Properties

Label 25.36.b.a.24.5
Level $25$
Weight $36$
Character 25.24
Analytic conductor $193.988$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(193.987826584\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial: \(x^{6} + 24844388 x^{4} + 154310903773636 x^{2} + 6999547445919666225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{6}\cdot 5^{8}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.5
Root \(-213.765i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.36.b.a.24.2

$q$-expansion

\(f(q)\) \(=\) \(q+165109. i q^{2} -3.45913e8i q^{3} +7.09870e9 q^{4} +5.71135e13 q^{6} +1.25160e14i q^{7} +6.84517e15i q^{8} -6.96245e16 q^{9} +O(q^{10})\) \(q+165109. i q^{2} -3.45913e8i q^{3} +7.09870e9 q^{4} +5.71135e13 q^{6} +1.25160e14i q^{7} +6.84517e15i q^{8} -6.96245e16 q^{9} -1.70842e18 q^{11} -2.45554e18i q^{12} -4.94986e19i q^{13} -2.06651e19 q^{14} -8.86291e20 q^{16} -1.32045e21i q^{17} -1.14956e22i q^{18} -3.94388e21 q^{19} +4.32945e22 q^{21} -2.82076e23i q^{22} -3.48881e23i q^{23} +2.36784e24 q^{24} +8.17268e24 q^{26} +6.77748e24i q^{27} +8.88473e23i q^{28} -3.21628e25 q^{29} +3.41044e25 q^{31} +8.88635e25i q^{32} +5.90965e26i q^{33} +2.18018e26 q^{34} -4.94244e26 q^{36} +4.03624e27i q^{37} -6.51171e26i q^{38} -1.71222e28 q^{39} +8.65857e27 q^{41} +7.14832e27i q^{42} +9.89389e26i q^{43} -1.21276e28 q^{44} +5.76034e28 q^{46} -1.95852e29i q^{47} +3.06580e29i q^{48} +3.63154e29 q^{49} -4.56760e29 q^{51} -3.51376e29i q^{52} -9.96909e29i q^{53} -1.11902e30 q^{54} -8.56741e29 q^{56} +1.36424e30i q^{57} -5.31037e30i q^{58} -3.91953e30 q^{59} +7.64909e29 q^{61} +5.63096e30i q^{62} -8.71420e30i q^{63} -4.51249e31 q^{64} -9.75738e31 q^{66} +1.64301e32i q^{67} -9.37346e30i q^{68} -1.20682e32 q^{69} +7.65930e31 q^{71} -4.76592e32i q^{72} -7.08063e32i q^{73} -6.66421e32 q^{74} -2.79964e31 q^{76} -2.13826e32i q^{77} -2.82704e33i q^{78} -2.34599e33 q^{79} -1.13900e33 q^{81} +1.42961e33i q^{82} +5.18971e33i q^{83} +3.07335e32 q^{84} -1.63357e32 q^{86} +1.11255e34i q^{87} -1.16944e34i q^{88} -1.44578e34 q^{89} +6.19525e33 q^{91} -2.47660e33i q^{92} -1.17972e34i q^{93} +3.23369e34 q^{94} +3.07391e34 q^{96} +2.87399e34i q^{97} +5.99600e34i q^{98} +1.18948e35 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 69682524288q^{4} - 9573061128768q^{6} - 300183957752487102q^{9} + O(q^{10}) \) \( 6q - 69682524288q^{4} - 9573061128768q^{6} - 300183957752487102q^{9} - 2315890857099974088q^{11} - \)\(27\!\cdots\!56\)\(q^{14} + \)\(43\!\cdots\!96\)\(q^{16} + \)\(64\!\cdots\!80\)\(q^{19} + \)\(44\!\cdots\!52\)\(q^{21} + \)\(75\!\cdots\!40\)\(q^{24} + \)\(85\!\cdots\!72\)\(q^{26} + \)\(76\!\cdots\!20\)\(q^{29} + \)\(20\!\cdots\!12\)\(q^{31} + \)\(11\!\cdots\!24\)\(q^{34} - \)\(12\!\cdots\!04\)\(q^{36} - \)\(37\!\cdots\!24\)\(q^{39} + \)\(47\!\cdots\!12\)\(q^{41} + \)\(16\!\cdots\!24\)\(q^{44} + \)\(62\!\cdots\!12\)\(q^{46} + \)\(11\!\cdots\!42\)\(q^{49} - \)\(36\!\cdots\!08\)\(q^{51} - \)\(88\!\cdots\!20\)\(q^{54} + \)\(11\!\cdots\!80\)\(q^{56} - \)\(87\!\cdots\!60\)\(q^{59} + \)\(47\!\cdots\!12\)\(q^{61} - \)\(18\!\cdots\!68\)\(q^{64} - \)\(15\!\cdots\!36\)\(q^{66} + \)\(65\!\cdots\!96\)\(q^{69} + \)\(69\!\cdots\!12\)\(q^{71} - \)\(42\!\cdots\!16\)\(q^{74} - \)\(54\!\cdots\!40\)\(q^{76} + \)\(85\!\cdots\!20\)\(q^{79} + \)\(37\!\cdots\!26\)\(q^{81} + \)\(26\!\cdots\!04\)\(q^{84} + \)\(29\!\cdots\!92\)\(q^{86} - \)\(61\!\cdots\!40\)\(q^{89} + \)\(20\!\cdots\!92\)\(q^{91} - \)\(38\!\cdots\!36\)\(q^{94} - \)\(64\!\cdots\!68\)\(q^{96} - \)\(60\!\cdots\!04\)\(q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 165109.i 0.890730i 0.895349 + 0.445365i \(0.146926\pi\)
−0.895349 + 0.445365i \(0.853074\pi\)
\(3\) − 3.45913e8i − 1.54648i −0.634111 0.773242i \(-0.718634\pi\)
0.634111 0.773242i \(-0.281366\pi\)
\(4\) 7.09870e9 0.206599
\(5\) 0 0
\(6\) 5.71135e13 1.37750
\(7\) 1.25160e14i 0.203353i 0.994818 + 0.101676i \(0.0324206\pi\)
−0.994818 + 0.101676i \(0.967579\pi\)
\(8\) 6.84517e15i 1.07475i
\(9\) −6.96245e16 −1.39161
\(10\) 0 0
\(11\) −1.70842e18 −1.01911 −0.509557 0.860437i \(-0.670191\pi\)
−0.509557 + 0.860437i \(0.670191\pi\)
\(12\) − 2.45554e18i − 0.319503i
\(13\) − 4.94986e19i − 1.58703i −0.608552 0.793514i \(-0.708249\pi\)
0.608552 0.793514i \(-0.291751\pi\)
\(14\) −2.06651e19 −0.181132
\(15\) 0 0
\(16\) −8.86291e20 −0.750717
\(17\) − 1.32045e21i − 0.387137i −0.981087 0.193569i \(-0.937994\pi\)
0.981087 0.193569i \(-0.0620062\pi\)
\(18\) − 1.14956e22i − 1.23955i
\(19\) −3.94388e21 −0.165096 −0.0825479 0.996587i \(-0.526306\pi\)
−0.0825479 + 0.996587i \(0.526306\pi\)
\(20\) 0 0
\(21\) 4.32945e22 0.314482
\(22\) − 2.82076e23i − 0.907756i
\(23\) − 3.48881e23i − 0.515750i −0.966178 0.257875i \(-0.916978\pi\)
0.966178 0.257875i \(-0.0830224\pi\)
\(24\) 2.36784e24 1.66209
\(25\) 0 0
\(26\) 8.17268e24 1.41361
\(27\) 6.77748e24i 0.605623i
\(28\) 8.88473e23i 0.0420125i
\(29\) −3.21628e25 −0.822979 −0.411489 0.911415i \(-0.634991\pi\)
−0.411489 + 0.911415i \(0.634991\pi\)
\(30\) 0 0
\(31\) 3.41044e25 0.271632 0.135816 0.990734i \(-0.456634\pi\)
0.135816 + 0.990734i \(0.456634\pi\)
\(32\) 8.88635e25i 0.406068i
\(33\) 5.90965e26i 1.57604i
\(34\) 2.18018e26 0.344835
\(35\) 0 0
\(36\) −4.94244e26 −0.287506
\(37\) 4.03624e27i 1.45361i 0.686845 + 0.726804i \(0.258995\pi\)
−0.686845 + 0.726804i \(0.741005\pi\)
\(38\) − 6.51171e26i − 0.147056i
\(39\) −1.71222e28 −2.45431
\(40\) 0 0
