Properties

Label 25.34.b.b.24.5
Level $25$
Weight $34$
Character 25.24
Analytic conductor $172.457$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} + 2 x^{8} + 325405868686 x^{7} + 327807761582101561 x^{6} + 230788916575056333044 x^{5} + 52482256238971860814088 x^{4} - 28868477926481009935781561728 x^{3} + 5099072964771111558047480848223296 x^{2} - 596428985476749456366946929757065984 x + 34881588984360865981321269538813997568\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{10}\cdot 5^{24}\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.5
Root \(58.5034 + 58.5034i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.34.b.b.24.6

$q$-expansion

\(f(q)\) \(=\) \(q-5627.97i q^{2} -1.24605e8i q^{3} +8.55826e9 q^{4} -7.01272e11 q^{6} -7.01560e13i q^{7} -9.65096e13i q^{8} -9.96727e15 q^{9} +O(q^{10})\) \(q-5627.97i q^{2} -1.24605e8i q^{3} +8.55826e9 q^{4} -7.01272e11 q^{6} -7.01560e13i q^{7} -9.65096e13i q^{8} -9.96727e15 q^{9} +9.39806e16 q^{11} -1.06640e18i q^{12} +9.95093e17i q^{13} -3.94836e17 q^{14} +7.29717e19 q^{16} +2.59085e20i q^{17} +5.60955e19i q^{18} +1.31117e21 q^{19} -8.74176e21 q^{21} -5.28920e20i q^{22} +5.44854e22i q^{23} -1.20255e22 q^{24} +5.60036e21 q^{26} +5.49284e23i q^{27} -6.00413e23i q^{28} +9.77301e23 q^{29} +6.82815e24 q^{31} -1.23969e24i q^{32} -1.17104e25i q^{33} +1.45812e24 q^{34} -8.53025e25 q^{36} +7.50351e25i q^{37} -7.37925e24i q^{38} +1.23993e26 q^{39} -2.94248e25 q^{41} +4.91984e25i q^{42} -1.00868e27i q^{43} +8.04311e26 q^{44} +3.06642e26 q^{46} -1.91365e27i q^{47} -9.09262e27i q^{48} +2.80913e27 q^{49} +3.22832e28 q^{51} +8.51627e27i q^{52} +4.33344e28i q^{53} +3.09135e27 q^{54} -6.77072e27 q^{56} -1.63379e29i q^{57} -5.50022e27i q^{58} +1.21137e29 q^{59} +4.03792e29 q^{61} -3.84287e28i q^{62} +6.99264e29i q^{63} +6.19846e29 q^{64} -6.59060e28 q^{66} +1.33311e30i q^{67} +2.21732e30i q^{68} +6.78914e30 q^{69} -4.10877e30 q^{71} +9.61937e29i q^{72} +8.64070e30i q^{73} +4.22295e29 q^{74} +1.12214e31 q^{76} -6.59330e30i q^{77} -6.97831e29i q^{78} +1.76553e31 q^{79} +1.30347e31 q^{81} +1.65602e29i q^{82} -6.43615e30i q^{83} -7.48143e31 q^{84} -5.67680e30 q^{86} -1.21776e32i q^{87} -9.07003e30i q^{88} -1.32441e32 q^{89} +6.98117e31 q^{91} +4.66300e32i q^{92} -8.50820e32i q^{93} -1.07700e31 q^{94} -1.54472e32 q^{96} -2.90988e32i q^{97} -1.58097e31i q^{98} -9.36730e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 2282622720q^{4} + 25850126231520q^{6} - 29082501980817930q^{9} + O(q^{10}) \) \( 10q - 2282622720q^{4} + 25850126231520q^{6} - 29082501980817930q^{9} - 575603603755953280q^{11} - 23352899832544964640q^{14} - \)\(17\!\cdots\!40\)\(q^{16} - \)\(18\!\cdots\!00\)\(q^{19} - \)\(21\!\cdots\!80\)\(q^{21} + \)\(14\!\cdots\!00\)\(q^{24} - \)\(84\!\cdots\!80\)\(q^{26} + \)\(18\!\cdots\!00\)\(q^{29} - \)\(26\!\cdots\!80\)\(q^{31} + \)\(13\!\cdots\!60\)\(q^{34} - \)\(36\!\cdots\!40\)\(q^{36} + \)\(15\!\cdots\!40\)\(q^{39} - \)\(26\!\cdots\!80\)\(q^{41} + \)\(37\!\cdots\!60\)\(q^{44} - \)\(57\!\cdots\!80\)\(q^{46} - \)\(82\!\cdots\!70\)\(q^{49} + \)\(14\!\cdots\!20\)\(q^{51} - \)\(12\!\cdots\!00\)\(q^{54} + \)\(28\!\cdots\!00\)\(q^{56} - \)\(51\!\cdots\!00\)\(q^{59} + \)\(14\!\cdots\!20\)\(q^{61} + \)\(20\!\cdots\!80\)\(q^{64} - \)\(66\!\cdots\!60\)\(q^{66} + \)\(55\!\cdots\!40\)\(q^{69} - \)\(11\!\cdots\!80\)\(q^{71} + \)\(72\!\cdots\!60\)\(q^{74} + \)\(14\!\cdots\!00\)\(q^{76} + \)\(20\!\cdots\!00\)\(q^{79} + \)\(15\!\cdots\!10\)\(q^{81} - \)\(34\!\cdots\!40\)\(q^{84} + \)\(13\!\cdots\!20\)\(q^{86} + \)\(12\!\cdots\!00\)\(q^{89} - \)\(53\!\cdots\!80\)\(q^{91} + \)\(72\!\cdots\!60\)\(q^{94} - \)\(68\!\cdots\!80\)\(q^{96} + \)\(35\!\cdots\!40\)\(q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5627.97i − 0.0607235i −0.999539 0.0303618i \(-0.990334\pi\)
0.999539 0.0303618i \(-0.00966594\pi\)
\(3\) − 1.24605e8i − 1.67122i −0.549323 0.835610i \(-0.685114\pi\)
0.549323 0.835610i \(-0.314886\pi\)
\(4\) 8.55826e9 0.996313
\(5\) 0 0
\(6\) −7.01272e11 −0.101482
\(7\) − 7.01560e13i − 0.797897i −0.916973 0.398949i \(-0.869375\pi\)
0.916973 0.398949i \(-0.130625\pi\)
\(8\) − 9.65096e13i − 0.121223i
\(9\) −9.96727e15 −1.79298
\(10\) 0 0
\(11\) 9.39806e16 0.616679 0.308339 0.951276i \(-0.400227\pi\)
0.308339 + 0.951276i \(0.400227\pi\)
\(12\) − 1.06640e18i − 1.66506i
\(13\) 9.95093e17i 0.414761i 0.978260 + 0.207381i \(0.0664939\pi\)
−0.978260 + 0.207381i \(0.933506\pi\)
\(14\) −3.94836e17 −0.0484511
\(15\) 0 0
\(16\) 7.29717e19 0.988952
\(17\) 2.59085e20i 1.29132i 0.763624 + 0.645662i \(0.223418\pi\)
−0.763624 + 0.645662i \(0.776582\pi\)
\(18\) 5.60955e19i 0.108876i
\(19\) 1.31117e21 1.04286 0.521430 0.853294i \(-0.325399\pi\)
0.521430 + 0.853294i \(0.325399\pi\)
\(20\) 0 0
\(21\) −8.74176e21 −1.33346
\(22\) − 5.28920e20i − 0.0374469i
\(23\) 5.44854e22i 1.85255i 0.376843 + 0.926277i \(0.377010\pi\)
−0.376843 + 0.926277i \(0.622990\pi\)
\(24\) −1.20255e22 −0.202591
\(25\) 0 0
\(26\) 5.60036e21 0.0251858
\(27\) 5.49284e23i 1.32524i
\(28\) − 6.00413e23i − 0.794955i
\(29\) 9.77301e23 0.725206 0.362603 0.931944i \(-0.381888\pi\)
0.362603 + 0.931944i \(0.381888\pi\)
\(30\) 0 0
\(31\) 6.82815e24 1.68591 0.842957 0.537981i \(-0.180813\pi\)
0.842957 + 0.537981i \(0.180813\pi\)
\(32\) − 1.23969e24i − 0.181276i
\(33\) − 1.17104e25i − 1.03061i
\(34\) 1.45812e24 0.0784137
\(35\) 0 0
\(36\) −8.53025e25 −1.78637
\(37\) 7.50351e25i 0.999852i 0.866068 + 0.499926i \(0.166640\pi\)
