Properties

Label 25.10.a.c.1.3
Level $25$
Weight $10$
Character 25.1
Self dual yes
Analytic conductor $12.876$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.8758959041\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 652 x + 4000\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(6.48955\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-expansion

\(f(q)\) \(=\) \(q+20.2014 q^{2} -30.5073 q^{3} -103.903 q^{4} -616.291 q^{6} +4010.25 q^{7} -12442.1 q^{8} -18752.3 q^{9} +O(q^{10})\) \(q+20.2014 q^{2} -30.5073 q^{3} -103.903 q^{4} -616.291 q^{6} +4010.25 q^{7} -12442.1 q^{8} -18752.3 q^{9} -42110.0 q^{11} +3169.79 q^{12} -123743. q^{13} +81012.8 q^{14} -198150. q^{16} -319945. q^{17} -378823. q^{18} +1.08733e6 q^{19} -122342. q^{21} -850682. q^{22} -1.50672e6 q^{23} +379575. q^{24} -2.49979e6 q^{26} +1.17256e6 q^{27} -416676. q^{28} -2.62160e6 q^{29} +3.27023e6 q^{31} +2.36745e6 q^{32} +1.28466e6 q^{33} -6.46335e6 q^{34} +1.94841e6 q^{36} +2.51034e6 q^{37} +2.19655e7 q^{38} +3.77508e6 q^{39} +2.95349e7 q^{41} -2.47148e6 q^{42} +1.42413e7 q^{43} +4.37534e6 q^{44} -3.04378e7 q^{46} -1.35318e6 q^{47} +6.04503e6 q^{48} -2.42715e7 q^{49} +9.76067e6 q^{51} +1.28573e7 q^{52} -9.73342e7 q^{53} +2.36873e7 q^{54} -4.98960e7 q^{56} -3.31714e7 q^{57} -5.29599e7 q^{58} -7.48924e6 q^{59} -9.11752e7 q^{61} +6.60633e7 q^{62} -7.52015e7 q^{63} +1.49279e8 q^{64} +2.59520e7 q^{66} -2.94376e8 q^{67} +3.32432e7 q^{68} +4.59659e7 q^{69} +1.56193e8 q^{71} +2.33318e8 q^{72} +2.82539e8 q^{73} +5.07124e7 q^{74} -1.12976e8 q^{76} -1.68872e8 q^{77} +7.62619e7 q^{78} -5.55294e8 q^{79} +3.33330e8 q^{81} +5.96647e8 q^{82} -6.48378e6 q^{83} +1.27117e7 q^{84} +2.87694e8 q^{86} +7.99778e7 q^{87} +5.23937e8 q^{88} -5.99001e8 q^{89} -4.96242e8 q^{91} +1.56552e8 q^{92} -9.97660e7 q^{93} -2.73361e7 q^{94} -7.22244e7 q^{96} -9.25317e8 q^{97} -4.90319e8 q^{98} +7.89660e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 33q^{2} - 89q^{3} + 341q^{4} + 3421q^{6} - 5258q^{7} + 105q^{8} + 58234q^{9} + O(q^{10}) \) \( 3q - 33q^{2} - 89q^{3} + 341q^{4} + 3421q^{6} - 5258q^{7} + 105q^{8} + 58234q^{9} - 54699q^{11} - 141853q^{12} - 215884q^{13} + 272922q^{14} - 699247q^{16} - 334983q^{17} - 2571854q^{18} + 818845q^{19} - 2375394q^{21} - 72761q^{22} - 3526854q^{23} + 2805735q^{24} - 1280004q^{26} - 6633395q^{27} + 428134q^{28} + 2175480q^{29} + 4274066q^{31} + 9464577q^{32} + 22122137q^{33} - 6838963q^{34} + 26795798q^{36} + 10305042q^{37} + 24180495q^{38} - 50414092q^{39} + 5926311q^{41} + 47560254q^{42} + 24429956q^{43} - 21995703q^{44} + 14223246q^{46} - 66858708q^{47} + 29557871q^{48} - 6453929q^{49} - 25634699q^{51} + 57862932q^{52} - 132620514q^{53} + 282250495q^{54} - 169538130q^{56} - 252946415q^{57} - 201908320q^{58} + 5670960q^{59} + 125306926q^{61} + 39831174q^{62} - 284323404q^{63} + 167542401q^{64} - 556915843q^{66} - 88829483q^{67} + 71162559q^{68} - 314274942q^{69} + 297550596q^{71} + 552918990q^{72} - 181321729q^{73} - 251507358q^{74} + 89414865q^{76} - 561214086q^{77} + 1439424692q^{78} - 310025170q^{79} + 1398847363q^{81} + 1368322979q^{82} + 731088801q^{83} + 38267082q^{84} + 33947196q^{86} - 1046385560q^{87} + 943671285q^{88} - 1103860035q^{89} + 1183187656q^{91} + 190024242q^{92} - 107386758q^{93} + 1727891132q^{94} - 2567180449q^{96} + 332236842q^{97} - 457927581q^{98} - 892234522q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 20.2014 0.892785 0.446393 0.894837i \(-0.352709\pi\)
0.446393 + 0.894837i \(0.352709\pi\)
\(3\) −30.5073 −0.217449 −0.108725 0.994072i \(-0.534677\pi\)
−0.108725 + 0.994072i \(0.534677\pi\)
\(4\) −103.903 −0.202935
\(5\) 0 0
\(6\) −616.291 −0.194136
\(7\) 4010.25 0.631292 0.315646 0.948877i \(-0.397779\pi\)
0.315646 + 0.948877i \(0.397779\pi\)
\(8\) −12442.1 −1.07396
\(9\) −18752.3 −0.952716
\(10\) 0 0
\(11\) −42110.0 −0.867198 −0.433599 0.901106i \(-0.642757\pi\)
−0.433599 + 0.901106i \(0.642757\pi\)
\(12\) 3169.79 0.0441281
\(13\) −123743. −1.20165 −0.600824 0.799382i \(-0.705161\pi\)
−0.600824 + 0.799382i \(0.705161\pi\)
\(14\) 81012.8 0.563608
\(15\) 0 0
\(16\) −198150. −0.755883
\(17\) −319945. −0.929085 −0.464543 0.885551i \(-0.653781\pi\)
−0.464543 + 0.885551i \(0.653781\pi\)
\(18\) −378823. −0.850570
\(19\) 1.08733e6 1.91412 0.957059 0.289893i \(-0.0936197\pi\)
0.957059 + 0.289893i \(0.0936197\pi\)
\(20\) 0 0
\(21\) −122342. −0.137274
\(22\) −850682. −0.774221
\(23\) −1.50672e6 −1.12268 −0.561341 0.827585i \(-0.689714\pi\)
−0.561341 + 0.827585i \(0.689714\pi\)
\(24\) 379575. 0.233532
\(25\) 0 0
\(26\) −2.49979e6 −1.07281
\(27\) 1.17256e6 0.424617
\(28\) −416676. −0.128111
\(29\) −2.62160e6 −0.688295 −0.344148 0.938916i \(-0.611832\pi\)
−0.344148 + 0.938916i \(0.611832\pi\)
\(30\) 0 0
\(31\) 3.27023e6 0.635991 0.317996 0.948092i \(-0.396990\pi\)
0.317996 + 0.948092i \(0.396990\pi\)
\(32\) 2.36745e6 0.399122
\(33\) 1.28466e6 0.188572
\(34\) −6.46335e6 −0.829473
\(35\) 0 0
\(36\) 1.94841e6 0.193339
\(37\) 2.51034e6 0.220204 0.110102 0.993920i \(-0.464882\pi\)
0.110102 + 0.993920i \(0.464882\pi\)
\(38\) 2.19655e7 1.70890
\(39\) 3.77508e6 0.261297
\(40\) 0 0
\(41\) 2.95349e7 1.63233 0.816165 0.577819i \(-0.196096\pi\)
0.816165 + 0.577819i \(0.196096\pi\)
\(42\) −2.47148e6 −0.122556
