Properties

Label 414.8.a.i.1.1
Level $414$
Weight $8$
Character 414.1
Self dual yes
Analytic conductor $129.327$
Analytic rank $0$
Dimension $4$
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] = [414,8,Mod(1,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("414.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 414.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: \(129.327400550\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-56.1267\) of defining polynomial
Character \(\chi\) \(=\) 414.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} +64.0000 q^{4} -366.876 q^{5} -561.721 q^{7} -512.000 q^{8} +O(q^{10})\) \(q-8.00000 q^{2} +64.0000 q^{4} -366.876 q^{5} -561.721 q^{7} -512.000 q^{8} +2935.00 q^{10} +1096.20 q^{11} +6925.11 q^{13} +4493.77 q^{14} +4096.00 q^{16} -13654.6 q^{17} +29540.8 q^{19} -23480.0 q^{20} -8769.58 q^{22} +12167.0 q^{23} +56472.7 q^{25} -55400.9 q^{26} -35950.1 q^{28} -150266. q^{29} -204841. q^{31} -32768.0 q^{32} +109237. q^{34} +206082. q^{35} -492163. q^{37} -236327. q^{38} +187840. q^{40} -416521. q^{41} +371500. q^{43} +70156.6 q^{44} -97336.0 q^{46} -930660. q^{47} -508013. q^{49} -451782. q^{50} +443207. q^{52} -1.01318e6 q^{53} -402168. q^{55} +287601. q^{56} +1.20213e6 q^{58} +2.28957e6 q^{59} +2.95586e6 q^{61} +1.63873e6 q^{62} +262144. q^{64} -2.54066e6 q^{65} -1.46138e6 q^{67} -873896. q^{68} -1.64865e6 q^{70} +1.82588e6 q^{71} -40352.1 q^{73} +3.93731e6 q^{74} +1.89061e6 q^{76} -615757. q^{77} -2.36229e6 q^{79} -1.50272e6 q^{80} +3.33217e6 q^{82} -1.27885e6 q^{83} +5.00955e6 q^{85} -2.97200e6 q^{86} -561253. q^{88} -3.88462e6 q^{89} -3.88998e6 q^{91} +778688. q^{92} +7.44528e6 q^{94} -1.08378e7 q^{95} -8.46502e6 q^{97} +4.06410e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{2} + 256 q^{4} - 270 q^{5} + 2022 q^{7} - 2048 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{2} + 256 q^{4} - 270 q^{5} + 2022 q^{7} - 2048 q^{8} + 2160 q^{10} - 4120 q^{11} + 8036 q^{13} - 16176 q^{14} + 16384 q^{16} - 37182 q^{17} + 5702 q^{19} - 17280 q^{20} + 32960 q^{22} + 48668 q^{23} + 121480 q^{25} - 64288 q^{26} + 129408 q^{28} - 217716 q^{29} + 222852 q^{31} - 131072 q^{32} + 297456 q^{34} - 68440 q^{35} + 486428 q^{37} - 45616 q^{38} + 138240 q^{40} - 338336 q^{41} + 730974 q^{43} - 263680 q^{44} - 389344 q^{46} - 338248 q^{47} - 310552 q^{49} - 971840 q^{50} + 514304 q^{52} + 375502 q^{53} + 424840 q^{55} - 1035264 q^{56} + 1741728 q^{58} - 71392 q^{59} + 2101164 q^{61} - 1782816 q^{62} + 