Properties

Label 175.10.a.d.1.3
Level $175$
Weight $10$
Character 175.1
Self dual yes
Analytic conductor $90.131$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,10,Mod(1,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(90.1312713287\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 426x + 2016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(18.2745\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+34.1627 q^{2} +79.6469 q^{3} +655.088 q^{4} +2720.95 q^{6} -2401.00 q^{7} +4888.28 q^{8} -13339.4 q^{9} +69354.4 q^{11} +52175.7 q^{12} -105959. q^{13} -82024.6 q^{14} -168409. q^{16} -568267. q^{17} -455709. q^{18} -396405. q^{19} -191232. q^{21} +2.36933e6 q^{22} +620765. q^{23} +389336. q^{24} -3.61984e6 q^{26} -2.63013e6 q^{27} -1.57287e6 q^{28} +4.87652e6 q^{29} -1.42482e6 q^{31} -8.25609e6 q^{32} +5.52386e6 q^{33} -1.94135e7 q^{34} -8.73846e6 q^{36} -1.31092e7 q^{37} -1.35423e7 q^{38} -8.43931e6 q^{39} -2.03049e7 q^{41} -6.53300e6 q^{42} +1.11768e7 q^{43} +4.54332e7 q^{44} +2.12070e7 q^{46} +1.99352e7 q^{47} -1.34132e7 q^{48} +5.76480e6 q^{49} -4.52607e7 q^{51} -6.94125e7 q^{52} -5.65007e7 q^{53} -8.98523e7 q^{54} -1.17367e7 q^{56} -3.15725e7 q^{57} +1.66595e8 q^{58} -1.09340e8 q^{59} +3.20008e7 q^{61} -4.86755e7 q^{62} +3.20278e7 q^{63} -1.95825e8 q^{64} +1.88710e8 q^{66} -8.02869e7 q^{67} -3.72265e8 q^{68} +4.94420e7 q^{69} +2.07893e8 q^{71} -6.52065e7 q^{72} +2.70274e8 q^{73} -4.47844e8 q^{74} -2.59680e8 q^{76} -1.66520e8 q^{77} -2.88309e8 q^{78} -5.16196e8 q^{79} +5.30772e7 q^{81} -6.93671e8 q^{82} +6.82693e8 q^{83} -1.25274e8 q^{84} +3.81830e8 q^{86} +3.88400e8 q^{87} +3.39023e8 q^{88} -1.47150e8 q^{89} +2.54408e8 q^{91} +4.06656e8 q^{92} -1.13482e8 q^{93} +6.81040e8 q^{94} -6.57572e8 q^{96} -1.09643e9 q^{97} +1.96941e8 q^{98} -9.25144e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 21 q^{2} - 84 q^{3} + 1557 q^{4} + 4914 q^{6} - 7203 q^{7} - 14055 q^{8} - 26001 q^{9} - 3444 q^{11} + 106386 q^{12} + 19782 q^{13} + 50421 q^{14} + 482961 q^{16} - 1016694 q^{17} + 273267 q^{18}+ \cdots - 1900979172 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 34.1627 1.50979 0.754896 0.655845i \(-0.227688\pi\)
0.754896 + 0.655845i \(0.227688\pi\)
\(3\) 79.6469 0.567706 0.283853 0.958868i \(-0.408387\pi\)
0.283853 + 0.958868i \(0.408387\pi\)
\(4\) 655.088 1.27947
\(5\) 0 0
\(6\) 2720.95 0.857117
\(7\) −2401.00 −0.377964
\(8\) 4888.28 0.421940
\(9\) −13339.4 −0.677710
\(10\) 0 0
\(11\) 69354.4 1.42826 0.714129 0.700014i \(-0.246823\pi\)
0.714129 + 0.700014i \(0.246823\pi\)
\(12\) 52175.7 0.726362
\(13\) −105959. −1.02895 −0.514473 0.857506i \(-0.672013\pi\)
−0.514473 + 0.857506i \(0.672013\pi\)
\(14\) −82024.6 −0.570647
\(15\) 0 0
\(16\) −168409. −0.642428
\(17\) −568267. −1.65018 −0.825092 0.564998i \(-0.808877\pi\)
−0.825092 + 0.564998i \(0.808877\pi\)
\(18\) −455709. −1.02320
\(19\) −396405. −0.697828 −0.348914 0.937155i \(-0.613449\pi\)
−0.348914 + 0.937155i \(0.613449\pi\)
\(20\) 0 0
\(21\) −191232. −0.214573
\(22\) 2.36933e6 2.15637
\(23\) 620765. 0.462543 0.231271 0.972889i \(-0.425712\pi\)
0.231271 + 0.972889i \(0.425712\pi\)
\(24\) 389336. 0.239538
\(25\) 0 0
\(26\) −3.61984e6 −1.55349
\(27\) −2.63013e6 −0.952446
\(28\) −1.57287e6 −0.483594
\(29\) 4.87652e6 1.28032 0.640161 0.768241i \(-0.278868\pi\)
0.640161 + 0.768241i \(0.278868\pi\)
\(30\) 0 0
\(31\) −1.42482e6 −0.277096 −0.138548 0.990356i \(-0.544244\pi\)
−0.138548 + 0.990356i \(0.544244\pi\)
\(32\) −8.25609e6 −1.39187
\(33\) 5.52386e6 0.810830
\(34\) −1.94135e7 −2.49143
\(35\) 0 0
\(36\) −8.73846e6 −0.867109
\(37\) −1.31092e7 −1.14992 −0.574960 0.818182i \(-0.694982\pi\)
−0.574960 + 0.818182i \(0.694982\pi\)
\(38\) −1.35423e7 −1.05357
\(39\) −8.43931e6 −0.584139
\(40\) 0 0
\(41\) −2.03049e7 −1.12221 −0.561105 0.827744i \(-0.689624\pi\)
−0.561105 + 0.827744i \(0.689624\pi\)
\(42\) −6.53300e6 −0.323960
\(43\) 1.11768e7 0.498551 0.249276 0.968433i \(-0.419807\pi\)
0.249276 + 0.968433i \(0.419807\pi\)
\(44\) 4.54332e7 1.82741
\(45\) 0 0
\(46\) 2.12070e7 0.698343
\(47\) 1.99352e7 0.595909 0.297955 0.954580i \(-0.403696\pi\)
0.297955 + 0.954580i \(0.403696\pi\)
\(48\) −1.34132e7 −0.364710
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) −4.52607e7 −0.936819
\(52\) −6.94125e7 −1.31651
\(53\) −5.65007e7 −0.983586 −0.491793 0.870712i \(-0.663658\pi\)
−0.491793 + 0.870712i \(0.663658\pi\)
\(54\) −8.98523e7 −1.43799
\(55\) 0 0
\(56\) −1.17367e7 −0.159478
\(57\) −3.15725e7 −0.396161
\(58\) 1.66595e8 1.93302
\(59\) −1.09340e8 −1.17475 −0.587375 0.809315i \(-0.699839\pi\)
−0.587375 + 0.809315i \(0.699839\pi\)
\(60\) 0 0
\(61\) 3.20008e7 0.295921 0.147961 0.988993i \(-0.452729\pi\)
0.147961 + 0.988993i \(0.452729\pi\)
\(62\) −4.86755e7 −0.418358
\(63\) 3.20278e7 0.256150
\(64\) −1.95825e8 −1.45901
\(65\) 0 0
\(66\) 1.88710e8 1.22418
\(67\) −8.02869e7 −0.486752 −0.243376 0.969932i \(-0.578255\pi\)
−0.243376 + 0.969932i \(0.578255\pi\)
\(68\) −3.72265e8 −2.11136
\(69\) 4.94420e7 0.262588
\(70\) 0 0
\(71\) 2.07893e8 0.970906 0.485453 0.874263i \(-0.338655\pi\)