\(41\) 8.65857e27 0.517283 0.258642 0.965973i \(-0.416725\pi\)
0.258642 + 0.965973i \(0.416725\pi\)
\(42\) 7.14832e27i 0.280118i
\(43\) 9.89389e26i 0.0256844i 0.999918 + 0.0128422i \(0.00408791\pi\)
−0.999918 + 0.0128422i \(0.995912\pi\)
\(44\) −1.21276e28 −0.210548
\(45\) 0 0
\(46\) 5.76034e28 0.459395
\(47\) − 1.95852e29i − 1.07205i −0.844202 0.536025i \(-0.819925\pi\)
0.844202 0.536025i \(-0.180075\pi\)
\(48\) 3.06580e29i 1.16097i
\(49\) 3.63154e29 0.958648
\(50\) 0 0
\(51\) −4.56760e29 −0.598702
\(52\) − 3.51376e29i − 0.327879i
\(53\) − 9.96909e29i − 0.666543i −0.942831 0.333271i \(-0.891847\pi\)
0.942831 0.333271i \(-0.108153\pi\)
\(54\) −1.11902e30 −0.539447
\(55\) 0 0
\(56\) −8.56741e29 −0.218554
\(57\) 1.36424e30i 0.255318i
\(58\) − 5.31037e30i − 0.733052i
\(59\) −3.91953e30 −0.401166 −0.200583 0.979677i \(-0.564284\pi\)
−0.200583 + 0.979677i \(0.564284\pi\)
\(60\) 0 0
\(61\) 7.64909e29 0.0436855 0.0218428 0.999761i \(-0.493047\pi\)
0.0218428 + 0.999761i \(0.493047\pi\)
\(62\) 5.63096e30i 0.241951i
\(63\) − 8.71420e30i − 0.282988i
\(64\) −4.51249e31 −1.11241
\(65\) 0 0
\(66\) −9.75738e31 −1.40383
\(67\) 1.64301e32i 1.81690i 0.417992 + 0.908451i \(0.362734\pi\)
−0.417992 + 0.908451i \(0.637266\pi\)
\(68\) − 9.37346e30i − 0.0799823i
\(69\) −1.20682e32 −0.797600
\(70\) 0 0
\(71\) 7.65930e31 0.307021 0.153510 0.988147i \(-0.450942\pi\)
0.153510 + 0.988147i \(0.450942\pi\)
\(72\) − 4.76592e32i − 1.49564i
\(73\) − 7.08063e32i − 1.74551i −0.488160 0.872754i \(-0.662332\pi\)
0.488160 0.872754i \(-0.337668\pi\)
\(74\) −6.66421e32 −1.29477
\(75\) 0 0
\(76\) −2.79964e31 −0.0341087
\(77\) − 2.13826e32i − 0.207239i
\(78\) − 2.82704e33i − 2.18613i
\(79\) −2.34599e33 −1.45162 −0.725809 0.687896i \(-0.758534\pi\)
−0.725809 + 0.687896i \(0.758534\pi\)
\(80\) 0 0
\(81\) −1.13900e33 −0.455027
\(82\) 1.42961e33i 0.460760i
\(83\) 5.18971e33i 1.35293i 0.736473 + 0.676467i \(0.236490\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(84\) 3.07335e32 0.0649717
\(85\) 0 0
\(86\) −1.63357e32 −0.0228779
\(87\) 1.11255e34i 1.27272i
\(88\) − 1.16944e34i − 1.09530i
\(89\) −1.44578e34 −1.11116 −0.555582 0.831461i \(-0.687505\pi\)
−0.555582 + 0.831461i \(0.687505\pi\)
\(90\) 0 0
\(91\) 6.19525e33 0.322726
\(92\) − 2.47660e33i − 0.106554i
\(93\) − 1.17972e34i − 0.420075i
\(94\) 3.23369e34 0.954907
\(95\) 0 0
\(96\) 3.07391e34 0.627978
\(97\) 2.87399e34i 0.489756i 0.969554 + 0.244878i \(0.0787479\pi\)
−0.969554 + 0.244878i \(0.921252\pi\)
\(98\) 5.99600e34i 0.853897i
\(99\) 1.18948e35 1.41821
\(100\) 0 0
\(101\) 1.13198e35 0.951079 0.475539 0.879694i \(-0.342253\pi\)
0.475539 + 0.879694i \(0.342253\pi\)
\(102\) − 7.54153e34i − 0.533282i
\(103\) 2.53635e35i 1.51202i 0.654559 + 0.756011i \(0.272855\pi\)
−0.654559 + 0.756011i \(0.727145\pi\)
\(104\) 3.38826e35 1.70567
\(105\) 0 0
\(106\) 1.64599e35 0.593710
\(107\) 3.63939e35i 1.11381i 0.830575 + 0.556907i \(0.188012\pi\)
−0.830575 + 0.556907i \(0.811988\pi\)
\(108\) 4.81113e34i 0.125121i
\(109\) 4.65371e34 0.103000 0.0514998 0.998673i \(-0.483600\pi\)
0.0514998 + 0.998673i \(0.483600\pi\)
\(110\) 0 0
\(111\) 1.39619e36 2.24798
\(112\) − 1.10928e35i − 0.152660i
\(113\) − 2.18346e35i − 0.257201i −0.991696 0.128601i \(-0.958951\pi\)
0.991696 0.128601i \(-0.0410485\pi\)
\(114\) −2.25249e35 −0.227419
\(115\) 0 0
\(116\) −2.28314e35 −0.170027
\(117\) 3.44632e36i 2.20853i
\(118\) − 6.47150e35i − 0.357330i
\(119\) 1.65267e35 0.0787254
\(120\) 0 0
\(121\) 1.08456e35 0.0385932
\(122\) 1.26293e35i 0.0389120i
\(123\) − 2.99511e36i − 0.799970i
\(124\) 2.42097e35 0.0561191
\(125\) 0 0
\(126\) 1.43880e36 0.252066
\(127\) − 4.19558e36i − 0.640070i −0.947406 0.320035i \(-0.896305\pi\)
0.947406 0.320035i \(-0.103695\pi\)
\(128\) − 4.39721e36i − 0.584793i
\(129\) 3.42243e35 0.0397205
\(130\) 0 0
\(131\) 2.40631e36 0.213355 0.106678 0.994294i \(-0.465979\pi\)
0.106678 + 0.994294i \(0.465979\pi\)
\(132\) 4.19509e36i 0.325610i
\(133\) − 4.93616e35i − 0.0335727i
\(134\) −2.71276e37 −1.61837
\(135\) 0 0
\(136\) 9.03869e36 0.416078
\(137\) 2.14718e37i 0.869477i 0.900557 + 0.434739i \(0.143159\pi\)
−0.900557 + 0.434739i \(0.856841\pi\)
\(138\) − 1.99258e37i − 0.710446i
\(139\) −2.63209e37 −0.827068 −0.413534 0.910489i \(-0.635706\pi\)
−0.413534 + 0.910489i \(0.635706\pi\)
\(140\) 0 0
\(141\) −6.77477e37 −1.65791
\(142\) 1.26462e37i 0.273473i
\(143\) 8.45645e37i 1.61736i
\(144\) 6.17076e37 1.04471
\(145\) 0 0
\(146\) 1.16908e38 1.55478
\(147\) − 1.25620e38i − 1.48253i
\(148\) 2.86521e37i 0.300314i
\(149\) 1.74389e38 1.62465 0.812324 0.583206i \(-0.198202\pi\)
0.812324 + 0.583206i \(0.198202\pi\)
\(150\) 0 0
\(151\) −1.40947e38 −1.03982 −0.519912 0.854220i \(-0.674035\pi\)
−0.519912 + 0.854220i \(0.674035\pi\)
\(152\) − 2.69965e37i − 0.177437i
\(153\) 9.19355e37i 0.538745i
\(154\) 3.53046e37 0.184595
\(155\) 0 0
\(156\) −1.21546e38 −0.507059
\(157\) − 1.49924e38i − 0.559277i −0.960105 0.279639i \(-0.909785\pi\)
0.960105 0.279639i \(-0.0902147\pi\)
\(158\) − 3.87345e38i − 1.29300i
\(159\) −3.44844e38 −1.03080
\(160\) 0 0
\(161\) 4.36659e37 0.104879
\(162\) − 1.88060e38i − 0.405306i
\(163\) 8.60787e38i 1.66576i 0.553453 + 0.832880i \(0.313310\pi\)
−0.553453 + 0.832880i \(0.686690\pi\)
\(164\) 6.14646e37 0.106870
\(165\) 0 0
\(166\) −8.56869e38 −1.20510
\(167\) 9.23344e38i 1.16903i 0.811383 + 0.584515i \(0.198715\pi\)
−0.811383 + 0.584515i \(0.801285\pi\)
\(168\) 2.96358e38i 0.337991i