1048576 q^{64} - 1578780 q^{65} + 4337162 q^{67} - 2379648 q^{68} + 547520 q^{70} - 2288016 q^{71} - 1107328 q^{73} - 3891424 q^{74} + 364928 q^{76} - 5826200 q^{77} + 60610 q^{79} - 1105920 q^{80} + 2706688 q^{82} - 1485464 q^{83} - 8843820 q^{85} - 5847792 q^{86} + 2109440 q^{88} - 1485090 q^{89} - 2898412 q^{91} + 3114752 q^{92} + 2705984 q^{94} - 8545200 q^{95} + 1935444 q^{97} + 2484416 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) 0 0
\(4\) 64.0000 0.500000
\(5\) −366.876 −1.31257 −0.656287 0.754511i \(-0.727874\pi\)
−0.656287 + 0.754511i \(0.727874\pi\)
\(6\) 0 0
\(7\) −561.721 −0.618981 −0.309491 0.950903i \(-0.600159\pi\)
−0.309491 + 0.950903i \(0.600159\pi\)
\(8\) −512.000 −0.353553
\(9\) 0 0
\(10\) 2935.00 0.928130
\(11\) 1096.20 0.248322 0.124161 0.992262i \(-0.460376\pi\)
0.124161 + 0.992262i \(0.460376\pi\)
\(12\) 0 0
\(13\) 6925.11 0.874229 0.437115 0.899406i \(-0.356000\pi\)
0.437115 + 0.899406i \(0.356000\pi\)
\(14\) 4493.77 0.437686
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) −13654.6 −0.674075 −0.337038 0.941491i \(-0.609425\pi\)
−0.337038 + 0.941491i \(0.609425\pi\)
\(18\) 0 0
\(19\) 29540.8 0.988064 0.494032 0.869444i \(-0.335523\pi\)
0.494032 + 0.869444i \(0.335523\pi\)
\(20\) −23480.0 −0.656287
\(21\) 0 0
\(22\) −8769.58 −0.175590
\(23\) 12167.0 0.208514
\(24\) 0 0
\(25\) 56472.7 0.722851
\(26\) −55400.9 −0.618173
\(27\) 0 0
\(28\) −35950.1 −0.309491
\(29\) −150266. −1.14411 −0.572054 0.820216i \(-0.693853\pi\)
−0.572054 + 0.820216i \(0.693853\pi\)
\(30\) 0 0
\(31\) −204841. −1.23496 −0.617479 0.786588i \(-0.711846\pi\)
−0.617479 + 0.786588i \(0.711846\pi\)
\(32\) −32768.0 −0.176777
\(33\) 0 0
\(34\) 109237. 0.476643
\(35\) 206082. 0.812459
\(36\) 0 0
\(37\) −492163. −1.59736 −0.798681 0.601755i \(-0.794468\pi\)
−0.798681 + 0.601755i \(0.794468\pi\)
\(38\) −236327. −0.698667
\(39\) 0 0
\(40\) 187840. 0.464065
\(41\) −416521. −0.943829 −0.471915 0.881644i \(-0.656437\pi\)
−0.471915 + 0.881644i \(0.656437\pi\)
\(42\) 0 0
\(43\) 371500. 0.712557 0.356279 0.934380i \(-0.384045\pi\)
0.356279 + 0.934380i \(0.384045\pi\)
\(44\) 70156.6 0.124161
\(45\) 0 0
\(46\) −97336.0 −0.147442
\(47\) −930660. −1.30752 −0.653760 0.756702i \(-0.726809\pi\)
−0.653760 + 0.756702i \(0.726809\pi\)
\(48\) 0 0
\(49\) −508013. −0.616862
\(50\) −451782. −0.511133
\(51\) 0 0
\(52\) 443207. 0.437115
\(53\) −1.01318e6 −0.934804 −0.467402 0.884045i \(-0.654810\pi\)
−0.467402 + 0.884045i \(0.654810\pi\)
\(54\) 0 0
\(55\) −402168. −0.325940
