Properties

Label 25.34.b.c.24.1
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 24.1
Root \(-89431.3i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.34.b.c.24.12

$q$-expansion

\(f(q)\) \(=\) \(q-178863. i q^{2} +8.55112e7i q^{3} -2.34019e10 q^{4} +1.52948e13 q^{6} +8.09549e13i q^{7} +2.64931e15i q^{8} -1.75311e15 q^{9} +O(q^{10})\) \(q-178863. i q^{2} +8.55112e7i q^{3} -2.34019e10 q^{4} +1.52948e13 q^{6} +8.09549e13i q^{7} +2.64931e15i q^{8} -1.75311e15 q^{9} +1.95800e17 q^{11} -2.00113e18i q^{12} -2.59207e18i q^{13} +1.44798e19 q^{14} +2.72841e20 q^{16} +1.52177e20i q^{17} +3.13565e20i q^{18} -1.39526e21 q^{19} -6.92255e21 q^{21} -3.50214e22i q^{22} -3.40316e22i q^{23} -2.26546e23 q^{24} -4.63625e23 q^{26} +3.25452e23i q^{27} -1.89450e24i q^{28} +1.49246e24 q^{29} -1.38398e24 q^{31} -2.60438e25i q^{32} +1.67431e25i q^{33} +2.72188e25 q^{34} +4.10261e25 q^{36} -1.15380e26i q^{37} +2.49559e26i q^{38} +2.21651e26 q^{39} -6.35087e25 q^{41} +1.23819e27i q^{42} -2.64294e26i q^{43} -4.58211e27 q^{44} -6.08698e27 q^{46} -1.03700e27i q^{47} +2.33310e28i q^{48} +1.17730e27 q^{49} -1.30128e28 q^{51} +6.06594e28i q^{52} +5.20280e28i q^{53} +5.82111e28 q^{54} -2.14475e29 q^{56} -1.19310e29i q^{57} -2.66946e29i q^{58} +6.17754e28 q^{59} -1.21654e29 q^{61} +2.47543e29i q^{62} -1.41923e29i q^{63} -2.31456e30 q^{64} +2.99472e30 q^{66} +2.87484e29i q^{67} -3.56123e30i q^{68} +2.91008e30 q^{69} -2.84409e30 q^{71} -4.64452e30i q^{72} -2.67045e30i q^{73} -2.06372e31 q^{74} +3.26517e31 q^{76} +1.58510e31i q^{77} -3.96451e31i q^{78} +7.84640e30 q^{79} -3.75754e31 q^{81} +1.13593e31i q^{82} +1.28723e31i q^{83} +1.62001e32 q^{84} -4.72724e31 q^{86} +1.27622e32i q^{87} +5.18736e32i q^{88} +1.56902e32 q^{89} +2.09841e32 q^{91} +7.96404e32i q^{92} -1.18346e32i q^{93} -1.85481e32 q^{94} +2.22703e33 q^{96} -4.84738e32i q^{97} -2.10574e32i q^{98} -3.43259e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + O(q^{10}) \) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + 301521793111780384q^{11} + 14172220017979782912q^{14} + \)\(12\!\cdots\!32\)\(q^{16} - \)\(34\!\cdots\!00\)\(q^{19} - \)\(21\!\cdots\!56\)\(q^{21} - \)\(42\!\cdots\!00\)\(q^{24} - \)\(72\!\cdots\!76\)\(q^{26} - \)\(26\!\cdots\!00\)\(q^{29} + \)\(21\!\cdots\!04\)\(q^{31} - \)\(17\!\cdots\!48\)\(q^{34} + \)\(33\!\cdots\!92\)\(q^{36} - \)\(11\!\cdots\!12\)\(q^{39} + \)\(89\!\cdots\!64\)\(q^{41} - \)\(14\!\cdots\!28\)\(q^{44} + \)\(69\!\cdots\!44\)\(q^{46} + \)\(14\!\cdots\!16\)\(q^{49} - \)\(10\!\cdots\!76\)\(q^{51} + \)\(36\!\cdots\!00\)\(q^{54} - \)\(78\!\cdots\!00\)\(q^{56} + \)\(35\!\cdots\!00\)\(q^{59} - \)\(56\!\cdots\!16\)\(q^{61} - \)\(38\!\cdots\!44\)\(q^{64} + \)\(10\!\cdots\!28\)\(q^{66} - \)\(63\!\cdots\!72\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} - \)\(40\!\cdots\!68\)\(q^{74} + \)\(58\!\cdots\!00\)\(q^{76} + \)\(44\!\cdots\!00\)\(q^{79} - \)\(26\!\cdots\!48\)\(q^{81} + \)\(47\!\cdots\!52\)\(q^{84} - \)\(25\!\cdots\!16\)\(q^{86} + \)\(67\!\cdots\!00\)\(q^{89} + \)\(38\!\cdots\!64\)\(q^{91} - \)\(29\!\cdots\!28\)\(q^{94} + \)\(63\!\cdots\!44\)\(q^{96} - \)\(33\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 178863.i − 1.92986i −0.262516 0.964928i \(-0.584552\pi\)
0.262516 0.964928i \(-0.415448\pi\)
\(3\) 8.55112e7i 1.14689i 0.819243 + 0.573446i \(0.194394\pi\)
−0.819243 + 0.573446i \(0.805606\pi\)
\(4\) −2.34019e10 −2.72434
\(5\) 0 0
\(6\) 1.52948e13 2.21333
\(7\) 8.09549e13i 0.920716i 0.887733 + 0.460358i \(0.152279\pi\)
−0.887733 + 0.460358i \(0.847721\pi\)
\(8\) 2.64931e15i 3.32773i
\(9\) −1.75311e15 −0.315360
\(10\) 0 0
\(11\) 1.95800e17 1.28480 0.642398 0.766371i \(-0.277939\pi\)
0.642398 + 0.766371i \(0.277939\pi\)
\(12\) − 2.00113e18i − 3.12452i
\(13\) − 2.59207e18i − 1.08039i −0.841539 0.540197i \(-0.818350\pi\)
0.841539 0.540197i \(-0.181650\pi\)
\(14\) 1.44798e19 1.77685
\(15\) 0 0
\(16\) 2.72841e20 3.69769
\(17\) 1.52177e20i 0.758475i 0.925299 + 0.379238i \(0.123814\pi\)
−0.925299 + 0.379238i \(0.876186\pi\)
\(18\) 3.13565e20i 0.608600i
\(19\) −1.39526e21 −1.10974 −0.554868 0.831938i \(-0.687231\pi\)
−0.554868 + 0.831938i \(0.687231\pi\)
\(20\) 0 0
\(21\) −6.92255e21 −1.05596
\(22\) − 3.50214e22i − 2.47947i
\(23\) − 3.40316e22i − 1.15711i −0.815645 0.578553i \(-0.803618\pi\)
0.815645 0.578553i \(-0.196382\pi\)
\(24\) −2.26546e23 −3.81654
\(25\) 0 0
\(26\) −4.63625e23 −2.08500
\(27\) 3.25452e23i 0.785207i
\(28\) − 1.89450e24i − 2.50834i
\(29\) 1.49246e24 1.10748 0.553741 0.832689i \(-0.313200\pi\)
0.553741 + 0.832689i \(0.313200\pi\)
\(30\) 0 0
\(31\) −1.38398e24 −0.341714 −0.170857 0.985296i \(-0.554654\pi\)
−0.170857 + 0.985296i \(0.554654\pi\)
\(32\) − 2.60438e25i − 3.80828i
\(33\) 1.67431e25i 1.47352i
\(34\) 2.72188e25 1.46375
\(35\) 0 0
\(36\) 4.10261e25 0.859149
\(37\) − 1.15380e26i − 1.53745i −0.639577 0.768727i \(-0.720891\pi\)
0.639577 0.768727i \(-0.279109\pi\)
\(38\) 2.49559e26i 2.14163i
\(39\) 2.21651e26 1.23909
\(40\) 0 0
\(41\) −6.35087e25 −0.155561 −0.0777804 0.996971i \(-0.524783\pi\)
−0.0777804 + 0.996971i \(0.524783\pi\)
\(42\) 1.23819e27i 2.03785i
\(43\) − 2.64294e26i − 0.295025i −0.989060 0.147512i \(-0.952873\pi\)
0.989060 0.147512i \(-0.0471266\pi\)
\(44\) −4.58211e27 −3.50022
\(45\) 0 0
\(46\) −6.08698e27 −2.23305