0.485453 + 0.874263i \(0.338655\pi\)
\(72\) −6.52065e7 −0.285953
\(73\) 2.70274e8 1.11391 0.556957 0.830541i \(-0.311969\pi\)
0.556957 + 0.830541i \(0.311969\pi\)
\(74\) −4.47844e8 −1.73614
\(75\) 0 0
\(76\) −2.59680e8 −0.892849
\(77\) −1.66520e8 −0.539831
\(78\) −2.88309e8 −0.881928
\(79\) −5.16196e8 −1.49105 −0.745526 0.666477i \(-0.767802\pi\)
−0.745526 + 0.666477i \(0.767802\pi\)
\(80\) 0 0
\(81\) 5.30772e7 0.137002
\(82\) −6.93671e8 −1.69430
\(83\) 6.82693e8 1.57897 0.789486 0.613769i \(-0.210347\pi\)
0.789486 + 0.613769i \(0.210347\pi\)
\(84\) −1.25274e8 −0.274539
\(85\) 0 0
\(86\) 3.81830e8 0.752708
\(87\) 3.88400e8 0.726846
\(88\) 3.39023e8 0.602639
\(89\) −1.47150e8 −0.248602 −0.124301 0.992245i \(-0.539669\pi\)
−0.124301 + 0.992245i \(0.539669\pi\)
\(90\) 0 0
\(91\) 2.54408e8 0.388905
\(92\) 4.06656e8 0.591809
\(93\) −1.13482e8 −0.157309
\(94\) 6.81040e8 0.899699
\(95\) 0 0
\(96\) −6.57572e8 −0.790174
\(97\) −1.09643e9 −1.25750 −0.628750 0.777608i \(-0.716433\pi\)
−0.628750 + 0.777608i \(0.716433\pi\)
\(98\) 1.96941e8 0.215684
\(99\) −9.25144e8 −0.967945
\(100\) 0 0
\(101\) 2.08683e8 0.199545 0.0997727 0.995010i \(-0.468188\pi\)
0.0997727 + 0.995010i \(0.468188\pi\)
\(102\) −1.54623e9 −1.41440
\(103\) −6.78194e8 −0.593727 −0.296863 0.954920i \(-0.595941\pi\)
−0.296863 + 0.954920i \(0.595941\pi\)
\(104\) −5.17957e8 −0.434154
\(105\) 0 0
\(106\) −1.93021e9 −1.48501
\(107\) 4.59542e8 0.338921 0.169461 0.985537i \(-0.445797\pi\)
0.169461 + 0.985537i \(0.445797\pi\)
\(108\) −1.72297e9 −1.21862
\(109\) 5.21086e8 0.353582 0.176791 0.984248i \(-0.443428\pi\)
0.176791 + 0.984248i \(0.443428\pi\)
\(110\) 0 0
\(111\) −1.04410e9 −0.652816
\(112\) 4.04349e8 0.242815
\(113\) −4.45612e8 −0.257101 −0.128551 0.991703i \(-0.541032\pi\)
−0.128551 + 0.991703i \(0.541032\pi\)
\(114\) −1.07860e9 −0.598120
\(115\) 0 0
\(116\) 3.19455e9 1.63813
\(117\) 1.41343e9 0.697328
\(118\) −3.73535e9 −1.77363
\(119\) 1.36441e9 0.623711
\(120\) 0 0
\(121\) 2.45208e9 1.03992
\(122\) 1.09323e9 0.446779
\(123\) −1.61723e9 −0.637086
\(124\) −9.33380e8 −0.354536
\(125\) 0 0
\(126\) 1.09416e9 0.386734
\(127\) −9.28626e8 −0.316755 −0.158378 0.987379i \(-0.550626\pi\)
−0.158378 + 0.987379i \(0.550626\pi\)
\(128\) −2.46278e9 −0.810925
\(129\) 8.90198e8 0.283030
\(130\) 0 0
\(131\) 2.57694e9 0.764509 0.382255 0.924057i \(-0.375148\pi\)
0.382255 + 0.924057i \(0.375148\pi\)
\(132\) 3.61862e9 1.03743
\(133\) 9.51769e8 0.263754
\(134\) −2.74281e9 −0.734894
\(135\) 0 0
\(136\) −2.77785e9 −0.696279
\(137\) 4.44116e9 1.07710 0.538548 0.842595i \(-0.318973\pi\)
0.538548 + 0.842595i \(0.318973\pi\)
\(138\) 1.68907e9 0.396453
\(139\) −7.28389e9 −1.65499 −0.827497 0.561470i \(-0.810236\pi\)
−0.827497 + 0.561470i \(0.810236\pi\)
\(140\) 0 0
\(141\) 1.58778e9 0.338301
\(142\) 7.10218e9 1.46587
\(143\) −7.34872e9 −1.46960
\(144\) 2.24647e9 0.435380
\(145\) 0 0
\(146\) 9.23328e9 1.68178
\(147\) 4.59149e8 0.0811008
\(148\) −8.58766e9 −1.47129
\(149\) 4.87355e9 0.810042 0.405021 0.914307i \(-0.367264\pi\)
0.405021 + 0.914307i \(0.367264\pi\)
\(150\) 0 0
\(151\) −8.63776e9 −1.35209 −0.676044 0.736861i \(-0.736307\pi\)
−0.676044 + 0.736861i \(0.736307\pi\)
\(152\) −1.93774e9 −0.294442
\(153\) 7.58033e9 1.11835
\(154\) −5.68876e9 −0.815032
\(155\) 0 0
\(156\) −5.52849e9 −0.747388
\(157\) 9.34170e9 1.22709 0.613546 0.789659i \(-0.289742\pi\)
0.613546 + 0.789659i \(0.289742\pi\)
\(158\) −1.76346e10 −2.25118
\(159\) −4.50010e9 −0.558387
\(160\) 0 0
\(161\) −1.49046e9 −0.174825
\(162\) 1.81326e9 0.206844
\(163\) 2.10680e9 0.233765 0.116883 0.993146i \(-0.462710\pi\)
0.116883 + 0.993146i \(0.462710\pi\)
\(164\) −1.33015e10 −1.43583
\(165\) 0 0
\(166\) 2.33226e10 2.38392
\(167\) −1.39800e10 −1.39086 −0.695431 0.718593i \(-0.744787\pi\)
−0.695431 + 0.718593i \(0.744787\pi\)
\(168\) −9.34796e8 −0.0905367
\(169\) 6.22820e8 0.0587316
\(170\) 0 0
\(171\) 5.28780e9 0.472925
\(172\) 7.32180e9 0.637881
\(173\) 1.24435e10 1.05618 0.528088 0.849190i \(-0.322909\pi\)
0.528088 + 0.849190i \(0.322909\pi\)
\(174\) 1.32688e10 1.09739
\(175\) 0 0
\(176\) −1.16799e10 −0.917553
\(177\) −8.70861e9 −0.666913
\(178\) −5.02704e9 −0.375338
\(179\) 7.30178e9 0.531606 0.265803 0.964027i \(-0.414363\pi\)
0.265803 + 0.964027i \(0.414363\pi\)
\(180\) 0 0
\(181\) −1.27074e10 −0.880038 −0.440019 0.897988i \(-0.645028\pi\)
−0.440019 + 0.897988i \(0.645028\pi\)
\(182\) 8.69125e9 0.587166
\(183\) 2.54876e9 0.167996
\(184\) 3.03447e9 0.195165
\(185\) 0 0
\(186\) −3.87685e9 −0.237504
\(187\) −3.94118e10 −2.35689
\(188\) 1.30593e10 0.762448
\(189\) 6.31494e9 0.359991
\(190\) 0 0
\(191\) 1.61547e10 0.878311 0.439155 0.898411i \(-0.355278\pi\)
0.439155 + 0.898411i \(0.355278\pi\)
\(192\) −1.55968e10 −0.828287
\(193\) 1.52841e10 0.792924 0.396462 0.918051i \(-0.370238\pi\)
0.396462 + 0.918051i \(0.370238\pi\)
\(194\) −3.74570e10 −1.89856
\(195\) 0 0
\(196\) 3.77645e9 0.182781
\(197\) −2.33886e10 −1.10639 −0.553193 0.833053i \(-0.686591\pi\)
−0.553193 + 0.833053i \(0.686591\pi\)
\(198\) −3.16054e10 −1.46139