\(56\) 287601. 0.218843
\(57\) 0 0
\(58\) 1.20213e6 0.809007
\(59\) 2.28957e6 1.45135 0.725676 0.688036i \(-0.241527\pi\)
0.725676 + 0.688036i \(0.241527\pi\)
\(60\) 0 0
\(61\) 2.95586e6 1.66736 0.833681 0.552246i \(-0.186229\pi\)
0.833681 + 0.552246i \(0.186229\pi\)
\(62\) 1.63873e6 0.873246
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −2.54066e6 −1.14749
\(66\) 0 0
\(67\) −1.46138e6 −0.593609 −0.296804 0.954938i \(-0.595921\pi\)
−0.296804 + 0.954938i \(0.595921\pi\)
\(68\) −873896. −0.337038
\(69\) 0 0
\(70\) −1.64865e6 −0.574495
\(71\) 1.82588e6 0.605437 0.302718 0.953080i \(-0.402106\pi\)
0.302718 + 0.953080i \(0.402106\pi\)
\(72\) 0 0
\(73\) −40352.1 −0.0121405 −0.00607024 0.999982i \(-0.501932\pi\)
−0.00607024 + 0.999982i \(0.501932\pi\)
\(74\) 3.93731e6 1.12951
\(75\) 0 0
\(76\) 1.89061e6 0.494032
\(77\) −615757. −0.153706
\(78\) 0 0
\(79\) −2.36229e6 −0.539061 −0.269531 0.962992i \(-0.586869\pi\)
−0.269531 + 0.962992i \(0.586869\pi\)
\(80\) −1.50272e6 −0.328144
\(81\) 0 0
\(82\) 3.33217e6 0.667388
\(83\) −1.27885e6 −0.245497 −0.122749 0.992438i \(-0.539171\pi\)
−0.122749 + 0.992438i \(0.539171\pi\)
\(84\) 0 0
\(85\) 5.00955e6 0.884774
\(86\) −2.97200e6 −0.503854
\(87\) 0 0
\(88\) −561253. −0.0877949
\(89\) −3.88462e6 −0.584095 −0.292047 0.956404i \(-0.594336\pi\)
−0.292047 + 0.956404i \(0.594336\pi\)
\(90\) 0 0
\(91\) −3.88998e6 −0.541131
\(92\) 778688. 0.104257
\(93\) 0 0
\(94\) 7.44528e6 0.924556
\(95\) −1.08378e7 −1.29691
\(96\) 0 0
\(97\) −8.46502e6 −0.941731 −0.470865 0.882205i \(-0.656058\pi\)
−0.470865 + 0.882205i \(0.656058\pi\)
\(98\) 4.06410e6 0.436188
\(99\) 0 0
\(100\) 3.61425e6 0.361425
\(101\) −5.79563e6 −0.559727 −0.279863 0.960040i \(-0.590289\pi\)
−0.279863 + 0.960040i \(0.590289\pi\)
\(102\) 0 0
\(103\) 1.34033e6 0.120860 0.0604298 0.998172i \(-0.480753\pi\)
0.0604298 + 0.998172i \(0.480753\pi\)
\(104\) −3.54566e6 −0.309087
\(105\) 0 0
\(106\) 8.10543e6 0.661006
\(107\) 1.37815e7 1.08756 0.543779 0.839229i \(-0.316993\pi\)
0.543779 + 0.839229i \(0.316993\pi\)
\(108\) 0 0
\(109\) 9.40692e6 0.695752 0.347876 0.937541i \(-0.386903\pi\)
0.347876 + 0.937541i \(0.386903\pi\)
\(110\) 3.21734e6 0.230475
\(111\) 0 0
\(112\) −2.30081e6 −0.154745
\(113\) 776740. 0.0506409 0.0253204 0.999679i \(-0.491939\pi\)
0.0253204 + 0.999679i \(0.491939\pi\)
\(114\) 0 0
\(115\) −4.46378e6 −0.273691
\(116\) −9.61701e6 −0.572054
\(117\) 0 0
\(118\) −1.83166e7 −1.02626
\(119\) 7.67008e6 0.417240
\(120\) 0 0
\(121\) −1.82855e7 −0.938336