−0.866068 + 0.499926i \(0.833360\pi\)
\(38\) − 7.37925e24i − 0.0633262i
\(39\) 1.23993e26 0.693158
\(40\) 0 0
\(41\) −2.94248e25 −0.0720741 −0.0360371 0.999350i \(-0.511473\pi\)
−0.0360371 + 0.999350i \(0.511473\pi\)
\(42\) 4.91984e25i 0.0809725i
\(43\) − 1.00868e27i − 1.12596i −0.826471 0.562979i \(-0.809655\pi\)
0.826471 0.562979i \(-0.190345\pi\)
\(44\) 8.04311e26 0.614405
\(45\) 0 0
\(46\) 3.06642e26 0.112494
\(47\) − 1.91365e27i − 0.492321i −0.969229 0.246160i \(-0.920831\pi\)
0.969229 0.246160i \(-0.0791690\pi\)
\(48\) − 9.09262e27i − 1.65276i
\(49\) 2.80913e27 0.363360
\(50\) 0 0
\(51\) 3.22832e28 2.15809
\(52\) 8.51627e27i 0.413232i
\(53\) 4.33344e28i 1.53561i 0.640684 + 0.767804i \(0.278651\pi\)
−0.640684 + 0.767804i \(0.721349\pi\)
\(54\) 3.09135e27 0.0804733
\(55\) 0 0
\(56\) −6.77072e27 −0.0967236
\(57\) − 1.63379e29i − 1.74285i
\(58\) − 5.50022e27i − 0.0440371i
\(59\) 1.21137e29 0.731507 0.365754 0.930712i \(-0.380811\pi\)
0.365754 + 0.930712i \(0.380811\pi\)
\(60\) 0 0
\(61\) 4.03792e29 1.40674 0.703372 0.710822i \(-0.251677\pi\)
0.703372 + 0.710822i \(0.251677\pi\)
\(62\) − 3.84287e28i − 0.102375i
\(63\) 6.99264e29i 1.43061i
\(64\) 6.19846e29 0.977944
\(65\) 0 0
\(66\) −6.59060e28 −0.0625821
\(67\) 1.33311e30i 0.987713i 0.869543 + 0.493856i \(0.164413\pi\)
−0.869543 + 0.493856i \(0.835587\pi\)
\(68\) 2.21732e30i 1.28656i
\(69\) 6.78914e30 3.09603
\(70\) 0 0
\(71\) −4.10877e30 −1.16936 −0.584681 0.811263i \(-0.698780\pi\)
−0.584681 + 0.811263i \(0.698780\pi\)
\(72\) 9.61937e29i 0.217350i
\(73\) 8.64070e30i 1.55497i 0.628902 + 0.777484i \(0.283505\pi\)
−0.628902 + 0.777484i \(0.716495\pi\)
\(74\) 4.22295e29 0.0607146
\(75\) 0 0
\(76\) 1.12214e31 1.03901
\(77\) − 6.59330e30i − 0.492046i
\(78\) − 6.97831e29i − 0.0420910i
\(79\) 1.76553e31 0.863033 0.431516 0.902105i \(-0.357979\pi\)
0.431516 + 0.902105i \(0.357979\pi\)
\(80\) 0 0
\(81\) 1.30347e31 0.421790
\(82\) 1.65602e29i 0.00437659i
\(83\) − 6.43615e30i − 0.139264i −0.997573 0.0696318i \(-0.977818\pi\)
0.997573 0.0696318i \(-0.0221824\pi\)
\(84\) −7.48143e31 −1.32855
\(85\) 0 0
\(86\) −5.67680e30 −0.0683722
\(87\) − 1.21776e32i − 1.21198i
\(88\) − 9.07003e30i − 0.0747558i
\(89\) −1.32441e32 −0.905911 −0.452956 0.891533i \(-0.649630\pi\)
−0.452956 + 0.891533i \(0.649630\pi\)
\(90\) 0 0
\(91\) 6.98117e31 0.330937
\(92\) 4.66300e32i 1.84572i
\(93\) − 8.50820e32i − 2.81753i
\(94\) −1.07700e31 −0.0298955
\(95\) 0 0
\(96\) −1.54472e32 −0.302952
\(97\) − 2.90988e32i − 0.480996i −0.970650 0.240498i \(-0.922689\pi\)
0.970650 0.240498i \(-0.0773107\pi\)
\(98\) − 1.58097e31i − 0.0220645i
\(99\) −9.36730e32 −1.10569
\(100\) 0 0
\(101\) −2.11490e33 −1.79468 −0.897341 0.441338i \(-0.854504\pi\)
−0.897341 + 0.441338i \(0.854504\pi\)
\(102\) − 1.81689e32i − 0.131047i
\(103\) 9.19643e32i 0.564684i 0.959314 + 0.282342i \(0.0911112\pi\)
−0.959314 + 0.282342i \(0.908889\pi\)
\(104\) 9.60360e31 0.0502787
\(105\) 0 0
\(106\) 2.43885e32 0.0932476
\(107\) 3.10653e33i 1.01729i 0.860977 + 0.508644i \(0.169853\pi\)
−0.860977 + 0.508644i \(0.830147\pi\)
\(108\) 4.70091e33i 1.32035i
\(109\) 4.33103e33 1.04485 0.522425 0.852685i \(-0.325028\pi\)
0.522425 + 0.852685i \(0.325028\pi\)
\(110\) 0 0
\(111\) 9.34972e33 1.67097
\(112\) − 5.11940e33i − 0.789082i
\(113\) 1.60843e33i 0.214096i 0.994254 + 0.107048i \(0.0341398\pi\)
−0.994254 + 0.107048i \(0.965860\pi\)
\(114\) −9.19490e32 −0.105832
\(115\) 0 0
\(116\) 8.36400e33 0.722532
\(117\) − 9.91836e33i − 0.743658i
\(118\) − 6.81756e32i − 0.0444197i
\(119\) 1.81764e34 1.03034
\(120\) 0 0
\(121\) −1.43928e34 −0.619707
\(122\) − 2.27253e33i − 0.0854225i
\(123\) 3.66646e33i 0.120452i
\(124\) 5.84371e34 1.67970
\(125\) 0 0
\(126\) 3.93544e33 0.0868718
\(127\) 6.65683e34i 1.28975i 0.764287 + 0.644876i \(0.223091\pi\)
−0.764287 + 0.644876i \(0.776909\pi\)
\(128\) − 1.41374e34i − 0.240660i
\(129\) −1.25686e35 −1.88173
\(130\) 0 0
\(131\) −3.69750e34 −0.429469 −0.214734 0.976672i \(-0.568889\pi\)
−0.214734 + 0.976672i \(0.568889\pi\)
\(132\) − 1.00221e35i − 1.02681i
\(133\) − 9.19867e34i − 0.832095i
\(134\) 7.50269e33 0.0599774
\(135\) 0 0
\(136\) 2.50042e34 0.156538
\(137\) 1.76237e34i 0.0977705i 0.998804 + 0.0488853i \(0.0155669\pi\)
−0.998804 + 0.0488853i \(0.984433\pi\)
\(138\) − 3.82091e34i − 0.188002i
\(139\) −1.00540e35 −0.439130 −0.219565 0.975598i \(-0.570464\pi\)
−0.219565 + 0.975598i \(0.570464\pi\)
\(140\) 0 0
\(141\) −2.38450e35 −0.822776
\(142\) 2.31240e34i 0.0710078i
\(143\) 9.35195e34i 0.255775i
\(144\) −7.27329e35 −1.77317
\(145\) 0 0
\(146\) 4.86296e34 0.0944232
\(147\) − 3.50031e35i − 0.607254i
\(148\) 6.42170e35i 0.996165i
\(149\) 1.14557e36 1.59019 0.795093 0.606487i \(-0.207422\pi\)
0.795093 + 0.606487i \(0.207422\pi\)
\(150\) 0 0
\(151\) −6.40257e35 −0.713239 −0.356619 0.934250i \(-0.616071\pi\)
−0.356619 + 0.934250i \(0.616071\pi\)
\(152\) − 1.26541e35i − 0.126419i
\(153\) − 2.58237e36i − 2.31531i
\(154\) −3.71069e34 −0.0298788
\(155\) 0 0
\(156\) 1.06117e36 0.690602
\(157\) − 1.01110e36i − 0.592172i −0.955161 0.296086i \(-0.904319\pi\)
0.955161 0.296086i \(-0.0956815\pi\)
\(158\) − 9.93633e34i − 0.0524064i
\(159\) 5.39967e36 2.56634
\(160\) 0 0
\(161\) 3.82248e36 1.47815
\(162\) − 7.33587e34i − 0.0256126i
\(163\) 4.84374e36i 1.52786i 0.645296 + 0.763932i \(0.276734\pi\)
−0.645296 + 0.763932i \(0.723266\pi\)
\(164\) −2.51825e35 −0.0718083
\(165\) 0 0
\(166\) −3.62225e34 −0.00845658