\(199\) −2.53610e10 −1.14638 −0.573189 0.819423i \(-0.694294\pi\)
−0.573189 + 0.819423i \(0.694294\pi\)
\(200\) 0 0
\(201\) −6.39460e9 −0.276332
\(202\) 7.12918e9 0.301272
\(203\) −1.17085e10 −0.483916
\(204\) −2.96498e10 −1.19863
\(205\) 0 0
\(206\) −2.31689e10 −0.896403
\(207\) −8.28061e9 −0.313470
\(208\) 1.78444e10 0.661024
\(209\) −2.74924e10 −0.996678
\(210\) 0 0
\(211\) −9.23757e8 −0.0320838 −0.0160419 0.999871i \(-0.505107\pi\)
−0.0160419 + 0.999871i \(0.505107\pi\)
\(212\) −3.70129e10 −1.25847
\(213\) 1.65580e10 0.551189
\(214\) 1.56992e10 0.511700
\(215\) 0 0
\(216\) −1.28568e10 −0.401875
\(217\) 3.42098e9 0.104733
\(218\) 1.78017e10 0.533835
\(219\) 2.15265e10 0.632375
\(220\) 0 0
\(221\) 6.02130e10 1.69795
\(222\) −3.56694e10 −0.985615
\(223\) −6.68635e9 −0.181058 −0.0905290 0.995894i \(-0.528856\pi\)
−0.0905290 + 0.995894i \(0.528856\pi\)
\(224\) 1.98229e10 0.526078
\(225\) 0 0
\(226\) −1.52233e10 −0.388169
\(227\) 4.82876e10 1.20703 0.603516 0.797351i \(-0.293766\pi\)
0.603516 + 0.797351i \(0.293766\pi\)
\(228\) −2.06827e10 −0.506876
\(229\) −2.34264e10 −0.562918 −0.281459 0.959573i \(-0.590818\pi\)
−0.281459 + 0.959573i \(0.590818\pi\)
\(230\) 0 0
\(231\) −1.32628e10 −0.306465
\(232\) 2.38378e10 0.540219
\(233\) 3.29140e10 0.731609 0.365805 0.930692i \(-0.380794\pi\)
0.365805 + 0.930692i \(0.380794\pi\)
\(234\) 4.82864e10 1.05282
\(235\) 0 0
\(236\) −7.16275e10 −1.50306
\(237\) −4.11134e10 −0.846479
\(238\) 4.66119e10 0.941673
\(239\) 2.51973e10 0.499533 0.249766 0.968306i \(-0.419646\pi\)
0.249766 + 0.968306i \(0.419646\pi\)
\(240\) 0 0
\(241\) 7.00815e10 1.33822 0.669109 0.743165i \(-0.266676\pi\)
0.669109 + 0.743165i \(0.266676\pi\)
\(242\) 8.37696e10 1.57006
\(243\) 5.59963e10 1.03022
\(244\) 2.09633e10 0.378622
\(245\) 0 0
\(246\) −5.52488e10 −0.961866
\(247\) 4.20027e10 0.718028
\(248\) −6.96489e9 −0.116918
\(249\) 5.43744e10 0.896391
\(250\) 0 0
\(251\) 4.19681e10 0.667401 0.333701 0.942679i \(-0.391703\pi\)
0.333701 + 0.942679i \(0.391703\pi\)
\(252\) 2.09811e10 0.327737
\(253\) 4.30527e10 0.660630
\(254\) −3.17243e10 −0.478235
\(255\) 0 0
\(256\) 1.61271e10 0.234680
\(257\) −8.37595e10 −1.19766 −0.598832 0.800875i \(-0.704368\pi\)
−0.598832 + 0.800875i \(0.704368\pi\)
\(258\) 3.04116e10 0.427317
\(259\) 3.14751e10 0.434629
\(260\) 0 0
\(261\) −6.50497e10 −0.867687
\(262\) 8.80350e10 1.15425
\(263\) 2.34604e9 0.0302367 0.0151184 0.999886i \(-0.495187\pi\)
0.0151184 + 0.999886i \(0.495187\pi\)
\(264\) 2.70021e10 0.342122
\(265\) 0 0
\(266\) 3.25150e10 0.398214
\(267\) −1.17200e10 −0.141133
\(268\) −5.25950e10 −0.622784
\(269\) −6.44659e10 −0.750663 −0.375332 0.926891i \(-0.622471\pi\)
−0.375332 + 0.926891i \(0.622471\pi\)
\(270\) 0 0
\(271\) −1.40530e10 −0.158273 −0.0791365 0.996864i \(-0.525216\pi\)
−0.0791365 + 0.996864i \(0.525216\pi\)
\(272\) 9.57011e10 1.06012
\(273\) 2.02628e10 0.220784
\(274\) 1.51722e11 1.62619
\(275\) 0 0
\(276\) 3.23889e10 0.335973
\(277\) 7.52768e10 0.768249 0.384124 0.923281i \(-0.374503\pi\)
0.384124 + 0.923281i \(0.374503\pi\)
\(278\) −2.48837e11 −2.49870
\(279\) 1.90061e10 0.187791
\(280\) 0 0
\(281\) 9.56085e10 0.914783 0.457391 0.889266i \(-0.348784\pi\)
0.457391 + 0.889266i \(0.348784\pi\)
\(282\) 5.42427e10 0.510764
\(283\) −4.82806e10 −0.447439 −0.223719 0.974654i \(-0.571820\pi\)
−0.223719 + 0.974654i \(0.571820\pi\)
\(284\) 1.36188e11 1.24224
\(285\) 0 0
\(286\) −2.51052e11 −2.21879
\(287\) 4.87522e10 0.424156
\(288\) 1.10131e11 0.943286
\(289\) 2.04340e11 1.72311
\(290\) 0 0
\(291\) −8.73272e10 −0.713890
\(292\) 1.77053e11 1.42522
\(293\) 7.07439e10 0.560770 0.280385 0.959888i \(-0.409538\pi\)
0.280385 + 0.959888i \(0.409538\pi\)
\(294\) 1.56857e10 0.122445
\(295\) 0 0
\(296\) −6.40812e10 −0.485197
\(297\) −1.82411e11 −1.36034
\(298\) 1.66494e11 1.22299
\(299\) −6.57756e10 −0.475932
\(300\) 0 0
\(301\) −2.68355e10 −0.188435
\(302\) −2.95089e11 −2.04137
\(303\) 1.66210e10 0.113283
\(304\) 6.67581e10 0.448304
\(305\) 0 0
\(306\) 2.58964e11 1.68847
\(307\) −1.30493e11 −0.838429 −0.419214 0.907887i \(-0.637694\pi\)
−0.419214 + 0.907887i \(0.637694\pi\)
\(308\) −1.09085e11 −0.690697
\(309\) −5.40161e10 −0.337062
\(310\) 0 0
\(311\) 8.51715e10 0.516265 0.258132 0.966110i \(-0.416893\pi\)
0.258132 + 0.966110i \(0.416893\pi\)
\(312\) −4.12537e10 −0.246472
\(313\) 1.21745e11 0.716969 0.358484 0.933536i \(-0.383294\pi\)
0.358484 + 0.933536i \(0.383294\pi\)
\(314\) 3.19137e11 1.85265
\(315\) 0 0
\(316\) −3.38154e11 −1.90775
\(317\) 2.14595e11 1.19358 0.596791 0.802397i \(-0.296442\pi\)
0.596791 + 0.802397i \(0.296442\pi\)
\(318\) −1.53736e11 −0.843048
\(319\) 3.38208e11 1.82863
\(320\) 0 0
\(321\) 3.66011e10 0.192407
\(322\) −5.09180e10 −0.263949
\(323\) 2.25264e11 1.15154
\(324\) 3.47703e10 0.175289
\(325\) 0 0
\(326\) 7.19740e10 0.352936
\(327\) 4.15029e10 0.200731
\(328\) −9.92562e10 −0.473506
\(329\) −4.78644e10 −0.225233
\(330\) 0 0
\(331\) −5.48000e10 −0.250931 −0.125466 0.992098i \(-0.540042\pi\)
−0.125466 + 0.992098i \(0.540042\pi\)
\(332\) 4.47224e11 2.02025
\(333\) 1.74868e11 0.779312
