0.0835398\pi\)
−0.965758 + 0.259446i \(0.916460\pi\)
\(858\) 2.41437e48i 0.955572i
\(859\) 3.42147e48 1.32839 0.664195 0.747559i \(-0.268774\pi\)
0.664195 + 0.747559i \(0.268774\pi\)
\(860\) 0 0
\(861\) −6.76173e48 −2.52642
\(862\) − 2.24816e48i − 0.824057i
\(863\) − 1.37063e48i − 0.492882i −0.969158 0.246441i \(-0.920739\pi\)
0.969158 0.246441i \(-0.0792612\pi\)
\(864\) −4.38816e46 −0.0154812
\(865\) 0 0
\(866\) 1.55157e48 0.526899
\(867\) − 3.92347e48i − 1.30724i
\(868\) 1.50723e47i 0.0492723i
\(869\) −6.63769e47 −0.212907
\(870\) 0 0
\(871\) 4.87706e48 1.50612
\(872\) − 2.81606e48i − 0.853335i
\(873\) − 5.96919e48i − 1.77493i
\(874\) −2.71821e48 −0.793131
\(875\) 0 0
\(876\) −2.01439e47 −0.0566011
\(877\) − 1.64670e48i − 0.454069i −0.973887 0.227035i \(-0.927097\pi\)
0.973887 0.227035i \(-0.0729031\pi\)
\(878\) 4.42531e48i 1.19753i
\(879\) −8.47711e46 −0.0225129
\(880\) 0 0
\(881\) −3.42124e48 −0.875148 −0.437574 0.899182i \(-0.644162\pi\)
−0.437574 + 0.899182i \(0.644162\pi\)
\(882\) 1.78839e47i 0.0448983i
\(883\) − 7.49779e48i − 1.84749i −0.383012 0.923743i \(-0.625113\pi\)
0.383012 0.923743i \(-0.374887\pi\)
\(884\) 9.80950e46 0.0237238
\(885\) 0 0
\(886\) 5.69842e48 1.32769
\(887\) 1.29073e48i 0.295186i 0.989048 + 0.147593i \(0.0471526\pi\)
−0.989048 + 0.147593i \(0.952847\pi\)
\(888\) 9.22581e48i 2.07105i
\(889\) −3.32317e48 −0.732273
\(890\) 0 0
\(891\) −2.73613e48 −0.580972
\(892\) − 2.78343e47i − 0.0580176i
\(893\) 4.92994e48i 1.00877i
\(894\) −4.91682e48 −0.987678
\(895\) 0 0
\(896\) 5.56415e48 1.07725
\(897\) 4.93630e48i 0.938266i
\(898\) − 1.58139e48i − 0.295106i
\(899\) −7.07538e48 −1.29633
\(900\) 0 0
\(901\) −2.03557e48 −0.359524
\(902\) − 6.73713e48i − 1.16833i
\(903\) 3.59311e48i 0.611818i
\(904\) 9.45743e48 1.58123
\(905\) 0 0
\(906\) −3.06683e48 −0.494395
\(907\) − 4.05607e48i − 0.642074i −0.947067 0.321037i \(-0.895969\pi\)
0.947067 0.321037i \(-0.104031\pi\)
\(908\) 1.51969e47i 0.0236232i
\(909\) 4.89325e47 0.0746953
\(910\) 0 0
\(911\) −7.15538e48 −1.05337 −0.526683 0.850062i \(-0.676564\pi\)
−0.526683 + 0.850062i \(0.676564\pi\)
\(912\) 1.27294e49i 1.84032i
\(913\) − 3.91828e48i − 0.556324i
\(914\) 2.59210e47 0.0361443
\(915\) 0 0
\(916\) −8.11802e47 −0.109188
\(917\) − 6.27408e48i − 0.828812i
\(918\) − 2.43633e47i − 0.0316106i
\(919\) 1.51037e49 1.92476 0.962382 0.271700i \(-0.0875859\pi\)
0.962382 + 0.271700i \(0.0875859\pi\)
\(920\) 0 0
\(921\) 1.64678e49 2.02465
\(922\) 8.57521e48i 1.03558i
\(923\) 2.82220e48i 0.334780i
\(924\) 5.84906e47 0.0681551
\(925\) 0 0
\(926\) 7.51412e48 0.844884
\(927\) 1.07635e49i 1.18888i
\(928\) − 2.81040e48i − 0.304948i
\(929\) −1.54374e49 −1.64557 −0.822783 0.568356i \(-0.807580\pi\)
−0.822783 + 0.568356i \(0.807580\pi\)
\(930\) 0 0
\(931\) 4.69338e47 0.0482854
\(932\) 1.39812e48i 0.141313i
\(933\) 2.15709e49i 2.14201i