\(43\) 1.42413e7 0.635244 0.317622 0.948217i \(-0.397116\pi\)
0.317622 + 0.948217i \(0.397116\pi\)
\(44\) 4.37534e6 0.175985
\(45\) 0 0
\(46\) −3.04378e7 −1.00231
\(47\) −1.35318e6 −0.0404496 −0.0202248 0.999795i \(-0.506438\pi\)
−0.0202248 + 0.999795i \(0.506438\pi\)
\(48\) 6.04503e6 0.164366
\(49\) −2.42715e7 −0.601470
\(50\) 0 0
\(51\) 9.76067e6 0.202029
\(52\) 1.28573e7 0.243856
\(53\) −9.73342e7 −1.69443 −0.847216 0.531249i \(-0.821723\pi\)
−0.847216 + 0.531249i \(0.821723\pi\)
\(54\) 2.36873e7 0.379092
\(55\) 0 0
\(56\) −4.98960e7 −0.677984
\(57\) −3.31714e7 −0.416224
\(58\) −5.29599e7 −0.614500
\(59\) −7.48924e6 −0.0804644 −0.0402322 0.999190i \(-0.512810\pi\)
−0.0402322 + 0.999190i \(0.512810\pi\)
\(60\) 0 0
\(61\) −9.11752e7 −0.843126 −0.421563 0.906799i \(-0.638518\pi\)
−0.421563 + 0.906799i \(0.638518\pi\)
\(62\) 6.60633e7 0.567803
\(63\) −7.52015e7 −0.601442
\(64\) 1.49279e8 1.11221
\(65\) 0 0
\(66\) 2.59520e7 0.168354
\(67\) −2.94376e8 −1.78470 −0.892350 0.451344i \(-0.850945\pi\)
−0.892350 + 0.451344i \(0.850945\pi\)
\(68\) 3.32432e7 0.188544
\(69\) 4.59659e7 0.244127
\(70\) 0 0
\(71\) 1.56193e8 0.729455 0.364728 0.931114i \(-0.381162\pi\)
0.364728 + 0.931114i \(0.381162\pi\)
\(72\) 2.33318e8 1.02318
\(73\) 2.82539e8 1.16446 0.582232 0.813023i \(-0.302180\pi\)
0.582232 + 0.813023i \(0.302180\pi\)
\(74\) 5.07124e7 0.196594
\(75\) 0 0
\(76\) −1.12976e8 −0.388441
\(77\) −1.68872e8 −0.547455
\(78\) 7.62619e7 0.233282
\(79\) −5.55294e8 −1.60399 −0.801994 0.597332i \(-0.796228\pi\)
−0.801994 + 0.597332i \(0.796228\pi\)
\(80\) 0 0
\(81\) 3.33330e8 0.860383
\(82\) 5.96647e8 1.45732
\(83\) −6.48378e6 −0.0149960 −0.00749802 0.999972i \(-0.502387\pi\)
−0.00749802 + 0.999972i \(0.502387\pi\)
\(84\) 1.27117e7 0.0278577
\(85\) 0 0
\(86\) 2.87694e8 0.567137
\(87\) 7.99778e7 0.149669
\(88\) 5.23937e8 0.931338
\(89\) −5.99001e8 −1.01198 −0.505990 0.862539i \(-0.668873\pi\)
−0.505990 + 0.862539i \(0.668873\pi\)
\(90\) 0 0
\(91\) −4.96242e8 −0.758591
\(92\) 1.56552e8 0.227831
\(93\) −9.97660e7 −0.138296
\(94\) −2.73361e7 −0.0361128
\(95\) 0 0
\(96\) −7.22244e7 −0.0867887
\(97\) −9.25317e8 −1.06125 −0.530625 0.847606i \(-0.678043\pi\)
−0.530625 + 0.847606i \(0.678043\pi\)
\(98\) −4.90319e8 −0.536984
\(99\) 7.89660e8 0.826193
\(100\) 0 0
\(101\) 9.58959e8 0.916967 0.458483 0.888703i \(-0.348393\pi\)
0.458483 + 0.888703i \(0.348393\pi\)
\(102\) 1.97179e8 0.180368
\(103\) −1.60441e8 −0.140458 −0.0702292 0.997531i \(-0.522373\pi\)
−0.0702292 + 0.997531i \(0.522373\pi\)
\(104\) 1.53963e9 1.29052
\(105\) 0 0
\(106\) −1.96629e9 −1.51276
\(107\) −9.60457e8 −0.708355 −0.354178 0.935178i \(-0.615239\pi\)
−0.354178 + 0.935178i \(0.615239\pi\)
\(108\) −1.21832e8 −0.0861696
\(109\) 9.98912e8 0.677810 0.338905 0.940821i \(-0.389943\pi\)
0.338905 + 0.940821i \(0.389943\pi\)
\(110\) 0 0
\(111\) −7.65836e7 −0.0478831
\(112\) −7.94632e8 −0.477183
\(113\) −2.50705e9 −1.44647 −0.723236 0.690601i \(-0.757346\pi\)
−0.723236 + 0.690601i \(0.757346\pi\)
\(114\) −6.70109e8 −0.371598
\(115\) 0 0
\(116\) 2.72391e8 0.139679
\(117\) 2.32047e9 1.14483
\(118\) −1.51293e8 −0.0718374
\(119\) −1.28306e9 −0.586524
\(120\) 0 0
\(121\) −5.84695e8 −0.247968
\(122\) −1.84187e9 −0.752730
\(123\) −9.01030e8 −0.354949
\(124\) −3.39786e8 −0.129065
\(125\) 0 0
\(126\) −1.51918e9 −0.536958
\(127\) 2.47541e9 0.844364 0.422182 0.906511i \(-0.361264\pi\)
0.422182 + 0.906511i \(0.361264\pi\)
\(128\) 1.80351e9 0.593845
\(129\) −4.34463e8 −0.138134
\(130\) 0 0
\(131\) 1.92402e9 0.570808 0.285404 0.958407i \(-0.407872\pi\)
0.285404 + 0.958407i \(0.407872\pi\)
\(132\) −1.33480e8 −0.0382678
\(133\) 4.36045e9 1.20837
\(134\) −5.94680e9 −1.59335
\(135\) 0 0
\(136\) 3.98079e9 0.997802
\(137\) −4.48594e8 −0.108796 −0.0543978 0.998519i \(-0.517324\pi\)
−0.0543978 + 0.998519i \(0.517324\pi\)
\(138\) 9.28577e8 0.217953
\(139\) 4.48415e9 1.01886 0.509429 0.860513i \(-0.329857\pi\)
0.509429 + 0.860513i \(0.329857\pi\)
\(140\) 0 0
\(141\) 4.12818e7 0.00879575
\(142\) 3.15532e9 0.651247
\(143\) 5.21084e9 1.04207
\(144\) 3.71577e9 0.720141
\(145\) 0 0
\(146\) 5.70769e9 1.03962
\(147\) 7.40458e8 0.130789
\(148\) −2.60831e8 −0.0446870
\(149\) −2.20480e9 −0.366463 −0.183232 0.983070i \(-0.558656\pi\)
−0.183232 + 0.983070i \(0.558656\pi\)
\(150\) 0 0
\(151\) −3.21248e9 −0.502857 −0.251428 0.967876i \(-0.580900\pi\)
−0.251428 + 0.967876i \(0.580900\pi\)
\(152\) −1.35286e10 −2.05569
\(153\) 5.99971e9 0.885154
\(154\) −3.41145e9 −0.488760
\(155\) 0 0
\(156\) −3.92241e8 −0.0530264
\(157\) 1.08870e10 1.43007 0.715036 0.699088i \(-0.246410\pi\)
0.715036 + 0.699088i \(0.246410\pi\)
\(158\) −1.12177e10 −1.43202
\(159\) 2.96940e9 0.368453
\(160\) 0 0
\(161\) −6.04232e9 −0.708740
\(162\) 6.73374e9 0.768137
\(163\) −1.19994e10 −1.33142 −0.665708 0.746212i \(-0.731870\pi\)
−0.665708 + 0.746212i \(0.731870\pi\)
\(164\) −3.06875e9 −0.331257
\(165\) 0 0
\(166\) −1.30982e8 −0.0133882
\(167\) −9.68608e9 −0.963660 −0.481830 0.876265i \(-0.660028\pi\)
−0.481830 + 0.876265i \(0.660028\pi\)
\(168\) 1.52219e9 0.147427
\(169\) 4.70793e9 0.443956
\(170\) 0 0
\(171\) −2.03899e10 −1.82361
\(172\) −1.47971e9 −0.128913
\(173\) −7.35665e9 −0.624414 −0.312207 0.950014i \(-0.601068\pi\)
−0.312207 + 0.950014i \(0.601068\pi\)
\(174\) 1.61567e9 0.133623