\(334\) −4.77595e11 −2.09991
\(335\) 0 0
\(336\) 3.22051e10 0.137847
\(337\) 2.34297e11 0.989538 0.494769 0.869025i \(-0.335253\pi\)
0.494769 + 0.869025i \(0.335253\pi\)
\(338\) 2.12772e10 0.0886725
\(339\) −3.54916e10 −0.145958
\(340\) 0 0
\(341\) −9.88172e10 −0.395765
\(342\) 1.80645e11 0.714018
\(343\) −1.38413e10 −0.0539949
\(344\) 5.46353e10 0.210359
\(345\) 0 0
\(346\) 4.25104e11 1.59460
\(347\) −3.43449e10 −0.127169 −0.0635843 0.997976i \(-0.520253\pi\)
−0.0635843 + 0.997976i \(0.520253\pi\)
\(348\) 2.54436e11 0.929977
\(349\) −2.13485e11 −0.770288 −0.385144 0.922856i \(-0.625848\pi\)
−0.385144 + 0.922856i \(0.625848\pi\)
\(350\) 0 0
\(351\) 2.78686e11 0.980016
\(352\) −5.72595e11 −1.98795
\(353\) −2.75882e11 −0.945664 −0.472832 0.881153i \(-0.656768\pi\)
−0.472832 + 0.881153i \(0.656768\pi\)
\(354\) −2.97509e11 −1.00690
\(355\) 0 0
\(356\) −9.63962e10 −0.318079
\(357\) 1.08671e11 0.354084
\(358\) 2.49448e11 0.802614
\(359\) −3.46238e11 −1.10015 −0.550073 0.835117i \(-0.685400\pi\)
−0.550073 + 0.835117i \(0.685400\pi\)
\(360\) 0 0
\(361\) −1.65550e11 −0.513036
\(362\) −4.34117e11 −1.32867
\(363\) 1.95300e11 0.590369
\(364\) 1.66659e11 0.497592
\(365\) 0 0
\(366\) 8.70725e10 0.253639
\(367\) −3.56842e11 −1.02678 −0.513391 0.858155i \(-0.671611\pi\)
−0.513391 + 0.858155i \(0.671611\pi\)
\(368\) −1.04542e11 −0.297150
\(369\) 2.70855e11 0.760534
\(370\) 0 0
\(371\) 1.35658e11 0.371760
\(372\) −7.43408e10 −0.201272
\(373\) −6.73833e11 −1.80245 −0.901224 0.433354i \(-0.857330\pi\)
−0.901224 + 0.433354i \(0.857330\pi\)
\(374\) −1.34641e12 −3.55841
\(375\) 0 0
\(376\) 9.74487e10 0.251438
\(377\) −5.16711e11 −1.31738
\(378\) 2.15735e11 0.543511
\(379\) 5.90163e11 1.46925 0.734625 0.678473i \(-0.237358\pi\)
0.734625 + 0.678473i \(0.237358\pi\)
\(380\) 0 0
\(381\) −7.39622e10 −0.179824
\(382\) 5.51887e11 1.32607
\(383\) 1.58931e11 0.377412 0.188706 0.982034i \(-0.439571\pi\)
0.188706 + 0.982034i \(0.439571\pi\)
\(384\) −1.96153e11 −0.460367
\(385\) 0 0
\(386\) 5.22145e11 1.19715
\(387\) −1.49092e11 −0.337873
\(388\) −7.18258e11 −1.60893
\(389\) −3.75434e11 −0.831304 −0.415652 0.909524i \(-0.636447\pi\)
−0.415652 + 0.909524i \(0.636447\pi\)
\(390\) 0 0
\(391\) −3.52760e11 −0.763280
\(392\) 2.81799e10 0.0602771
\(393\) 2.05245e11 0.434016
\(394\) −7.99018e11 −1.67041
\(395\) 0 0
\(396\) −6.06051e11 −1.23846
\(397\) 4.33507e11 0.875869 0.437935 0.899007i \(-0.355710\pi\)
0.437935 + 0.899007i \(0.355710\pi\)
\(398\) −8.66400e11 −1.73079
\(399\) 7.58055e10 0.149735
\(400\) 0 0
\(401\) −1.77805e11 −0.343395 −0.171698 0.985150i \(-0.554925\pi\)
−0.171698 + 0.985150i \(0.554925\pi\)
\(402\) −2.18457e11 −0.417203
\(403\) 1.50972e11 0.285118
\(404\) 1.36706e11 0.255312
\(405\) 0 0
\(406\) −3.99994e11 −0.730612
\(407\) −9.09178e11 −1.64238
\(408\) −2.21247e11 −0.395281
\(409\) −7.67870e11 −1.35685 −0.678427 0.734668i \(-0.737338\pi\)
−0.678427 + 0.734668i \(0.737338\pi\)
\(410\) 0 0
\(411\) 3.53725e11 0.611473
\(412\) −4.44277e11 −0.759655
\(413\) 2.62526e11 0.444014
\(414\) −2.82888e11 −0.473274
\(415\) 0 0
\(416\) 8.74807e11 1.43216
\(417\) −5.80139e11 −0.939550
\(418\) −9.39215e11 −1.50478
\(419\) 4.96552e11 0.787048 0.393524 0.919314i \(-0.371256\pi\)
0.393524 + 0.919314i \(0.371256\pi\)
\(420\) 0 0
\(421\) −4.48514e11 −0.695835 −0.347917 0.937525i \(-0.613111\pi\)
−0.347917 + 0.937525i \(0.613111\pi\)
\(422\) −3.15580e10 −0.0484399
\(423\) −2.65923e11 −0.403854
\(424\) −2.76191e11 −0.415014
\(425\) 0 0
\(426\) 5.65667e11 0.832180
\(427\) −7.68338e10 −0.111848
\(428\) 3.01041e11 0.433639
\(429\) −5.85303e11 −0.834301
\(430\) 0 0
\(431\) −1.65940e11 −0.231635 −0.115817 0.993271i \(-0.536949\pi\)
−0.115817 + 0.993271i \(0.536949\pi\)
\(432\) 4.42936e11 0.611878
\(433\) 4.10674e11 0.561438 0.280719 0.959790i \(-0.409427\pi\)
0.280719 + 0.959790i \(0.409427\pi\)
\(434\) 1.16870e11 0.158124
\(435\) 0 0
\(436\) 3.41358e11 0.452398
\(437\) −2.46074e11 −0.322775
\(438\) 7.35402e11 0.954754
\(439\) −3.29593e11 −0.423534 −0.211767 0.977320i \(-0.567922\pi\)
−0.211767 + 0.977320i \(0.567922\pi\)
\(440\) 0 0
\(441\) −7.68988e10 −0.0968158
\(442\) 2.05704e12 2.56355
\(443\) −5.09091e11 −0.628027 −0.314014 0.949418i \(-0.601674\pi\)
−0.314014 + 0.949418i \(0.601674\pi\)
\(444\) −6.83981e11 −0.835258
\(445\) 0 0
\(446\) −2.28424e11 −0.273360
\(447\) 3.88163e11 0.459865
\(448\) 4.70175e11 0.551453
\(449\) 1.49596e12 1.73705 0.868525 0.495646i \(-0.165069\pi\)
0.868525 + 0.495646i \(0.165069\pi\)
\(450\) 0 0
\(451\) −1.40824e12 −1.60281
\(452\) −2.91915e11 −0.328953
\(453\) −6.87971e11 −0.767588
\(454\) 1.64963e12 1.82237
\(455\) 0 0
\(456\) −1.54335e11 −0.167156
\(457\) −1.43920e12 −1.54347 −0.771735 0.635944i \(-0.780611\pi\)
−0.771735 + 0.635944i \(0.780611\pi\)
\(458\) −8.00307e11 −0.849889
\(459\) 1.49462e12 1.57171
\(460\) 0 0
\(461\) −1.37741e12 −1.42039 −0.710195 0.704005i \(-0.751393\pi\)
−0.710195 + 0.704005i \(0.751393\pi\)
\(462\) −4.53092e11 −0.462698
\(463\) −1.76612e12 −1.78610 −0.893049 0.449960i \(-0.851438\pi\)
−0.893049 + 0.449960i \(0.851438\pi\)