\(122\) −2.36469e7 −1.17900
\(123\) 0 0
\(124\) −1.31099e7 −0.617479
\(125\) 7.94369e6 0.363779
\(126\) 0 0
\(127\) −2.02779e7 −0.878434 −0.439217 0.898381i \(-0.644744\pi\)
−0.439217 + 0.898381i \(0.644744\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 0 0
\(130\) 2.03252e7 0.811398
\(131\) 5.73353e6 0.222829 0.111415 0.993774i \(-0.464462\pi\)
0.111415 + 0.993774i \(0.464462\pi\)
\(132\) 0 0
\(133\) −1.65937e7 −0.611593
\(134\) 1.16910e7 0.419745
\(135\) 0 0
\(136\) 6.99116e6 0.238322
\(137\) 2.96971e7 0.986716 0.493358 0.869826i \(-0.335769\pi\)
0.493358 + 0.869826i \(0.335769\pi\)
\(138\) 0 0
\(139\) 2.70200e7 0.853362 0.426681 0.904402i \(-0.359683\pi\)
0.426681 + 0.904402i \(0.359683\pi\)
\(140\) 1.31892e7 0.406229
\(141\) 0 0
\(142\) −1.46071e7 −0.428108
\(143\) 7.59129e6 0.217090
\(144\) 0 0
\(145\) 5.51289e7 1.50173
\(146\) 322816. 0.00858461
\(147\) 0 0
\(148\) −3.14984e7 −0.798681
\(149\) 1.22268e7 0.302803 0.151401 0.988472i \(-0.451621\pi\)
0.151401 + 0.988472i \(0.451621\pi\)
\(150\) 0 0
\(151\) −7.91442e7 −1.87068 −0.935341 0.353747i \(-0.884907\pi\)
−0.935341 + 0.353747i \(0.884907\pi\)
\(152\) −1.51249e7 −0.349333
\(153\) 0 0
\(154\) 4.92606e6 0.108687
\(155\) 7.51513e7 1.62097
\(156\) 0 0
\(157\) −6.65855e7 −1.37319 −0.686595 0.727040i \(-0.740895\pi\)
−0.686595 + 0.727040i \(0.740895\pi\)
\(158\) 1.88983e7 0.381174
\(159\) 0 0
\(160\) 1.20218e7 0.232033
\(161\) −6.83446e6 −0.129066
\(162\) 0 0
\(163\) 5.57898e7 1.00902 0.504508 0.863407i \(-0.331674\pi\)
0.504508 + 0.863407i \(0.331674\pi\)
\(164\) −2.66573e7 −0.471915
\(165\) 0 0
\(166\) 1.02308e7 0.173593
\(167\) 2.85952e7 0.475100 0.237550 0.971375i \(-0.423656\pi\)
0.237550 + 0.971375i \(0.423656\pi\)
\(168\) 0 0
\(169\) −1.47913e7 −0.235724
\(170\) −4.00764e7 −0.625629
\(171\) 0 0
\(172\) 2.37760e7 0.356279
\(173\) 5.94429e7 0.872847 0.436424 0.899741i \(-0.356245\pi\)
0.436424 + 0.899741i \(0.356245\pi\)
\(174\) 0 0
\(175\) −3.17219e7 −0.447431
\(176\) 4.49002e6 0.0620804
\(177\) 0 0
\(178\) 3.10769e7 0.413017
\(179\) 1.09127e8 1.42216 0.711079 0.703112i \(-0.248207\pi\)
0.711079 + 0.703112i \(0.248207\pi\)
\(180\) 0 0
\(181\) −9.89431e7 −1.24025 −0.620127 0.784501i \(-0.712919\pi\)
−0.620127 + 0.784501i \(0.712919\pi\)
\(182\) 3.11199e7 0.382638
\(183\) 0 0
\(184\) −6.22950e6 −0.0737210
\(185\) 1.80563e8 2.09666
\(186\) 0 0
\(187\) −1.49682e7 −0.167387
\(188\) −5.95622e7 −0.653760
\(189\) 0 0
\(190\) 8.67025e7 0.917052
\(191\) 6.19114e7 0.642916 0.321458 0.946924i \(-0.395827\pi\)