\(175\) 0 0
\(176\) 8.34410e9 0.655500
\(177\) 2.28477e8 0.0174969
\(178\) −1.21007e10 −0.903481
\(179\) −2.00351e9 −0.145866 −0.0729329 0.997337i \(-0.523236\pi\)
−0.0729329 + 0.997337i \(0.523236\pi\)
\(180\) 0 0
\(181\) 5.63414e9 0.390188 0.195094 0.980785i \(-0.437499\pi\)
0.195094 + 0.980785i \(0.437499\pi\)
\(182\) −1.00248e10 −0.677258
\(183\) 2.78151e9 0.183337
\(184\) 1.87467e10 1.20572
\(185\) 0 0
\(186\) −2.01541e9 −0.123469
\(187\) 1.34729e10 0.805701
\(188\) 1.40599e8 0.00820864
\(189\) 4.70225e9 0.268057
\(190\) 0 0
\(191\) 9.16925e9 0.498521 0.249261 0.968436i \(-0.419812\pi\)
0.249261 + 0.968436i \(0.419812\pi\)
\(192\) −4.55409e9 −0.241850
\(193\) 3.16327e10 1.64107 0.820536 0.571594i \(-0.193675\pi\)
0.820536 + 0.571594i \(0.193675\pi\)
\(194\) −1.86927e10 −0.947469
\(195\) 0 0
\(196\) 2.52187e9 0.122059
\(197\) −2.59858e10 −1.22924 −0.614621 0.788822i \(-0.710691\pi\)
−0.614621 + 0.788822i \(0.710691\pi\)
\(198\) 1.59522e10 0.737613
\(199\) 1.05766e10 0.478088 0.239044 0.971009i \(-0.423166\pi\)
0.239044 + 0.971009i \(0.423166\pi\)
\(200\) 0 0
\(201\) 8.98061e9 0.388082
\(202\) 1.93723e10 0.818654
\(203\) −1.05133e10 −0.434515
\(204\) −1.01416e9 −0.0409987
\(205\) 0 0
\(206\) −3.24113e9 −0.125399
\(207\) 2.82544e10 1.06960
\(208\) 2.45198e10 0.908304
\(209\) −4.57873e10 −1.65992
\(210\) 0 0
\(211\) 1.80228e10 0.625965 0.312983 0.949759i \(-0.398672\pi\)
0.312983 + 0.949759i \(0.398672\pi\)
\(212\) 1.01133e10 0.343859
\(213\) −4.76502e9 −0.158620
\(214\) −1.94026e10 −0.632409
\(215\) 0 0
\(216\) −1.45891e10 −0.456023
\(217\) 1.31145e10 0.401496
\(218\) 2.01794e10 0.605139
\(219\) −8.61951e9 −0.253212
\(220\) 0 0
\(221\) 3.95911e10 1.11643
\(222\) −1.54710e9 −0.0427493
\(223\) −4.44522e10 −1.20371 −0.601855 0.798606i \(-0.705571\pi\)
−0.601855 + 0.798606i \(0.705571\pi\)
\(224\) 9.49405e9 0.251962
\(225\) 0 0
\(226\) −5.06460e10 −1.29139
\(227\) −5.59677e10 −1.39901 −0.699505 0.714627i \(-0.746596\pi\)
−0.699505 + 0.714627i \(0.746596\pi\)
\(228\) 3.44660e9 0.0844663
\(229\) −1.47705e9 −0.0354923 −0.0177462 0.999843i \(-0.505649\pi\)
−0.0177462 + 0.999843i \(0.505649\pi\)
\(230\) 0 0
\(231\) 5.15182e9 0.119044
\(232\) 3.26182e10 0.739203
\(233\) 7.83279e9 0.174106 0.0870532 0.996204i \(-0.472255\pi\)
0.0870532 + 0.996204i \(0.472255\pi\)
\(234\) 4.68769e10 1.02209
\(235\) 0 0
\(236\) 7.78152e8 0.0163290
\(237\) 1.69405e10 0.348786
\(238\) −2.59197e10 −0.523640
\(239\) 5.56371e10 1.10300 0.551498 0.834176i \(-0.314057\pi\)
0.551498 + 0.834176i \(0.314057\pi\)
\(240\) 0 0
\(241\) −1.16053e10 −0.221606 −0.110803 0.993842i \(-0.535342\pi\)
−0.110803 + 0.993842i \(0.535342\pi\)
\(242\) −1.18117e10 −0.221382
\(243\) −3.32485e10 −0.611707
\(244\) 9.47334e9 0.171100
\(245\) 0 0
\(246\) −1.82021e10 −0.316893
\(247\) −1.34549e11 −2.30009
\(248\) −4.06886e10 −0.683030
\(249\) 1.97803e8 0.00326088
\(250\) 0 0
\(251\) −3.45974e10 −0.550189 −0.275094 0.961417i \(-0.588709\pi\)
−0.275094 + 0.961417i \(0.588709\pi\)
\(252\) 7.81363e9 0.122054
\(253\) 6.34479e10 0.973587
\(254\) 5.00067e10 0.753836
\(255\) 0 0
\(256\) −3.99972e10 −0.582036
\(257\) 3.67735e10 0.525818 0.262909 0.964821i \(-0.415318\pi\)
0.262909 + 0.964821i \(0.415318\pi\)
\(258\) −8.77677e9 −0.123324
\(259\) 1.00671e10 0.139013
\(260\) 0 0
\(261\) 4.91609e10 0.655750
\(262\) 3.88680e10 0.509609
\(263\) −1.33758e11 −1.72392 −0.861962 0.506974i \(-0.830764\pi\)
−0.861962 + 0.506974i \(0.830764\pi\)
\(264\) −1.59839e10 −0.202519
\(265\) 0 0
\(266\) 8.80873e10 1.07881
\(267\) 1.82739e10 0.220055
\(268\) 3.05864e10 0.362178
\(269\) −1.42461e11 −1.65886 −0.829429 0.558612i \(-0.811334\pi\)
−0.829429 + 0.558612i \(0.811334\pi\)
\(270\) 0 0
\(271\) −1.11046e11 −1.25067 −0.625333 0.780358i \(-0.715037\pi\)
−0.625333 + 0.780358i \(0.715037\pi\)
\(272\) 6.33972e10 0.702279
\(273\) 1.51390e10 0.164955
\(274\) −9.06224e9 −0.0971310
\(275\) 0 0
\(276\) −4.77598e9 −0.0495418
\(277\) −2.80726e10 −0.286500 −0.143250 0.989687i \(-0.545755\pi\)
−0.143250 + 0.989687i \(0.545755\pi\)
\(278\) 9.05861e10 0.909620
\(279\) −6.13244e10 −0.605919
\(280\) 0 0
\(281\) 5.47143e10 0.523507 0.261753 0.965135i \(-0.415699\pi\)
0.261753 + 0.965135i \(0.415699\pi\)
\(282\) 8.33951e8 0.00785271
\(283\) 1.09950e11 1.01895 0.509477 0.860484i \(-0.329839\pi\)
0.509477 + 0.860484i \(0.329839\pi\)
\(284\) −1.62288e10 −0.148032
\(285\) 0 0
\(286\) 1.05266e11 0.930341
\(287\) 1.18442e11 1.03048
\(288\) −4.43951e10 −0.380249
\(289\) −1.62229e10 −0.136801
\(290\) 0 0
\(291\) 2.82289e10 0.230768
\(292\) −2.93566e10 −0.236310
\(293\) 2.30453e11 1.82675 0.913373 0.407124i \(-0.133468\pi\)
0.913373 + 0.407124i \(0.133468\pi\)
\(294\) 1.49583e10 0.116767
\(295\) 0 0
\(296\) −3.12339e10 −0.236490
\(297\) −4.93764e10 −0.368227
\(298\) −4.45400e10 −0.327173
\(299\) 1.86446e11 1.34907
\(300\) 0 0
\(301\) 5.71111e10 0.401025
\(302\) −6.48966e10 −0.448943
\(303\) −2.92552e10 −0.199394
\(304\) −2.15454e11 −1.44685
\(305\) 0 0
\(306\) 1.21203e11 0.790252
\(307\) −6.85036e10 −0.440140 −0.220070 0.975484i \(-0.570629\pi\)
−0.220070 + 0.975484i \(0.570629\pi\)
\(308\) 1.75462e10 0.111098
\(309\) 4.89462e9 0.0305426
\(310\) 0 0
\(311\) −1.98797e11 −1.20500 −0.602502 0.798118i \(-0.705829\pi\)
−0.602502 + 0.798118i \(0.705829\pi\)