\(167\) − 1.75726e36i − 0.371548i −0.982593 0.185774i \(-0.940521\pi\)
0.982593 0.185774i \(-0.0594792\pi\)
\(168\) 8.43664e35i 0.161647i
\(169\) 4.76592e36 0.827973
\(170\) 0 0
\(171\) −1.30688e37 −1.86982
\(172\) − 8.63252e36i − 1.12181i
\(173\) 1.04480e36i 0.123388i 0.998095 + 0.0616942i \(0.0196503\pi\)
−0.998095 + 0.0616942i \(0.980350\pi\)
\(174\) −6.85354e35 −0.0735957
\(175\) 0 0
\(176\) 6.85793e36 0.609866
\(177\) − 1.50942e37i − 1.22251i
\(178\) 7.45372e35i 0.0550101i
\(179\) −6.89417e36 −0.463881 −0.231941 0.972730i \(-0.574508\pi\)
−0.231941 + 0.972730i \(0.574508\pi\)
\(180\) 0 0
\(181\) 2.73403e36 0.153146 0.0765732 0.997064i \(-0.475602\pi\)
0.0765732 + 0.997064i \(0.475602\pi\)
\(182\) − 3.92898e35i − 0.0200957i
\(183\) − 5.03143e37i − 2.35098i
\(184\) 5.25836e36 0.224572
\(185\) 0 0
\(186\) −4.78839e36 −0.171091
\(187\) 2.43490e37i 0.796332i
\(188\) − 1.63775e37i − 0.490505i
\(189\) 3.85355e37 1.05741
\(190\) 0 0
\(191\) −7.33764e37 −1.69241 −0.846206 0.532856i \(-0.821119\pi\)
−0.846206 + 0.532856i \(0.821119\pi\)
\(192\) − 7.72357e37i − 1.63436i
\(193\) − 3.39286e37i − 0.658978i −0.944159 0.329489i \(-0.893124\pi\)
0.944159 0.329489i \(-0.106876\pi\)
\(194\) −1.63767e36 −0.0292078
\(195\) 0 0
\(196\) 2.40413e37 0.362020
\(197\) − 1.22669e38i − 1.69841i −0.528063 0.849205i \(-0.677081\pi\)
0.528063 0.849205i \(-0.322919\pi\)
\(198\) 5.27189e36i 0.0671415i
\(199\) 4.90395e37 0.574738 0.287369 0.957820i \(-0.407219\pi\)
0.287369 + 0.957820i \(0.407219\pi\)
\(200\) 0 0
\(201\) 1.66111e38 1.65069
\(202\) 1.19026e37i 0.108979i
\(203\) − 6.85635e37i − 0.578640i
\(204\) 2.76288e38 2.15013
\(205\) 0 0
\(206\) 5.17573e36 0.0342896
\(207\) − 5.43071e38i − 3.32159i
\(208\) 7.26137e37i 0.410179i
\(209\) 1.23225e38 0.643110
\(210\) 0 0
\(211\) −2.26597e38 −1.01063 −0.505316 0.862934i \(-0.668624\pi\)
−0.505316 + 0.862934i \(0.668624\pi\)
\(212\) 3.70867e38i 1.52995i
\(213\) 5.11972e38i 1.95426i
\(214\) 1.74835e37 0.0617733
\(215\) 0 0
\(216\) 5.30111e37 0.160650
\(217\) − 4.79036e38i − 1.34519i
\(218\) − 2.43749e37i − 0.0634469i
\(219\) 1.07667e39 2.59870
\(220\) 0 0
\(221\) −2.57814e38 −0.535591
\(222\) − 5.26200e37i − 0.101467i
\(223\) − 7.70736e38i − 1.37999i −0.723815 0.689994i \(-0.757613\pi\)
0.723815 0.689994i \(-0.242387\pi\)
\(224\) −8.69719e37 −0.144639
\(225\) 0 0
\(226\) 9.05221e36 0.0130007
\(227\) − 3.75970e38i − 0.502026i −0.967984 0.251013i \(-0.919236\pi\)
0.967984 0.251013i \(-0.0807637\pi\)
\(228\) − 1.39824e39i − 1.73642i
\(229\) −8.87055e38 −1.02486 −0.512431 0.858729i \(-0.671255\pi\)
−0.512431 + 0.858729i \(0.671255\pi\)
\(230\) 0 0
\(231\) −8.21557e38 −0.822318
\(232\) − 9.43189e37i − 0.0879118i
\(233\) − 9.06227e38i − 0.786800i −0.919367 0.393400i \(-0.871299\pi\)
0.919367 0.393400i \(-0.128701\pi\)
\(234\) −5.58203e37 −0.0451575
\(235\) 0 0
\(236\) 1.03672e39 0.728810
\(237\) − 2.19993e39i − 1.44232i
\(238\) − 1.02296e38i − 0.0625661i
\(239\) −2.67931e39 −1.52918 −0.764589 0.644518i \(-0.777058\pi\)
−0.764589 + 0.644518i \(0.777058\pi\)
\(240\) 0 0
\(241\) 4.69772e38 0.233672 0.116836 0.993151i \(-0.462725\pi\)
0.116836 + 0.993151i \(0.462725\pi\)
\(242\) 8.10022e37i 0.0376308i
\(243\) 1.42932e39i 0.620335i
\(244\) 3.45575e39 1.40156
\(245\) 0 0
\(246\) 2.06348e37 0.00731425
\(247\) 1.30474e39i 0.432538i
\(248\) − 6.58982e38i − 0.204372i
\(249\) −8.01974e38 −0.232740
\(250\) 0 0
\(251\) −1.31521e38 −0.0334486 −0.0167243 0.999860i \(-0.505324\pi\)
−0.0167243 + 0.999860i \(0.505324\pi\)
\(252\) 5.98448e39i 1.42534i
\(253\) 5.12057e39i 1.14243i
\(254\) 3.74645e38 0.0783183
\(255\) 0 0
\(256\) 5.24487e39 0.963330
\(257\) − 4.96495e39i − 0.855104i −0.903991 0.427552i \(-0.859376\pi\)
0.903991 0.427552i \(-0.140624\pi\)
\(258\) 7.07356e38i 0.114265i
\(259\) 5.26416e39 0.797779
\(260\) 0 0
\(261\) −9.74103e39 −1.30028
\(262\) 2.08094e38i 0.0260789i
\(263\) 6.41705e39i 0.755206i 0.925968 + 0.377603i \(0.123252\pi\)
−0.925968 + 0.377603i \(0.876748\pi\)
\(264\) −1.13017e39 −0.124933
\(265\) 0 0
\(266\) −5.17699e38 −0.0505278
\(267\) 1.65027e40i 1.51398i
\(268\) 1.14091e40i 0.984071i
\(269\) 1.97544e39 0.160233 0.0801164 0.996786i \(-0.474471\pi\)
0.0801164 + 0.996786i \(0.474471\pi\)
\(270\) 0 0
\(271\) −1.48531e40 −1.06616 −0.533082 0.846063i \(-0.678966\pi\)
−0.533082 + 0.846063i \(0.678966\pi\)
\(272\) 1.89059e40i 1.27706i
\(273\) − 8.69887e39i − 0.553069i
\(274\) 9.91858e37 0.00593697
\(275\) 0 0
\(276\) 5.81032e40 3.08461
\(277\) − 6.46494e39i − 0.323332i −0.986846 0.161666i \(-0.948313\pi\)
0.986846 0.161666i \(-0.0516866\pi\)
\(278\) 5.65835e38i 0.0266655i
\(279\) −6.80581e40 −3.02281
\(280\) 0 0
\(281\) −1.46554e40 −0.578554 −0.289277 0.957245i \(-0.593415\pi\)
−0.289277 + 0.957245i \(0.593415\pi\)
\(282\) 1.34199e39i 0.0499619i
\(283\) − 2.68962e40i − 0.944529i −0.881457 0.472265i \(-0.843437\pi\)
0.881457 0.472265i \(-0.156563\pi\)
\(284\) −3.51639e40 −1.16505
\(285\) 0 0
\(286\) 5.26325e38 0.0155315
\(287\) 2.06432e39i 0.0575077i
\(288\) 1.23564e40i 0.325023i
\(289\) −2.68705e40 −0.667516
\(290\) 0 0
\(291\) −3.62584e40 −0.803850
\(292\) 7.39493e40i 1.54923i
\(293\) − 4.27715e40i − 0.846912i −0.905917 0.423456i \(-0.860817\pi\)
0.905917 0.423456i \(-0.139183\pi\)
\(294\) −1.96997e39 −0.0368746
\(295\) 0 0
\(296\) 7.24160e39 0.121205
\(297\) 5.16220e40i 0.817248i
\(298\) − 6.44723e39i − 0.0965617i
\(299\) −5.42181e40 −0.768368
\(300\) 0 0
\(301\) −7.07647e40 −0.898399