\(169\) −1.47733e39 −1.51866
\(170\) 0 0
\(171\) 2.74591e38 0.229749
\(172\) 7.02337e36i 0.00530638i
\(173\) − 1.33161e39i − 0.909012i −0.890744 0.454506i \(-0.849816\pi\)
0.890744 0.454506i \(-0.150184\pi\)
\(174\) −1.83693e39 −1.13365
\(175\) 0 0
\(176\) 1.51416e39 0.765066
\(177\) 1.35582e39i 0.620396i
\(178\) − 2.38712e39i − 0.989748i
\(179\) −6.27202e38 −0.235765 −0.117883 0.993028i \(-0.537611\pi\)
−0.117883 + 0.993028i \(0.537611\pi\)
\(180\) 0 0
\(181\) −2.39965e39 −0.742629 −0.371315 0.928507i \(-0.621093\pi\)
−0.371315 + 0.928507i \(0.621093\pi\)
\(182\) 1.02289e39i 0.287462i
\(183\) − 2.64592e38i − 0.0675590i
\(184\) 2.38815e39 0.554305
\(185\) 0 0
\(186\) 1.94782e39 0.374174
\(187\) 2.25588e39i 0.394537i
\(188\) − 1.39029e39i − 0.221485i
\(189\) −8.48269e38 −0.123155
\(190\) 0 0
\(191\) 8.14518e39 0.983596 0.491798 0.870709i \(-0.336340\pi\)
0.491798 + 0.870709i \(0.336340\pi\)
\(192\) 1.56093e40i 1.72033i
\(193\) 1.38549e40i 1.39428i 0.716936 + 0.697139i \(0.245544\pi\)
−0.716936 + 0.697139i \(0.754456\pi\)
\(194\) −4.74522e39 −0.436240
\(195\) 0 0
\(196\) 2.57792e39 0.198056
\(197\) 1.15539e40i 0.812026i 0.913867 + 0.406013i \(0.133081\pi\)
−0.913867 + 0.406013i \(0.866919\pi\)
\(198\) 1.96394e40i 1.26324i
\(199\) 4.76691e39 0.280742 0.140371 0.990099i \(-0.455170\pi\)
0.140371 + 0.990099i \(0.455170\pi\)
\(200\) 0 0
\(201\) 5.68340e40 2.80981
\(202\) 1.86901e40i 0.847155i
\(203\) − 4.02549e39i − 0.167355i
\(204\) −3.24241e39 −0.123691
\(205\) 0 0
\(206\) −4.18775e40 −1.34680
\(207\) 2.42906e40i 0.717725i
\(208\) 4.38702e40i 1.19141i
\(209\) 6.73781e39 0.168251
\(210\) 0 0
\(211\) −1.87354e40 −0.396022 −0.198011 0.980200i \(-0.563448\pi\)
−0.198011 + 0.980200i \(0.563448\pi\)
\(212\) − 7.07676e39i − 0.137707i
\(213\) − 2.64945e40i − 0.474803i
\(214\) −6.00896e40 −0.992107
\(215\) 0 0
\(216\) −4.63930e40 −0.650896
\(217\) 4.26851e39i 0.0552371i
\(218\) 7.68370e39i 0.0917448i
\(219\) −2.44929e41 −2.69940
\(220\) 0 0
\(221\) −6.53603e40 −0.614398
\(222\) 2.30524e41i 2.00234i
\(223\) − 7.12100e40i − 0.571750i −0.958267 0.285875i \(-0.907716\pi\)
0.958267 0.285875i \(-0.0922841\pi\)
\(224\) −1.11222e40 −0.0825750
\(225\) 0 0
\(226\) 3.60510e40 0.229097
\(227\) 1.04005e41i 0.611787i 0.952066 + 0.305893i \(0.0989551\pi\)
−0.952066 + 0.305893i \(0.901045\pi\)
\(228\) 9.68434e39i 0.0527485i
\(229\) −1.82859e41 −0.922564 −0.461282 0.887254i \(-0.652610\pi\)
−0.461282 + 0.887254i \(0.652610\pi\)
\(230\) 0 0
\(231\) −7.39652e40 −0.320493
\(232\) − 2.20160e41i − 0.884501i
\(233\) 1.57847e41i 0.588177i 0.955778 + 0.294088i \(0.0950160\pi\)
−0.955778 + 0.294088i \(0.904984\pi\)
\(234\) −5.69019e41 −1.96720
\(235\) 0 0
\(236\) −2.78236e40 −0.0828806
\(237\) 8.11510e41i 2.24490i
\(238\) 2.72871e40i 0.0701231i
\(239\) −5.00984e41 −1.19636 −0.598178 0.801363i \(-0.704108\pi\)
−0.598178 + 0.801363i \(0.704108\pi\)
\(240\) 0 0
\(241\) −3.33164e41 −0.687638 −0.343819 0.939036i \(-0.611721\pi\)
−0.343819 + 0.939036i \(0.611721\pi\)
\(242\) 1.79072e40i 0.0343762i
\(243\) 7.33084e41i 1.30931i
\(244\) 5.42986e39 0.00902540
\(245\) 0 0
\(246\) 4.94521e41 0.712558
\(247\) 1.95217e41i 0.262011i
\(248\) 2.33451e41i 0.291938i
\(249\) 1.79519e42 2.09229
\(250\) 0 0
\(251\) 1.52760e42 1.54782 0.773909 0.633297i \(-0.218299\pi\)
0.773909 + 0.633297i \(0.218299\pi\)
\(252\) − 6.18595e40i − 0.0584652i
\(253\) 5.96035e41i 0.525608i
\(254\) 6.92730e41 0.570130
\(255\) 0 0
\(256\) −8.24460e41 −0.591521
\(257\) 2.72482e42i 1.82603i 0.407925 + 0.913016i \(0.366253\pi\)
−0.407925 + 0.913016i \(0.633747\pi\)
\(258\) 5.65074e40i 0.0353802i
\(259\) −5.05176e41 −0.295595
\(260\) 0 0
\(261\) 2.23932e42 1.14527
\(262\) 3.97303e41i 0.190042i
\(263\) 4.22055e40i 0.0188861i 0.999955 + 0.00944307i \(0.00300587\pi\)
−0.999955 + 0.00944307i \(0.996994\pi\)
\(264\) −4.04526e42 −1.69386
\(265\) 0 0
\(266\) 8.15005e40 0.0299042
\(267\) 5.00116e42i 1.71840i
\(268\) 1.16632e42i 0.375371i
\(269\) 1.30618e42 0.393858 0.196929 0.980418i \(-0.436903\pi\)
0.196929 + 0.980418i \(0.436903\pi\)
\(270\) 0 0
\(271\) −4.11918e41 −0.109106 −0.0545529 0.998511i \(-0.517373\pi\)
−0.0545529 + 0.998511i \(0.517373\pi\)
\(272\) 1.17030e42i 0.290631i
\(273\) − 2.14302e42i − 0.499091i
\(274\) −3.54519e42 −0.774470
\(275\) 0 0
\(276\) −8.56689e41 −0.164784
\(277\) − 7.38232e42i − 1.33290i −0.745551 0.666449i \(-0.767813\pi\)
0.745551 0.666449i \(-0.232187\pi\)
\(278\) − 4.34582e42i − 0.736694i
\(279\) −2.37451e42 −0.378007
\(280\) 0 0
\(281\) 4.08223e42 0.573505 0.286752 0.958005i \(-0.407424\pi\)
0.286752 + 0.958005i \(0.407424\pi\)
\(282\) − 1.11858e43i − 1.47675i
\(283\) − 1.06503e43i − 1.32159i −0.750565 0.660796i \(-0.770219\pi\)
0.750565 0.660796i \(-0.229781\pi\)
\(284\) 5.43711e41 0.0634303
\(285\) 0 0
\(286\) −1.39624e43 −1.44063
\(287\) 1.08371e42i 0.105191i
\(288\) − 6.18708e42i − 0.565089i
\(289\) 9.88997e42 0.850125
\(290\) 0 0
\(291\) 9.94151e42 0.757399
\(292\) − 5.02633e42i − 0.360621i
\(293\) 1.90474e43i 1.28722i 0.765353 + 0.643610i \(0.222564\pi\)
−0.765353 + 0.643610i \(0.777436\pi\)
\(294\) 2.07410e43 1.32054
\(295\) 0 0
\(296\) −2.76288e43 −1.56227
\(297\) − 1.15788e43i − 0.617199i
\(298\) 2.87932e43i 1.44712i
\(299\) −1.72691e43 −0.818510
\(300\) 0 0
\(301\) −1.23832e41 −0.00522298
\(302\) − 2.32717e43i − 0.926203i
\(303\) − 3.91569e43i − 1.47083i
\(304\) 3.49543e42 0.123940
\(305\) 0 0
\(306\) −1.51794e43 −0.479877