\(47\) − 1.03700e27i − 0.266788i −0.991063 0.133394i \(-0.957412\pi\)
0.991063 0.133394i \(-0.0425875\pi\)
\(48\) 2.33310e28i 4.24085i
\(49\) 1.17730e27 0.152283
\(50\) 0 0
\(51\) −1.30128e28 −0.869889
\(52\) 6.06594e28i 2.94336i
\(53\) 5.20280e28i 1.84368i 0.387574 + 0.921839i \(0.373313\pi\)
−0.387574 + 0.921839i \(0.626687\pi\)
\(54\) 5.82111e28 1.51534
\(55\) 0 0
\(56\) −2.14475e29 −3.06389
\(57\) − 1.19310e29i − 1.27275i
\(58\) − 2.66946e29i − 2.13728i
\(59\) 6.17754e28 0.373042 0.186521 0.982451i \(-0.440279\pi\)
0.186521 + 0.982451i \(0.440279\pi\)
\(60\) 0 0
\(61\) −1.21654e29 −0.423822 −0.211911 0.977289i \(-0.567969\pi\)
−0.211911 + 0.977289i \(0.567969\pi\)
\(62\) 2.47543e29i 0.659458i
\(63\) − 1.41923e29i − 0.290357i
\(64\) −2.31456e30 −3.65174
\(65\) 0 0
\(66\) 2.99472e30 2.84369
\(67\) 2.87484e29i 0.213000i 0.994313 + 0.106500i \(0.0339644\pi\)
−0.994313 + 0.106500i \(0.966036\pi\)
\(68\) − 3.56123e30i − 2.06634i
\(69\) 2.91008e30 1.32707
\(70\) 0 0
\(71\) −2.84409e30 −0.809432 −0.404716 0.914442i \(-0.632630\pi\)
−0.404716 + 0.914442i \(0.632630\pi\)
\(72\) − 4.64452e30i − 1.04943i
\(73\) − 2.67045e30i − 0.480570i −0.970702 0.240285i \(-0.922759\pi\)
0.970702 0.240285i \(-0.0772410\pi\)
\(74\) −2.06372e31 −2.96706
\(75\) 0 0
\(76\) 3.26517e31 3.02330
\(77\) 1.58510e31i 1.18293i
\(78\) − 3.96451e31i − 2.39127i
\(79\) 7.84640e30 0.383551 0.191776 0.981439i \(-0.438575\pi\)
0.191776 + 0.981439i \(0.438575\pi\)
\(80\) 0 0
\(81\) −3.75754e31 −1.21591
\(82\) 1.13593e31i 0.300210i
\(83\) 1.28723e31i 0.278527i 0.990255 + 0.139263i \(0.0444735\pi\)
−0.990255 + 0.139263i \(0.955527\pi\)
\(84\) 1.62001e32 2.87680
\(85\) 0 0
\(86\) −4.72724e31 −0.569355
\(87\) 1.27622e32i 1.27016i
\(88\) 5.18736e32i 4.27545i
\(89\) 1.56902e32 1.07323 0.536616 0.843826i \(-0.319702\pi\)
0.536616 + 0.843826i \(0.319702\pi\)
\(90\) 0 0
\(91\) 2.09841e32 0.994735
\(92\) 7.96404e32i 3.15235i
\(93\) − 1.18346e32i − 0.391909i
\(94\) −1.85481e32 −0.514862
\(95\) 0 0
\(96\) 2.22703e33 4.36769
\(97\) − 4.84738e32i − 0.801259i −0.916240 0.400630i \(-0.868791\pi\)
0.916240 0.400630i \(-0.131209\pi\)
\(98\) − 2.10574e32i − 0.293883i
\(99\) −3.43259e32 −0.405174
\(100\) 0 0
\(101\) 2.51128e32 0.213105 0.106552 0.994307i \(-0.466019\pi\)
0.106552 + 0.994307i \(0.466019\pi\)
\(102\) 2.32751e33i 1.67876i
\(103\) − 2.60164e33i − 1.59747i −0.601681 0.798737i \(-0.705502\pi\)
0.601681 0.798737i \(-0.294498\pi\)
\(104\) 6.86720e33 3.59525
\(105\) 0 0
\(106\) 9.30586e33 3.55803
\(107\) − 3.74478e33i − 1.22629i −0.789970 0.613146i \(-0.789904\pi\)
0.789970 0.613146i \(-0.210096\pi\)
\(108\) − 7.61619e33i − 2.13917i
\(109\) 2.76732e33 0.667609 0.333805 0.942642i \(-0.391667\pi\)
0.333805 + 0.942642i \(0.391667\pi\)
\(110\) 0 0
\(111\) 9.86628e33 1.76329
\(112\) 2.20879e34i 3.40452i
\(113\) 1.26307e34i 1.68125i 0.541617 + 0.840626i \(0.317812\pi\)
−0.541617 + 0.840626i \(0.682188\pi\)
\(114\) −2.13401e34 −2.45622
\(115\) 0 0
\(116\) −3.49265e34 −3.01716
\(117\) 4.54418e33i 0.340713i
\(118\) − 1.10493e34i − 0.719917i
\(119\) −1.23195e34 −0.698340
\(120\) 0 0
\(121\) 1.51127e34 0.650703
\(122\) 2.17593e34i 0.817915i
\(123\) − 5.43071e33i − 0.178411i
\(124\) 3.23878e34 0.930945
\(125\) 0 0
\(126\) −2.53847e34 −0.560348
\(127\) 2.24824e33i 0.0435594i 0.999763 + 0.0217797i \(0.00693324\pi\)
−0.999763 + 0.0217797i \(0.993067\pi\)
\(128\) 1.90275e35i 3.23905i
\(129\) 2.26001e34 0.338361
\(130\) 0 0
\(131\) 7.19207e34 0.835368 0.417684 0.908592i \(-0.362842\pi\)
0.417684 + 0.908592i \(0.362842\pi\)
\(132\) − 3.91821e35i − 4.01438i
\(133\) − 1.12953e35i − 1.02175i
\(134\) 5.14202e34 0.411059
\(135\) 0 0
\(136\) −4.03163e35 −2.52400
\(137\) − 2.56636e35i − 1.42373i −0.702314 0.711867i \(-0.747850\pi\)
0.702314 0.711867i \(-0.252150\pi\)
\(138\) − 5.20505e35i − 2.56106i
\(139\) −7.81097e34 −0.341162 −0.170581 0.985344i \(-0.554564\pi\)
−0.170581 + 0.985344i \(0.554564\pi\)
\(140\) 0 0
\(141\) 8.86755e34 0.305977
\(142\) 5.08701e35i 1.56209i
\(143\) − 5.07529e35i − 1.38809i
\(144\) −4.78320e35 −1.16611
\(145\) 0 0
\(146\) −4.77643e35 −0.927431
\(147\) 1.00672e35i 0.174652i
\(148\) 2.70011e36i 4.18855i
\(149\) 1.40823e35 0.195479 0.0977397 0.995212i \(-0.468839\pi\)
0.0977397 + 0.995212i \(0.468839\pi\)
\(150\) 0 0
\(151\) 4.46232e35 0.497097 0.248548 0.968619i \(-0.420046\pi\)
0.248548 + 0.968619i \(0.420046\pi\)
\(152\) − 3.69647e36i − 3.69290i
\(153\) − 2.66782e35i − 0.239193i
\(154\) 2.83515e36 2.28289
\(155\) 0 0
\(156\) −5.18706e36 −3.37571
\(157\) − 7.61070e35i − 0.445739i −0.974848 0.222869i \(-0.928458\pi\)
0.974848 0.222869i \(-0.0715424\pi\)
\(158\) − 1.40343e36i − 0.740199i
\(159\) −4.44897e36 −2.11450
\(160\) 0 0
\(161\) 2.75502e36 1.06537
\(162\) 6.72084e36i 2.34653i
\(163\) 1.95336e36i 0.616152i 0.951362 + 0.308076i \(0.0996851\pi\)
−0.951362 + 0.308076i \(0.900315\pi\)
\(164\) 1.48623e36 0.423800
\(165\) 0 0
\(166\) 2.30237e36 0.537516
\(167\) − 2.87170e36i − 0.607180i −0.952803 0.303590i \(-0.901815\pi\)
0.952803 0.303590i \(-0.0981854\pi\)
\(168\) − 1.83400e37i − 3.51395i
\(169\) −9.62710e35 −0.167249
\(170\) 0 0
\(171\) 2.44604e36 0.349967
\(172\) 6.18499e36i 0.803748i
\(173\) 1.61105e37i 1.90261i 0.308258 + 0.951303i \(0.400254\pi\)
−0.308258 + 0.951303i \(0.599746\pi\)
\(174\) 2.28269e37 2.45123
\(175\) 0 0
\(176\) 5.34225e37 4.75078