\(302\) 3.60335e39i 0.0433104i
\(303\) 2.63527e41i 2.99931i
\(304\) 9.56787e40 1.03134
\(305\) 0 0
\(306\) −1.45335e40 −0.140594
\(307\) 5.26223e40i 0.482377i 0.970478 + 0.241189i \(0.0775372\pi\)
−0.970478 + 0.241189i \(0.922463\pi\)
\(308\) − 5.64272e40i − 0.490232i
\(309\) 1.14592e41 0.943711
\(310\) 0 0
\(311\) 3.83170e40 0.283691 0.141845 0.989889i \(-0.454696\pi\)
0.141845 + 0.989889i \(0.454696\pi\)
\(312\) − 1.19665e40i − 0.0840268i
\(313\) − 8.13055e40i − 0.541550i −0.962643 0.270775i \(-0.912720\pi\)
0.962643 0.270775i \(-0.0872799\pi\)
\(314\) −5.69042e39 −0.0359588
\(315\) 0 0
\(316\) 1.51098e41 0.859850
\(317\) − 7.49503e40i − 0.404852i −0.979298 0.202426i \(-0.935118\pi\)
0.979298 0.202426i \(-0.0648825\pi\)
\(318\) − 3.03892e40i − 0.155837i
\(319\) 9.18474e40 0.447219
\(320\) 0 0
\(321\) 3.87089e41 1.70011
\(322\) − 2.15128e40i − 0.0897584i
\(323\) 3.39706e41i 1.34667i
\(324\) 1.11554e41 0.420235
\(325\) 0 0
\(326\) 2.72604e40 0.0927774
\(327\) − 5.39666e41i − 1.74617i
\(328\) 2.83977e39i 0.00873705i
\(329\) −1.34254e41 −0.392821
\(330\) 0 0
\(331\) −5.91449e41 −1.56587 −0.782935 0.622103i \(-0.786278\pi\)
−0.782935 + 0.622103i \(0.786278\pi\)
\(332\) − 5.50822e40i − 0.138750i
\(333\) − 7.47895e41i − 1.79271i
\(334\) −9.88982e39 −0.0225617
\(335\) 0 0
\(336\) −6.37902e41 −1.31873
\(337\) − 1.33169e41i − 0.262126i −0.991374 0.131063i \(-0.958161\pi\)
0.991374 0.131063i \(-0.0418391\pi\)
\(338\) − 2.68225e40i − 0.0502774i
\(339\) 2.00418e41 0.357801
\(340\) 0 0
\(341\) 6.41714e41 1.03967
\(342\) 7.35510e40i 0.113542i
\(343\) − 7.39453e41i − 1.08782i
\(344\) −9.73469e40 −0.136492
\(345\) 0 0
\(346\) 5.88013e39 0.00749257
\(347\) − 1.89271e41i − 0.229957i −0.993368 0.114979i \(-0.963320\pi\)
0.993368 0.114979i \(-0.0366800\pi\)
\(348\) − 1.04219e42i − 1.20751i
\(349\) 5.82757e41 0.643974 0.321987 0.946744i \(-0.395649\pi\)
0.321987 + 0.946744i \(0.395649\pi\)
\(350\) 0 0
\(351\) −5.46588e41 −0.549658
\(352\) − 1.16507e41i − 0.111789i
\(353\) 5.33261e41i 0.488267i 0.969742 + 0.244134i \(0.0785035\pi\)
−0.969742 + 0.244134i \(0.921496\pi\)
\(354\) −8.49500e40 −0.0742351
\(355\) 0 0
\(356\) −1.13346e42 −0.902571
\(357\) − 2.26486e42i − 1.72193i
\(358\) 3.88002e40i 0.0281685i
\(359\) 1.28621e42 0.891772 0.445886 0.895090i \(-0.352889\pi\)
0.445886 + 0.895090i \(0.352889\pi\)
\(360\) 0 0
\(361\) 1.38408e41 0.0875575
\(362\) − 1.53871e40i − 0.00929959i
\(363\) 1.79341e42i 1.03567i
\(364\) 5.97467e41 0.329717
\(365\) 0 0
\(366\) −2.83168e41 −0.142760
\(367\) 1.99600e42i 0.961988i 0.876724 + 0.480994i \(0.159724\pi\)
−0.876724 + 0.480994i \(0.840276\pi\)
\(368\) 3.97589e42i 1.83209i
\(369\) 2.93284e41 0.129227
\(370\) 0 0
\(371\) 3.04017e42 1.22526
\(372\) − 7.28154e42i − 2.80714i
\(373\) − 4.88676e42i − 1.80229i −0.433516 0.901146i \(-0.642727\pi\)
0.433516 0.901146i \(-0.357273\pi\)
\(374\) 1.37035e41 0.0483561
\(375\) 0 0
\(376\) −1.84686e41 −0.0596807
\(377\) 9.72506e41i 0.300788i
\(378\) − 2.16877e41i − 0.0642094i
\(379\) 6.55646e42 1.85833 0.929164 0.369667i \(-0.120528\pi\)
0.929164 + 0.369667i \(0.120528\pi\)
\(380\) 0 0
\(381\) 8.29473e42 2.15546
\(382\) 4.12961e41i 0.102769i
\(383\) − 8.72699e41i − 0.208010i −0.994577 0.104005i \(-0.966834\pi\)
0.994577 0.104005i \(-0.0331658\pi\)
\(384\) −1.76158e42 −0.402196
\(385\) 0 0
\(386\) −1.90949e41 −0.0400155
\(387\) 1.00538e43i 2.01882i
\(388\) − 2.49035e42i − 0.479222i
\(389\) −1.58040e42 −0.291474 −0.145737 0.989323i \(-0.546555\pi\)
−0.145737 + 0.989323i \(0.546555\pi\)
\(390\) 0 0
\(391\) −1.41164e43 −2.39225
\(392\) − 2.71108e41i − 0.0440476i
\(393\) 4.60726e42i 0.717737i
\(394\) −6.90379e41 −0.103133
\(395\) 0 0
\(396\) −8.01678e42 −1.10161
\(397\) 5.01270e42i 0.660736i 0.943852 + 0.330368i \(0.107173\pi\)
−0.943852 + 0.330368i \(0.892827\pi\)
\(398\) − 2.75993e41i − 0.0349001i
\(399\) −1.14620e43 −1.39061
\(400\) 0 0
\(401\) 8.66109e42 0.967588 0.483794 0.875182i \(-0.339258\pi\)
0.483794 + 0.875182i \(0.339258\pi\)
\(402\) − 9.34870e41i − 0.100235i
\(403\) 6.79465e42i 0.699252i
\(404\) −1.80999e43 −1.78806
\(405\) 0 0
\(406\) −3.85874e41 −0.0351371
\(407\) 7.05184e42i 0.616588i
\(408\) − 3.11564e42i − 0.261610i
\(409\) −6.15808e42 −0.496605 −0.248302 0.968683i \(-0.579873\pi\)
−0.248302 + 0.968683i \(0.579873\pi\)
\(410\) 0 0
\(411\) 2.19600e42 0.163396
\(412\) 7.87055e42i 0.562601i
\(413\) − 8.49849e42i − 0.583668i
\(414\) −3.05639e42 −0.201699
\(415\) 0 0
\(416\) 1.23361e42 0.0751862
\(417\) 1.25277e43i 0.733883i
\(418\) − 6.93507e41i − 0.0390519i
\(419\) −1.55362e43 −0.841031 −0.420516 0.907285i \(-0.638151\pi\)
−0.420516 + 0.907285i \(0.638151\pi\)
\(420\) 0 0
\(421\) 1.08104e43 0.540987 0.270494 0.962722i \(-0.412813\pi\)
0.270494 + 0.962722i \(0.412813\pi\)
\(422\) 1.27528e42i 0.0613691i
\(423\) 1.90739e43i 0.882720i
\(424\) 4.18218e42 0.186151
\(425\) 0 0
\(426\) 2.88136e42 0.118670
\(427\) − 2.83284e43i − 1.12244i
\(428\) 2.65865e43i 1.01354i
\(429\) 1.16530e43 0.427456
\(430\) 0 0
\(431\) −4.98547e42 −0.169368 −0.0846840 0.996408i \(-0.526988\pi\)
−0.0846840 + 0.996408i \(0.526988\pi\)
\(432\) 4.00822e43i 1.31060i
\(433\) − 1.01098e43i − 0.318194i −0.987263 0.159097i \(-0.949142\pi\)
0.987263 0.159097i \(-0.0508583\pi\)
\(434\) −2.69600e42 −0.0816845
\(435\) 0 0
\(436\) 3.70661e43 1.04100
\(437\) 7.14399e43i 1.93195i
\(438\) − 6.05948e42i − 0.157802i
\(439\) 4.36398e43 1.09451 0.547253 0.836967i \(-0.315674\pi\)