\(307\) 1.39752e43i 0.417288i 0.977992 + 0.208644i \(0.0669050\pi\)
−0.977992 + 0.208644i \(0.933095\pi\)
\(308\) − 1.51789e42i − 0.0428156i
\(309\) 8.77359e43 2.33832
\(310\) 0 0
\(311\) 2.22377e43 0.529399 0.264700 0.964331i \(-0.414727\pi\)
0.264700 + 0.964331i \(0.414727\pi\)
\(312\) − 1.17205e44i − 2.63778i
\(313\) 3.63171e43i 0.772833i 0.922324 + 0.386417i \(0.126287\pi\)
−0.922324 + 0.386417i \(0.873713\pi\)
\(314\) 2.47538e43 0.498165
\(315\) 0 0
\(316\) −1.66535e43 −0.299903
\(317\) − 5.29385e43i − 0.902058i −0.892509 0.451029i \(-0.851057\pi\)
0.892509 0.451029i \(-0.148943\pi\)
\(318\) − 5.69370e43i − 0.918163i
\(319\) 5.49476e43 0.838709
\(320\) 0 0
\(321\) 1.25891e44 1.72249
\(322\) 7.20964e42i 0.0934191i
\(323\) 5.20769e42i 0.0639147i
\(324\) −8.08544e42 −0.0940083
\(325\) 0 0
\(326\) −1.42124e44 −1.48374
\(327\) − 1.60978e43i − 0.159287i
\(328\) 5.92693e43i 0.555953i
\(329\) 2.45128e43 0.218004
\(330\) 0 0
\(331\) 6.25551e43 0.500349 0.250174 0.968201i \(-0.419512\pi\)
0.250174 + 0.968201i \(0.419512\pi\)
\(332\) 3.68402e43i 0.279515i
\(333\) − 2.81022e44i − 2.02286i
\(334\) −1.52453e44 −1.04129
\(335\) 0 0
\(336\) −3.83715e43 −0.236087
\(337\) 1.04185e44i 0.608529i 0.952588 + 0.304264i \(0.0984106\pi\)
−0.952588 + 0.304264i \(0.901589\pi\)
\(338\) − 2.43920e44i − 1.35271i
\(339\) −7.55289e43 −0.397757
\(340\) 0 0
\(341\) −5.82647e43 −0.276824
\(342\) 4.53375e43i 0.204645i
\(343\) 9.28652e43i 0.398296i
\(344\) −6.77253e42 −0.0276044
\(345\) 0 0
\(346\) 2.19861e44 0.809685
\(347\) 3.07657e44i 1.07721i 0.842557 + 0.538606i \(0.181049\pi\)
−0.842557 + 0.538606i \(0.818951\pi\)
\(348\) 7.89769e43i 0.262944i
\(349\) 2.62767e44 0.832005 0.416003 0.909363i \(-0.363431\pi\)
0.416003 + 0.909363i \(0.363431\pi\)
\(350\) 0 0
\(351\) 3.35476e44 0.961140
\(352\) − 1.51816e44i − 0.413830i
\(353\) 7.06671e44i 1.83299i 0.400048 + 0.916494i \(0.368994\pi\)
−0.400048 + 0.916494i \(0.631006\pi\)
\(354\) −2.23858e44 −0.552606
\(355\) 0 0
\(356\) −1.02632e44 −0.229566
\(357\) − 5.71681e43i − 0.121748i
\(358\) − 1.03557e44i − 0.210003i
\(359\) 5.35788e44 1.03476 0.517380 0.855756i \(-0.326907\pi\)
0.517380 + 0.855756i \(0.326907\pi\)
\(360\) 0 0
\(361\) −5.55104e44 −0.972743
\(362\) − 3.96204e44i − 0.661483i
\(363\) − 3.75165e43i − 0.0596838i
\(364\) 4.39782e43 0.0666750
\(365\) 0 0
\(366\) 4.36866e43 0.0601768
\(367\) 3.00056e44i 0.394045i 0.980399 + 0.197022i \(0.0631272\pi\)
−0.980399 + 0.197022i \(0.936873\pi\)
\(368\) 3.09210e44i 0.387183i
\(369\) −6.02849e44 −0.719858
\(370\) 0 0
\(371\) 1.24773e44 0.135543
\(372\) − 8.37447e43i − 0.0867872i
\(373\) 1.08201e45i 1.06986i 0.844895 + 0.534932i \(0.179663\pi\)
−0.844895 + 0.534932i \(0.820337\pi\)
\(374\) −3.72466e44 −0.351426
\(375\) 0 0
\(376\) 1.34064e45 1.15219
\(377\) 1.59201e45i 1.30609i
\(378\) − 1.40057e44i − 0.109698i
\(379\) −1.21847e45 −0.911229 −0.455614 0.890177i \(-0.650580\pi\)
−0.455614 + 0.890177i \(0.650580\pi\)
\(380\) 0 0
\(381\) −1.45131e45 −0.989858
\(382\) 1.34484e45i 0.876119i
\(383\) − 9.53201e43i − 0.0593207i −0.999560 0.0296603i \(-0.990557\pi\)
0.999560 0.0296603i \(-0.00944256\pi\)
\(384\) −1.52105e45 −0.904373
\(385\) 0 0
\(386\) −2.28757e45 −1.24193
\(387\) − 6.88857e43i − 0.0357427i
\(388\) 2.04016e44i 0.101183i
\(389\) −1.90493e45 −0.903152 −0.451576 0.892233i \(-0.649138\pi\)
−0.451576 + 0.892233i \(0.649138\pi\)
\(390\) 0 0
\(391\) −4.60678e44 −0.199666
\(392\) 2.48585e45i 1.03031i
\(393\) − 8.32373e44i − 0.329950i
\(394\) −1.90766e45 −0.723296
\(395\) 0 0
\(396\) 8.44376e44 0.293002
\(397\) − 3.96399e45i − 1.31613i −0.752961 0.658065i \(-0.771375\pi\)
0.752961 0.658065i \(-0.228625\pi\)
\(398\) 7.87060e44i 0.250066i
\(399\) −1.70748e44 −0.0519196
\(400\) 0 0
\(401\) −5.53039e44 −0.154074 −0.0770369 0.997028i \(-0.524546\pi\)
−0.0770369 + 0.997028i \(0.524546\pi\)
\(402\) 9.38381e45i 2.50278i
\(403\) − 1.68812e45i − 0.431088i
\(404\) 8.03562e44 0.196492
\(405\) 0 0
\(406\) 6.64646e44 0.149068
\(407\) − 6.89560e45i − 1.48139i
\(408\) − 3.12660e45i − 0.643457i
\(409\) −3.35495e45 −0.661498 −0.330749 0.943719i \(-0.607301\pi\)
−0.330749 + 0.943719i \(0.607301\pi\)
\(410\) 0 0
\(411\) 7.42738e45 1.34463
\(412\) 1.80048e45i 0.312383i
\(413\) − 4.90568e44i − 0.0815781i
\(414\) −4.01061e45 −0.639299
\(415\) 0 0
\(416\) 4.39862e45 0.644441
\(417\) 9.10474e45i 1.27905i
\(418\) 1.11247e45i 0.149867i
\(419\) 7.69138e45 0.993708 0.496854 0.867834i \(-0.334488\pi\)
0.496854 + 0.867834i \(0.334488\pi\)
\(420\) 0 0
\(421\) −2.77229e45 −0.329535 −0.164767 0.986332i \(-0.552687\pi\)
−0.164767 + 0.986332i \(0.552687\pi\)
\(422\) − 3.09338e45i − 0.352749i
\(423\) 1.36361e46i 1.49188i
\(424\) 6.82401e45 0.716370
\(425\) 0 0
\(426\) 4.37449e45 0.422921
\(427\) 9.57360e43i 0.00888357i
\(428\) 2.58349e45i 0.230113i
\(429\) 2.92520e46 2.50122
\(430\) 0 0
\(431\) 6.18484e45 0.487502 0.243751 0.969838i \(-0.421622\pi\)
0.243751 + 0.969838i \(0.421622\pi\)
\(432\) − 6.00681e45i − 0.454651i
\(433\) 2.21783e46i 1.61209i 0.591852 + 0.806047i \(0.298397\pi\)
−0.591852 + 0.806047i \(0.701603\pi\)
\(434\) −7.04770e44 −0.0492014
\(435\) 0 0
\(436\) 3.30353e44 0.0212796
\(437\) 1.37594e45i 0.0851482i
\(438\) − 4.04399e46i − 2.40444i
\(439\) −1.91766e45 −0.109557 −0.0547787 0.998499i \(-0.517445\pi\)
−0.0547787 + 0.998499i \(0.517445\pi\)
\(440\) 0 0
\(441\) −2.52844e46 −1.33407
\(442\) − 1.07916e46i − 0.547263i