\(312\) −4.69699e10 −0.280624
\(313\) 6.15202e10 0.362300 0.181150 0.983455i \(-0.442018\pi\)
0.181150 + 0.983455i \(0.442018\pi\)
\(314\) 2.19932e11 1.27675
\(315\) 0 0
\(316\) 5.76965e10 0.325505
\(317\) 2.25932e11 1.25664 0.628320 0.777955i \(-0.283743\pi\)
0.628320 + 0.777955i \(0.283743\pi\)
\(318\) 5.99862e10 0.328950
\(319\) 1.10395e11 0.596888
\(320\) 0 0
\(321\) 2.93010e10 0.154031
\(322\) −1.22063e11 −0.632753
\(323\) −3.47885e11 −1.77838
\(324\) −3.46339e10 −0.174602
\(325\) 0 0
\(326\) −2.42404e11 −1.18867
\(327\) −3.04741e10 −0.147389
\(328\) −3.67476e11 −1.75306
\(329\) −5.42658e9 −0.0255355
\(330\) 0 0
\(331\) −8.38825e10 −0.384101 −0.192050 0.981385i \(-0.561514\pi\)
−0.192050 + 0.981385i \(0.561514\pi\)
\(332\) 6.73682e8 0.00304322
\(333\) −4.70746e10 −0.209791
\(334\) −1.95673e11 −0.860341
\(335\) 0 0
\(336\) 2.42421e10 0.103763
\(337\) 2.19457e11 0.926862 0.463431 0.886133i \(-0.346618\pi\)
0.463431 + 0.886133i \(0.346618\pi\)
\(338\) 9.51069e10 0.396357
\(339\) 7.64834e10 0.314535
\(340\) 0 0
\(341\) −1.37710e11 −0.551530
\(342\) −4.11904e11 −1.62809
\(343\) −2.59163e11 −1.01100
\(344\) −1.77191e11 −0.682228
\(345\) 0 0
\(346\) −1.48615e11 −0.557467
\(347\) 1.85960e11 0.688551 0.344276 0.938869i \(-0.388125\pi\)
0.344276 + 0.938869i \(0.388125\pi\)
\(348\) −8.30990e9 −0.0303731
\(349\) 2.73237e11 0.985881 0.492940 0.870063i \(-0.335922\pi\)
0.492940 + 0.870063i \(0.335922\pi\)
\(350\) 0 0
\(351\) −1.45096e11 −0.510240
\(352\) −9.96932e10 −0.346117
\(353\) 3.02861e11 1.03814 0.519072 0.854731i \(-0.326278\pi\)
0.519072 + 0.854731i \(0.326278\pi\)
\(354\) 4.61555e9 0.0156210
\(355\) 0 0
\(356\) 6.22378e10 0.205366
\(357\) 3.91427e10 0.127539
\(358\) −4.04738e10 −0.130227
\(359\) 2.47660e11 0.786919 0.393460 0.919342i \(-0.371278\pi\)
0.393460 + 0.919342i \(0.371278\pi\)
\(360\) 0 0
\(361\) 8.59591e11 2.66385
\(362\) 1.13818e11 0.348354
\(363\) 1.78375e10 0.0539205
\(364\) 5.15609e10 0.153944
\(365\) 0 0
\(366\) 5.61904e10 0.163681
\(367\) −2.66354e11 −0.766412 −0.383206 0.923663i \(-0.625180\pi\)
−0.383206 + 0.923663i \(0.625180\pi\)
\(368\) 2.98556e11 0.848616
\(369\) −5.53847e11 −1.55515
\(370\) 0 0
\(371\) −3.90335e11 −1.06968
\(372\) 1.03660e10 0.0280651
\(373\) 5.20850e11 1.39323 0.696616 0.717445i \(-0.254688\pi\)
0.696616 + 0.717445i \(0.254688\pi\)
\(374\) 2.72172e11 0.719318
\(375\) 0 0
\(376\) 1.68364e10 0.0434414
\(377\) 3.24405e11 0.827088
\(378\) 9.49921e10 0.239318
\(379\) 3.28308e10 0.0817344 0.0408672 0.999165i \(-0.486988\pi\)
0.0408672 + 0.999165i \(0.486988\pi\)
\(380\) 0 0
\(381\) −7.55180e10 −0.183607
\(382\) 1.85232e11 0.445072
\(383\) 7.15293e10 0.169859 0.0849297 0.996387i \(-0.472933\pi\)
0.0849297 + 0.996387i \(0.472933\pi\)
\(384\) −5.50202e10 −0.129131
\(385\) 0 0
\(386\) 6.39025e11 1.46513
\(387\) −2.67057e11 −0.605207
\(388\) 9.61429e10 0.215365
\(389\) −2.02681e11 −0.448787 −0.224394 0.974499i \(-0.572040\pi\)
−0.224394 + 0.974499i \(0.572040\pi\)
\(390\) 0 0
\(391\) 4.82067e11 1.04307
\(392\) 3.01989e11 0.645956
\(393\) −5.86968e10 −0.124122
\(394\) −5.24949e11 −1.09745
\(395\) 0 0
\(396\) −8.20477e10 −0.167663
\(397\) −1.63266e10 −0.0329867 −0.0164934 0.999864i \(-0.505250\pi\)
−0.0164934 + 0.999864i \(0.505250\pi\)
\(398\) 2.13663e11 0.426830
\(399\) −1.33026e11 −0.262759
\(400\) 0 0
\(401\) −8.20766e11 −1.58515 −0.792574 0.609776i \(-0.791259\pi\)
−0.792574 + 0.609776i \(0.791259\pi\)
\(402\) 1.81421e11 0.346474
\(403\) −4.04670e11 −0.764237
\(404\) −9.96383e10 −0.186085
\(405\) 0 0
\(406\) −2.12383e11 −0.387929
\(407\) −1.05710e11 −0.190960
\(408\) −1.21443e11 −0.216972
\(409\) −4.24628e11 −0.750332 −0.375166 0.926958i \(-0.622414\pi\)
−0.375166 + 0.926958i \(0.622414\pi\)
\(410\) 0 0
\(411\) 1.36854e10 0.0236575
\(412\) 1.66702e10 0.0285039
\(413\) −3.00337e10 −0.0507966
\(414\) 5.70780e11 0.954920
\(415\) 0 0
\(416\) −2.92956e11 −0.479603
\(417\) −1.36799e11 −0.221550
\(418\) −9.24969e11 −1.48195
\(419\) −1.26375e10 −0.0200308 −0.0100154 0.999950i \(-0.503188\pi\)
−0.0100154 + 0.999950i \(0.503188\pi\)
\(420\) 0 0
\(421\) −3.04545e10 −0.0472479 −0.0236240 0.999721i \(-0.507520\pi\)
−0.0236240 + 0.999721i \(0.507520\pi\)
\(422\) 3.64085e11 0.558853
\(423\) 2.53752e10 0.0385370
\(424\) 1.21104e12 1.81976
\(425\) 0 0
\(426\) −9.62602e10 −0.141613
\(427\) −3.65635e11 −0.532259
\(428\) 9.97940e10 0.143750
\(429\) −1.58969e11 −0.226597
\(430\) 0 0
\(431\) 4.05830e11 0.566495 0.283248 0.959047i \(-0.408588\pi\)
0.283248 + 0.959047i \(0.408588\pi\)
\(432\) −2.32342e11 −0.320961
\(433\) −1.36978e11 −0.187265 −0.0936324 0.995607i \(-0.529848\pi\)
−0.0936324 + 0.995607i \(0.529848\pi\)
\(434\) 2.64931e11 0.358450
\(435\) 0 0
\(436\) −1.03790e11 −0.137551
\(437\) −1.63829e12 −2.14895
\(438\) −1.74126e11 −0.226064
\(439\) 7.84981e11 1.00872 0.504358 0.863495i \(-0.331729\pi\)
0.504358 + 0.863495i \(0.331729\pi\)
\(440\) 0 0
\(441\) 4.55146e11 0.573030
\(442\) 7.99797e11 0.996734
\(443\) −8.87799e11 −1.09521 −0.547605 0.836737i \(-0.684460\pi\)
−0.547605 + 0.836737i \(0.684460\pi\)
\(444\) 7.95724e9 0.00971716
\(445\) 0 0
\(446\) −8.97998e11 −1.07465
\(447\) 6.72624e10 0.0796872
\(448\) 5.98645e11 0.702131
\(449\) 8.35477e11 0.970122 0.485061 0.874480i \(-0.338797\pi\)