\(177\) 5.28249e36i 0.427839i
\(178\) − 2.80639e37i − 2.07118i
\(179\) −9.06671e33 −0.000610063 0 −0.000305031 1.00000i \(-0.500097\pi\)
−0.000305031 1.00000i \(0.500097\pi\)
\(180\) 0 0
\(181\) 1.89723e36 0.106273 0.0531364 0.998587i \(-0.483078\pi\)
0.0531364 + 0.998587i \(0.483078\pi\)
\(182\) − 3.75327e37i − 1.91969i
\(183\) − 1.04028e37i − 0.486078i
\(184\) 9.01602e37 3.85053
\(185\) 0 0
\(186\) −2.11677e37 −0.756327
\(187\) 2.97963e37i 0.974487i
\(188\) 2.42679e37i 0.726821i
\(189\) −2.63469e37 −0.722953
\(190\) 0 0
\(191\) −6.88314e37 −1.58758 −0.793790 0.608192i \(-0.791895\pi\)
−0.793790 + 0.608192i \(0.791895\pi\)
\(192\) − 1.97921e38i − 4.18815i
\(193\) 5.10528e37i 0.991571i 0.868445 + 0.495786i \(0.165120\pi\)
−0.868445 + 0.495786i \(0.834880\pi\)
\(194\) −8.67015e37 −1.54631
\(195\) 0 0
\(196\) −2.75510e37 −0.414869
\(197\) 6.02889e37i 0.834727i 0.908740 + 0.417364i \(0.137046\pi\)
−0.908740 + 0.417364i \(0.862954\pi\)
\(198\) 6.13963e37i 0.781927i
\(199\) 1.42678e38 1.67218 0.836089 0.548594i \(-0.184837\pi\)
0.836089 + 0.548594i \(0.184837\pi\)
\(200\) 0 0
\(201\) −2.45831e37 −0.244288
\(202\) − 4.49175e37i − 0.411261i
\(203\) 1.20822e38i 1.01968i
\(204\) 3.04525e38 2.36987
\(205\) 0 0
\(206\) −4.65337e38 −3.08289
\(207\) 5.96610e37i 0.364905i
\(208\) − 7.07225e38i − 3.99496i
\(209\) −2.73192e38 −1.42579
\(210\) 0 0
\(211\) 3.27612e38 1.46116 0.730582 0.682825i \(-0.239249\pi\)
0.730582 + 0.682825i \(0.239249\pi\)
\(212\) − 1.21755e39i − 5.02280i
\(213\) − 2.43201e38i − 0.928331i
\(214\) −6.69801e38 −2.36656
\(215\) 0 0
\(216\) −8.62222e38 −2.61296
\(217\) − 1.12040e38i − 0.314621i
\(218\) − 4.94971e38i − 1.28839i
\(219\) 2.28353e38 0.551162
\(220\) 0 0
\(221\) 3.94453e38 0.819451
\(222\) − 1.76471e39i − 3.40290i
\(223\) − 3.22330e38i − 0.577126i −0.957461 0.288563i \(-0.906823\pi\)
0.957461 0.288563i \(-0.0931774\pi\)
\(224\) 2.10837e39 3.50634
\(225\) 0 0
\(226\) 2.25916e39 3.24457
\(227\) 6.02003e38i 0.803845i 0.915674 + 0.401922i \(0.131658\pi\)
−0.915674 + 0.401922i \(0.868342\pi\)
\(228\) 2.79209e39i 3.46740i
\(229\) 3.71091e38 0.428741 0.214370 0.976752i \(-0.431230\pi\)
0.214370 + 0.976752i \(0.431230\pi\)
\(230\) 0 0
\(231\) −1.35544e39 −1.35670
\(232\) 3.95400e39i 3.68540i
\(233\) − 3.81211e38i − 0.330973i −0.986212 0.165486i \(-0.947081\pi\)
0.986212 0.165486i \(-0.0529194\pi\)
\(234\) 8.12784e38 0.657527
\(235\) 0 0
\(236\) −1.44566e39 −1.01629
\(237\) 6.70955e38i 0.439892i
\(238\) 2.20349e39i 1.34770i
\(239\) −8.52350e38 −0.486466 −0.243233 0.969968i \(-0.578208\pi\)
−0.243233 + 0.969968i \(0.578208\pi\)
\(240\) 0 0
\(241\) 3.41820e39 1.70027 0.850133 0.526568i \(-0.176522\pi\)
0.850133 + 0.526568i \(0.176522\pi\)
\(242\) − 2.70309e39i − 1.25576i
\(243\) − 1.40391e39i − 0.609308i
\(244\) 2.84693e39 1.15464
\(245\) 0 0
\(246\) −9.71351e38 −0.344308
\(247\) 3.61661e39i 1.19895i
\(248\) − 3.66660e39i − 1.13713i
\(249\) −1.10072e39 −0.319440
\(250\) 0 0
\(251\) 2.52321e39 0.641706 0.320853 0.947129i \(-0.396030\pi\)
0.320853 + 0.947129i \(0.396030\pi\)
\(252\) 3.32126e39i 0.791032i
\(253\) − 6.66340e39i − 1.48665i
\(254\) 4.02127e38 0.0840633
\(255\) 0 0
\(256\) 1.41511e40 2.59915
\(257\) − 4.18511e39i − 0.720793i −0.932799 0.360396i \(-0.882641\pi\)
0.932799 0.360396i \(-0.117359\pi\)
\(258\) − 4.04232e39i − 0.652988i
\(259\) 9.34058e39 1.41556
\(260\) 0 0
\(261\) −2.61645e39 −0.349256
\(262\) − 1.28639e40i − 1.61214i
\(263\) 1.15657e40i 1.36114i 0.732683 + 0.680570i \(0.238268\pi\)
−0.732683 + 0.680570i \(0.761732\pi\)
\(264\) −4.43577e40 −4.90348
\(265\) 0 0
\(266\) −2.02031e40 −1.97183
\(267\) 1.34169e40i 1.23088i
\(268\) − 6.72768e39i − 0.580284i
\(269\) 7.21991e39 0.585624 0.292812 0.956170i \(-0.405409\pi\)
0.292812 + 0.956170i \(0.405409\pi\)
\(270\) 0 0
\(271\) 2.28804e40 1.64237 0.821183 0.570665i \(-0.193315\pi\)
0.821183 + 0.570665i \(0.193315\pi\)
\(272\) 4.15202e40i 2.80461i
\(273\) 1.79438e40i 1.14085i
\(274\) −4.59026e40 −2.74760
\(275\) 0 0
\(276\) −6.81015e40 −3.61540
\(277\) − 2.67522e40i − 1.33796i −0.743279 0.668981i \(-0.766731\pi\)
0.743279 0.668981i \(-0.233269\pi\)
\(278\) 1.39709e40i 0.658394i
\(279\) 2.42627e39 0.107763
\(280\) 0 0
\(281\) −2.51780e40 −0.993955 −0.496977 0.867763i \(-0.665557\pi\)
−0.496977 + 0.867763i \(0.665557\pi\)
\(282\) − 1.58607e40i − 0.590490i
\(283\) 4.17854e40i 1.46740i 0.679475 + 0.733699i \(0.262208\pi\)
−0.679475 + 0.733699i \(0.737792\pi\)
\(284\) 6.65571e40 2.20517
\(285\) 0 0
\(286\) −9.07780e40 −2.67880
\(287\) − 5.14135e39i − 0.143227i
\(288\) 4.56575e40i 1.20098i
\(289\) 1.70967e40 0.424715
\(290\) 0 0
\(291\) 4.14505e40 0.918958
\(292\) 6.24936e40i 1.30924i
\(293\) − 1.81801e40i − 0.359981i −0.983668 0.179991i \(-0.942393\pi\)
0.983668 0.179991i \(-0.0576068\pi\)
\(294\) 1.80064e40 0.337052
\(295\) 0 0
\(296\) 3.05677e41 5.11623
\(297\) 6.37236e40i 1.00883i
\(298\) − 2.51880e40i − 0.377247i
\(299\) −8.82124e40 −1.25013
\(300\) 0 0
\(301\) 2.13959e40 0.271634
\(302\) − 7.98142e40i − 0.959325i
\(303\) 2.14743e40i 0.244408i
\(304\) −3.80684e41 −4.10346
\(305\) 0 0
\(306\) −4.77174e40 −0.461608
\(307\) 1.04678e40i 0.0959564i 0.998848 + 0.0479782i \(0.0152778\pi\)
−0.998848 + 0.0479782i \(0.984722\pi\)
\(308\) − 3.70944e41i − 3.22271i
\(309\) 2.22470e41 1.83213
\(310\) 0 0