\(443\) − 3.00864e46i − 1.46658i −0.679916 0.733290i \(-0.737984\pi\)
0.679916 0.733290i \(-0.262016\pi\)
\(444\) 9.91114e45 0.464431
\(445\) 0 0
\(446\) 1.17574e46 0.509275
\(447\) − 6.03235e46i − 2.51249i
\(448\) − 5.64783e45i − 0.226212i
\(449\) −5.68788e45 −0.219099 −0.109549 0.993981i \(-0.534941\pi\)
−0.109549 + 0.993981i \(0.534941\pi\)
\(450\) 0 0
\(451\) −1.47925e46 −0.527171
\(452\) − 1.54997e45i − 0.0531376i
\(453\) 4.87555e46i 1.60807i
\(454\) −1.71721e46 −0.544937
\(455\) 0 0
\(456\) −9.33846e45 −0.274404
\(457\) − 4.67626e46i − 1.32241i −0.750206 0.661204i \(-0.770046\pi\)
0.750206 0.661204i \(-0.229954\pi\)
\(458\) − 3.01918e46i − 0.821755i
\(459\) 8.94930e45 0.234459
\(460\) 0 0
\(461\) −3.72978e46 −0.905560 −0.452780 0.891622i \(-0.649568\pi\)
−0.452780 + 0.891622i \(0.649568\pi\)
\(462\) − 1.22123e46i − 0.285472i
\(463\) − 3.67649e46i − 0.827497i −0.910391 0.413748i \(-0.864219\pi\)
0.910391 0.413748i \(-0.135781\pi\)
\(464\) 2.85056e46 0.617825
\(465\) 0 0
\(466\) −2.60620e46 −0.523907
\(467\) 2.37743e46i 0.460323i 0.973152 + 0.230161i \(0.0739254\pi\)
−0.973152 + 0.230161i \(0.926075\pi\)
\(468\) 2.44644e46i 0.456280i
\(469\) −2.05639e46 −0.369472
\(470\) 0 0
\(471\) −5.18607e46 −0.864914
\(472\) − 2.68298e46i − 0.431155i
\(473\) − 1.69029e45i − 0.0261753i
\(474\) −1.33988e47 −1.99960
\(475\) 0 0
\(476\) 1.17318e45 0.0162646
\(477\) 6.94094e46i 0.927569i
\(478\) − 8.27170e46i − 1.06563i
\(479\) 8.80315e46 1.09337 0.546684 0.837339i \(-0.315890\pi\)
0.546684 + 0.837339i \(0.315890\pi\)
\(480\) 0 0
\(481\) 1.99788e47 2.30691
\(482\) − 5.50084e46i − 0.612500i
\(483\) − 1.51046e46i − 0.162194i
\(484\) 7.69900e44 0.00797334
\(485\) 0 0
\(486\) −1.21039e47 −1.16625
\(487\) − 7.13443e46i − 0.663136i −0.943431 0.331568i \(-0.892422\pi\)
0.943431 0.331568i \(-0.107578\pi\)
\(488\) 5.23593e45i 0.0469512i
\(489\) 2.97758e47 2.57607
\(490\) 0 0
\(491\) 8.49115e46 0.683975 0.341988 0.939704i \(-0.388900\pi\)
0.341988 + 0.939704i \(0.388900\pi\)
\(492\) − 2.12614e46i − 0.165273i
\(493\) 4.24693e46i 0.318606i
\(494\) −3.22321e46 −0.233382
\(495\) 0 0
\(496\) −3.02265e46 −0.203919
\(497\) 9.58638e45i 0.0624335i
\(498\) 2.96402e47i 1.86367i
\(499\) 1.75876e47 1.06770 0.533848 0.845581i \(-0.320745\pi\)
0.533848 + 0.845581i \(0.320745\pi\)
\(500\) 0 0
\(501\) 3.19397e47 1.80789
\(502\) 2.52221e47i 1.37869i
\(503\) 1.24051e47i 0.654880i 0.944872 + 0.327440i \(0.106186\pi\)
−0.944872 + 0.327440i \(0.893814\pi\)
\(504\) 5.96502e46 0.304143
\(505\) 0 0
\(506\) −9.84108e46 −0.468175
\(507\) 5.11027e47i 2.34858i
\(508\) − 2.97832e46i − 0.132238i
\(509\) −3.71512e47 −1.59372 −0.796859 0.604166i \(-0.793507\pi\)
−0.796859 + 0.604166i \(0.793507\pi\)
\(510\) 0 0
\(511\) 8.86212e46 0.354954
\(512\) − 2.87213e47i − 1.11168i
\(513\) − 2.67296e46i − 0.0999857i
\(514\) −4.49893e47 −1.62650
\(515\) 0 0
\(516\) 2.42948e45 0.00820623
\(517\) 3.34597e47i 1.09254i
\(518\) − 8.34092e46i − 0.263295i
\(519\) −4.60621e47 −1.40577
\(520\) 0 0
\(521\) 7.89659e45 0.0225309 0.0112655 0.999937i \(-0.496414\pi\)
0.0112655 + 0.999937i \(0.496414\pi\)
\(522\) 3.69732e47i 1.02012i
\(523\) 1.18122e47i 0.315175i 0.987505 + 0.157588i \(0.0503717\pi\)
−0.987505 + 0.157588i \(0.949628\pi\)
\(524\) 1.70816e46 0.0440790
\(525\) 0 0
\(526\) −6.96851e45 −0.0168225
\(527\) − 4.50331e46i − 0.105159i
\(528\) − 5.23767e47i − 1.18316i
\(529\) 3.35870e47 0.734001
\(530\) 0 0
\(531\) 2.72895e47 0.558267
\(532\) − 3.50403e45i − 0.00693609i
\(533\) − 4.28587e47i − 0.820943i
\(534\) −8.25738e47 −1.53063
\(535\) 0 0
\(536\) −1.12467e48 −1.95272
\(537\) 2.16958e47i 0.364607i
\(538\) 2.15663e47i 0.350821i
\(539\) −6.20419e47 −0.976971
\(540\) 0 0
\(541\) −6.62658e47 −0.977997 −0.488998 0.872285i \(-0.662638\pi\)
−0.488998 + 0.872285i \(0.662638\pi\)
\(542\) − 6.80114e46i − 0.0971839i
\(543\) 8.30070e47i 1.14846i
\(544\) 1.17340e47 0.157204
\(545\) 0 0
\(546\) 3.53832e47 0.444555
\(547\) 1.12424e47i 0.136798i 0.997658 + 0.0683990i \(0.0217891\pi\)
−0.997658 + 0.0683990i \(0.978211\pi\)
\(548\) 1.52422e47i 0.179633i
\(549\) −5.32564e46 −0.0607933
\(550\) 0 0
\(551\) 1.26846e47 0.135870
\(552\) − 8.26092e47i − 0.857224i
\(553\) − 2.93624e47i − 0.295190i
\(554\) 1.21889e48 1.18725
\(555\) 0 0
\(556\) −1.86844e47 −0.170872
\(557\) − 6.68090e47i − 0.592064i −0.955178 0.296032i \(-0.904336\pi\)
0.955178 0.296032i \(-0.0956636\pi\)
\(558\) − 3.92053e47i − 0.336702i
\(559\) 4.89734e46 0.0407618
\(560\) 0 0
\(561\) 7.80339e47 0.610145
\(562\) 6.74014e47i 0.510838i
\(563\) 2.47733e48i 1.82007i 0.414536 + 0.910033i \(0.363944\pi\)
−0.414536 + 0.910033i \(0.636056\pi\)
\(564\) −4.80921e47 −0.342523
\(565\) 0 0
\(566\) 1.75846e48 1.17718
\(567\) − 1.42558e47i − 0.0925309i
\(568\) 5.24292e47i 0.329972i
\(569\) −8.59661e47 −0.524641 −0.262321 0.964981i \(-0.584488\pi\)
−0.262321 + 0.964981i \(0.584488\pi\)
\(570\) 0 0
\(571\) −3.39185e48 −1.94673 −0.973364 0.229266i \(-0.926367\pi\)
−0.973364 + 0.229266i \(0.926367\pi\)
\(572\) 6.00298e47i 0.334146i
\(573\) − 2.81753e48i − 1.52112i
\(574\) −1.78930e47 −0.0936968
\(575\) 0 0
\(576\) 3.14180e48 1.54805
\(577\) − 2.12534e48i − 1.01590i −0.861386 0.507951i \(-0.830403\pi\)
0.861386 0.507951i \(-0.169597\pi\)
\(578\) 1.63292e48i 0.757232i
\(579\) 4.79258e48 2.15623
\(580\) 0 0
\(581\) −6.49544e47 −0.275123
\(582\) 1.64143e48i 0.674639i
\(583\) 1.70314e48i 0.679283i
\(584\) 4.84681e48 1.87599
\(585\) 0 0