\(934\) 1.29210e49 1.26059
\(935\) 0 0
\(936\) 1.10903e49 1.04446
\(937\) − 8.72693e48i − 0.807530i −0.914863 0.403765i \(-0.867701\pi\)
0.914863 0.403765i \(-0.132299\pi\)
\(938\) − 1.65258e49i − 1.50250i
\(939\) 7.39687e48 0.660793
\(940\) 0 0
\(941\) −6.99024e48 −0.602925 −0.301463 0.953478i \(-0.597475\pi\)
−0.301463 + 0.953478i \(0.597475\pi\)
\(942\) − 2.33537e49i − 1.97932i
\(943\) − 1.37744e49i − 1.14717i
\(944\) 2.31912e49 1.89795
\(945\) 0 0
\(946\) −3.58003e48 −0.282933
\(947\) 2.43383e48i 0.189024i 0.995524 + 0.0945118i \(0.0301290\pi\)
−0.995524 + 0.0945118i \(0.969871\pi\)
\(948\) − 4.92076e47i − 0.0375573i
\(949\) 6.94104e48 0.520634
\(950\) 0 0
\(951\) −2.42246e49 −1.75501
\(952\) 3.98435e48i 0.283692i
\(953\) 2.16685e48i 0.151633i 0.997122 + 0.0758167i \(0.0241564\pi\)
−0.997122 + 0.0758167i \(0.975844\pi\)
\(954\) 1.91989e49 1.32047
\(955\) 0 0
\(956\) −6.01405e47 −0.0399587
\(957\) 2.74573e49i 1.79313i
\(958\) 5.85321e48i 0.375719i
\(959\) −1.12869e49 −0.712141
\(960\) 0 0
\(961\) −9.40382e48 −0.573282
\(962\) 2.65204e49i 1.58924i
\(963\) 1.40886e49i 0.829915i
\(964\) 1.13485e48 0.0657157
\(965\) 0 0
\(966\) 1.67265e49 0.936014
\(967\) 3.10853e49i 1.71009i 0.518556 + 0.855044i \(0.326470\pi\)
−0.518556 + 0.855044i \(0.673530\pi\)
\(968\) 1.07257e49i 0.580072i
\(969\) −9.77053e48 −0.519488
\(970\) 0 0
\(971\) −1.05532e49 −0.542334 −0.271167 0.962532i \(-0.587410\pi\)
−0.271167 + 0.962532i \(0.587410\pi\)
\(972\) − 2.18193e48i − 0.110242i
\(973\) − 2.46375e49i − 1.22387i
\(974\) −2.20790e49 −1.07834
\(975\) 0 0
\(976\) −5.59292e48 −0.264069
\(977\) 1.91994e49i 0.891306i 0.895206 + 0.445653i \(0.147029\pi\)
−0.895206 + 0.445653i \(0.852971\pi\)
\(978\) 1.53402e48i 0.0700230i
\(979\) 1.43409e49 0.643670
\(980\) 0 0
\(981\) −2.19655e49 −0.953242
\(982\) 4.82639e48i 0.205960i
\(983\) 5.26490e48i 0.220931i 0.993880 + 0.110466i \(0.0352342\pi\)
−0.993880 + 0.110466i \(0.964766\pi\)
\(984\) −5.98682e49 −2.47046
\(985\) 0 0
\(986\) 1.56035e49 0.622663
\(987\) − 3.03363e49i − 1.19050i
\(988\) 2.42807e48i 0.0937069i
\(989\) −7.31956e48 −0.277809
\(990\) 0 0
\(991\) −1.70362e48 −0.0625398 −0.0312699 0.999511i \(-0.509955\pi\)
−0.0312699 + 0.999511i \(0.509955\pi\)
\(992\) 2.78032e48i 0.100381i
\(993\) 1.86663e49i 0.662819i
\(994\) 9.56295e48 0.333977
\(995\) 0 0
\(996\) 2.90476e48 0.0981367
\(997\) − 3.60379e49i − 1.19754i −0.800921 0.598770i \(-0.795656\pi\)
0.800921 0.598770i \(-0.204344\pi\)
\(998\) − 1.23504e49i − 0.403672i
\(999\) 4.70921e48 0.151397
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.24.4 12
5.2 odd 4 25.34.a.c.1.5 6
5.3 odd 4 5.34.a.b.1.2 6
5.4 even 2 inner 25.34.b.c.24.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.a.b.1.2 6 5.3 odd 4
25.34.a.c.1.5 6 5.2 odd 4
25.34.b.c.24.4 12 1.1 even 1 trivial
25.34.b.c.24.9 12 5.4 even 2 inner