\(311\) 1.01794e41 0.753660 0.376830 0.926282i \(-0.377014\pi\)
0.376830 + 0.926282i \(0.377014\pi\)
\(312\) 5.87223e41i 4.12337i
\(313\) − 2.12567e41i − 1.41584i −0.706292 0.707921i \(-0.749633\pi\)
0.706292 0.707921i \(-0.250367\pi\)
\(314\) −1.36127e41 −0.860212
\(315\) 0 0
\(316\) −1.83621e41 −1.04492
\(317\) − 5.12986e40i − 0.277094i −0.990356 0.138547i \(-0.955757\pi\)
0.990356 0.138547i \(-0.0442433\pi\)
\(318\) 7.95755e41i 4.08067i
\(319\) 2.92225e41 1.42289
\(320\) 0 0
\(321\) 3.20220e41 1.40642
\(322\) − 4.92771e41i − 2.05600i
\(323\) − 2.12326e41i − 0.841708i
\(324\) 8.79336e41 3.31255
\(325\) 0 0
\(326\) 3.49384e41 1.18908
\(327\) 2.36637e41i 0.765676i
\(328\) − 1.68254e41i − 0.517664i
\(329\) 8.39506e40 0.245636
\(330\) 0 0
\(331\) −4.71143e41 −1.24736 −0.623679 0.781681i \(-0.714363\pi\)
−0.623679 + 0.781681i \(0.714363\pi\)
\(332\) − 3.01236e41i − 0.758802i
\(333\) 2.02274e41i 0.484852i
\(334\) −5.13640e41 −1.17177
\(335\) 0 0
\(336\) −1.88876e42 −3.90462
\(337\) − 8.62495e40i − 0.169771i −0.996391 0.0848854i \(-0.972948\pi\)
0.996391 0.0848854i \(-0.0270524\pi\)
\(338\) 1.72193e41i 0.322767i
\(339\) −1.08007e42 −1.92821
\(340\) 0 0
\(341\) −2.70984e41 −0.439033
\(342\) − 4.37505e41i − 0.675386i
\(343\) 7.21170e41i 1.06092i
\(344\) 7.00197e41 0.981762
\(345\) 0 0
\(346\) 2.88157e42 3.67175
\(347\) − 2.78000e41i − 0.337760i −0.985637 0.168880i \(-0.945985\pi\)
0.985637 0.168880i \(-0.0540151\pi\)
\(348\) − 2.98661e42i − 3.46036i
\(349\) 1.12911e42 1.24772 0.623860 0.781536i \(-0.285563\pi\)
0.623860 + 0.781536i \(0.285563\pi\)
\(350\) 0 0
\(351\) 8.43594e41 0.848333
\(352\) − 5.09938e42i − 4.89287i
\(353\) 1.75490e42i 1.60683i 0.595422 + 0.803413i \(0.296985\pi\)
−0.595422 + 0.803413i \(0.703015\pi\)
\(354\) 9.44841e41 0.825667
\(355\) 0 0
\(356\) −3.67181e42 −2.92385
\(357\) − 1.05345e42i − 0.800920i
\(358\) 1.62170e39i 0.00117733i
\(359\) 2.42990e42 1.68473 0.842365 0.538907i \(-0.181163\pi\)
0.842365 + 0.538907i \(0.181163\pi\)
\(360\) 0 0
\(361\) 3.65974e41 0.231516
\(362\) − 3.39343e41i − 0.205091i
\(363\) 1.29230e42i 0.746286i
\(364\) −4.91068e42 −2.71000
\(365\) 0 0
\(366\) −1.86067e42 −0.938060
\(367\) 2.37509e42i 1.14470i 0.820010 + 0.572349i \(0.193968\pi\)
−0.820010 + 0.572349i \(0.806032\pi\)
\(368\) − 9.28523e42i − 4.27862i
\(369\) 1.11338e41 0.0490577
\(370\) 0 0
\(371\) −4.21192e42 −1.69750
\(372\) 2.76952e42i 1.06769i
\(373\) 1.58778e42i 0.585592i 0.956175 + 0.292796i \(0.0945856\pi\)
−0.956175 + 0.292796i \(0.905414\pi\)
\(374\) 5.32944e42 1.88062
\(375\) 0 0
\(376\) 2.74734e42 0.887797
\(377\) − 3.86858e42i − 1.19652i
\(378\) 4.71248e42i 1.39519i
\(379\) 2.74032e42 0.776703 0.388351 0.921511i \(-0.373045\pi\)
0.388351 + 0.921511i \(0.373045\pi\)
\(380\) 0 0
\(381\) −1.92250e41 −0.0499579
\(382\) 1.23114e43i 3.06380i
\(383\) 3.57273e42i 0.851570i 0.904824 + 0.425785i \(0.140002\pi\)
−0.904824 + 0.425785i \(0.859998\pi\)
\(384\) −1.62706e43 −3.71483
\(385\) 0 0
\(386\) 9.13144e42 1.91359
\(387\) 4.63336e41i 0.0930391i
\(388\) 1.13438e43i 2.18290i
\(389\) 2.24258e42 0.413599 0.206799 0.978383i \(-0.433695\pi\)
0.206799 + 0.978383i \(0.433695\pi\)
\(390\) 0 0
\(391\) 5.17882e42 0.877636
\(392\) 3.11902e42i 0.506755i
\(393\) 6.15003e42i 0.958077i
\(394\) 1.07834e43 1.61090
\(395\) 0 0
\(396\) 8.03292e42 1.10383
\(397\) 8.69665e41i 0.114633i 0.998356 + 0.0573164i \(0.0182544\pi\)
−0.998356 + 0.0573164i \(0.981746\pi\)
\(398\) − 2.55198e43i − 3.22706i
\(399\) 9.65875e42 1.17184
\(400\) 0 0
\(401\) 1.79590e41 0.0200632 0.0100316 0.999950i \(-0.496807\pi\)
0.0100316 + 0.999950i \(0.496807\pi\)
\(402\) 4.39700e42i 0.471440i
\(403\) 3.58738e42i 0.369185i
\(404\) −5.87688e42 −0.580569
\(405\) 0 0
\(406\) 2.16106e43 1.96783
\(407\) − 2.25915e43i − 1.97532i
\(408\) − 3.44750e43i − 2.89475i
\(409\) −1.72895e43 −1.39427 −0.697136 0.716939i \(-0.745543\pi\)
−0.697136 + 0.716939i \(0.745543\pi\)
\(410\) 0 0
\(411\) 2.19453e43 1.63287
\(412\) 6.08834e43i 4.35206i
\(413\) 5.00103e42i 0.343466i
\(414\) 1.06711e43 0.704214
\(415\) 0 0
\(416\) −6.75073e43 −4.11444
\(417\) − 6.67926e42i − 0.391276i
\(418\) 4.88639e43i 2.75156i
\(419\) 3.07423e43 1.66420 0.832099 0.554628i \(-0.187139\pi\)
0.832099 + 0.554628i \(0.187139\pi\)
\(420\) 0 0
\(421\) −1.46291e43 −0.732090 −0.366045 0.930597i \(-0.619288\pi\)
−0.366045 + 0.930597i \(0.619288\pi\)
\(422\) − 5.85975e43i − 2.81983i
\(423\) 1.81798e42i 0.0841343i
\(424\) −1.37838e44 −6.13526
\(425\) 0 0
\(426\) −4.34997e43 −1.79154
\(427\) − 9.84848e42i − 0.390220i
\(428\) 8.76349e43i 3.34083i
\(429\) 4.33994e43 1.59198
\(430\) 0 0
\(431\) 3.24328e43 1.10182 0.550909 0.834565i \(-0.314281\pi\)
0.550909 + 0.834565i \(0.314281\pi\)
\(432\) 8.87967e43i 2.90345i
\(433\) − 2.61817e42i − 0.0824040i −0.999151 0.0412020i \(-0.986881\pi\)
0.999151 0.0412020i \(-0.0131187\pi\)
\(434\) −2.00398e43 −0.607174
\(435\) 0 0
\(436\) −6.47606e43 −1.81880
\(437\) 4.74828e43i 1.28408i
\(438\) − 4.08439e43i − 1.06366i
\(439\) 6.39695e43 1.60438 0.802192 0.597066i \(-0.203667\pi\)
0.802192 + 0.597066i \(0.203667\pi\)
\(440\) 0 0
\(441\) −2.06393e42 −0.0480239
\(442\) − 7.05530e43i − 1.58142i
\(443\) 1.30321e43i 0.281418i 0.990051 + 0.140709i \(0.0449382\pi\)
−0.990051 + 0.140709i \(0.955062\pi\)
\(444\) −2.30890e44 −4.80381
\(445\) 0 0