\(586\) −3.14491e48 −1.14657
\(587\) − 1.06269e48i − 0.376044i −0.982165 0.188022i \(-0.939792\pi\)
0.982165 0.188022i \(-0.0602076\pi\)
\(588\) − 8.91737e47i − 0.306290i
\(589\) −1.34504e47 −0.0448453
\(590\) 0 0
\(591\) 3.99665e48 1.25578
\(592\) − 3.57728e48i − 1.09125i
\(593\) − 3.98260e48i − 1.17953i −0.807574 0.589766i \(-0.799220\pi\)
0.807574 0.589766i \(-0.200780\pi\)
\(594\) 1.91176e48 0.549758
\(595\) 0 0
\(596\) 1.23794e48 0.335651
\(597\) − 1.64894e48i − 0.434163i
\(598\) − 2.85129e48i − 0.729072i
\(599\) −5.14535e48 −1.27775 −0.638874 0.769311i \(-0.720599\pi\)
−0.638874 + 0.769311i \(0.720599\pi\)
\(600\) 0 0
\(601\) −1.83878e48 −0.430751 −0.215375 0.976531i \(-0.569098\pi\)
−0.215375 + 0.976531i \(0.569098\pi\)
\(602\) − 2.04458e46i − 0.00465227i
\(603\) − 1.14394e49i − 2.52842i
\(604\) −1.00054e48 −0.214827
\(605\) 0 0
\(606\) 6.46516e48 1.31011
\(607\) − 2.06846e48i − 0.407236i −0.979050 0.203618i \(-0.934730\pi\)
0.979050 0.203618i \(-0.0652700\pi\)
\(608\) − 3.50467e47i − 0.0670401i
\(609\) −1.39247e48 −0.258812
\(610\) 0 0
\(611\) −9.69439e48 −1.70137
\(612\) 6.52623e47i 0.111304i
\(613\) 1.02050e48i 0.169143i 0.996417 + 0.0845717i \(0.0269522\pi\)
−0.996417 + 0.0845717i \(0.973048\pi\)
\(614\) −2.30743e48 −0.371691
\(615\) 0 0
\(616\) 1.46367e48 0.222732
\(617\) 3.46313e48i 0.512246i 0.966644 + 0.256123i \(0.0824452\pi\)
−0.966644 + 0.256123i \(0.917555\pi\)
\(618\) 1.44860e49i 2.08281i
\(619\) −6.56428e48 −0.917488 −0.458744 0.888568i \(-0.651701\pi\)
−0.458744 + 0.888568i \(0.651701\pi\)
\(620\) 0 0
\(621\) 2.36453e48 0.312350
\(622\) 3.67165e48i 0.471552i
\(623\) − 1.80954e48i − 0.225958i
\(624\) 1.51753e49 1.84249
\(625\) 0 0
\(626\) −5.99629e48 −0.688386
\(627\) − 2.33070e48i − 0.260198i
\(628\) − 1.06427e48i − 0.115546i
\(629\) 5.32965e48 0.562746
\(630\) 0 0
\(631\) 5.18692e48 0.518079 0.259039 0.965867i \(-0.416594\pi\)
0.259039 + 0.965867i \(0.416594\pi\)
\(632\) − 1.60587e49i − 1.56013i
\(633\) 6.48082e48i 0.612442i
\(634\) 8.74063e48 0.803490
\(635\) 0 0
\(636\) −2.44795e48 −0.212962
\(637\) − 1.79756e49i − 1.52140i
\(638\) 9.07235e48i 0.747064i
\(639\) −5.33275e48 −0.427254
\(640\) 0 0
\(641\) −6.15569e48 −0.466940 −0.233470 0.972364i \(-0.575008\pi\)
−0.233470 + 0.972364i \(0.575008\pi\)
\(642\) 2.07858e49i 1.53428i
\(643\) − 6.50853e48i − 0.467510i −0.972296 0.233755i \(-0.924899\pi\)
0.972296 0.233755i \(-0.0751013\pi\)
\(644\) 3.09971e47 0.0216680
\(645\) 0 0
\(646\) −8.59837e47 −0.0569308
\(647\) − 2.43533e49i − 1.56940i −0.619878 0.784698i \(-0.712818\pi\)
0.619878 0.784698i \(-0.287182\pi\)
\(648\) − 7.79667e48i − 0.489042i
\(649\) 6.69620e48 0.408833
\(650\) 0 0
\(651\) 1.47654e48 0.0854233
\(652\) 6.11047e48i 0.344145i
\(653\) 4.33343e48i 0.237602i 0.992918 + 0.118801i \(0.0379051\pi\)
−0.992918 + 0.118801i \(0.962095\pi\)
\(654\) 2.65789e48 0.141882
\(655\) 0 0
\(656\) −7.67401e48 −0.388334
\(657\) 4.92986e49i 2.42907i
\(658\) 4.04729e48i 0.194183i
\(659\) −2.26798e49 −1.05960 −0.529802 0.848121i \(-0.677734\pi\)
−0.529802 + 0.848121i \(0.677734\pi\)
\(660\) 0 0
\(661\) −2.84364e49 −1.25994 −0.629969 0.776620i \(-0.716932\pi\)
−0.629969 + 0.776620i \(0.716932\pi\)
\(662\) 1.03284e49i 0.445676i
\(663\) 2.26090e49i 0.950156i
\(664\) −3.55245e49 −1.45407
\(665\) 0 0
\(666\) 4.63992e49 1.80182
\(667\) 1.12210e49i 0.424452i
\(668\) 6.55455e48i 0.241521i
\(669\) −2.46325e49 −0.884202
\(670\) 0 0
\(671\) −1.30679e48 −0.0445205
\(672\) 3.84730e48i 0.127701i
\(673\) 1.14434e49i 0.370077i 0.982731 + 0.185039i \(0.0592410\pi\)
−0.982731 + 0.185039i \(0.940759\pi\)
\(674\) −1.72019e49 −0.542035
\(675\) 0 0
\(676\) −1.04871e49 −0.313753
\(677\) 3.00594e49i 0.876351i 0.898890 + 0.438175i \(0.144375\pi\)
−0.898890 + 0.438175i \(0.855625\pi\)
\(678\) − 1.24705e49i − 0.354295i
\(679\) −3.59708e48 −0.0995931
\(680\) 0 0
\(681\) 3.59766e49 0.946119
\(682\) − 9.62004e48i − 0.246576i
\(683\) − 7.78736e49i − 1.94549i −0.231885 0.972743i \(-0.574489\pi\)
0.231885 0.972743i \(-0.425511\pi\)
\(684\) 1.94924e48 0.0474661
\(685\) 0 0
\(686\) −1.53329e49 −0.354774
\(687\) 6.32535e49i 1.42673i
\(688\) − 8.76886e47i − 0.0192817i
\(689\) −4.93456e49 −1.05782
\(690\) 0 0
\(691\) 9.61614e49 1.95945 0.979726 0.200340i \(-0.0642046\pi\)
0.979726 + 0.200340i \(0.0642046\pi\)
\(692\) − 9.45269e48i − 0.187801i
\(693\) 1.48875e49i 0.288397i
\(694\) −5.07971e49 −0.959506
\(695\) 0 0
\(696\) −7.61562e49 −1.36787
\(697\) − 1.14332e49i − 0.200260i
\(698\) 4.33853e49i 0.741093i
\(699\) 5.46014e49 0.909606
\(700\) 0 0
\(701\) 1.19741e49 0.189748 0.0948742 0.995489i \(-0.469755\pi\)
0.0948742 + 0.995489i \(0.469755\pi\)
\(702\) 5.53901e49i 0.856116i
\(703\) − 1.59185e49i − 0.239984i
\(704\) 7.70923e49 1.13368
\(705\) 0 0
\(706\) −1.16678e50 −1.63270
\(707\) 1.41679e49i 0.193404i
\(708\) 9.62454e48i 0.128173i
\(709\) 4.80032e49 0.623680 0.311840 0.950135i \(-0.399055\pi\)
0.311840 + 0.950135i \(0.399055\pi\)
\(710\) 0 0
\(711\) 1.63339e50 2.02009
\(712\) − 9.89664e49i − 1.19423i
\(713\) − 1.18984e49i − 0.140094i
\(714\) 9.43898e48 0.108444
\(715\) 0 0
\(716\) −4.45232e48 −0.0487089
\(717\) 1.73297e50i 1.85015i
\(718\) 8.84635e49i 0.921693i
\(719\) 1.45387e49 0.147832 0.0739162 0.997264i \(-0.476450\pi\)
0.0739162 + 0.997264i \(0.476450\pi\)
\(720\) 0 0
\(721\) −3.17450e49 −0.307474
\(722\) − 9.16528e49i − 0.866452i
\(723\) 1.15246e50i 1.06342i
\(724\) −1.70344e49 −0.153427
\(725\) 0 0