0.485061 + 0.874480i \(0.338797\pi\)
\(450\) 0 0
\(451\) −1.24371e12 −1.41555
\(452\) 2.60489e11 0.293540
\(453\) 9.80041e10 0.109346
\(454\) −1.13063e12 −1.24902
\(455\) 0 0
\(456\) 4.12722e11 0.447009
\(457\) −6.01172e11 −0.644727 −0.322364 0.946616i \(-0.604477\pi\)
−0.322364 + 0.946616i \(0.604477\pi\)
\(458\) −2.98384e10 −0.0316870
\(459\) −3.75154e11 −0.394505
\(460\) 0 0
\(461\) 1.52807e12 1.57576 0.787879 0.615829i \(-0.211179\pi\)
0.787879 + 0.615829i \(0.211179\pi\)
\(462\) 1.04074e11 0.106281
\(463\) 7.80402e11 0.789231 0.394615 0.918846i \(-0.370878\pi\)
0.394615 + 0.918846i \(0.370878\pi\)
\(464\) 5.19469e11 0.520270
\(465\) 0 0
\(466\) 1.58234e11 0.155440
\(467\) 4.04751e11 0.393788 0.196894 0.980425i \(-0.436915\pi\)
0.196894 + 0.980425i \(0.436915\pi\)
\(468\) −2.41103e11 −0.232326
\(469\) −1.18052e12 −1.12667
\(470\) 0 0
\(471\) −3.32132e11 −0.310968
\(472\) 9.31820e10 0.0864158
\(473\) −5.99700e11 −0.550883
\(474\) 3.42223e11 0.311391
\(475\) 0 0
\(476\) 1.33313e11 0.119026
\(477\) 1.82524e12 1.61431
\(478\) 1.12395e12 0.984738
\(479\) −2.06972e12 −1.79640 −0.898199 0.439588i \(-0.855124\pi\)
−0.898199 + 0.439588i \(0.855124\pi\)
\(480\) 0 0
\(481\) −3.10638e11 −0.264607
\(482\) −2.34444e11 −0.197846
\(483\) 1.84335e11 0.154115
\(484\) 6.07514e10 0.0503213
\(485\) 0 0
\(486\) −6.71666e11 −0.546123
\(487\) 2.41184e11 0.194298 0.0971490 0.995270i \(-0.469028\pi\)
0.0971490 + 0.995270i \(0.469028\pi\)
\(488\) 1.13441e12 0.905485
\(489\) 3.66068e11 0.289516
\(490\) 0 0
\(491\) 2.27883e12 1.76948 0.884739 0.466088i \(-0.154337\pi\)
0.884739 + 0.466088i \(0.154337\pi\)
\(492\) 9.36194e10 0.0720316
\(493\) 8.38767e11 0.639485
\(494\) −2.71809e12 −2.05349
\(495\) 0 0
\(496\) −6.47997e11 −0.480735
\(497\) 6.26373e11 0.460499
\(498\) 3.99589e9 0.00291127
\(499\) 9.88752e11 0.713896 0.356948 0.934124i \(-0.383817\pi\)
0.356948 + 0.934124i \(0.383817\pi\)
\(500\) 0 0
\(501\) 2.95496e11 0.209547
\(502\) −6.98916e11 −0.491200
\(503\) −1.22385e12 −0.852455 −0.426228 0.904616i \(-0.640158\pi\)
−0.426228 + 0.904616i \(0.640158\pi\)
\(504\) 9.35665e11 0.645926
\(505\) 0 0
\(506\) 1.28174e12 0.869204
\(507\) −1.43626e11 −0.0965380
\(508\) −2.57201e11 −0.171351
\(509\) 1.58447e12 1.04629 0.523146 0.852243i \(-0.324758\pi\)
0.523146 + 0.852243i \(0.324758\pi\)
\(510\) 0 0
\(511\) 1.13305e12 0.735117
\(512\) −1.73140e12 −1.11348
\(513\) 1.27495e12 0.812767
\(514\) 7.42877e11 0.469443
\(515\) 0 0
\(516\) 4.51418e10 0.0280321
\(517\) 5.69823e10 0.0350778
\(518\) 2.03369e11 0.124109
\(519\) 2.24432e11 0.135778
\(520\) 0 0
\(521\) −2.71561e12 −1.61472 −0.807362 0.590057i \(-0.799105\pi\)
−0.807362 + 0.590057i \(0.799105\pi\)
\(522\) 9.93121e11 0.585443
\(523\) 2.16171e12 1.26340 0.631698 0.775214i \(-0.282358\pi\)
0.631698 + 0.775214i \(0.282358\pi\)
\(524\) −1.99911e11 −0.115837
\(525\) 0 0
\(526\) −2.70210e12 −1.53909
\(527\) −1.04630e12 −0.590890
\(528\) −2.54556e11 −0.142538
\(529\) 4.69046e11 0.260415
\(530\) 0 0
\(531\) 1.40441e11 0.0766597
\(532\) −4.53062e11 −0.245220
\(533\) −3.65475e12 −1.96149
\(534\) 3.69159e11 0.196462
\(535\) 0 0
\(536\) 3.66265e12 1.91670
\(537\) 6.11217e10 0.0317184
\(538\) −2.87791e12 −1.48100
\(539\) 1.02207e12 0.521594
\(540\) 0 0
\(541\) −2.29090e12 −1.14979 −0.574895 0.818227i \(-0.694957\pi\)
−0.574895 + 0.818227i \(0.694957\pi\)
\(542\) −2.24329e12 −1.11658
\(543\) −1.71882e11 −0.0848462
\(544\) −7.57453e11 −0.370818
\(545\) 0 0
\(546\) 3.05830e11 0.147269
\(547\) 3.39447e12 1.62117 0.810586 0.585620i \(-0.199149\pi\)
0.810586 + 0.585620i \(0.199149\pi\)
\(548\) 4.66101e10 0.0220784
\(549\) 1.70974e12 0.803259
\(550\) 0 0
\(551\) −2.85053e12 −1.31748
\(552\) −5.71913e11 −0.262183
\(553\) −2.22687e12 −1.01259
\(554\) −5.67107e11 −0.255783
\(555\) 0 0
\(556\) −4.65915e11 −0.206762
\(557\) −1.74706e12 −0.769060 −0.384530 0.923113i \(-0.625636\pi\)
−0.384530 + 0.923113i \(0.625636\pi\)
\(558\) −1.23884e12 −0.540955
\(559\) −1.76226e12 −0.763340
\(560\) 0 0
\(561\) −4.11022e11 −0.175199
\(562\) 1.10531e12 0.467379
\(563\) 2.58864e12 1.08588 0.542942 0.839770i \(-0.317310\pi\)
0.542942 + 0.839770i \(0.317310\pi\)
\(564\) −4.28929e9 −0.00178496
\(565\) 0 0
\(566\) 2.22114e12 0.909707
\(567\) 1.33674e12 0.543153
\(568\) −1.94337e12 −0.783407
\(569\) −1.99294e12 −0.797055 −0.398527 0.917156i \(-0.630479\pi\)
−0.398527 + 0.917156i \(0.630479\pi\)
\(570\) 0 0
\(571\) 3.50761e10 0.0138086 0.00690428 0.999976i \(-0.497802\pi\)
0.00690428 + 0.999976i \(0.497802\pi\)
\(572\) −5.41420e11 −0.211471
\(573\) −2.79729e11 −0.108403
\(574\) 2.39270e12 0.919995
\(575\) 0 0
\(576\) −2.79932e12 −1.05962
\(577\) 7.75900e11 0.291417 0.145708 0.989328i \(-0.453454\pi\)
0.145708 + 0.989328i \(0.453454\pi\)
\(578\) −3.27726e11 −0.122134
\(579\) −9.65027e11 −0.356850
\(580\) 0 0
\(581\) −2.60016e10 −0.00946689
\(582\) 5.70265e11 0.206027
\(583\) 4.09874e12 1.46941
\(584\) −3.51538e12 −1.25059
\(585\) 0 0
\(586\) 4.65548e12 1.63089
\(587\) −1.24477e12 −0.432730 −0.216365 0.976313i \(-0.569420\pi\)
−0.216365 + 0.976313i \(0.569420\pi\)
\(588\) −7.69355e10 −0.0265417
\(589\) 3.55581e12 1.21736
\(590\) 0 0
\(591\) 7.92756e11 0.267298
\(592\) −4.97424e11 −0.166448
\(593\) 2.57794e11 0.0856103 0.0428052 0.999083i \(-0.486371\pi\)