\(446\) −5.76528e43 −1.11377
\(447\) 1.20420e43i 0.224194i
\(448\) − 1.87375e44i − 3.36221i
\(449\) 9.56472e43 1.65428 0.827138 0.561999i \(-0.189968\pi\)
0.827138 + 0.561999i \(0.189968\pi\)
\(450\) 0 0
\(451\) −1.24350e43 −0.199864
\(452\) − 2.95582e44i − 4.58030i
\(453\) 3.81578e43i 0.570116i
\(454\) 1.07676e44 1.55130
\(455\) 0 0
\(456\) 3.16090e44 4.23536
\(457\) 3.78578e42i 0.0489258i 0.999701 + 0.0244629i \(0.00778755\pi\)
−0.999701 + 0.0244629i \(0.992212\pi\)
\(458\) − 6.63743e43i − 0.827408i
\(459\) −4.95262e43 −0.595560
\(460\) 0 0
\(461\) 7.04190e43 0.788179 0.394089 0.919072i \(-0.371060\pi\)
0.394089 + 0.919072i \(0.371060\pi\)
\(462\) 2.42437e44i 2.61823i
\(463\) 1.41714e44i 1.47682i 0.674353 + 0.738409i \(0.264423\pi\)
−0.674353 + 0.738409i \(0.735577\pi\)
\(464\) 4.07206e44 4.09513
\(465\) 0 0
\(466\) −6.81843e43 −0.638729
\(467\) 1.83277e44i 1.65721i 0.559831 + 0.828606i \(0.310866\pi\)
−0.559831 + 0.828606i \(0.689134\pi\)
\(468\) − 1.06343e44i − 0.928219i
\(469\) −2.32732e43 −0.196112
\(470\) 0 0
\(471\) 6.50801e43 0.511214
\(472\) 1.63662e44i 1.24138i
\(473\) − 5.17490e43i − 0.379047i
\(474\) 1.20009e44 0.848928
\(475\) 0 0
\(476\) 2.88299e44 1.90252
\(477\) − 9.12106e43i − 0.581423i
\(478\) 1.52454e44i 0.938809i
\(479\) 6.67837e43 0.397314 0.198657 0.980069i \(-0.436342\pi\)
0.198657 + 0.980069i \(0.436342\pi\)
\(480\) 0 0
\(481\) −2.99073e44 −1.66105
\(482\) − 6.11389e44i − 3.28127i
\(483\) 2.35586e44i 1.22186i
\(484\) −3.53666e44 −1.77274
\(485\) 0 0
\(486\) −2.51108e44 −1.17588
\(487\) 1.75263e44i 0.793345i 0.917960 + 0.396673i \(0.129835\pi\)
−0.917960 + 0.396673i \(0.870165\pi\)
\(488\) − 3.22299e44i − 1.41036i
\(489\) −1.67035e44 −0.706659
\(490\) 0 0
\(491\) 3.92240e44 1.55134 0.775670 0.631139i \(-0.217412\pi\)
0.775670 + 0.631139i \(0.217412\pi\)
\(492\) 1.27089e44i 0.486053i
\(493\) 2.27118e44i 0.839998i
\(494\) 6.46876e44 2.31380
\(495\) 0 0
\(496\) −3.77608e44 −1.26355
\(497\) − 2.30243e44i − 0.745257i
\(498\) 1.96878e44i 0.616473i
\(499\) −7.64433e43 −0.231569 −0.115785 0.993274i \(-0.536938\pi\)
−0.115785 + 0.993274i \(0.536938\pi\)
\(500\) 0 0
\(501\) 2.45563e44 0.696370
\(502\) − 4.51307e44i − 1.23840i
\(503\) − 4.88916e44i − 1.29826i −0.760676 0.649132i \(-0.775132\pi\)
0.760676 0.649132i \(-0.224868\pi\)
\(504\) 3.75997e44 0.966230
\(505\) 0 0
\(506\) −1.19183e45 −2.86901
\(507\) − 8.23225e43i − 0.191817i
\(508\) − 5.26132e43i − 0.118671i
\(509\) 3.35020e44 0.731522 0.365761 0.930709i \(-0.380809\pi\)
0.365761 + 0.930709i \(0.380809\pi\)
\(510\) 0 0
\(511\) 2.16186e44 0.442469
\(512\) − 8.96657e44i − 1.77694i
\(513\) − 4.54089e44i − 0.871374i
\(514\) −7.48560e44 −1.39103
\(515\) 0 0
\(516\) −5.28886e44 −0.921812
\(517\) − 2.03046e44i − 0.342768i
\(518\) − 1.67068e45i − 2.73182i
\(519\) −1.37763e45 −2.18208
\(520\) 0 0
\(521\) −5.22367e44 −0.776521 −0.388261 0.921550i \(-0.626924\pi\)
−0.388261 + 0.921550i \(0.626924\pi\)
\(522\) 4.67985e44i 0.674014i
\(523\) − 1.10438e45i − 1.54113i −0.637360 0.770566i \(-0.719974\pi\)
0.637360 0.770566i \(-0.280026\pi\)
\(524\) −1.68308e45 −2.27583
\(525\) 0 0
\(526\) 2.06868e45 2.62680
\(527\) − 2.10610e44i − 0.259182i
\(528\) 4.56822e45i 5.44863i
\(529\) −2.93144e44 −0.338893
\(530\) 0 0
\(531\) −1.08299e44 −0.117643
\(532\) 2.64332e45i 2.78360i
\(533\) 1.64619e44i 0.168067i
\(534\) 2.39978e45 2.37542
\(535\) 0 0
\(536\) −7.61634e44 −0.708806
\(537\) − 7.75305e41i 0 0.000699676i
\(538\) − 1.29137e45i − 1.13017i
\(539\) 2.30515e44 0.195652
\(540\) 0 0
\(541\) −1.40716e45 −1.12354 −0.561772 0.827292i \(-0.689880\pi\)
−0.561772 + 0.827292i \(0.689880\pi\)
\(542\) − 4.09244e45i − 3.16953i
\(543\) 1.62234e44i 0.121883i
\(544\) 3.96326e45 2.88849
\(545\) 0 0
\(546\) 3.20947e45 2.20168
\(547\) 1.00599e45i 0.669579i 0.942293 + 0.334789i \(0.108665\pi\)
−0.942293 + 0.334789i \(0.891335\pi\)
\(548\) 6.00578e45i 3.87874i
\(549\) 2.13272e44 0.133657
\(550\) 0 0
\(551\) −2.08237e45 −1.22901
\(552\) 7.70971e45i 4.41614i
\(553\) 6.35205e44i 0.353142i
\(554\) −4.78498e45 −2.58207
\(555\) 0 0
\(556\) 1.82792e45 0.929442
\(557\) − 6.35386e44i − 0.313637i −0.987627 0.156818i \(-0.949876\pi\)
0.987627 0.156818i \(-0.0501238\pi\)
\(558\) − 4.33969e44i − 0.207967i
\(559\) −6.85070e44 −0.318743
\(560\) 0 0
\(561\) −2.54792e45 −1.11763
\(562\) 4.50340e45i 1.91819i
\(563\) − 1.73714e45i − 0.718534i −0.933235 0.359267i \(-0.883027\pi\)
0.933235 0.359267i \(-0.116973\pi\)
\(564\) −2.07518e45 −0.833584
\(565\) 0 0
\(566\) 7.47384e45 2.83187
\(567\) − 3.04191e45i − 1.11951i
\(568\) − 7.53487e45i − 2.69357i
\(569\) −3.27076e45 −1.13579 −0.567894 0.823102i \(-0.692241\pi\)
−0.567894 + 0.823102i \(0.692241\pi\)
\(570\) 0 0
\(571\) 2.07979e45 0.681591 0.340795 0.940137i \(-0.389304\pi\)
0.340795 + 0.940137i \(0.389304\pi\)
\(572\) 1.18771e46i 3.78162i
\(573\) − 5.88585e45i − 1.82078i
\(574\) −9.19595e44 −0.276408
\(575\) 0 0
\(576\) 4.05768e45 1.15161
\(577\) 2.30164e45i 0.634800i 0.948292 + 0.317400i \(0.102810\pi\)
−0.948292 + 0.317400i \(0.897190\pi\)
\(578\) − 3.05796e45i − 0.819639i
\(579\) −4.36559e45 −1.13722
\(580\) 0 0
\(581\) −1.04207e45 −0.256444
\(582\) − 7.41395e45i − 1.77346i
\(583\) 1.01871e46i 2.36875i
\(584\) 7.07484e45 1.59921
\(585\) 0 0
\(586\) −3.25174e45 −0.694712
\(587\) − 1.99372e45i − 0.414128i −0.978327 0.207064i \(-0.933609\pi\)