0.321458 + 0.946924i \(0.395827\pi\)
\(192\) 0 0
\(193\) 1.03484e8 1.03615 0.518076 0.855334i \(-0.326648\pi\)
0.518076 + 0.855334i \(0.326648\pi\)
\(194\) 6.77201e7 0.665904
\(195\) 0 0
\(196\) −3.25128e7 −0.308431
\(197\) 1.11986e8 1.04360 0.521798 0.853069i \(-0.325261\pi\)
0.521798 + 0.853069i \(0.325261\pi\)
\(198\) 0 0
\(199\) −1.09861e7 −0.0988228 −0.0494114 0.998779i \(-0.515735\pi\)
−0.0494114 + 0.998779i \(0.515735\pi\)
\(200\) −2.89140e7 −0.255566
\(201\) 0 0
\(202\) 4.63651e7 0.395787
\(203\) 8.44074e7 0.708182
\(204\) 0 0
\(205\) 1.52811e8 1.23885
\(206\) −1.07226e7 −0.0854606
\(207\) 0 0
\(208\) 2.83653e7 0.218557
\(209\) 3.23826e7 0.245357
\(210\) 0 0
\(211\) 7.14052e7 0.523289 0.261644 0.965164i \(-0.415735\pi\)
0.261644 + 0.965164i \(0.415735\pi\)
\(212\) −6.48434e7 −0.467402
\(213\) 0 0
\(214\) −1.10252e8 −0.769019
\(215\) −1.36294e8 −0.935284
\(216\) 0 0
\(217\) 1.15064e8 0.764415
\(218\) −7.52553e7 −0.491971
\(219\) 0 0
\(220\) −2.57388e7 −0.162970
\(221\) −9.45598e7 −0.589296
\(222\) 0 0
\(223\) 1.21124e8 0.731415 0.365707 0.930730i \(-0.380827\pi\)
0.365707 + 0.930730i \(0.380827\pi\)
\(224\) 1.84065e7 0.109421
\(225\) 0 0
\(226\) −6.21392e6 −0.0358085
\(227\) −3.23693e7 −0.183672 −0.0918360 0.995774i \(-0.529274\pi\)
−0.0918360 + 0.995774i \(0.529274\pi\)
\(228\) 0 0
\(229\) 2.14673e8 1.18128 0.590642 0.806934i \(-0.298875\pi\)
0.590642 + 0.806934i \(0.298875\pi\)
\(230\) 3.57102e7 0.193528
\(231\) 0 0
\(232\) 7.69361e7 0.404503
\(233\) 2.34580e8 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(234\) 0 0
\(235\) 3.41436e8 1.71622
\(236\) 1.46533e8 0.725676
\(237\) 0 0
\(238\) −6.13607e7 −0.295033
\(239\) 1.73526e8 0.822189 0.411095 0.911593i \(-0.365147\pi\)
0.411095 + 0.911593i \(0.365147\pi\)
\(240\) 0 0
\(241\) 1.89649e8 0.872752 0.436376 0.899764i \(-0.356262\pi\)
0.436376 + 0.899764i \(0.356262\pi\)
\(242\) 1.46284e8 0.663504
\(243\) 0 0
\(244\) 1.89175e8 0.833681
\(245\) 1.86377e8 0.809678
\(246\) 0 0
\(247\) 2.04574e8 0.863794
\(248\) 1.04879e8 0.436623
\(249\) 0 0
\(250\) −6.35495e7 −0.257230
\(251\) −2.71105e8 −1.08213 −0.541064 0.840981i \(-0.681978\pi\)
−0.541064 + 0.840981i \(0.681978\pi\)
\(252\) 0 0
\(253\) 1.33374e7 0.0517786
\(254\) 1.62223e8 0.621147
\(255\) 0 0
\(256\) 1.67772e7 0.0625000
\(257\) −4.36260e8 −1.60317 −0.801584 0.597882i \(-0.796009\pi\)
−0.801584 + 0.597882i \(0.796009\pi\)
\(258\) 0 0
\(259\) 2.76458e8 0.988737
\(260\) −1.62602e8 −0.573745
\(261\) 0 0
\(262\) −4.58682e7 −0.157564