0.0428052 + 0.999083i \(0.486371\pi\)
\(594\) −9.97474e11 −0.328747
\(595\) 0 0
\(596\) 2.29084e11 0.0743681
\(597\) −3.22664e11 −0.103960
\(598\) 3.76648e12 1.20443
\(599\) 2.32089e12 0.736604 0.368302 0.929706i \(-0.379939\pi\)
0.368302 + 0.929706i \(0.379939\pi\)
\(600\) 0 0
\(601\) 1.78665e12 0.558605 0.279302 0.960203i \(-0.409897\pi\)
0.279302 + 0.960203i \(0.409897\pi\)
\(602\) 1.15373e12 0.358029
\(603\) 5.52022e12 1.70031
\(604\) 3.33785e11 0.102047
\(605\) 0 0
\(606\) −5.90997e11 −0.178016
\(607\) 1.45787e12 0.435883 0.217941 0.975962i \(-0.430066\pi\)
0.217941 + 0.975962i \(0.430066\pi\)
\(608\) 2.57419e12 0.763966
\(609\) 3.20731e11 0.0944851
\(610\) 0 0
\(611\) 1.67447e11 0.0486062
\(612\) −6.23386e11 −0.179629
\(613\) −1.42075e12 −0.406394 −0.203197 0.979138i \(-0.565133\pi\)
−0.203197 + 0.979138i \(0.565133\pi\)
\(614\) −1.38387e12 −0.392950
\(615\) 0 0
\(616\) 2.10112e12 0.587946
\(617\) −1.20441e12 −0.334573 −0.167286 0.985908i \(-0.553500\pi\)
−0.167286 + 0.985908i \(0.553500\pi\)
\(618\) 9.88782e10 0.0272680
\(619\) −4.91349e12 −1.34519 −0.672593 0.740013i \(-0.734819\pi\)
−0.672593 + 0.740013i \(0.734819\pi\)
\(620\) 0 0
\(621\) −1.76671e12 −0.476710
\(622\) −4.01598e12 −1.07581
\(623\) −2.40214e12 −0.638856
\(624\) −7.48032e11 −0.197510
\(625\) 0 0
\(626\) 1.24280e12 0.323456
\(627\) 1.39685e12 0.360948
\(628\) −1.13118e12 −0.290211
\(629\) −8.03171e11 −0.204588
\(630\) 0 0
\(631\) 4.58663e12 1.15176 0.575879 0.817535i \(-0.304660\pi\)
0.575879 + 0.817535i \(0.304660\pi\)
\(632\) 6.90903e12 1.72262
\(633\) −5.49826e11 −0.136116
\(634\) 4.56414e12 1.12191
\(635\) 0 0
\(636\) −3.08529e11 −0.0747720
\(637\) 3.00344e12 0.722755
\(638\) 2.23014e12 0.532893
\(639\) −2.92898e12 −0.694963
\(640\) 0 0
\(641\) −2.30636e12 −0.539593 −0.269796 0.962917i \(-0.586956\pi\)
−0.269796 + 0.962917i \(0.586956\pi\)
\(642\) 5.91921e11 0.137517
\(643\) −2.59493e12 −0.598654 −0.299327 0.954151i \(-0.596762\pi\)
−0.299327 + 0.954151i \(0.596762\pi\)
\(644\) 6.27813e11 0.143828
\(645\) 0 0
\(646\) −7.02777e12 −1.58771
\(647\) −5.14811e12 −1.15499 −0.577495 0.816394i \(-0.695970\pi\)
−0.577495 + 0.816394i \(0.695970\pi\)
\(648\) −4.14733e12 −0.924019
\(649\) 3.15372e11 0.0697786
\(650\) 0 0
\(651\) −4.00087e11 −0.0873051
\(652\) 1.24676e12 0.270191
\(653\) −4.42559e12 −0.952492 −0.476246 0.879312i \(-0.658003\pi\)
−0.476246 + 0.879312i \(0.658003\pi\)
\(654\) −6.15621e11 −0.131587
\(655\) 0 0
\(656\) −5.85234e12 −1.23385
\(657\) −5.29826e12 −1.10940
\(658\) −1.09625e11 −0.0227977
\(659\) 1.09827e12 0.226842 0.113421 0.993547i \(-0.463819\pi\)
0.113421 + 0.993547i \(0.463819\pi\)
\(660\) 0 0
\(661\) 7.99232e11 0.162842 0.0814209 0.996680i \(-0.474054\pi\)
0.0814209 + 0.996680i \(0.474054\pi\)
\(662\) −1.69455e12 −0.342919
\(663\) −1.20782e12 −0.242768
\(664\) 8.06719e10 0.0161052
\(665\) 0 0
\(666\) −9.50974e11 −0.187299
\(667\) 3.95000e12 0.772736
\(668\) 1.00641e12 0.195560
\(669\) 1.35612e12 0.261746
\(670\) 0 0
\(671\) 3.83939e12 0.731157
\(672\) −2.89638e11 −0.0547891
\(673\) −7.68445e12 −1.44393 −0.721963 0.691931i \(-0.756760\pi\)
−0.721963 + 0.691931i \(0.756760\pi\)
\(674\) 4.43334e12 0.827488
\(675\) 0 0
\(676\) −4.89167e11 −0.0900942
\(677\) −1.28854e12 −0.235749 −0.117874 0.993029i \(-0.537608\pi\)
−0.117874 + 0.993029i \(0.537608\pi\)
\(678\) 1.54507e12 0.280812
\(679\) −3.71076e12 −0.669959
\(680\) 0 0
\(681\) 1.70742e12 0.304214
\(682\) −2.78193e12 −0.492398
\(683\) 8.11444e11 0.142681 0.0713404 0.997452i \(-0.477272\pi\)
0.0713404 + 0.997452i \(0.477272\pi\)
\(684\) 2.11856e12 0.370074
\(685\) 0 0
\(686\) −5.23546e12 −0.902602
\(687\) 4.50607e10 0.00771778
\(688\) −2.82191e12 −0.480170
\(689\) 1.20445e13 2.03611
\(690\) 0 0
\(691\) 2.26741e12 0.378338 0.189169 0.981945i \(-0.439421\pi\)
0.189169 + 0.981945i \(0.439421\pi\)
\(692\) 7.64375e11 0.126715
\(693\) 3.16673e12 0.521569
\(694\) 3.75665e12 0.614728
\(695\) 0 0
\(696\) −9.95093e11 −0.160739
\(697\) −9.44955e12 −1.51657
\(698\) 5.51977e12 0.880180
\(699\) −2.38957e11 −0.0378594
\(700\) 0 0
\(701\) −4.92113e12 −0.769721 −0.384861 0.922975i \(-0.625751\pi\)
−0.384861 + 0.922975i \(0.625751\pi\)
\(702\) −2.93115e12 −0.455534
\(703\) 2.72956e12 0.421496
\(704\) −6.28612e12 −0.964508
\(705\) 0 0
\(706\) 6.11822e12 0.926839
\(707\) 3.84566e12 0.578874
\(708\) −2.37393e10 −0.00355074
\(709\) −6.12354e12 −0.910112 −0.455056 0.890463i \(-0.650381\pi\)
−0.455056 + 0.890463i \(0.650381\pi\)
\(710\) 0 0
\(711\) 1.04130e13 1.52815
\(712\) 7.45283e12 1.08683
\(713\) −4.92732e12 −0.714016
\(714\) 7.90739e11 0.113865
\(715\) 0 0
\(716\) 2.08170e11 0.0296012
\(717\) −1.69734e12 −0.239846
\(718\) 5.00307e12 0.702550
\(719\) −2.30376e11 −0.0321483 −0.0160741 0.999871i \(-0.505117\pi\)
−0.0160741 + 0.999871i \(0.505117\pi\)
\(720\) 0 0
\(721\) −6.43408e11 −0.0886702
\(722\) 1.73650e13 2.37824
\(723\) 3.54048e11 0.0481880
\(724\) −5.85402e11 −0.0791827
\(725\) 0 0
\(726\) 3.60342e11 0.0481394
\(727\) −9.25894e12 −1.22930 −0.614648 0.788801i \(-0.710702\pi\)
−0.614648 + 0.788801i \(0.710702\pi\)
\(728\) 6.17430e12 0.814698
\(729\) −5.54661e12 −0.727368
\(730\) 0 0
\(731\) −4.55643e12 −0.590196
\(732\) −2.89006e11 −0.0372055
\(733\) 7.22531e12 0.924461 0.462231 0.886760i \(-0.347049\pi\)