0.978327 0.207064i \(-0.0663909\pi\)
\(588\) − 2.35592e45i − 0.475810i
\(589\) 1.93101e45 0.379213
\(590\) 0 0
\(591\) −5.15538e45 −0.957341
\(592\) − 3.14804e46i − 5.68503i
\(593\) − 6.54556e45i − 1.14959i −0.818296 0.574797i \(-0.805081\pi\)
0.818296 0.574797i \(-0.194919\pi\)
\(594\) 1.13978e46 1.94690
\(595\) 0 0
\(596\) −3.29553e45 −0.532553
\(597\) 1.22006e46i 1.91781i
\(598\) 1.57779e46i 2.41257i
\(599\) −1.89735e45 −0.282231 −0.141115 0.989993i \(-0.545069\pi\)
−0.141115 + 0.989993i \(0.545069\pi\)
\(600\) 0 0
\(601\) −8.66013e45 −1.21926 −0.609629 0.792687i \(-0.708681\pi\)
−0.609629 + 0.792687i \(0.708681\pi\)
\(602\) − 3.82693e45i − 0.524214i
\(603\) − 5.03991e44i − 0.0671718i
\(604\) −1.04427e46 −1.35426
\(605\) 0 0
\(606\) 3.84095e45 0.471672
\(607\) − 8.81789e45i − 1.05378i −0.849932 0.526892i \(-0.823357\pi\)
0.849932 0.526892i \(-0.176643\pi\)
\(608\) 3.63378e46i 4.22619i
\(609\) −1.03317e46 −1.16946
\(610\) 0 0
\(611\) −2.68799e45 −0.288236
\(612\) 6.24322e45i 0.651643i
\(613\) − 1.91712e46i − 1.94783i −0.226912 0.973915i \(-0.572863\pi\)
0.226912 0.973915i \(-0.427137\pi\)
\(614\) 1.87231e45 0.185182
\(615\) 0 0
\(616\) −4.19942e46 −3.93648
\(617\) 1.53369e46i 1.39969i 0.714295 + 0.699844i \(0.246747\pi\)
−0.714295 + 0.699844i \(0.753253\pi\)
\(618\) − 3.97915e46i − 3.53574i
\(619\) −8.04204e45 −0.695778 −0.347889 0.937536i \(-0.613101\pi\)
−0.347889 + 0.937536i \(0.613101\pi\)
\(620\) 0 0
\(621\) 1.10756e46 0.908568
\(622\) − 1.82071e46i − 1.45446i
\(623\) 1.27020e46i 0.988142i
\(624\) 6.04757e46 4.58179
\(625\) 0 0
\(626\) −3.80203e46 −2.73237
\(627\) − 2.33610e46i − 1.63522i
\(628\) 1.78105e46i 1.21434i
\(629\) 1.75582e46 1.16612
\(630\) 0 0
\(631\) 2.52921e46 1.59405 0.797023 0.603949i \(-0.206407\pi\)
0.797023 + 0.603949i \(0.206407\pi\)
\(632\) 2.07875e46i 1.27635i
\(633\) 2.80145e46i 1.67580i
\(634\) −9.17540e45 −0.534752
\(635\) 0 0
\(636\) 1.04114e47 5.76061
\(637\) − 3.05163e45i − 0.164525i
\(638\) − 5.22682e46i − 2.74597i
\(639\) 4.98599e45 0.255263
\(640\) 0 0
\(641\) 2.51177e46 1.22130 0.610651 0.791900i \(-0.290908\pi\)
0.610651 + 0.791900i \(0.290908\pi\)
\(642\) − 5.72755e46i − 2.71419i
\(643\) − 1.47547e46i − 0.681473i −0.940159 0.340736i \(-0.889324\pi\)
0.940159 0.340736i \(-0.110676\pi\)
\(644\) −6.44728e46 −2.90242
\(645\) 0 0
\(646\) −3.79772e46 −1.62437
\(647\) 4.64548e46i 1.93691i 0.249182 + 0.968457i \(0.419838\pi\)
−0.249182 + 0.968457i \(0.580162\pi\)
\(648\) − 9.95488e46i − 4.04621i
\(649\) 1.20957e46 0.479283
\(650\) 0 0
\(651\) 9.58069e45 0.360837
\(652\) − 4.57124e46i − 1.67861i
\(653\) 3.12135e46i 1.11757i 0.829312 + 0.558786i \(0.188733\pi\)
−0.829312 + 0.558786i \(0.811267\pi\)
\(654\) 4.23255e46 1.47764
\(655\) 0 0
\(656\) −1.73278e46 −0.575215
\(657\) 4.68158e45i 0.151553i
\(658\) − 1.50156e46i − 0.474041i
\(659\) −1.16512e46 −0.358724 −0.179362 0.983783i \(-0.557403\pi\)
−0.179362 + 0.983783i \(0.557403\pi\)
\(660\) 0 0
\(661\) −1.14126e46 −0.334241 −0.167121 0.985936i \(-0.553447\pi\)
−0.167121 + 0.985936i \(0.553447\pi\)
\(662\) 8.42699e46i 2.40722i
\(663\) 3.37302e46i 0.939822i
\(664\) −3.41026e46 −0.926861
\(665\) 0 0
\(666\) 3.61792e46 0.935694
\(667\) − 5.07909e46i − 1.28147i
\(668\) 6.72033e46i 1.65417i
\(669\) 2.75628e46 0.661901
\(670\) 0 0
\(671\) −2.38199e46 −0.544525
\(672\) 1.80289e47i 4.02140i
\(673\) 4.26138e45i 0.0927475i 0.998924 + 0.0463737i \(0.0147665\pi\)
−0.998924 + 0.0463737i \(0.985233\pi\)
\(674\) −1.54268e46 −0.327633
\(675\) 0 0
\(676\) 2.25292e46 0.455644
\(677\) 6.45770e46i 1.27457i 0.770627 + 0.637287i \(0.219943\pi\)
−0.770627 + 0.637287i \(0.780057\pi\)
\(678\) 1.93183e47i 3.72117i
\(679\) 3.92419e46 0.737732
\(680\) 0 0
\(681\) −5.14780e46 −0.921923
\(682\) 4.84690e46i 0.847270i
\(683\) 3.35691e46i 0.572794i 0.958111 + 0.286397i \(0.0924577\pi\)
−0.958111 + 0.286397i \(0.907542\pi\)
\(684\) −5.72419e46 −0.953430
\(685\) 0 0
\(686\) 1.28990e47 2.04743
\(687\) 3.17324e46i 0.491719i
\(688\) − 7.21105e46i − 1.09091i
\(689\) 1.34860e47 1.99190
\(690\) 0 0
\(691\) −4.55000e46 −0.640654 −0.320327 0.947307i \(-0.603793\pi\)
−0.320327 + 0.947307i \(0.603793\pi\)
\(692\) − 3.77017e47i − 5.18334i
\(693\) − 2.77885e46i − 0.373050i
\(694\) −4.97238e46 −0.651828
\(695\) 0 0
\(696\) −3.38111e47 −4.22676
\(697\) − 9.66456e45i − 0.117989i
\(698\) − 2.01956e47i − 2.40792i
\(699\) 3.25978e46 0.379590
\(700\) 0 0
\(701\) −1.37611e47 −1.52864 −0.764320 0.644837i \(-0.776925\pi\)
−0.764320 + 0.644837i \(0.776925\pi\)
\(702\) − 1.50888e47i − 1.63716i
\(703\) 1.60985e47i 1.70617i
\(704\) −4.53193e47 −4.69174
\(705\) 0 0
\(706\) 3.13885e47 3.10094
\(707\) 2.03301e46i 0.196209i
\(708\) − 1.23620e47i − 1.16558i
\(709\) 1.72099e47 1.58531 0.792656 0.609670i \(-0.208698\pi\)
0.792656 + 0.609670i \(0.208698\pi\)
\(710\) 0 0
\(711\) −1.37556e46 −0.120957
\(712\) 4.15682e47i 3.57143i
\(713\) 4.70991e46i 0.395399i
\(714\) −1.88423e47 −1.54566
\(715\) 0 0
\(716\) 2.12178e44 0.00166202
\(717\) − 7.28855e46i − 0.557924i
\(718\) − 4.34619e47i − 3.25128i
\(719\) −1.00489e47 −0.734667 −0.367334 0.930089i \(-0.619729\pi\)
−0.367334 + 0.930089i \(0.619729\pi\)
\(720\) 0 0
\(721\) 2.10616e47 1.47082
\(722\) − 6.54591e46i − 0.446793i
\(723\) 2.92295e47i 1.95002i
\(724\) −4.43987e46 −0.289524