\(726\) 6.19432e48 0.0531622
\(727\) − 1.25503e50i − 1.05148i −0.850644 0.525742i \(-0.823788\pi\)
0.850644 0.525742i \(-0.176212\pi\)
\(728\) 4.24075e49i 0.346851i
\(729\) 1.96598e50 1.56981
\(730\) 0 0
\(731\) 1.30644e48 0.00994338
\(732\) − 1.87826e48i − 0.0139576i
\(733\) − 1.40440e50i − 1.01899i −0.860473 0.509495i \(-0.829832\pi\)
0.860473 0.509495i \(-0.170168\pi\)
\(734\) −4.95419e49 −0.350988
\(735\) 0 0
\(736\) 3.10027e49 0.209430
\(737\) − 2.80695e50i − 1.85163i
\(738\) − 9.95358e49i − 0.641199i
\(739\) −2.93572e49 −0.184687 −0.0923435 0.995727i \(-0.529436\pi\)
−0.0923435 + 0.995727i \(0.529436\pi\)
\(740\) 0 0
\(741\) 6.75281e49 0.405197
\(742\) 2.06012e49i 0.120732i
\(743\) 1.10795e50i 0.634185i 0.948395 + 0.317093i \(0.102707\pi\)
−0.948395 + 0.317093i \(0.897293\pi\)
\(744\) 8.07537e49 0.451478
\(745\) 0 0
\(746\) −1.78650e50 −0.952960
\(747\) − 3.61331e50i − 1.88276i
\(748\) 1.60138e49i 0.0815111i
\(749\) −4.55506e49 −0.226497
\(750\) 0 0
\(751\) −4.13601e50 −1.96283 −0.981416 0.191892i \(-0.938538\pi\)
−0.981416 + 0.191892i \(0.938538\pi\)
\(752\) 1.73582e50i 0.804806i
\(753\) − 5.28418e50i − 2.39368i
\(754\) −2.62856e50 −1.16337
\(755\) 0 0
\(756\) −6.02161e48 −0.0254437
\(757\) − 3.15399e50i − 1.30221i −0.758987 0.651106i \(-0.774305\pi\)
0.758987 0.651106i \(-0.225695\pi\)
\(758\) − 2.01180e50i − 0.811659i
\(759\) 2.06176e50 0.812845
\(760\) 0 0
\(761\) −3.65841e50 −1.37740 −0.688702 0.725045i \(-0.741819\pi\)
−0.688702 + 0.725045i \(0.741819\pi\)
\(762\) − 2.39624e50i − 0.881696i
\(763\) 5.82458e48i 0.0209452i
\(764\) 5.78202e49 0.203210
\(765\) 0 0
\(766\) 1.57382e49 0.0528387
\(767\) 1.94011e50i 0.636661i
\(768\) 2.85192e50i 0.914778i
\(769\) 2.07264e50 0.649849 0.324925 0.945740i \(-0.394661\pi\)
0.324925 + 0.945740i \(0.394661\pi\)
\(770\) 0 0
\(771\) 9.42552e50 2.82393
\(772\) 9.83516e49i 0.288057i
\(773\) 2.04692e50i 0.586084i 0.956100 + 0.293042i \(0.0946675\pi\)
−0.956100 + 0.293042i \(0.905332\pi\)
\(774\) 1.13737e49 0.0318371
\(775\) 0 0
\(776\) −1.96729e50 −0.526367
\(777\) 1.74747e50i 0.457133i
\(778\) − 3.14521e50i − 0.804465i
\(779\) −3.41484e49 −0.0854013
\(780\) 0 0
\(781\) −1.30853e50 −0.312889
\(782\) − 7.60622e49i − 0.177849i
\(783\) − 2.17983e50i − 0.498415i
\(784\) −3.21860e50 −0.719673
\(785\) 0 0
\(786\) 1.37432e50 0.293897
\(787\) 2.74398e50i 0.573882i 0.957948 + 0.286941i \(0.0926383\pi\)
−0.957948 + 0.286941i \(0.907362\pi\)
\(788\) 8.20177e49i 0.167764i
\(789\) 1.45994e49 0.0292071
\(790\) 0 0
\(791\) 2.73282e49 0.0523025
\(792\) 8.14219e50i 1.52423i
\(793\) − 3.78619e49i − 0.0693301i
\(794\) 6.54491e50 1.17232
\(795\) 0 0
\(796\) 3.38388e49 0.0580012
\(797\) − 3.73868e50i − 0.626899i −0.949605 0.313450i \(-0.898515\pi\)
0.949605 0.313450i \(-0.101485\pi\)
\(798\) − 2.81921e49i − 0.0462463i
\(799\) −2.58612e50 −0.415030
\(800\) 0 0
\(801\) 1.00662e51 1.54631
\(802\) − 9.13117e49i − 0.137238i
\(803\) 1.20967e51i 1.77887i
\(804\) 4.03447e50 0.580505
\(805\) 0 0
\(806\) 2.78725e50 0.383983
\(807\) − 4.51827e50i − 0.609095i
\(808\) 7.74862e50i 1.02218i
\(809\) 3.87407e50 0.500114 0.250057 0.968231i \(-0.419551\pi\)
0.250057 + 0.968231i \(0.419551\pi\)
\(810\) 0 0
\(811\) −9.21816e50 −1.13967 −0.569835 0.821759i \(-0.692993\pi\)
−0.569835 + 0.821759i \(0.692993\pi\)
\(812\) − 2.85758e49i − 0.0345754i
\(813\) 1.42488e50i 0.168730i
\(814\) 1.13853e51 1.31952
\(815\) 0 0
\(816\) 4.04823e50 0.449456
\(817\) − 3.90203e48i − 0.00424038i
\(818\) − 5.53933e50i − 0.589216i
\(819\) −4.31341e50 −0.449110
\(820\) 0 0
\(821\) 1.64737e51 1.64357 0.821783 0.569801i \(-0.192980\pi\)
0.821783 + 0.569801i \(0.192980\pi\)
\(822\) 1.22633e51i 1.19771i
\(823\) − 6.02163e50i − 0.575727i −0.957671 0.287864i \(-0.907055\pi\)
0.957671 0.287864i \(-0.0929449\pi\)
\(824\) −1.73618e51 −1.62505
\(825\) 0 0
\(826\) 8.09973e49 0.0726641
\(827\) 4.30476e50i 0.378097i 0.981968 + 0.189048i \(0.0605403\pi\)
−0.981968 + 0.189048i \(0.939460\pi\)
\(828\) 1.72432e50i 0.148282i
\(829\) −8.88751e50 −0.748300 −0.374150 0.927368i \(-0.622065\pi\)
−0.374150 + 0.927368i \(0.622065\pi\)
\(830\) 0 0
\(831\) −2.55364e51 −2.06131
\(832\) 2.23362e51i 1.76543i
\(833\) − 4.79525e50i − 0.371128i
\(834\) −1.50328e51 −1.13929
\(835\) 0 0
\(836\) 4.78297e49 0.0347606
\(837\) 2.31142e50i 0.164507i
\(838\) 1.26992e51i 0.885126i
\(839\) −3.76912e50 −0.257280 −0.128640 0.991691i \(-0.541061\pi\)
−0.128640 + 0.991691i \(0.541061\pi\)
\(840\) 0 0
\(841\) −4.92875e50 −0.322706
\(842\) − 4.57730e50i − 0.293527i
\(843\) − 1.41210e51i − 0.886916i
\(844\) −1.32997e50 −0.0818179
\(845\) 0 0
\(846\) −2.25144e51 −1.32886
\(847\) 1.35744e49i 0.00784804i
\(848\) 8.83551e50i 0.500385i
\(849\) −3.68407e51 −2.04382
\(850\) 0 0
\(851\) 1.40817e51 0.749699
\(852\) − 1.88077e50i − 0.0980940i
\(853\) − 3.00248e51i − 1.53417i −0.641547 0.767083i \(-0.721707\pi\)
0.641547 0.767083i \(-0.278293\pi\)
\(854\) −1.58069e49 −0.00791286
\(855\) 0 0
\(856\) −2.49122e51 −1.19708
\(857\) 2.46088e50i 0.115858i 0.998321 + 0.0579291i \(0.0184497\pi\)
−0.998321 + 0.0579291i \(0.981550\pi\)
\(858\) 4.82977e51i 2.22792i
\(859\) −2.64506e51 −1.19551 −0.597757 0.801677i \(-0.703941\pi\)
−0.597757 + 0.801677i \(0.703941\pi\)
\(860\) 0 0
\(861\) 3.74868e50 0.162676
\(862\) 1.02117e51i 0.434233i
\(863\) − 9.93861e50i − 0.414130i −0.978327 0.207065i \(-0.933609\pi\)
0.978327 0.207065i \(-0.0663911\pi\)
\(864\) −6.02270e50 −0.245924