0.462231 + 0.886760i \(0.347049\pi\)
\(734\) −5.38074e12 −0.684241
\(735\) 0 0
\(736\) −3.56707e12 −0.448086
\(737\) 1.23962e13 1.54769
\(738\) −1.11885e13 −1.38841
\(739\) −1.24725e13 −1.53834 −0.769170 0.639044i \(-0.779330\pi\)
−0.769170 + 0.639044i \(0.779330\pi\)
\(740\) 0 0
\(741\) 4.10474e12 0.500154
\(742\) −7.88531e12 −0.954996
\(743\) 1.15645e13 1.39212 0.696058 0.717986i \(-0.254936\pi\)
0.696058 + 0.717986i \(0.254936\pi\)
\(744\) 1.24130e12 0.148525
\(745\) 0 0
\(746\) 1.05219e13 1.24386
\(747\) 1.21586e11 0.0142870
\(748\) −1.39987e12 −0.163505
\(749\) −3.85167e12 −0.447179
\(750\) 0 0
\(751\) −3.70732e12 −0.425285 −0.212642 0.977130i \(-0.568207\pi\)
−0.212642 + 0.977130i \(0.568207\pi\)
\(752\) 2.68132e11 0.0305752
\(753\) 1.05547e12 0.119638
\(754\) 6.55344e12 0.738412
\(755\) 0 0
\(756\) −4.88576e11 −0.0543982
\(757\) 4.86635e11 0.0538607 0.0269303 0.999637i \(-0.491427\pi\)
0.0269303 + 0.999637i \(0.491427\pi\)
\(758\) 6.63228e11 0.0729712
\(759\) −1.93562e12 −0.211706
\(760\) 0 0
\(761\) −4.28395e12 −0.463035 −0.231518 0.972831i \(-0.574369\pi\)
−0.231518 + 0.972831i \(0.574369\pi\)
\(762\) −1.52557e12 −0.163921
\(763\) 4.00589e12 0.427896
\(764\) −9.52709e11 −0.101167
\(765\) 0 0
\(766\) 1.44499e12 0.151648
\(767\) 9.26745e11 0.0966899
\(768\) 1.22021e12 0.126563
\(769\) −4.03625e12 −0.416207 −0.208103 0.978107i \(-0.566729\pi\)
−0.208103 + 0.978107i \(0.566729\pi\)
\(770\) 0 0
\(771\) −1.12186e12 −0.114339
\(772\) −3.28672e12 −0.333031
\(773\) −1.21916e13 −1.22815 −0.614076 0.789247i \(-0.710471\pi\)
−0.614076 + 0.789247i \(0.710471\pi\)
\(774\) −5.39492e12 −0.540320
\(775\) 0 0
\(776\) 1.15129e13 1.13974
\(777\) −3.07120e11 −0.0302282
\(778\) −4.09445e12 −0.400671
\(779\) 3.21141e13 3.12447
\(780\) 0 0
\(781\) −6.57728e12 −0.632582
\(782\) 9.73844e12 0.931235
\(783\) −3.07397e12 −0.292262
\(784\) 4.80940e12 0.454641
\(785\) 0 0
\(786\) −1.18576e12 −0.110814
\(787\) 1.20299e12 0.111783 0.0558913 0.998437i \(-0.482200\pi\)
0.0558913 + 0.998437i \(0.482200\pi\)
\(788\) 2.69999e12 0.249456
\(789\) 4.08059e12 0.374866
\(790\) 0 0
\(791\) −1.00539e13 −0.913147
\(792\) −9.82503e12 −0.887300
\(793\) 1.12823e13 1.01314
\(794\) −3.29821e11 −0.0294501
\(795\) 0 0
\(796\) −1.09894e12 −0.0970208
\(797\) −1.72611e13 −1.51533 −0.757663 0.652646i \(-0.773659\pi\)
−0.757663 + 0.652646i \(0.773659\pi\)
\(798\) −2.68731e12 −0.234587
\(799\) 4.32943e11 0.0375811
\(800\) 0 0
\(801\) 1.12326e13 0.964130
\(802\) −1.65806e13 −1.41520
\(803\) −1.18977e13 −1.00982
\(804\) −9.33109e11 −0.0787553
\(805\) 0 0
\(806\) −8.17490e12 −0.682299
\(807\) 4.34609e12 0.360718
\(808\) −1.19315e13 −0.984788
\(809\) 7.44408e12 0.611002 0.305501 0.952192i \(-0.401176\pi\)
0.305501 + 0.952192i \(0.401176\pi\)
\(810\) 0 0
\(811\) −9.59955e12 −0.779214 −0.389607 0.920981i \(-0.627389\pi\)
−0.389607 + 0.920981i \(0.627389\pi\)
\(812\) 1.09235e12 0.0881783
\(813\) 3.38772e12 0.271957
\(814\) −2.13550e12 −0.170486
\(815\) 0 0
\(816\) −1.93408e12 −0.152710
\(817\) 1.54849e13 1.21593
\(818\) −8.57808e12 −0.669885
\(819\) 9.30568e12 0.722721
\(820\) 0 0
\(821\) −5.04043e12 −0.387189 −0.193595 0.981082i \(-0.562015\pi\)
−0.193595 + 0.981082i \(0.562015\pi\)
\(822\) 2.76464e11 0.0211211
\(823\) −1.62323e13 −1.23333 −0.616666 0.787225i \(-0.711517\pi\)
−0.616666 + 0.787225i \(0.711517\pi\)
\(824\) 1.99622e12 0.150847
\(825\) 0 0
\(826\) −6.06724e11 −0.0453504
\(827\) 1.74898e13 1.30020 0.650099 0.759850i \(-0.274727\pi\)
0.650099 + 0.759850i \(0.274727\pi\)
\(828\) −2.93571e12 −0.217058
\(829\) −1.04102e13 −0.765535 −0.382768 0.923845i \(-0.625029\pi\)
−0.382768 + 0.923845i \(0.625029\pi\)
\(830\) 0 0
\(831\) 8.56420e11 0.0622992
\(832\) −1.84722e13 −1.33649
\(833\) 7.76555e12 0.558817
\(834\) −2.76354e12 −0.197796
\(835\) 0 0
\(836\) 4.75742e12 0.336855
\(837\) 3.83454e12 0.270053
\(838\) −2.55296e11 −0.0178832
\(839\) −2.66219e13 −1.85485 −0.927426 0.374006i \(-0.877984\pi\)
−0.927426 + 0.374006i \(0.877984\pi\)
\(840\) 0 0
\(841\) −7.63439e12 −0.526250
\(842\) −6.15225e11 −0.0421822
\(843\) −1.66919e12 −0.113836
\(844\) −1.87261e12 −0.127030
\(845\) 0 0
\(846\) 5.12615e11 0.0344052
\(847\) −2.34477e12 −0.156540
\(848\) 1.92868e13 1.28079
\(849\) −3.35427e12 −0.221571
\(850\) 0 0
\(851\) −3.78237e12 −0.247219
\(852\) 4.95098e11 0.0321894
\(853\) 7.30064e12 0.472161 0.236080 0.971734i \(-0.424137\pi\)
0.236080 + 0.971734i \(0.424137\pi\)
\(854\) −7.38635e12 −0.475192
\(855\) 0 0
\(856\) 1.19501e13 0.760747
\(857\) 1.18598e13 0.751041 0.375520 0.926814i \(-0.377464\pi\)
0.375520 + 0.926814i \(0.377464\pi\)
\(858\) −3.21139e12 −0.202302
\(859\) 1.67692e13 1.05085 0.525427 0.850839i \(-0.323906\pi\)
0.525427 + 0.850839i \(0.323906\pi\)
\(860\) 0 0
\(861\) −3.61336e12 −0.224077
\(862\) 8.19834e12 0.505759
\(863\) −6.71596e12 −0.412154 −0.206077 0.978536i \(-0.566070\pi\)
−0.206077 + 0.978536i \(0.566070\pi\)
\(864\) 2.77597e12 0.169474
\(865\) 0 0
\(866\) −2.76716e12 −0.167187
\(867\) 4.94917e11 0.0297472
\(868\) −1.36263e12 −0.0814776
\(869\) 2.33834e13 1.39098
\(870\) 0 0
\(871\) 3.64270e13 2.14458
\(872\) −1.24286e13 −0.727943
\(873\) 1.73518e13 1.01107
\(874\) −3.30959e13 −1.91855
\(875\) 0 0
\(876\) 8.95590e11 0.0513855
\(877\) 2.79757e13 1.59692 0.798458 0.602051i \(-0.205649\pi\)