\(725\) 0 0
\(726\) 2.31145e47 1.44022
\(727\) − 1.79204e47i − 1.09151i −0.837944 0.545757i \(-0.816242\pi\)
0.837944 0.545757i \(-0.183758\pi\)
\(728\) 5.55934e47i 3.31021i
\(729\) −8.88336e46 −0.517098
\(730\) 0 0
\(731\) 4.02195e46 0.223769
\(732\) 2.43445e47i 1.32424i
\(733\) 1.91032e46i 0.101599i 0.998709 + 0.0507996i \(0.0161770\pi\)
−0.998709 + 0.0507996i \(0.983823\pi\)
\(734\) 4.24816e47 2.20910
\(735\) 0 0
\(736\) −8.86310e47 −4.40658
\(737\) 5.62895e46i 0.273662i
\(738\) − 1.99141e46i − 0.0946742i
\(739\) −2.81727e47 −1.30977 −0.654886 0.755728i \(-0.727283\pi\)
−0.654886 + 0.755728i \(0.727283\pi\)
\(740\) 0 0
\(741\) −3.09261e47 −1.37507
\(742\) 7.53355e47i 3.27593i
\(743\) − 5.77441e46i − 0.245580i −0.992433 0.122790i \(-0.960816\pi\)
0.992433 0.122790i \(-0.0391841\pi\)
\(744\) 3.13535e47 1.30417
\(745\) 0 0
\(746\) 2.83995e47 1.13011
\(747\) − 2.25665e46i − 0.0878363i
\(748\) − 6.97290e47i − 2.65483i
\(749\) 3.03158e47 1.12907
\(750\) 0 0
\(751\) 2.71539e47 0.967772 0.483886 0.875131i \(-0.339225\pi\)
0.483886 + 0.875131i \(0.339225\pi\)
\(752\) − 2.82938e47i − 0.986498i
\(753\) 2.15762e47i 0.735968i
\(754\) −6.91944e47 −2.30910
\(755\) 0 0
\(756\) 6.16568e47 1.96957
\(757\) − 2.34370e47i − 0.732521i −0.930512 0.366260i \(-0.880638\pi\)
0.930512 0.366260i \(-0.119362\pi\)
\(758\) − 4.90141e47i − 1.49892i
\(759\) 5.69796e47 1.70502
\(760\) 0 0
\(761\) −3.85769e47 −1.10530 −0.552651 0.833413i \(-0.686384\pi\)
−0.552651 + 0.833413i \(0.686384\pi\)
\(762\) 3.43863e46i 0.0964115i
\(763\) 2.24028e47i 0.614679i
\(764\) 1.61079e48 4.32511
\(765\) 0 0
\(766\) 6.39028e47 1.64341
\(767\) − 1.60126e47i − 0.403032i
\(768\) 1.21008e48i 2.98094i
\(769\) −5.88677e47 −1.41936 −0.709679 0.704525i \(-0.751160\pi\)
−0.709679 + 0.704525i \(0.751160\pi\)
\(770\) 0 0
\(771\) 3.57874e47 0.826671
\(772\) − 1.19473e48i − 2.70138i
\(773\) − 4.68346e47i − 1.03659i −0.855203 0.518293i \(-0.826568\pi\)
0.855203 0.518293i \(-0.173432\pi\)
\(774\) 8.28736e46 0.179552
\(775\) 0 0
\(776\) 1.28422e48 2.66637
\(777\) 7.98724e47i 1.62349i
\(778\) − 4.01113e47i − 0.798185i
\(779\) 8.86111e46 0.172631
\(780\) 0 0
\(781\) −5.56874e47 −1.03996
\(782\) − 9.26297e47i − 1.69371i
\(783\) 4.85725e47i 0.869604i
\(784\) 3.21215e47 0.563094
\(785\) 0 0
\(786\) 1.10001e48 1.84895
\(787\) 3.57150e47i 0.587851i 0.955828 + 0.293926i \(0.0949618\pi\)
−0.955828 + 0.293926i \(0.905038\pi\)
\(788\) − 1.41087e48i − 2.27408i
\(789\) −9.88999e47 −1.56108
\(790\) 0 0
\(791\) −1.02252e48 −1.54795
\(792\) − 9.09400e47i − 1.34831i
\(793\) 3.15336e47i 0.457895i
\(794\) 1.55551e47 0.221225
\(795\) 0 0
\(796\) −3.33895e48 −4.55558
\(797\) 3.36343e47i 0.449490i 0.974418 + 0.224745i \(0.0721549\pi\)
−0.974418 + 0.224745i \(0.927845\pi\)
\(798\) − 1.72759e48i − 2.26148i
\(799\) 1.57808e47 0.202352
\(800\) 0 0
\(801\) −2.75066e47 −0.338455
\(802\) − 3.21219e46i − 0.0387190i
\(803\) − 5.22875e47i − 0.617435i
\(804\) 5.75292e47 0.665523
\(805\) 0 0
\(806\) 6.41649e47 0.712474
\(807\) 6.17383e47i 0.671647i
\(808\) 6.65316e47i 0.709154i
\(809\) −8.08353e47 −0.844210 −0.422105 0.906547i \(-0.638709\pi\)
−0.422105 + 0.906547i \(0.638709\pi\)
\(810\) 0 0
\(811\) −1.18729e48 −1.19045 −0.595225 0.803559i \(-0.702937\pi\)
−0.595225 + 0.803559i \(0.702937\pi\)
\(812\) − 2.82747e48i − 2.77795i
\(813\) 1.95653e48i 1.88362i
\(814\) −4.04077e48 −3.81207
\(815\) 0 0
\(816\) −3.55044e48 −3.21658
\(817\) 3.68759e47i 0.327400i
\(818\) 3.09244e48i 2.69074i
\(819\) −3.67874e47 −0.313700
\(820\) 0 0
\(821\) −1.55006e48 −1.26966 −0.634829 0.772653i \(-0.718929\pi\)
−0.634829 + 0.772653i \(0.718929\pi\)
\(822\) − 3.92519e48i − 3.15120i
\(823\) 2.91534e47i 0.229399i 0.993400 + 0.114700i \(0.0365905\pi\)
−0.993400 + 0.114700i \(0.963409\pi\)
\(824\) 6.89256e48 5.31596
\(825\) 0 0
\(826\) 8.94497e47 0.662839
\(827\) 1.33839e48i 0.972166i 0.873913 + 0.486083i \(0.161575\pi\)
−0.873913 + 0.486083i \(0.838425\pi\)
\(828\) − 1.39618e48i − 0.994126i
\(829\) 2.06685e47 0.144264 0.0721321 0.997395i \(-0.477020\pi\)
0.0721321 + 0.997395i \(0.477020\pi\)
\(830\) 0 0
\(831\) 2.28762e48 1.53450
\(832\) 5.99952e48i 3.94531i
\(833\) 1.79157e47i 0.115502i
\(834\) −1.19467e48 −0.755106
\(835\) 0 0
\(836\) 6.39322e48 3.88433
\(837\) − 4.50419e47i − 0.268316i
\(838\) − 5.49865e48i − 3.21166i
\(839\) 1.80225e48 1.03215 0.516075 0.856544i \(-0.327393\pi\)
0.516075 + 0.856544i \(0.327393\pi\)
\(840\) 0 0
\(841\) 4.11374e47 0.226518
\(842\) 2.61661e48i 1.41283i
\(843\) − 2.15300e48i − 1.13996i
\(844\) −7.66674e48 −3.98071
\(845\) 0 0
\(846\) 3.25169e47 0.162367
\(847\) 1.22345e48i 0.599113i
\(848\) 1.41954e49i 6.81735i
\(849\) −3.57312e48 −1.68295
\(850\) 0 0
\(851\) −3.92656e48 −1.77900
\(852\) 5.69138e48i 2.52909i
\(853\) − 3.50343e48i − 1.52699i −0.645816 0.763493i \(-0.723483\pi\)
0.645816 0.763493i \(-0.276517\pi\)
\(854\) −1.76153e48 −0.753068
\(855\) 0 0
\(856\) 9.92107e48 4.08076
\(857\) − 4.03540e48i − 1.62818i −0.580737 0.814091i \(-0.697236\pi\)
0.580737 0.814091i \(-0.302764\pi\)
\(858\) − 7.76254e48i − 3.07230i
\(859\) 4.93399e46 0.0191563 0.00957813 0.999954i \(-0.496951\pi\)
0.00957813 + 0.999954i \(0.496951\pi\)
\(860\) 0 0
\(861\) 4.39643e47 0.164266
\(862\) − 5.80102e48i − 2.12635i
\(863\) − 5.86818e47i − 0.211021i −0.994418 0.105510i \(-0.966352\pi\)