\(865\) 0 0
\(866\) −3.66184e51 −1.43594
\(867\) − 3.42107e51i − 1.31470i
\(868\) 3.03009e49i 0.0114120i
\(869\) 4.00794e51 1.47936
\(870\) 0 0
\(871\) 8.13268e51 2.88347
\(872\) 3.18554e50i 0.110699i
\(873\) − 2.00100e51i − 0.681550i
\(874\) −2.27181e50 −0.0758441
\(875\) 0 0
\(876\) −1.73867e51 −0.557695
\(877\) 2.79832e50i 0.0879842i 0.999032 + 0.0439921i \(0.0140076\pi\)
−0.999032 + 0.0439921i \(0.985992\pi\)
\(878\) − 3.16623e50i − 0.0975861i
\(879\) 6.58876e51 1.99067
\(880\) 0 0
\(881\) −5.95050e50 −0.172773 −0.0863863 0.996262i \(-0.527532\pi\)
−0.0863863 + 0.996262i \(0.527532\pi\)
\(882\) − 4.17469e51i − 1.18829i
\(883\) − 3.85350e51i − 1.07533i −0.843157 0.537667i \(-0.819306\pi\)
0.843157 0.537667i \(-0.180694\pi\)
\(884\) −4.63973e50 −0.126934
\(885\) 0 0
\(886\) 4.96755e51 1.30633
\(887\) 4.56338e50i 0.117658i 0.998268 + 0.0588292i \(0.0187367\pi\)
−0.998268 + 0.0588292i \(0.981263\pi\)
\(888\) 9.55716e51i 2.41603i
\(889\) 5.25119e50 0.130160
\(890\) 0 0
\(891\) 1.94590e51 0.463724
\(892\) − 5.05498e50i − 0.118123i
\(893\) 7.72416e50i 0.176991i
\(894\) 9.95996e51 2.23795
\(895\) 0 0
\(896\) 5.50354e50 0.118919
\(897\) 5.97362e51i 1.26581i
\(898\) − 9.39122e50i − 0.195158i
\(899\) −1.09689e51 −0.223548
\(900\) 0 0
\(901\) −1.31637e51 −0.258044
\(902\) − 2.44237e51i − 0.469567i
\(903\) 4.28351e49i 0.00807726i
\(904\) 1.49462e51 0.276428
\(905\) 0 0
\(906\) −8.04998e51 −1.43236
\(907\) 4.12286e51i 0.719566i 0.933036 + 0.359783i \(0.117149\pi\)
−0.933036 + 0.359783i \(0.882851\pi\)
\(908\) 7.38298e50i 0.126395i
\(909\) −7.88139e51 −1.32353
\(910\) 0 0
\(911\) 6.95561e51 1.12399 0.561996 0.827140i \(-0.310034\pi\)
0.561996 + 0.827140i \(0.310034\pi\)
\(912\) − 1.20911e51i − 0.191672i
\(913\) − 8.86621e51i − 1.37879i
\(914\) 7.72094e51 1.17791
\(915\) 0 0
\(916\) −1.29806e51 −0.190601
\(917\) 3.01173e50i 0.0433863i
\(918\) 1.47761e51i 0.208840i
\(919\) −1.23335e51 −0.171027 −0.0855136 0.996337i \(-0.527253\pi\)
−0.0855136 + 0.996337i \(0.527253\pi\)
\(920\) 0 0
\(921\) 4.83420e51 0.645329
\(922\) − 6.15821e51i − 0.806610i
\(923\) − 3.79125e51i − 0.487251i
\(924\) −5.25057e50 −0.0662136
\(925\) 0 0
\(926\) 6.07023e51 0.737076
\(927\) − 1.76592e52i − 2.10415i
\(928\) − 2.85810e51i − 0.334185i
\(929\) 9.97208e51 1.14422 0.572112 0.820176i \(-0.306124\pi\)
0.572112 + 0.820176i \(0.306124\pi\)
\(930\) 0 0
\(931\) −1.43224e51 −0.158269
\(932\) 1.12051e51i 0.121517i
\(933\) − 7.69232e51i − 0.818707i
\(934\) −3.92536e51 −0.410024
\(935\) 0 0
\(936\) −2.35906e52 −2.37363
\(937\) − 6.01334e51i − 0.593844i −0.954902 0.296922i \(-0.904040\pi\)
0.954902 0.296922i \(-0.0959602\pi\)
\(938\) − 3.39529e51i − 0.329100i
\(939\) 1.25626e52 1.19517
\(940\) 0 0
\(941\) 1.38035e52 1.26523 0.632617 0.774465i \(-0.281981\pi\)
0.632617 + 0.774465i \(0.281981\pi\)
\(942\) − 8.56268e51i − 0.770405i
\(943\) − 3.02081e51i − 0.266789i
\(944\) 3.47384e51 0.301162
\(945\) 0 0
\(946\) 2.79083e50 0.0233151
\(947\) 9.01648e51i 0.739456i 0.929140 + 0.369728i \(0.120549\pi\)
−0.929140 + 0.369728i \(0.879451\pi\)
\(948\) 5.76067e51i 0.463796i
\(949\) −3.50482e52 −2.77017
\(950\) 0 0
\(951\) −1.83121e52 −1.39502
\(952\) 1.13128e51i 0.0846105i
\(953\) 4.32450e51i 0.317549i 0.987315 + 0.158774i \(0.0507542\pi\)
−0.987315 + 0.158774i \(0.949246\pi\)
\(954\) −1.14601e52 −0.826214
\(955\) 0 0
\(956\) −3.55633e51 −0.247166
\(957\) − 1.90071e52i − 1.29705i
\(958\) 1.45348e52i 0.973897i
\(959\) −2.68741e51 −0.176810
\(960\) 0 0
\(961\) −1.46006e52 −0.926216
\(962\) 3.29869e52i 2.05484i
\(963\) − 2.53391e52i − 1.55000i
\(964\) −2.36503e51 −0.142066
\(965\) 0 0
\(966\) 2.49391e51 0.144471
\(967\) 1.91085e51i 0.108709i 0.998522 + 0.0543543i \(0.0173101\pi\)
−0.998522 + 0.0543543i \(0.982690\pi\)
\(968\) 7.42403e50i 0.0414783i
\(969\) 1.80141e51 0.0988431
\(970\) 0 0
\(971\) 2.54698e52 1.34800 0.674000 0.738732i \(-0.264575\pi\)
0.674000 + 0.738732i \(0.264575\pi\)
\(972\) 5.20394e51i 0.270504i
\(973\) − 3.29432e51i − 0.168186i
\(974\) 1.17796e52 0.590675
\(975\) 0 0
\(976\) −6.77932e50 −0.0327955
\(977\) − 2.34465e52i − 1.11410i −0.830480 0.557048i \(-0.811934\pi\)
0.830480 0.557048i \(-0.188066\pi\)
\(978\) 4.91625e52i 2.29459i
\(979\) 2.47001e52 1.13240
\(980\) 0 0
\(981\) −3.24012e51 −0.143335
\(982\) 1.40197e52i 0.609238i
\(983\) 1.06892e52i 0.456308i 0.973625 + 0.228154i \(0.0732690\pi\)
−0.973625 + 0.228154i \(0.926731\pi\)
\(984\) 2.05021e52 0.859772
\(985\) 0 0
\(986\) −7.01207e51 −0.283792
\(987\) − 8.47931e51i − 0.337140i
\(988\) 1.38579e51i 0.0541314i
\(989\) 3.45178e50 0.0132467
\(990\) 0 0
\(991\) 1.97227e52 0.730597 0.365298 0.930891i \(-0.380967\pi\)
0.365298 + 0.930891i \(0.380967\pi\)
\(992\) 3.03064e51i 0.110301i
\(993\) − 2.16387e52i − 0.773782i
\(994\) −1.58280e51 −0.0556114
\(995\) 0 0
\(996\) 1.27435e52 0.432266
\(997\) 1.44359e52i 0.481149i 0.970631 + 0.240575i \(0.0773358\pi\)
−0.970631 + 0.240575i \(0.922664\pi\)
\(998\) 2.90388e52i 0.951029i
\(999\) −2.73555e52 −0.880338
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 25.36.b.a.24.5 6
5.2 odd 4 1.36.a.a.1.1 3
5.3 odd 4 25.36.a.a.1.3 3
5.4 even 2 inner 25.36.b.a.24.2 6
15.2 even 4 9.36.a.b.1.3 3
20.7 even 4 16.36.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.36.a.a.1.1 3 5.2 odd 4
9.36.a.b.1.3 3 15.2 even 4
16.36.a.d.1.3 3 20.7 even 4
25.36.a.a.1.3 3 5.3 odd 4
25.36.b.a.24.2 6 5.4 even 2 inner
25.36.b.a.24.5 6 1.1 even 1 trivial