0.798458 + 0.602051i \(0.205649\pi\)
\(878\) 1.58577e13 0.900566
\(879\) −7.03050e12 −0.397225
\(880\) 0 0
\(881\) 2.06225e13 1.15332 0.576660 0.816984i \(-0.304356\pi\)
0.576660 + 0.816984i \(0.304356\pi\)
\(882\) 9.19460e12 0.511593
\(883\) 2.02048e13 1.11849 0.559244 0.829003i \(-0.311091\pi\)
0.559244 + 0.829003i \(0.311091\pi\)
\(884\) −4.11362e12 −0.226563
\(885\) 0 0
\(886\) −1.79348e13 −0.977788
\(887\) −3.19954e13 −1.73553 −0.867763 0.496978i \(-0.834443\pi\)
−0.867763 + 0.496978i \(0.834443\pi\)
\(888\) 9.52862e11 0.0514247
\(889\) 9.92701e12 0.533041
\(890\) 0 0
\(891\) −1.40365e13 −0.746122
\(892\) 4.61870e12 0.244275
\(893\) −1.47135e12 −0.0774254
\(894\) 1.35880e12 0.0711436
\(895\) 0 0
\(896\) 7.23252e12 0.374890
\(897\) −5.68798e12 −0.293354
\(898\) 1.68778e13 0.866110
\(899\) −8.57323e12 −0.437750
\(900\) 0 0
\(901\) 3.11416e13 1.57427
\(902\) −2.51248e13 −1.26378
\(903\) −1.74231e12 −0.0872026
\(904\) 3.11930e13 1.55346
\(905\) 0 0
\(906\) 1.97982e12 0.0976223
\(907\) −2.40537e12 −0.118018 −0.0590091 0.998257i \(-0.518794\pi\)
−0.0590091 + 0.998257i \(0.518794\pi\)
\(908\) 5.81519e12 0.283908
\(909\) −1.79827e13 −0.873609
\(910\) 0 0
\(911\) −3.11773e13 −1.49970 −0.749852 0.661606i \(-0.769875\pi\)
−0.749852 + 0.661606i \(0.769875\pi\)
\(912\) 6.57292e12 0.314616
\(913\) 2.73032e11 0.0130045
\(914\) −1.21445e13 −0.575603
\(915\) 0 0
\(916\) 1.53469e11 0.00720263
\(917\) 7.71582e12 0.360346
\(918\) −7.57865e12 −0.352208
\(919\) 3.42677e13 1.58477 0.792383 0.610024i \(-0.208840\pi\)
0.792383 + 0.610024i \(0.208840\pi\)
\(920\) 0 0
\(921\) 2.08986e12 0.0957082
\(922\) 3.08692e13 1.40681
\(923\) −1.93278e13 −0.876548
\(924\) −5.35288e11 −0.0241581
\(925\) 0 0
\(926\) 1.57652e13 0.704614
\(927\) 3.00864e12 0.133817
\(928\) −6.20648e12 −0.274713
\(929\) −2.52782e13 −1.11346 −0.556730 0.830693i \(-0.687944\pi\)
−0.556730 + 0.830693i \(0.687944\pi\)
\(930\) 0 0
\(931\) −2.63910e13 −1.15129
\(932\) −8.13848e11 −0.0353323
\(933\) 6.06476e12 0.262027
\(934\) 8.17655e12 0.351568
\(935\) 0 0
\(936\) −2.88716e13 −1.22950
\(937\) −2.72636e13 −1.15546 −0.577731 0.816227i \(-0.696062\pi\)
−0.577731 + 0.816227i \(0.696062\pi\)
\(938\) −2.38482e13 −1.00587
\(939\) −1.87682e12 −0.0787819
\(940\) 0 0
\(941\) 1.57383e13 0.654340 0.327170 0.944966i \(-0.393905\pi\)
0.327170 + 0.944966i \(0.393905\pi\)
\(942\) −6.70953e12 −0.277628
\(943\) −4.45008e13 −1.83259
\(944\) 1.48399e12 0.0608217
\(945\) 0 0
\(946\) −1.21148e13 −0.491820
\(947\) 9.71128e12 0.392375 0.196188 0.980566i \(-0.437144\pi\)
0.196188 + 0.980566i \(0.437144\pi\)
\(948\) −1.76017e12 −0.0707809
\(949\) −3.49624e13 −1.39927
\(950\) 0 0
\(951\) −6.89257e12 −0.273256
\(952\) 1.59640e13 0.629905
\(953\) −4.15385e12 −0.163130 −0.0815648 0.996668i \(-0.525992\pi\)
−0.0815648 + 0.996668i \(0.525992\pi\)
\(954\) 3.68725e13 1.44123
\(955\) 0 0
\(956\) −5.78084e12 −0.223836
\(957\) −3.36787e12 −0.129793
\(958\) −4.18114e13 −1.60380
\(959\) −1.79897e12 −0.0686818
\(960\) 0 0
\(961\) −1.57452e13 −0.595515
\(962\) −6.27532e12 −0.236237
\(963\) 1.80108e13 0.674861
\(964\) 1.20582e12 0.0449715
\(965\) 0 0
\(966\) 3.72383e12 0.137592
\(967\) −1.84544e13 −0.678706 −0.339353 0.940659i \(-0.610208\pi\)
−0.339353 + 0.940659i \(0.610208\pi\)
\(968\) 7.27484e12 0.266308
\(969\) 1.06130e13 0.386707
\(970\) 0 0
\(971\) −2.95973e12 −0.106848 −0.0534238 0.998572i \(-0.517013\pi\)
−0.0534238 + 0.998572i \(0.517013\pi\)
\(972\) 3.45460e12 0.124137
\(973\) 1.79826e13 0.643197
\(974\) 4.87226e12 0.173466
\(975\) 0 0
\(976\) 1.80664e13 0.637304
\(977\) −4.44178e13 −1.55967 −0.779833 0.625988i \(-0.784696\pi\)
−0.779833 + 0.625988i \(0.784696\pi\)
\(978\) 7.39509e12 0.258475
\(979\) 2.52239e13 0.877588
\(980\) 0 0
\(981\) −1.87319e13 −0.645761
\(982\) 4.60356e13 1.57976
\(983\) −4.57558e13 −1.56299 −0.781493 0.623914i \(-0.785541\pi\)
−0.781493 + 0.623914i \(0.785541\pi\)
\(984\) 1.12107e13 0.381202
\(985\) 0 0
\(986\) 1.69443e13 0.570922
\(987\) 1.65550e11 0.00555269
\(988\) 1.39800e13 0.466769
\(989\) −2.14576e13 −0.713177
\(990\) 0 0
\(991\) 1.30469e13 0.429710 0.214855 0.976646i \(-0.431072\pi\)
0.214855 + 0.976646i \(0.431072\pi\)
\(992\) 7.74210e12 0.253838
\(993\) 2.55903e12 0.0835225
\(994\) 1.26536e13 0.411127
\(995\) 0 0
\(996\) −2.05522e10 −0.000661747 0
\(997\) 5.07578e12 0.162695 0.0813476 0.996686i \(-0.474078\pi\)
0.0813476 + 0.996686i \(0.474078\pi\)
\(998\) 1.99742e13 0.637356
\(999\) 2.94352e12 0.0935022
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 25.10.a.c.1.3 3
3.2 odd 2 225.10.a.p.1.1 3
4.3 odd 2 400.10.a.y.1.2 3
5.2 odd 4 25.10.b.c.24.4 6
5.3 odd 4 25.10.b.c.24.3 6
5.4 even 2 25.10.a.d.1.1 yes 3
15.2 even 4 225.10.b.m.199.3 6
15.8 even 4 225.10.b.m.199.4 6
15.14 odd 2 225.10.a.m.1.3 3
20.3 even 4 400.10.c.q.49.4 6
20.7 even 4 400.10.c.q.49.3 6
20.19 odd 2 400.10.a.u.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.10.a.c.1.3 3 1.1 even 1 trivial
25.10.a.d.1.1 yes 3 5.4 even 2
25.10.b.c.24.3 6 5.3 odd 4
25.10.b.c.24.4 6 5.2 odd 4
225.10.a.m.1.3 3 15.14 odd 2
225.10.a.p.1.1 3 3.2 odd 2
225.10.b.m.199.3 6 15.2 even 4
225.10.b.m.199.4 6 15.8 even 4
400.10.a.u.1.2 3 20.19 odd 2
400.10.a.y.1.2 3 4.3 odd 2
400.10.c.q.49.3 6 20.7 even 4
400.10.c.q.49.4 6 20.3 even 4