0.547253 + 0.836967i \(0.315674\pi\)
\(440\) 0 0
\(441\) −2.79994e43 −0.651496
\(442\) 1.45097e42i 0.0325230i
\(443\) 1.22624e43i 0.264798i 0.991196 + 0.132399i \(0.0422681\pi\)
−0.991196 + 0.132399i \(0.957732\pi\)
\(444\) 8.00174e43 1.66481
\(445\) 0 0
\(446\) −4.33768e42 −0.0837978
\(447\) − 1.42743e44i − 2.65755i
\(448\) − 4.34859e43i − 0.780299i
\(449\) −8.39293e43 −1.45161 −0.725803 0.687903i \(-0.758532\pi\)
−0.725803 + 0.687903i \(0.758532\pi\)
\(450\) 0 0
\(451\) −2.76536e42 −0.0444466
\(452\) 1.37654e43i 0.213306i
\(453\) 7.97791e43i 1.19198i
\(454\) −2.11595e42 −0.0304848
\(455\) 0 0
\(456\) −1.57676e43 −0.211274
\(457\) − 9.02636e42i − 0.116653i −0.998298 0.0583264i \(-0.981424\pi\)
0.998298 0.0583264i \(-0.0185764\pi\)
\(458\) 4.99232e42i 0.0622332i
\(459\) −1.42311e44 −1.71131
\(460\) 0 0
\(461\) −3.52096e42 −0.0394091 −0.0197045 0.999806i \(-0.506273\pi\)
−0.0197045 + 0.999806i \(0.506273\pi\)
\(462\) 4.62370e42i 0.0499341i
\(463\) 2.80824e42i 0.0292650i 0.999893 + 0.0146325i \(0.00465783\pi\)
−0.999893 + 0.0146325i \(0.995342\pi\)
\(464\) 7.13154e43 0.717194
\(465\) 0 0
\(466\) −5.10022e42 −0.0477773
\(467\) 1.17834e44i 1.06547i 0.846282 + 0.532736i \(0.178836\pi\)
−0.846282 + 0.532736i \(0.821164\pi\)
\(468\) − 8.48839e43i − 0.740916i
\(469\) 9.35254e43 0.788094
\(470\) 0 0
\(471\) −1.25987e44 −0.989650
\(472\) − 1.16909e43i − 0.0886756i
\(473\) − 9.47961e43i − 0.694355i
\(474\) −1.23811e43 −0.0875826
\(475\) 0 0
\(476\) 1.55558e44 1.02654
\(477\) − 4.31925e44i − 2.75331i
\(478\) 1.50791e43i 0.0928571i
\(479\) −2.89874e43 −0.172454 −0.0862271 0.996276i \(-0.527481\pi\)
−0.0862271 + 0.996276i \(0.527481\pi\)
\(480\) 0 0
\(481\) −7.46669e43 −0.414700
\(482\) − 2.64386e42i − 0.0141894i
\(483\) − 4.76299e44i − 2.47031i
\(484\) −1.23177e44 −0.617422
\(485\) 0 0
\(486\) 8.04418e42 0.0376689
\(487\) − 2.41311e44i − 1.09232i −0.837682 0.546158i \(-0.816090\pi\)
0.837682 0.546158i \(-0.183910\pi\)
\(488\) − 3.89698e43i − 0.170530i
\(489\) 6.03552e44 2.55340
\(490\) 0 0
\(491\) −2.36480e44 −0.935297 −0.467648 0.883915i \(-0.654899\pi\)
−0.467648 + 0.883915i \(0.654899\pi\)
\(492\) 3.13785e43i 0.120008i
\(493\) 2.53204e44i 0.936476i
\(494\) 7.34305e42 0.0262653
\(495\) 0 0
\(496\) 4.98262e44 1.66729
\(497\) 2.88255e44i 0.933031i
\(498\) 4.51349e42i 0.0141328i
\(499\) 4.17695e44 1.26532 0.632659 0.774430i \(-0.281963\pi\)
0.632659 + 0.774430i \(0.281963\pi\)
\(500\) 0 0
\(501\) −2.18963e44 −0.620938
\(502\) 7.40196e41i 0.00203112i
\(503\) 1.63646e44i 0.434543i 0.976111 + 0.217271i \(0.0697157\pi\)
−0.976111 + 0.217271i \(0.930284\pi\)
\(504\) 6.74856e43 0.173423
\(505\) 0 0
\(506\) 2.88184e43 0.0693724
\(507\) − 5.93856e44i − 1.38373i
\(508\) 5.69709e44i 1.28500i
\(509\) 9.46414e43 0.206651 0.103326 0.994648i \(-0.467052\pi\)
0.103326 + 0.994648i \(0.467052\pi\)
\(510\) 0 0
\(511\) 6.06197e44 1.24071
\(512\) − 1.50957e44i − 0.299157i
\(513\) 7.20207e44i 1.38204i
\(514\) −2.79426e43 −0.0519249
\(515\) 0 0
\(516\) −1.07565e45 −1.87479
\(517\) − 1.79846e44i − 0.303604i
\(518\) − 2.96265e43i − 0.0484440i
\(519\) 1.30187e44 0.206209
\(520\) 0 0
\(521\) 7.82192e44 1.16276 0.581381 0.813631i \(-0.302512\pi\)
0.581381 + 0.813631i \(0.302512\pi\)
\(522\) 5.48222e43i 0.0789575i
\(523\) − 8.02848e44i − 1.12035i −0.828373 0.560177i \(-0.810733\pi\)
0.828373 0.560177i \(-0.189267\pi\)
\(524\) −3.16442e44 −0.427885
\(525\) 0 0
\(526\) 3.61150e43 0.0458588
\(527\) 1.76907e45i 2.17706i
\(528\) − 8.54530e44i − 1.01922i
\(529\) −2.10365e45 −2.43196
\(530\) 0 0
\(531\) −1.20741e45 −1.31158
\(532\) − 7.87246e44i − 0.829027i
\(533\) − 2.92804e43i − 0.0298936i
\(534\) 9.28768e43 0.0919341
\(535\) 0 0
\(536\) 1.28658e44 0.119734
\(537\) 8.59046e44i 0.775248i
\(538\) − 1.11177e43i − 0.00972990i
\(539\) 2.64004e44 0.224076
\(540\) 0 0
\(541\) −2.90457e44 −0.231914 −0.115957 0.993254i \(-0.536994\pi\)
−0.115957 + 0.993254i \(0.536994\pi\)
\(542\) 8.35929e43i 0.0647413i
\(543\) − 3.40673e44i − 0.255941i
\(544\) 3.21186e44 0.234086
\(545\) 0 0
\(546\) −4.89570e43 −0.0335843
\(547\) 2.63690e45i 1.75511i 0.479480 + 0.877553i \(0.340825\pi\)
−0.479480 + 0.877553i \(0.659175\pi\)
\(548\) 1.50828e44i 0.0974100i
\(549\) −4.02470e45 −2.52226
\(550\) 0 0
\(551\) 1.28141e45 0.756289
\(552\) − 6.55217e44i − 0.375310i
\(553\) − 1.23862e45i − 0.688611i
\(554\) −3.63845e43 −0.0196338
\(555\) 0 0
\(556\) −8.60445e44 −0.437511
\(557\) 1.37430e45i 0.678378i 0.940718 + 0.339189i \(0.110153\pi\)
−0.940718 + 0.339189i \(0.889847\pi\)
\(558\) 3.83029e44i 0.183555i
\(559\) 1.00373e45 0.467004
\(560\) 0 0
\(561\) 3.03400e45 1.33085
\(562\) 8.24803e43i 0.0351319i
\(563\) 1.83620e45i 0.759505i 0.925088 + 0.379753i \(0.123991\pi\)
−0.925088 + 0.379753i \(0.876009\pi\)
\(564\) −2.04072e45 −0.819743
\(565\) 0 0
\(566\) −1.51371e44 −0.0573552
\(567\) − 9.14459e44i − 0.336546i
\(568\) 3.96535e44i 0.141754i
\(569\) 5.13991e45 1.78485 0.892427 0.451192i \(-0.149001\pi\)
0.892427 + 0.451192i \(0.149001\pi\)
\(570\) 0 0
\(571\) 5.28676e45 1.73258 0.866291 0.499540i \(-0.166498\pi\)
0.866291 + 0.499540i \(0.166498\pi\)
\(572\) 8.00364e44i 0.254832i
\(573\) 9.14305e45i 2.82839i
\(574\) 1.16180e43 0.00349207
\(575\) 0 0
\(576\) −6.17817e45 −1.75343
\(577\) 1.37030e45i 0.377934i 0.981983 + 0.188967i \(0.0605139\pi\)
−0.981983 + 0.188967i \(0.939486\pi\)
\(578\) 1.51227e44i 0.0405340i
\(579\) −4.22767e45 −1.10130
\(580\) 0 0
\(581\) −4.51534e44 −0.111118