\(464\) −8.21248e11 −0.822514
\(465\) 0 0
\(466\) 1.12443e12 1.10458
\(467\) 1.17323e12 1.14145 0.570727 0.821140i \(-0.306661\pi\)
0.570727 + 0.821140i \(0.306661\pi\)
\(468\) 9.25919e11 0.892209
\(469\) 1.92769e11 0.183975
\(470\) 0 0
\(471\) 7.44037e11 0.696627
\(472\) −5.34485e11 −0.495674
\(473\) 7.75161e11 0.712060
\(474\) −1.40454e12 −1.27801
\(475\) 0 0
\(476\) 8.93808e11 0.798019
\(477\) 7.53683e11 0.666586
\(478\) 8.60808e11 0.754190
\(479\) −1.98723e12 −1.72480 −0.862401 0.506225i \(-0.831040\pi\)
−0.862401 + 0.506225i \(0.831040\pi\)
\(480\) 0 0
\(481\) 1.38903e12 1.18321
\(482\) 2.39417e12 2.02043
\(483\) −1.18710e11 −0.0992489
\(484\) 1.60633e12 1.33055
\(485\) 0 0
\(486\) 1.91298e12 1.55542
\(487\) 1.23240e12 0.992825 0.496413 0.868087i \(-0.334650\pi\)
0.496413 + 0.868087i \(0.334650\pi\)
\(488\) 1.56428e11 0.124861
\(489\) 1.67800e11 0.132710
\(490\) 0 0
\(491\) −2.03763e12 −1.58219 −0.791095 0.611693i \(-0.790489\pi\)
−0.791095 + 0.611693i \(0.790489\pi\)
\(492\) −1.05943e12 −0.815131
\(493\) −2.77117e12 −2.11277
\(494\) 1.43493e12 1.08407
\(495\) 0 0
\(496\) 2.39951e11 0.178014
\(497\) −4.99151e11 −0.366968
\(498\) 1.85758e12 1.35336
\(499\) −3.26299e11 −0.235594 −0.117797 0.993038i \(-0.537583\pi\)
−0.117797 + 0.993038i \(0.537583\pi\)
\(500\) 0 0
\(501\) −1.11347e12 −0.789601
\(502\) 1.43374e12 1.00764
\(503\) 4.46869e11 0.311261 0.155630 0.987815i \(-0.450259\pi\)
0.155630 + 0.987815i \(0.450259\pi\)
\(504\) 1.56561e11 0.108080
\(505\) 0 0
\(506\) 1.47080e12 0.997413
\(507\) 4.96056e10 0.0333423
\(508\) −6.08332e11 −0.405279
\(509\) 1.34100e12 0.885523 0.442761 0.896639i \(-0.353999\pi\)
0.442761 + 0.896639i \(0.353999\pi\)
\(510\) 0 0
\(511\) −6.48928e11 −0.421020
\(512\) 1.81189e12 1.16524
\(513\) 1.04260e12 0.664643
\(514\) −2.86145e12 −1.80822
\(515\) 0 0
\(516\) 5.83158e11 0.362129
\(517\) 1.38259e12 0.851112
\(518\) 1.07527e12 0.656199
\(519\) 9.91089e11 0.599597
\(520\) 0 0
\(521\) 2.98523e12 1.77504 0.887520 0.460769i \(-0.152426\pi\)
0.887520 + 0.460769i \(0.152426\pi\)
\(522\) −2.22227e12 −1.31003
\(523\) −1.64651e12 −0.962289 −0.481145 0.876641i \(-0.659779\pi\)
−0.481145 + 0.876641i \(0.659779\pi\)
\(524\) 1.68812e12 0.978166
\(525\) 0 0
\(526\) 8.01470e10 0.0456511
\(527\) 8.09676e11 0.457260
\(528\) −9.30265e11 −0.520900
\(529\) −1.41580e12 −0.786054
\(530\) 0 0
\(531\) 1.45853e12 0.796141
\(532\) 6.23493e11 0.337465
\(533\) 2.15149e12 1.15470
\(534\) −4.00388e11 −0.213081
\(535\) 0 0
\(536\) −3.92464e11 −0.205380
\(537\) 5.81564e11 0.301796
\(538\) −2.20233e12 −1.13334
\(539\) 3.99814e11 0.204037
\(540\) 0 0
\(541\) −4.57968e11 −0.229852 −0.114926 0.993374i \(-0.536663\pi\)
−0.114926 + 0.993374i \(0.536663\pi\)
\(542\) −4.80087e11 −0.238959
\(543\) −1.01210e12 −0.499603
\(544\) 4.69166e12 2.29684
\(545\) 0 0
\(546\) 6.92231e11 0.333337
\(547\) −2.39624e12 −1.14443 −0.572213 0.820105i \(-0.693915\pi\)
−0.572213 + 0.820105i \(0.693915\pi\)
\(548\) 2.90935e12 1.37811
\(549\) −4.26870e11 −0.200549
\(550\) 0 0
\(551\) −1.93308e12 −0.893444
\(552\) 2.41686e11 0.110796
\(553\) 1.23939e12 0.563565
\(554\) 2.57166e12 1.15990
\(555\) 0 0
\(556\) −4.77159e12 −2.11751
\(557\) −1.07863e12 −0.474814 −0.237407 0.971410i \(-0.576298\pi\)
−0.237407 + 0.971410i \(0.576298\pi\)
\(558\) 6.49301e11 0.283525
\(559\) −1.18428e12 −0.512983
\(560\) 0 0
\(561\) −3.13903e12 −1.33802
\(562\) 3.26624e12 1.38113
\(563\) −2.33140e11 −0.0977976 −0.0488988 0.998804i \(-0.515571\pi\)
−0.0488988 + 0.998804i \(0.515571\pi\)
\(564\) 1.04013e12 0.432846
\(565\) 0 0
\(566\) −1.64939e12 −0.675539
\(567\) −1.27438e11 −0.0517817
\(568\) 1.01624e12 0.409664
\(569\) 4.50535e11 0.180187 0.0900934 0.995933i \(-0.471283\pi\)
0.0900934 + 0.995933i \(0.471283\pi\)
\(570\) 0 0
\(571\) −4.38839e12 −1.72760 −0.863800 0.503835i \(-0.831922\pi\)
−0.863800 + 0.503835i \(0.831922\pi\)
\(572\) −4.81406e12 −1.88031
\(573\) 1.28667e12 0.498622
\(574\) 1.66550e12 0.640387
\(575\) 0 0
\(576\) 2.61218e12 0.988785
\(577\) −3.13994e12 −1.17932 −0.589658 0.807653i \(-0.700737\pi\)
−0.589658 + 0.807653i \(0.700737\pi\)
\(578\) 6.98079e12 2.60153
\(579\) 1.21733e12 0.450148
\(580\) 0 0
\(581\) −1.63915e12 −0.596795
\(582\) −2.98333e12 −1.07782
\(583\) −3.91857e12 −1.40481
\(584\) 1.32117e12 0.470005
\(585\) 0 0
\(586\) 2.41680e12 0.846645
\(587\) −2.52512e12 −0.877829 −0.438915 0.898529i \(-0.644637\pi\)
−0.438915 + 0.898529i \(0.644637\pi\)
\(588\) 3.00783e11 0.103766
\(589\) 5.64805e11 0.193366
\(590\) 0 0
\(591\) −1.86283e12 −0.628102
\(592\) 2.20770e12 0.738740
\(593\) 9.35417e11 0.310641 0.155321 0.987864i \(-0.450359\pi\)
0.155321 + 0.987864i \(0.450359\pi\)
\(594\) −6.23165e12 −2.05383
\(595\) 0 0
\(596\) 3.19261e12 1.03642
\(597\) −2.01993e12 −0.650805
\(598\) −2.24707e12 −0.718557
\(599\) 4.73586e12 1.50307 0.751534 0.659694i \(-0.229314\pi\)
0.751534 + 0.659694i \(0.229314\pi\)
\(600\) 0 0
\(601\) −5.99855e12 −1.87548 −0.937738 0.347344i \(-0.887084\pi\)
−0.937738 + 0.347344i \(0.887084\pi\)
\(602\) −9.16773e11 −0.284497
\(603\) 1.07098e12 0.329877