0.994418 0.105510i \(-0.0336476\pi\)
\(864\) 8.47598e48 2.99029
\(865\) 0 0
\(866\) −4.68294e47 −0.159028
\(867\) 1.46196e48i 0.487103i
\(868\) 2.62195e48i 0.857136i
\(869\) 1.53633e48 0.492786
\(870\) 0 0
\(871\) 7.45180e47 0.230124
\(872\) 7.33149e48i 2.22162i
\(873\) 8.49797e47i 0.252686i
\(874\) 8.49291e48 2.47809
\(875\) 0 0
\(876\) −5.34390e48 −1.50155
\(877\) 5.15342e48i 1.42103i 0.703683 + 0.710514i \(0.251537\pi\)
−0.703683 + 0.710514i \(0.748463\pi\)
\(878\) − 1.14418e49i − 3.09623i
\(879\) 1.55460e48 0.412860
\(880\) 0 0
\(881\) 7.21570e48 1.84576 0.922882 0.385084i \(-0.125827\pi\)
0.922882 + 0.385084i \(0.125827\pi\)
\(882\) 3.69159e47i 0.0926791i
\(883\) 7.21502e48i 1.77781i 0.458089 + 0.888906i \(0.348534\pi\)
−0.458089 + 0.888906i \(0.651466\pi\)
\(884\) −9.23096e48 −2.23246
\(885\) 0 0
\(886\) 2.33095e48 0.543097
\(887\) 2.04483e48i 0.467647i 0.972279 + 0.233824i \(0.0751238\pi\)
−0.972279 + 0.233824i \(0.924876\pi\)
\(888\) 2.61388e49i 5.86776i
\(889\) −1.82006e47 −0.0401058
\(890\) 0 0
\(891\) −7.35728e48 −1.56220
\(892\) 7.54314e48i 1.57229i
\(893\) 1.44689e48i 0.296064i
\(894\) 2.15386e48 0.432661
\(895\) 0 0
\(896\) −1.54037e49 −2.98224
\(897\) − 7.54315e48i − 1.43376i
\(898\) − 1.71077e49i − 3.19251i
\(899\) −2.06554e48 −0.378442
\(900\) 0 0
\(901\) −7.91745e48 −1.39838
\(902\) 2.22416e48i 0.385708i
\(903\) 1.82959e48i 0.311535i
\(904\) −3.34626e49 −5.59475
\(905\) 0 0
\(906\) 6.82501e48 1.10024
\(907\) − 4.33823e48i − 0.686739i −0.939200 0.343370i \(-0.888432\pi\)
0.939200 0.343370i \(-0.111568\pi\)
\(908\) − 1.40880e49i − 2.18995i
\(909\) −4.40255e47 −0.0672048
\(910\) 0 0
\(911\) −1.54797e48 −0.227882 −0.113941 0.993488i \(-0.536348\pi\)
−0.113941 + 0.993488i \(0.536348\pi\)
\(912\) − 3.25528e49i − 4.70623i
\(913\) 2.52040e48i 0.357850i
\(914\) 6.77134e47 0.0944196
\(915\) 0 0
\(916\) −8.68424e48 −1.16804
\(917\) 5.82234e48i 0.769136i
\(918\) 8.85839e48i 1.14935i
\(919\) −4.44681e48 −0.566686 −0.283343 0.959019i \(-0.591443\pi\)
−0.283343 + 0.959019i \(0.591443\pi\)
\(920\) 0 0
\(921\) −8.95118e47 −0.110052
\(922\) − 1.25953e49i − 1.52107i
\(923\) 7.37208e48i 0.874505i
\(924\) 3.17199e49 3.69610
\(925\) 0 0
\(926\) 2.53474e49 2.85005
\(927\) 4.56096e48i 0.503780i
\(928\) − 3.88694e49i − 4.21760i
\(929\) −7.81292e48 −0.832826 −0.416413 0.909176i \(-0.636713\pi\)
−0.416413 + 0.909176i \(0.636713\pi\)
\(930\) 0 0
\(931\) −1.64263e48 −0.168994
\(932\) 8.92105e48i 0.901682i
\(933\) 8.70453e48i 0.864367i
\(934\) 3.27813e49 3.19818
\(935\) 0 0
\(936\) −1.20389e49 −1.13380
\(937\) − 1.29441e49i − 1.19776i −0.800839 0.598880i \(-0.795613\pi\)
0.800839 0.598880i \(-0.204387\pi\)
\(938\) 4.16271e48i 0.378469i
\(939\) 1.81769e49 1.62382
\(940\) 0 0
\(941\) −5.82381e47 −0.0502318 −0.0251159 0.999685i \(-0.507995\pi\)
−0.0251159 + 0.999685i \(0.507995\pi\)
\(942\) − 1.16404e49i − 0.986569i
\(943\) 2.16130e48i 0.180000i
\(944\) 1.68549e49 1.37939
\(945\) 0 0
\(946\) −9.25596e48 −0.731505
\(947\) − 1.87865e49i − 1.45905i −0.683953 0.729526i \(-0.739741\pi\)
0.683953 0.729526i \(-0.260259\pi\)
\(948\) − 1.57016e49i − 1.19842i
\(949\) −6.92200e48 −0.519205
\(950\) 0 0
\(951\) 4.38661e48 0.317797
\(952\) − 3.26381e49i − 2.32389i
\(953\) 1.71040e49i 1.19692i 0.801154 + 0.598459i \(0.204220\pi\)
−0.801154 + 0.598459i \(0.795780\pi\)
\(954\) −1.63142e49 −1.12206
\(955\) 0 0
\(956\) 1.99466e49 1.32530
\(957\) 2.49885e49i 1.63190i
\(958\) − 1.19451e49i − 0.766759i
\(959\) 2.07760e49 1.31085
\(960\) 0 0
\(961\) −1.44881e49 −0.883232
\(962\) 5.34930e49i 3.20559i
\(963\) 6.56500e48i 0.386724i
\(964\) −7.99925e49 −4.63210
\(965\) 0 0
\(966\) 4.21374e49 2.35801
\(967\) 3.31445e49i 1.82337i 0.410893 + 0.911683i \(0.365217\pi\)
−0.410893 + 0.911683i \(0.634783\pi\)
\(968\) 4.00382e49i 2.16536i
\(969\) 1.81562e49 0.965348
\(970\) 0 0
\(971\) −1.48405e49 −0.762662 −0.381331 0.924439i \(-0.624534\pi\)
−0.381331 + 0.924439i \(0.624534\pi\)
\(972\) 3.28542e49i 1.65996i
\(973\) − 6.32337e48i − 0.314113i
\(974\) 3.13480e49 1.53104
\(975\) 0 0
\(976\) −3.31922e49 −1.56716
\(977\) 7.11883e47i 0.0330482i 0.999863 + 0.0165241i \(0.00526003\pi\)
−0.999863 + 0.0165241i \(0.994740\pi\)
\(978\) 2.98762e49i 1.36375i
\(979\) 3.07215e49 1.37889
\(980\) 0 0
\(981\) −4.85141e48 −0.210538
\(982\) − 7.01570e49i − 2.99386i
\(983\) 3.13902e49i 1.31723i 0.752481 + 0.658614i \(0.228857\pi\)
−0.752481 + 0.658614i \(0.771143\pi\)
\(984\) 1.43876e49 0.593704
\(985\) 0 0
\(986\) 4.06230e49 1.62107
\(987\) 7.17872e48i 0.281717i
\(988\) − 8.46356e49i − 3.26635i
\(989\) −8.99436e48 −0.341375
\(990\) 0 0
\(991\) 2.53674e49 0.931239 0.465619 0.884985i \(-0.345832\pi\)
0.465619 + 0.884985i \(0.345832\pi\)
\(992\) 3.60441e49i 1.30134i
\(993\) − 4.02880e49i − 1.43058i
\(994\) −4.11819e49 −1.43824
\(995\) 0 0
\(996\) 2.57590e49 0.870263
\(997\) − 4.24313e48i − 0.140999i −0.997512 0.0704995i \(-0.977541\pi\)
0.997512 0.0704995i \(-0.0224593\pi\)
\(998\) 1.36729e49i 0.446895i
\(999\) 3.75506e49 1.20722
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.24.1 12
5.2 odd 4 5.34.a.b.1.6 6
5.3 odd 4 25.34.a.c.1.1 6
5.4 even 2 inner 25.34.b.c.24.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.a.b.1.6 6 5.2 odd 4
25.34.a.c.1.1 6 5.3 odd 4
25.34.b.c.24.1 12 1.1 even 1 trivial
25.34.b.c.24.12 12 5.4 even 2 inner