\(582\) 2.04062e44i 0.0488126i
\(583\) 4.07259e45i 0.946978i
\(584\) 8.33910e44 0.188498
\(585\) 0 0
\(586\) −2.40717e44 −0.0514275
\(587\) − 1.82654e45i − 0.379402i −0.981842 0.189701i \(-0.939248\pi\)
0.981842 0.189701i \(-0.0607518\pi\)
\(588\) − 2.99566e45i − 0.605015i
\(589\) 8.95290e45 1.75817
\(590\) 0 0
\(591\) −1.52852e46 −2.83842
\(592\) 5.47544e45i 0.988805i
\(593\) − 9.10503e45i − 1.59911i −0.600592 0.799556i \(-0.705068\pi\)
0.600592 0.799556i \(-0.294932\pi\)
\(594\) 2.90527e44 0.0496262
\(595\) 0 0
\(596\) 9.80408e45 1.58432
\(597\) − 6.11055e45i − 0.960514i
\(598\) 3.05138e44i 0.0466580i
\(599\) −1.51052e45 −0.224690 −0.112345 0.993669i \(-0.535836\pi\)
−0.112345 + 0.993669i \(0.535836\pi\)
\(600\) 0 0
\(601\) 1.17313e46 1.65165 0.825825 0.563927i \(-0.190710\pi\)
0.825825 + 0.563927i \(0.190710\pi\)
\(602\) 3.98262e44i 0.0545540i
\(603\) − 1.32874e46i − 1.77095i
\(604\) −5.47949e45 −0.710609
\(605\) 0 0
\(606\) 1.48312e45 0.182129
\(607\) 9.39432e45i 1.12267i 0.827589 + 0.561335i \(0.189712\pi\)
−0.827589 + 0.561335i \(0.810288\pi\)
\(608\) − 1.62546e45i − 0.189045i
\(609\) −8.54334e45 −0.967035
\(610\) 0 0
\(611\) 1.90426e45 0.204196
\(612\) − 2.21006e46i − 2.30678i
\(613\) − 9.71928e45i − 0.987498i −0.869604 0.493749i \(-0.835626\pi\)
0.869604 0.493749i \(-0.164374\pi\)
\(614\) 2.96157e44 0.0292916
\(615\) 0 0
\(616\) −6.36317e44 −0.0596474
\(617\) − 1.24293e46i − 1.13433i −0.823603 0.567166i \(-0.808040\pi\)
0.823603 0.567166i \(-0.191960\pi\)
\(618\) − 6.44920e44i − 0.0573054i
\(619\) 1.07266e46 0.928043 0.464021 0.885824i \(-0.346406\pi\)
0.464021 + 0.885824i \(0.346406\pi\)
\(620\) 0 0
\(621\) −2.99279e46 −2.45508
\(622\) − 2.15647e44i − 0.0172267i
\(623\) 9.29149e45i 0.722824i
\(624\) 9.04801e45 0.685499
\(625\) 0 0
\(626\) −4.57585e44 −0.0328848
\(627\) − 1.53544e46i − 1.07478i
\(628\) − 8.65322e45i − 0.589988i
\(629\) −1.94405e46 −1.29113
\(630\) 0 0
\(631\) −1.79249e46 −1.12972 −0.564862 0.825186i \(-0.691071\pi\)
−0.564862 + 0.825186i \(0.691071\pi\)
\(632\) − 1.70390e45i − 0.104620i
\(633\) 2.82350e46i 1.68899i
\(634\) −4.21818e44 −0.0245840
\(635\) 0 0
\(636\) 4.62117e46 2.55688
\(637\) 2.79535e45i 0.150708i
\(638\) − 5.16915e44i − 0.0271567i
\(639\) 4.09532e46 2.09664
\(640\) 0 0
\(641\) 2.31165e46 1.12400 0.561998 0.827138i \(-0.310033\pi\)
0.561998 + 0.827138i \(0.310033\pi\)
\(642\) − 2.17852e45i − 0.103237i
\(643\) − 3.13717e45i − 0.144896i −0.997372 0.0724480i \(-0.976919\pi\)
0.997372 0.0724480i \(-0.0230811\pi\)
\(644\) 3.27138e46 1.47270
\(645\) 0 0
\(646\) 1.91185e45 0.0817746
\(647\) 1.23833e46i 0.516318i 0.966102 + 0.258159i \(0.0831158\pi\)
−0.966102 + 0.258159i \(0.916884\pi\)
\(648\) − 1.25797e45i − 0.0511308i
\(649\) 1.13845e46 0.451105
\(650\) 0 0
\(651\) −5.96901e46 −2.24810
\(652\) 4.14540e46i 1.52223i
\(653\) − 9.15492e45i − 0.327784i −0.986478 0.163892i \(-0.947595\pi\)
0.986478 0.163892i \(-0.0524048\pi\)
\(654\) −3.03723e45 −0.106034
\(655\) 0 0
\(656\) −2.14718e45 −0.0712778
\(657\) − 8.61242e46i − 2.78802i
\(658\) 7.55578e44i 0.0238535i
\(659\) 4.32900e46 1.33284 0.666420 0.745577i \(-0.267826\pi\)
0.666420 + 0.745577i \(0.267826\pi\)
\(660\) 0 0
\(661\) 1.40663e46 0.411961 0.205980 0.978556i \(-0.433962\pi\)
0.205980 + 0.978556i \(0.433962\pi\)
\(662\) 3.32866e45i 0.0950852i
\(663\) 3.21248e46i 0.895091i
\(664\) −6.21150e44 −0.0168820
\(665\) 0 0
\(666\) −4.20913e45 −0.108860
\(667\) 5.32487e46i 1.34348i
\(668\) − 1.50391e46i − 0.370178i
\(669\) −9.60373e46 −2.30626
\(670\) 0 0
\(671\) 3.79486e46 0.867509
\(672\) 1.08371e46i 0.241724i
\(673\) 2.27399e46i 0.494926i 0.968897 + 0.247463i \(0.0795969\pi\)
−0.968897 + 0.247463i \(0.920403\pi\)
\(674\) −7.49473e44 −0.0159172
\(675\) 0 0
\(676\) 4.07880e46 0.824920
\(677\) − 4.02008e46i − 0.793453i −0.917937 0.396727i \(-0.870146\pi\)
0.917937 0.396727i \(-0.129854\pi\)
\(678\) − 1.12795e45i − 0.0217270i
\(679\) −2.04145e46 −0.383785
\(680\) 0 0
\(681\) −4.68476e46 −0.838996
\(682\) − 3.61155e45i − 0.0631323i
\(683\) − 8.62730e46i − 1.47209i −0.676935 0.736043i \(-0.736692\pi\)
0.676935 0.736043i \(-0.263308\pi\)
\(684\) −1.11846e47 −1.86293
\(685\) 0 0
\(686\) −4.16162e45 −0.0660563
\(687\) 1.10531e47i 1.71277i
\(688\) − 7.36049e46i − 1.11352i
\(689\) −4.31217e46 −0.636911
\(690\) 0 0
\(691\) 2.76452e46 0.389252 0.194626 0.980877i \(-0.437651\pi\)
0.194626 + 0.980877i \(0.437651\pi\)
\(692\) 8.94170e45i 0.122933i
\(693\) 6.57172e46i 0.882228i
\(694\) −1.06521e45 −0.0139638
\(695\) 0 0
\(696\) −1.17526e46 −0.146920
\(697\) − 7.62351e45i − 0.0930710i
\(698\) − 3.27974e45i − 0.0391043i
\(699\) −1.12920e47 −1.31492
\(700\) 0 0
\(701\) 5.83426e45 0.0648094 0.0324047 0.999475i \(-0.489683\pi\)
0.0324047 + 0.999475i \(0.489683\pi\)
\(702\) 3.07618e45i 0.0333772i
\(703\) 9.83841e46i 1.04271i
\(704\) 5.82535e46 0.603077
\(705\) 0 0
\(706\) 3.00118e45 0.0296493
\(707\) 1.48373e47i 1.43197i
\(708\) − 1.29180e47i − 1.21800i
\(709\) 3.21473e46 0.296130 0.148065 0.988978i \(-0.452696\pi\)
0.148065 + 0.988978i \(0.452696\pi\)
\(710\) 0 0
\(711\) −1.75975e47 −1.54740
\(712\) 1.27818e46i 0.109817i
\(713\) 3.72035e47i 3.12325i
\(714\) −1.27466e46 −0.104562
\(715\) 0 0
\(716\) −5.90021e46 −0.462171
\(717\) 3.33855e47i 2.55559i
\(718\) − 7.23876e45i − 0.0541515i
\(719\) 5.61687e46 0.410646 0.205323 0.978694i \(-0.434176\pi\)
0.205323 + 0.978694i \(0.434176\pi\)
\(720\) 0 0
\(721\) 6.45185e46 0.450560