\(604\) −5.65850e12 −1.72995
\(605\) 0 0
\(606\) 5.67817e11 0.171034
\(607\) −3.43715e12 −1.02766 −0.513830 0.857892i \(-0.671774\pi\)
−0.513830 + 0.857892i \(0.671774\pi\)
\(608\) 3.27276e12 0.971287
\(609\) −9.32548e11 −0.274722
\(610\) 0 0
\(611\) −2.11231e12 −0.613159
\(612\) 4.96578e12 1.43089
\(613\) −4.15273e12 −1.18785 −0.593925 0.804520i \(-0.702423\pi\)
−0.593925 + 0.804520i \(0.702423\pi\)
\(614\) −4.45801e12 −1.26585
\(615\) 0 0
\(616\) −8.13995e11 −0.227776
\(617\) 1.12196e12 0.311669 0.155834 0.987783i \(-0.450193\pi\)
0.155834 + 0.987783i \(0.450193\pi\)
\(618\) −1.84533e12 −0.508893
\(619\) −5.98940e12 −1.63974 −0.819871 0.572549i \(-0.805955\pi\)
−0.819871 + 0.572549i \(0.805955\pi\)
\(620\) 0 0
\(621\) −1.63269e12 −0.440547
\(622\) 2.90969e12 0.779452
\(623\) 3.53307e11 0.0939628
\(624\) 1.42125e12 0.375267
\(625\) 0 0
\(626\) 4.15912e12 1.08247
\(627\) −2.18969e12 −0.565820
\(628\) 6.11963e12 1.57003
\(629\) 7.44951e12 1.89758
\(630\) 0 0
\(631\) 4.97135e12 1.24837 0.624184 0.781277i \(-0.285432\pi\)
0.624184 + 0.781277i \(0.285432\pi\)
\(632\) −2.52331e12 −0.629134
\(633\) −7.35743e10 −0.0182142
\(634\) 7.33113e12 1.80206
\(635\) 0 0
\(636\) −2.94796e12 −0.714439
\(637\) −6.10833e11 −0.146992
\(638\) 1.15541e13 2.76085
\(639\) −2.77316e12 −0.657993
\(640\) 0 0
\(641\) −2.41181e12 −0.564263 −0.282131 0.959376i \(-0.591041\pi\)
−0.282131 + 0.959376i \(0.591041\pi\)
\(642\) 1.25039e12 0.290495
\(643\) −6.86804e12 −1.58447 −0.792234 0.610217i \(-0.791082\pi\)
−0.792234 + 0.610217i \(0.791082\pi\)
\(644\) −9.76380e11 −0.223683
\(645\) 0 0
\(646\) 7.69562e12 1.73859
\(647\) 1.73394e12 0.389013 0.194506 0.980901i \(-0.437690\pi\)
0.194506 + 0.980901i \(0.437690\pi\)
\(648\) 2.59456e11 0.0578064
\(649\) −7.58322e12 −1.67785
\(650\) 0 0
\(651\) 2.72471e11 0.0594573
\(652\) 1.38014e12 0.299095
\(653\) −2.86440e10 −0.00616488 −0.00308244 0.999995i \(-0.500981\pi\)
−0.00308244 + 0.999995i \(0.500981\pi\)
\(654\) 1.41785e12 0.303061
\(655\) 0 0
\(656\) 3.41953e12 0.720940
\(657\) −3.60529e12 −0.754911
\(658\) −1.63518e12 −0.340054
\(659\) −6.31728e12 −1.30481 −0.652403 0.757872i \(-0.726239\pi\)
−0.652403 + 0.757872i \(0.726239\pi\)
\(660\) 0 0
\(661\) −3.49558e12 −0.712217 −0.356109 0.934445i \(-0.615897\pi\)
−0.356109 + 0.934445i \(0.615897\pi\)
\(662\) −1.87212e12 −0.378854
\(663\) 4.79578e12 0.963937
\(664\) 3.33719e12 0.666231
\(665\) 0 0
\(666\) 5.97396e12 1.17660
\(667\) 3.02717e12 0.592203
\(668\) −9.15816e12 −1.77957
\(669\) −5.32547e11 −0.102788
\(670\) 0 0
\(671\) 2.21939e12 0.422652
\(672\) 1.57883e12 0.298658
\(673\) 8.02535e12 1.50798 0.753991 0.656885i \(-0.228126\pi\)
0.753991 + 0.656885i \(0.228126\pi\)
\(674\) 8.00422e12 1.49400
\(675\) 0 0
\(676\) 4.08002e11 0.0751453
\(677\) −1.17163e12 −0.214358 −0.107179 0.994240i \(-0.534182\pi\)
−0.107179 + 0.994240i \(0.534182\pi\)
\(678\) −1.21249e12 −0.220366
\(679\) 2.63253e12 0.475290
\(680\) 0 0
\(681\) 3.84596e12 0.685239
\(682\) −3.37586e12 −0.597523
\(683\) −4.69754e12 −0.825995 −0.412997 0.910732i \(-0.635518\pi\)
−0.412997 + 0.910732i \(0.635518\pi\)
\(684\) 3.46397e12 0.605093
\(685\) 0 0
\(686\) −4.72855e11 −0.0815211
\(687\) −1.86584e12 −0.319572
\(688\) −1.88227e12 −0.320283
\(689\) 5.98676e12 1.01206
\(690\) 0 0
\(691\) −7.83193e12 −1.30683 −0.653413 0.757001i \(-0.726664\pi\)
−0.653413 + 0.757001i \(0.726664\pi\)
\(692\) 8.15161e12 1.35134
\(693\) 2.22127e12 0.365849
\(694\) −1.17332e12 −0.191998
\(695\) 0 0
\(696\) 1.89860e12 0.306685
\(697\) 1.15386e13 1.85185
\(698\) −7.29322e12 −1.16297
\(699\) 2.62150e12 0.415339
\(700\) 0 0
\(701\) 7.13243e12 1.11559 0.557797 0.829977i \(-0.311647\pi\)
0.557797 + 0.829977i \(0.311647\pi\)
\(702\) 9.52066e12 1.47962
\(703\) 5.19655e12 0.802446
\(704\) −1.35813e13 −2.08384
\(705\) 0 0
\(706\) −9.42486e12 −1.42776
\(707\) −5.01049e11 −0.0754211
\(708\) −5.70491e12 −0.853294
\(709\) 8.65274e12 1.28601 0.643007 0.765861i \(-0.277687\pi\)
0.643007 + 0.765861i \(0.277687\pi\)
\(710\) 0 0
\(711\) 6.88573e12 1.01050
\(712\) −7.19310e11 −0.104895
\(713\) −8.84475e11 −0.128169
\(714\) 3.71249e12 0.534593
\(715\) 0 0
\(716\) 4.78331e12 0.680174
\(717\) 2.00689e12 0.283588
\(718\) −1.18284e13 −1.66099
\(719\) 4.58446e11 0.0639747 0.0319873 0.999488i \(-0.489816\pi\)
0.0319873 + 0.999488i \(0.489816\pi\)
\(720\) 0 0
\(721\) 1.62834e12 0.224408
\(722\) −5.65565e12 −0.774577
\(723\) 5.58177e12 0.759713
\(724\) −8.32444e12 −1.12598
\(725\) 0 0
\(726\) 6.67199e12 0.891334
\(727\) 2.53850e12 0.337033 0.168517 0.985699i \(-0.446102\pi\)
0.168517 + 0.985699i \(0.446102\pi\)
\(728\) 1.24361e12 0.164095
\(729\) 3.41521e12 0.447861
\(730\) 0 0
\(731\) −6.35141e12 −0.822701
\(732\) 1.66966e12 0.214946
\(733\) −1.09361e13 −1.39925 −0.699624 0.714511i \(-0.746649\pi\)
−0.699624 + 0.714511i \(0.746649\pi\)
\(734\) −1.21907e13 −1.55023
\(735\) 0 0
\(736\) −5.12509e12 −0.643800
\(737\) −5.56824e12 −0.695208
\(738\) 9.25314e12 1.14825
\(739\) 7.34996e12 0.906536 0.453268 0.891374i \(-0.350258\pi\)
0.453268 + 0.891374i \(0.350258\pi\)