\(722\) − 7.78958e44i − 0.00531680i
\(723\) − 5.85358e46i − 0.390517i
\(724\) 2.33986e46 0.152582
\(725\) 0 0
\(726\) 1.00933e46 0.0628894
\(727\) − 2.29582e47i − 1.39836i −0.714944 0.699181i \(-0.753548\pi\)
0.714944 0.699181i \(-0.246452\pi\)
\(728\) − 6.73750e45i − 0.0401172i
\(729\) 2.50561e47 1.45851
\(730\) 0 0
\(731\) 2.61333e47 1.45398
\(732\) − 4.30603e47i − 2.34231i
\(733\) 1.21401e46i 0.0645662i 0.999479 + 0.0322831i \(0.0102778\pi\)
−0.999479 + 0.0322831i \(0.989722\pi\)
\(734\) 1.12334e46 0.0584153
\(735\) 0 0
\(736\) 6.75452e46 0.335823
\(737\) 1.25286e47i 0.609102i
\(738\) − 1.65060e45i − 0.00784713i
\(739\) −3.32044e47 −1.54370 −0.771848 0.635807i \(-0.780667\pi\)
−0.771848 + 0.635807i \(0.780667\pi\)
\(740\) 0 0
\(741\) 1.62577e47 0.722867
\(742\) − 1.71100e46i − 0.0744020i
\(743\) − 1.84054e47i − 0.782762i −0.920229 0.391381i \(-0.871997\pi\)
0.920229 0.391381i \(-0.128003\pi\)
\(744\) −8.21123e46 −0.341550
\(745\) 0 0
\(746\) −2.75025e46 −0.109442
\(747\) 6.41508e46i 0.249696i
\(748\) 2.08385e47i 0.793396i
\(749\) 2.17942e47 0.811691
\(750\) 0 0
\(751\) −7.67452e45 −0.0273522 −0.0136761 0.999906i \(-0.504353\pi\)
−0.0136761 + 0.999906i \(0.504353\pi\)
\(752\) − 1.39642e47i − 0.486881i
\(753\) 1.63881e46i 0.0559000i
\(754\) 5.47324e45 0.0182649
\(755\) 0 0
\(756\) 3.29797e47 1.05351
\(757\) 1.14623e47i 0.358254i 0.983826 + 0.179127i \(0.0573273\pi\)
−0.983826 + 0.179127i \(0.942673\pi\)
\(758\) − 3.68996e46i − 0.112844i
\(759\) 6.38047e47 1.90925
\(760\) 0 0
\(761\) −6.74831e47 −1.93352 −0.966759 0.255690i \(-0.917697\pi\)
−0.966759 + 0.255690i \(0.917697\pi\)
\(762\) − 4.66825e46i − 0.130887i
\(763\) − 3.03847e47i − 0.833682i
\(764\) −6.27975e47 −1.68617
\(765\) 0 0
\(766\) −4.91152e45 −0.0126311
\(767\) 1.20543e47i 0.303401i
\(768\) − 6.53535e47i − 1.60994i
\(769\) −1.58437e47 −0.382007 −0.191004 0.981589i \(-0.561174\pi\)
−0.191004 + 0.981589i \(0.561174\pi\)
\(770\) 0 0
\(771\) −6.18657e47 −1.42907
\(772\) − 2.90370e47i − 0.656548i
\(773\) − 1.15351e47i − 0.255305i −0.991819 0.127652i \(-0.959256\pi\)
0.991819 0.127652i \(-0.0407442\pi\)
\(774\) 5.65822e46 0.122590
\(775\) 0 0
\(776\) −2.80831e46 −0.0583078
\(777\) − 6.55939e47i − 1.33327i
\(778\) 8.89447e45i 0.0176993i
\(779\) −3.85810e46 −0.0751632
\(780\) 0 0
\(781\) −3.86145e47 −0.721121
\(782\) 7.94464e46i 0.145266i
\(783\) 5.36816e47i 0.961072i
\(784\) 2.04987e47 0.359345
\(785\) 0 0
\(786\) 2.59295e46 0.0435835
\(787\) − 7.25057e47i − 1.19341i −0.802461 0.596704i \(-0.796477\pi\)
0.802461 0.596704i \(-0.203523\pi\)
\(788\) − 1.04983e48i − 1.69215i
\(789\) 7.99595e47 1.26212
\(790\) 0 0
\(791\) 1.12841e47 0.170827
\(792\) 9.04034e46i 0.134035i
\(793\) 4.01810e47i 0.583463i
\(794\) 2.82113e46 0.0401222
\(795\) 0 0
\(796\) 4.19693e47 0.572619
\(797\) 5.72072e47i 0.764519i 0.924055 + 0.382260i \(0.124854\pi\)
−0.924055 + 0.382260i \(0.875146\pi\)
\(798\) 6.45077e46i 0.0844430i
\(799\) 4.95798e47 0.635745
\(800\) 0 0
\(801\) 1.32007e48 1.62428
\(802\) − 4.87444e46i − 0.0587553i
\(803\) 8.12058e47i 0.958916i
\(804\) 1.42162e48 1.64460
\(805\) 0 0
\(806\) 3.82401e46 0.0424611
\(807\) − 2.46149e47i − 0.267784i
\(808\) 2.04108e47i 0.217557i
\(809\) −1.25233e48 −1.30788 −0.653939 0.756547i \(-0.726885\pi\)
−0.653939 + 0.756547i \(0.726885\pi\)
\(810\) 0 0
\(811\) −1.43044e48 −1.43425 −0.717126 0.696944i \(-0.754543\pi\)
−0.717126 + 0.696944i \(0.754543\pi\)
\(812\) − 5.86785e47i − 0.576506i
\(813\) 1.85077e48i 1.78180i
\(814\) 3.96876e46 0.0374414
\(815\) 0 0
\(816\) 2.35576e48 2.13424
\(817\) − 1.32255e48i − 1.17422i
\(818\) 3.46575e46i 0.0301556i
\(819\) −6.95832e47 −0.593363
\(820\) 0 0
\(821\) −7.52133e47 −0.616074 −0.308037 0.951374i \(-0.599672\pi\)
−0.308037 + 0.951374i \(0.599672\pi\)
\(822\) − 1.23590e46i − 0.00992199i
\(823\) − 2.11125e47i − 0.166128i −0.996544 0.0830638i \(-0.973529\pi\)
0.996544 0.0830638i \(-0.0264705\pi\)
\(824\) 8.87544e46 0.0684527
\(825\) 0 0
\(826\) −4.78292e46 −0.0354424
\(827\) 1.64488e47i 0.119480i 0.998214 + 0.0597399i \(0.0190271\pi\)
−0.998214 + 0.0597399i \(0.980973\pi\)
\(828\) − 4.64774e48i − 3.30934i
\(829\) 2.28060e48 1.59184 0.795919 0.605404i \(-0.206988\pi\)
0.795919 + 0.605404i \(0.206988\pi\)
\(830\) 0 0
\(831\) −8.05561e47 −0.540358
\(832\) 6.16804e47i 0.405613i
\(833\) 7.27804e47i 0.469215i
\(834\) 7.05056e46 0.0445640
\(835\) 0 0
\(836\) 1.05459e48 0.640739
\(837\) 3.75059e48i 2.23424i
\(838\) 8.74370e46i 0.0510704i
\(839\) −1.56509e48 −0.896327 −0.448163 0.893952i \(-0.647922\pi\)
−0.448163 + 0.893952i \(0.647922\pi\)
\(840\) 0 0
\(841\) −8.60958e47 −0.474076
\(842\) − 6.08406e46i − 0.0328506i
\(843\) 1.82613e48i 0.966892i
\(844\) −1.93927e48 −1.00691
\(845\) 0 0
\(846\) 1.07347e47 0.0536019
\(847\) 1.00974e48i 0.494463i
\(848\) 3.16218e48i 1.51864i
\(849\) −3.35140e48 −1.57852
\(850\) 0 0
\(851\) −4.08832e48 −1.85228
\(852\) 4.38159e48i 1.94706i
\(853\) − 1.60296e48i − 0.698657i −0.937000 0.349329i \(-0.886410\pi\)
0.937000 0.349329i \(-0.113590\pi\)
\(854\) −1.59431e47 −0.0681584
\(855\) 0 0
\(856\) 2.99810e47 0.123319
\(857\) − 3.00131e48i − 1.21095i −0.795864 0.605476i \(-0.792983\pi\)
0.795864 0.605476i \(-0.207017\pi\)
\(858\) − 6.55826e46i − 0.0259566i
\(859\) −4.48822e48 −1.74256 −0.871279 0.490788i \(-0.836709\pi\)
−0.871279 + 0.490788i \(0.836709\pi\)
\(860\) 0 0
\(861\) 2.57224e47 0.0961081
\(862\) 2.80581e46i 0.0102846i