\(740\) 0 0
\(741\) 3.34539e12 0.407628
\(742\) 4.63444e12 0.561281
\(743\) 1.60813e12 0.193584 0.0967922 0.995305i \(-0.469142\pi\)
0.0967922 + 0.995305i \(0.469142\pi\)
\(744\) −5.54732e11 −0.0663751
\(745\) 0 0
\(746\) −2.30199e13 −2.72132
\(747\) −9.10670e12 −1.07009
\(748\) −2.58182e13 −3.01557
\(749\) −1.10336e12 −0.128100
\(750\) 0 0
\(751\) −2.95903e12 −0.339446 −0.169723 0.985492i \(-0.554287\pi\)
−0.169723 + 0.985492i \(0.554287\pi\)
\(752\) −3.35726e12 −0.382829
\(753\) 3.34263e12 0.378888
\(754\) −1.76522e13 −1.98897
\(755\) 0 0
\(756\) 4.13684e12 0.460597
\(757\) 4.83223e12 0.534830 0.267415 0.963581i \(-0.413830\pi\)
0.267415 + 0.963581i \(0.413830\pi\)
\(758\) 2.01616e13 2.21826
\(759\) 3.42902e12 0.375043
\(760\) 0 0
\(761\) 3.15213e12 0.340701 0.170350 0.985384i \(-0.445510\pi\)
0.170350 + 0.985384i \(0.445510\pi\)
\(762\) −2.52675e12 −0.271496
\(763\) −1.25113e12 −0.133642
\(764\) 1.05827e13 1.12377
\(765\) 0 0
\(766\) 5.42952e12 0.569813
\(767\) 1.15856e13 1.20876
\(768\) 1.28447e12 0.133229
\(769\) −6.87650e12 −0.709086 −0.354543 0.935040i \(-0.615364\pi\)
−0.354543 + 0.935040i \(0.615364\pi\)
\(770\) 0 0
\(771\) −6.67118e12 −0.679920
\(772\) 1.00124e13 1.01452
\(773\) −1.16083e13 −1.16939 −0.584696 0.811253i \(-0.698786\pi\)
−0.584696 + 0.811253i \(0.698786\pi\)
\(774\) −5.09337e12 −0.510118
\(775\) 0 0
\(776\) −5.35965e12 −0.530590
\(777\) 2.50690e12 0.246741
\(778\) −1.28258e13 −1.25510
\(779\) 8.04899e12 0.783110
\(780\) 0 0
\(781\) 1.44183e13 1.38670
\(782\) −1.20512e13 −1.15239
\(783\) −1.28259e13 −1.21944
\(784\) −9.70842e11 −0.0917754
\(785\) 0 0
\(786\) 7.01171e12 0.655274
\(787\) 2.50005e12 0.232307 0.116154 0.993231i \(-0.462943\pi\)
0.116154 + 0.993231i \(0.462943\pi\)
\(788\) −1.53216e13 −1.41559
\(789\) 1.86855e11 0.0171656
\(790\) 0 0
\(791\) 1.06991e12 0.0971751
\(792\) −4.52236e12 −0.408415
\(793\) −3.39077e12 −0.304487
\(794\) 1.48098e13 1.32238
\(795\) 0 0
\(796\) −1.66137e13 −1.46675
\(797\) −2.77023e12 −0.243194 −0.121597 0.992580i \(-0.538802\pi\)
−0.121597 + 0.992580i \(0.538802\pi\)
\(798\) 2.58972e12 0.226068
\(799\) −1.13285e13 −0.983360
\(800\) 0 0
\(801\) 1.96289e12 0.168480
\(802\) −6.07429e12 −0.518455
\(803\) 1.87447e13 1.59096
\(804\) −4.18903e12 −0.353558
\(805\) 0 0
\(806\) 5.15761e12 0.430468
\(807\) −5.13451e12 −0.426156
\(808\) 1.02010e12 0.0841962
\(809\) −2.96640e12 −0.243479 −0.121739 0.992562i \(-0.538847\pi\)
−0.121739 + 0.992562i \(0.538847\pi\)
\(810\) 0 0
\(811\) −9.01447e12 −0.731722 −0.365861 0.930669i \(-0.619225\pi\)
−0.365861 + 0.930669i \(0.619225\pi\)
\(812\) −7.67012e12 −0.619156
\(813\) −1.11928e12 −0.0898524
\(814\) −3.10599e13 −2.47965
\(815\) 0 0
\(816\) 7.62229e12 0.601838
\(817\) −4.43055e12 −0.347903
\(818\) −2.62325e13 −2.04857
\(819\) −3.39364e12 −0.263565
\(820\) 0 0
\(821\) 1.25176e13 0.961563 0.480782 0.876840i \(-0.340353\pi\)
0.480782 + 0.876840i \(0.340353\pi\)
\(822\) 1.20842e13 0.923197
\(823\) 1.50348e13 1.14234 0.571172 0.820830i \(-0.306489\pi\)
0.571172 + 0.820830i \(0.306489\pi\)
\(824\) −3.31520e12 −0.250517
\(825\) 0 0
\(826\) 8.96858e12 0.670369
\(827\) −2.31522e13 −1.72115 −0.860574 0.509325i \(-0.829895\pi\)
−0.860574 + 0.509325i \(0.829895\pi\)
\(828\) −5.42453e12 −0.401075
\(829\) 4.88071e12 0.358912 0.179456 0.983766i \(-0.442566\pi\)
0.179456 + 0.983766i \(0.442566\pi\)
\(830\) 0 0
\(831\) 5.99556e12 0.436139
\(832\) 2.07494e13 1.50124
\(833\) −3.27595e12 −0.235741
\(834\) −1.98191e13 −1.41852
\(835\) 0 0
\(836\) −1.80100e13 −1.27522
\(837\) 3.74745e12 0.263919
\(838\) 1.69635e13 1.18828
\(839\) −1.18015e13 −0.822258 −0.411129 0.911577i \(-0.634865\pi\)
−0.411129 + 0.911577i \(0.634865\pi\)
\(840\) 0 0
\(841\) 9.27330e12 0.639223
\(842\) −1.53224e13 −1.05057
\(843\) 7.61492e12 0.519327
\(844\) −6.05142e11 −0.0410503
\(845\) 0 0
\(846\) −9.08464e12 −0.609735
\(847\) −5.88744e12 −0.393053
\(848\) 9.51520e12 0.631883
\(849\) −3.84540e12 −0.254014
\(850\) 0 0
\(851\) −8.13771e12 −0.531887
\(852\) 1.08470e13 0.705229
\(853\) 2.54707e13 1.64729 0.823646 0.567104i \(-0.191936\pi\)
0.823646 + 0.567104i \(0.191936\pi\)
\(854\) −2.62485e12 −0.168867
\(855\) 0 0
\(856\) 2.24637e12 0.143004
\(857\) −1.26584e13 −0.801615 −0.400807 0.916162i \(-0.631270\pi\)
−0.400807 + 0.916162i \(0.631270\pi\)
\(858\) −1.99955e13 −1.25962
\(859\) 2.01387e13 1.26201 0.631004 0.775779i \(-0.282643\pi\)
0.631004 + 0.775779i \(0.282643\pi\)
\(860\) 0 0
\(861\) 3.88296e12 0.240796
\(862\) −5.66896e12 −0.349720
\(863\) 1.44977e13 0.889716 0.444858 0.895601i \(-0.353254\pi\)
0.444858 + 0.895601i \(0.353254\pi\)
\(864\) 2.17146e13 1.32568
\(865\) 0 0
\(866\) 1.40297e13 0.847655
\(867\) 1.62750e13 0.978217
\(868\) 2.24104e12 0.134002
\(869\) −3.58004e13 −2.12961
\(870\) 0 0
\(871\) 8.50712e12 0.500842
\(872\) 2.54721e12 0.149191
\(873\) 1.46257e13 0.852221
\(874\) −8.40656e12 −0.487323
\(875\) 0 0
\(876\) 1.41018e13 0.809104
\(877\) −1.83297e13 −1.04630 −0.523150 0.852241i \(-0.675243\pi\)
−0.523150 + 0.852241i \(0.675243\pi\)
\(878\) −1.12598e13 −0.639448
\(879\) 5.63453e12 0.318352