\(863\) 5.23525e47i 0.188260i 0.995560 + 0.0941302i \(0.0300070\pi\)
−0.995560 + 0.0941302i \(0.969993\pi\)
\(864\) 6.80943e47 0.240234
\(865\) 0 0
\(866\) −5.68977e46 −0.0193219
\(867\) 3.34819e48i 1.11557i
\(868\) − 4.09971e48i − 1.34023i
\(869\) 1.65925e48 0.532214
\(870\) 0 0
\(871\) −1.32657e48 −0.409665
\(872\) − 4.17985e47i − 0.126660i
\(873\) 2.90035e48i 0.862414i
\(874\) 4.02062e47 0.117315
\(875\) 0 0
\(876\) 9.21444e48 2.58911
\(877\) 1.66421e48i 0.458898i 0.973321 + 0.229449i \(0.0736924\pi\)
−0.973321 + 0.229449i \(0.926308\pi\)
\(878\) − 2.45603e47i − 0.0664622i
\(879\) −5.32952e48 −1.41538
\(880\) 0 0
\(881\) 1.63041e48 0.417055 0.208528 0.978016i \(-0.433133\pi\)
0.208528 + 0.978016i \(0.433133\pi\)
\(882\) 1.57580e47i 0.0395611i
\(883\) 3.57816e48i 0.881674i 0.897587 + 0.440837i \(0.145318\pi\)
−0.897587 + 0.440837i \(0.854682\pi\)
\(884\) −2.20644e48 −0.533616
\(885\) 0 0
\(886\) 6.90126e46 0.0160795
\(887\) − 3.83241e48i − 0.876461i −0.898863 0.438230i \(-0.855605\pi\)
0.898863 0.438230i \(-0.144395\pi\)
\(888\) − 9.02338e47i − 0.202561i
\(889\) 4.67017e48 1.02909
\(890\) 0 0
\(891\) 1.22501e48 0.260109
\(892\) − 6.59616e48i − 1.37490i
\(893\) − 2.50913e48i − 0.513422i
\(894\) −8.03356e47 −0.161376
\(895\) 0 0
\(896\) −9.91821e47 −0.192022
\(897\) 6.75582e48i 1.28411i
\(898\) 4.72352e47i 0.0881467i
\(899\) 6.67316e48 1.22264
\(900\) 0 0
\(901\) −1.12273e49 −1.98297
\(902\) 1.55634e46i 0.00269895i
\(903\) 8.81761e48i 1.50142i
\(904\) 1.55229e47 0.0259534
\(905\) 0 0
\(906\) 4.48994e47 0.0723812
\(907\) 9.46008e48i 1.49753i 0.662838 + 0.748763i \(0.269352\pi\)
−0.662838 + 0.748763i \(0.730648\pi\)
\(908\) − 3.21765e48i − 0.500175i
\(909\) 2.10798e49 3.21782
\(910\) 0 0
\(911\) 7.92211e48 1.16624 0.583120 0.812386i \(-0.301832\pi\)
0.583120 + 0.812386i \(0.301832\pi\)
\(912\) − 1.19220e49i − 1.72359i
\(913\) − 6.04873e47i − 0.0858809i
\(914\) −5.08001e46 −0.00708357
\(915\) 0 0
\(916\) −7.59165e48 −1.02108
\(917\) 2.59402e48i 0.342672i
\(918\) 8.00923e47i 0.103917i
\(919\) −4.92680e48 −0.627854 −0.313927 0.949447i \(-0.601645\pi\)
−0.313927 + 0.949447i \(0.601645\pi\)
\(920\) 0 0
\(921\) 6.55699e48 0.806158
\(922\) 1.98159e46i 0.00239306i
\(923\) − 4.08861e48i − 0.485006i
\(924\) −7.03110e48 −0.819286
\(925\) 0 0
\(926\) 1.58047e46 0.00177707
\(927\) − 9.16633e48i − 1.01246i
\(928\) − 1.21155e48i − 0.131462i
\(929\) −2.38059e48 −0.253761 −0.126881 0.991918i \(-0.540497\pi\)
−0.126881 + 0.991918i \(0.540497\pi\)
\(930\) 0 0
\(931\) 3.68326e48 0.378934
\(932\) − 7.75573e48i − 0.783899i
\(933\) − 4.77448e48i − 0.474110i
\(934\) 6.63166e47 0.0646992
\(935\) 0 0
\(936\) −9.57217e47 −0.0901486
\(937\) 4.20946e47i 0.0389514i 0.999810 + 0.0194757i \(0.00619970\pi\)
−0.999810 + 0.0194757i \(0.993800\pi\)
\(938\) − 5.26358e47i − 0.0478558i
\(939\) −1.01310e49 −0.905049
\(940\) 0 0
\(941\) −1.57932e49 −1.36221 −0.681103 0.732188i \(-0.738499\pi\)
−0.681103 + 0.732188i \(0.738499\pi\)
\(942\) 7.09053e47i 0.0600950i
\(943\) − 1.60322e48i − 0.133521i
\(944\) 8.83958e48 0.723425
\(945\) 0 0
\(946\) −5.33510e47 −0.0421637
\(947\) − 1.67820e49i − 1.30337i −0.758490 0.651685i \(-0.774062\pi\)
0.758490 0.651685i \(-0.225938\pi\)
\(948\) − 1.88276e49i − 1.43700i
\(949\) −8.59830e48 −0.644941
\(950\) 0 0
\(951\) −9.33916e48 −0.676596
\(952\) − 1.75419e48i − 0.124902i
\(953\) 2.07777e49i 1.45400i 0.686639 + 0.726999i \(0.259085\pi\)
−0.686639 + 0.726999i \(0.740915\pi\)
\(954\) −2.43086e48 −0.167191
\(955\) 0 0
\(956\) −2.29303e49 −1.52354
\(957\) − 1.14446e49i − 0.747402i
\(958\) 1.63141e47i 0.0104720i
\(959\) 1.23641e48 0.0780109
\(960\) 0 0
\(961\) 3.02202e49 1.84231
\(962\) 4.20223e47i 0.0251821i
\(963\) − 3.09637e49i − 1.82397i
\(964\) 4.02043e48 0.232810
\(965\) 0 0
\(966\) −2.68060e48 −0.150006
\(967\) − 1.42054e48i − 0.0781480i −0.999236 0.0390740i \(-0.987559\pi\)
0.999236 0.0390740i \(-0.0124408\pi\)
\(968\) 1.38904e48i 0.0751229i
\(969\) 4.23289e49 2.25058
\(970\) 0 0
\(971\) 3.67703e49 1.88964 0.944821 0.327588i \(-0.106236\pi\)
0.944821 + 0.327588i \(0.106236\pi\)
\(972\) 1.22325e49i 0.618048i
\(973\) 7.05346e48i 0.350381i
\(974\) −1.35809e48 −0.0663293
\(975\) 0 0
\(976\) 2.94654e49 1.39120
\(977\) 5.91833e48i 0.274751i 0.990519 + 0.137375i \(0.0438667\pi\)
−0.990519 + 0.137375i \(0.956133\pi\)
\(978\) − 3.39678e48i − 0.155051i
\(979\) −1.24468e49 −0.558657
\(980\) 0 0
\(981\) −4.31685e49 −1.87339
\(982\) 1.33090e48i 0.0567945i
\(983\) − 4.20754e49i − 1.76561i −0.469740 0.882805i \(-0.655652\pi\)
0.469740 0.882805i \(-0.344348\pi\)
\(984\) 3.53849e47 0.0146015
\(985\) 0 0
\(986\) 1.42503e48 0.0568661
\(987\) 1.67287e49i 0.656491i
\(988\) 1.11663e49i 0.430943i
\(989\) 5.49582e49 2.08590
\(990\) 0 0
\(991\) −8.36780e48 −0.307182 −0.153591 0.988134i \(-0.549084\pi\)
−0.153591 + 0.988134i \(0.549084\pi\)
\(992\) − 8.46482e48i − 0.305615i
\(993\) 7.36974e49i 2.61691i
\(994\) 1.62229e48 0.0566569
\(995\) 0 0
\(996\) −6.86351e48 −0.231882
\(997\) − 4.11174e49i − 1.36633i −0.730264 0.683166i \(-0.760603\pi\)
0.730264 0.683166i \(-0.239397\pi\)
\(998\) − 2.35077e48i − 0.0768346i
\(999\) −4.12155e49 −1.32504
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.b.24.5 10
5.2 odd 4 5.34.a.a.1.3 5
5.3 odd 4 25.34.a.b.1.3 5
5.4 even 2 inner 25.34.b.b.24.6 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.a.a.1.3 5 5.2 odd 4
25.34.a.b.1.3 5 5.3 odd 4
25.34.b.b.24.5 10 1.1 even 1 trivial
25.34.b.b.24.6 10 5.4 even 2 inner