\(880\) 0 0
\(881\) 4.57369e12 0.255785 0.127892 0.991788i \(-0.459179\pi\)
0.127892 + 0.991788i \(0.459179\pi\)
\(882\) −2.62707e12 −0.146172
\(883\) −3.07854e13 −1.70420 −0.852101 0.523377i \(-0.824672\pi\)
−0.852101 + 0.523377i \(0.824672\pi\)
\(884\) 3.94448e13 2.17248
\(885\) 0 0
\(886\) −1.73919e13 −0.948190
\(887\) 2.58803e13 1.40382 0.701911 0.712264i \(-0.252330\pi\)
0.701911 + 0.712264i \(0.252330\pi\)
\(888\) −5.10387e12 −0.275449
\(889\) 2.22963e12 0.119722
\(890\) 0 0
\(891\) 3.68114e12 0.195674
\(892\) −4.38015e12 −0.231658
\(893\) −7.90242e12 −0.415842
\(894\) 1.32607e13 0.694301
\(895\) 0 0
\(896\) 5.91314e12 0.306501
\(897\) −5.23882e12 −0.270189
\(898\) 5.11061e13 2.62258
\(899\) −6.94814e12 −0.354773
\(900\) 0 0
\(901\) 3.21075e13 1.62310
\(902\) −4.81091e13 −2.41990
\(903\) −2.13737e12 −0.106975
\(904\) −2.17827e12 −0.108481
\(905\) 0 0
\(906\) −2.35029e13 −1.15890
\(907\) −9.64118e12 −0.473040 −0.236520 0.971627i \(-0.576007\pi\)
−0.236520 + 0.971627i \(0.576007\pi\)
\(908\) 3.16326e13 1.54436
\(909\) −2.78371e12 −0.135234
\(910\) 0 0
\(911\) 7.28746e12 0.350545 0.175272 0.984520i \(-0.443919\pi\)
0.175272 + 0.984520i \(0.443919\pi\)
\(912\) 5.31707e12 0.254505
\(913\) 4.73478e13 2.25518
\(914\) −4.91669e13 −2.33032
\(915\) 0 0
\(916\) −1.53463e13 −0.720236
\(917\) −6.18722e12 −0.288957
\(918\) 5.10601e13 2.37295
\(919\) 2.73276e13 1.26381 0.631906 0.775045i \(-0.282273\pi\)
0.631906 + 0.775045i \(0.282273\pi\)
\(920\) 0 0
\(921\) −1.03934e13 −0.475981
\(922\) −4.70559e13 −2.14449
\(923\) −2.20281e13 −0.999011
\(924\) −8.68829e12 −0.392112
\(925\) 0 0
\(926\) −6.03353e13 −2.69663
\(927\) 9.04668e12 0.402375
\(928\) −4.02610e13 −1.78204
\(929\) −7.21805e12 −0.317943 −0.158971 0.987283i \(-0.550818\pi\)
−0.158971 + 0.987283i \(0.550818\pi\)
\(930\) 0 0
\(931\) −2.28520e12 −0.0996897
\(932\) 2.15616e13 0.936071
\(933\) 6.78365e12 0.293087
\(934\) 4.00808e13 1.72336
\(935\) 0 0
\(936\) 6.90922e12 0.294231
\(937\) −1.41988e13 −0.601761 −0.300880 0.953662i \(-0.597281\pi\)
−0.300880 + 0.953662i \(0.597281\pi\)
\(938\) 6.58550e12 0.277764
\(939\) 9.69658e12 0.407027
\(940\) 0 0
\(941\) −2.53242e13 −1.05289 −0.526445 0.850209i \(-0.676475\pi\)
−0.526445 + 0.850209i \(0.676475\pi\)
\(942\) 2.54183e13 1.05176
\(943\) −1.26046e13 −0.519070
\(944\) 1.84138e13 0.754693
\(945\) 0 0
\(946\) 2.64816e13 1.07506
\(947\) 1.37160e13 0.554182 0.277091 0.960844i \(-0.410630\pi\)
0.277091 + 0.960844i \(0.410630\pi\)
\(948\) −2.69329e13 −1.08304
\(949\) −2.86380e13 −1.14616
\(950\) 0 0
\(951\) 1.70918e13 0.677604
\(952\) 6.66961e12 0.263169
\(953\) −2.44096e13 −0.958610 −0.479305 0.877648i \(-0.659111\pi\)
−0.479305 + 0.877648i \(0.659111\pi\)
\(954\) 2.57478e13 1.00641
\(955\) 0 0
\(956\) 1.65065e13 0.639137
\(957\) 2.69372e13 1.03812
\(958\) −6.78892e13 −2.60409
\(959\) −1.06632e13 −0.407104
\(960\) 0 0
\(961\) −2.44095e13 −0.923218
\(962\) 4.74531e13 1.78639
\(963\) −6.13001e12 −0.229690
\(964\) 4.59096e13 1.71221
\(965\) 0 0
\(966\) −4.05546e12 −0.149845
\(967\) 4.19850e13 1.54410 0.772049 0.635563i \(-0.219232\pi\)
0.772049 + 0.635563i \(0.219232\pi\)
\(968\) 1.19864e13 0.438784
\(969\) 1.79416e13 0.653738
\(970\) 0 0
\(971\) 3.13254e12 0.113086 0.0565432 0.998400i \(-0.481992\pi\)
0.0565432 + 0.998400i \(0.481992\pi\)
\(972\) 3.66825e13 1.31814
\(973\) 1.74886e13 0.625529
\(974\) 4.21022e13 1.49896
\(975\) 0 0
\(976\) −5.38920e12 −0.190108
\(977\) −4.53124e13 −1.59108 −0.795539 0.605902i \(-0.792812\pi\)
−0.795539 + 0.605902i \(0.792812\pi\)
\(978\) 5.73251e12 0.200364
\(979\) −1.02055e13 −0.355068
\(980\) 0 0
\(981\) −6.95097e12 −0.239626
\(982\) −6.96109e13 −2.38878
\(983\) 2.54481e13 0.869290 0.434645 0.900602i \(-0.356874\pi\)
0.434645 + 0.900602i \(0.356874\pi\)
\(984\) −7.90545e12 −0.268812
\(985\) 0 0
\(986\) −9.46704e13 −3.18983
\(987\) −3.81225e12 −0.127866
\(988\) 2.75155e13 0.918694
\(989\) 6.93817e12 0.230601
\(990\) 0 0
\(991\) −5.41691e13 −1.78410 −0.892052 0.451933i \(-0.850734\pi\)
−0.892052 + 0.451933i \(0.850734\pi\)
\(992\) 1.17634e13 0.385683
\(993\) −4.36465e12 −0.142455
\(994\) −1.70523e13 −0.554045
\(995\) 0 0
\(996\) 3.56200e13 1.14690
\(997\) 4.29329e13 1.37614 0.688069 0.725646i \(-0.258459\pi\)
0.688069 + 0.725646i \(0.258459\pi\)
\(998\) −1.11473e13 −0.355697
\(999\) 3.44788e13 1.09524
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 175.10.a.d.1.3 3
5.2 odd 4 175.10.b.d.99.5 6
5.3 odd 4 175.10.b.d.99.2 6
5.4 even 2 7.10.a.b.1.1 3
15.14 odd 2 63.10.a.e.1.3 3
20.19 odd 2 112.10.a.h.1.3 3
35.4 even 6 49.10.c.d.30.3 6
35.9 even 6 49.10.c.d.18.3 6
35.19 odd 6 49.10.c.e.18.3 6
35.24 odd 6 49.10.c.e.30.3 6
35.34 odd 2 49.10.a.c.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.a.b.1.1 3 5.4 even 2
49.10.a.c.1.1 3 35.34 odd 2
49.10.c.d.18.3 6 35.9 even 6
49.10.c.d.30.3 6 35.4 even 6
49.10.c.e.18.3 6 35.19 odd 6
49.10.c.e.30.3 6 35.24 odd 6
63.10.a.e.1.3 3 15.14 odd 2
112.10.a.h.1.3 3 20.19 odd 2
175.10.a.d.1.3 3 1.1 even 1 trivial
175.10.b.d.99.2 6 5.3 odd 4
175.10.b.d.99.5 6 5.2 odd 4