\(263\) −1.27332e8 −0.431609 −0.215805 0.976437i \(-0.569237\pi\)
−0.215805 + 0.976437i \(0.569237\pi\)
\(264\) 0 0
\(265\) 3.71711e8 1.22700
\(266\) 1.32750e8 0.432461
\(267\) 0 0
\(268\) −9.35281e7 −0.296804
\(269\) −2.50990e8 −0.786182 −0.393091 0.919500i \(-0.628594\pi\)
−0.393091 + 0.919500i \(0.628594\pi\)
\(270\) 0 0
\(271\) 1.19209e8 0.363844 0.181922 0.983313i \(-0.441768\pi\)
0.181922 + 0.983313i \(0.441768\pi\)
\(272\) −5.59293e7 −0.168519
\(273\) 0 0
\(274\) −2.37577e8 −0.697714
\(275\) 6.19052e7 0.179499
\(276\) 0 0
\(277\) −2.70183e8 −0.763799 −0.381899 0.924204i \(-0.624730\pi\)
−0.381899 + 0.924204i \(0.624730\pi\)
\(278\) −2.16160e8 −0.603418
\(279\) 0 0
\(280\) −1.05514e8 −0.287248
\(281\) 6.33010e8 1.70192 0.850958 0.525233i \(-0.176022\pi\)
0.850958 + 0.525233i \(0.176022\pi\)
\(282\) 0 0
\(283\) 6.68305e8 1.75276 0.876379 0.481622i \(-0.159952\pi\)
0.876379 + 0.481622i \(0.159952\pi\)
\(284\) 1.16857e8 0.302718
\(285\) 0 0
\(286\) −6.07303e7 −0.153506
\(287\) 2.33969e8 0.584213
\(288\) 0 0
\(289\) −2.23890e8 −0.545623
\(290\) −4.41031e8 −1.06188
\(291\) 0 0
\(292\) −2.58253e6 −0.00607024
\(293\) −5.36402e8 −1.24582 −0.622908 0.782295i \(-0.714049\pi\)
−0.622908 + 0.782295i \(0.714049\pi\)
\(294\) 0 0
\(295\) −8.39989e8 −1.90501
\(296\) 2.51988e8 0.564753
\(297\) 0 0
\(298\) −9.78141e7 −0.214114
\(299\) 8.42579e7 0.182289
\(300\) 0 0
\(301\) −2.08680e8 −0.441059
\(302\) 6.33154e8 1.32277
\(303\) 0 0
\(304\) 1.20999e8 0.247016
\(305\) −1.08443e9 −2.18854
\(306\) 0 0
\(307\) −1.07221e8 −0.211493 −0.105746 0.994393i \(-0.533723\pi\)
−0.105746 + 0.994393i \(0.533723\pi\)
\(308\) −3.94084e7 −0.0768532
\(309\) 0 0
\(310\) −6.01211e8 −1.14620
\(311\) 9.74611e8 1.83726 0.918629 0.395122i \(-0.129298\pi\)
0.918629 + 0.395122i \(0.129298\pi\)
\(312\) 0 0
\(313\) −6.47093e8 −1.19278 −0.596392 0.802693i \(-0.703400\pi\)
−0.596392 + 0.802693i \(0.703400\pi\)
\(314\) 5.32684e8 0.970992
\(315\) 0 0
\(316\) −1.51186e8 −0.269531
\(317\) −5.22962e8 −0.922068 −0.461034 0.887382i \(-0.652521\pi\)
−0.461034 + 0.887382i \(0.652521\pi\)
\(318\) 0 0
\(319\) −1.64721e8 −0.284107
\(320\) −9.61742e7 −0.164072
\(321\) 0 0
\(322\) 5.46757e7 0.0912638
\(323\) −4.03369e8 −0.666029
\(324\) 0 0
\(325\) 3.91080e8 0.631937
\(326\) −4.46319e8 −0.713483
\(327\) 0 0
\(328\) 2.13259e8 0.333694
\(329\) 5.22771e8 0.809330
\(330\) 0 0
\(331\) −1.00533e8 −0.152373 −0.0761867 0.997094i \(-0.524275\pi\)
−0.0761867 + 0.997094i \(0.524275\pi\)
\(332\) −8.18465e7 −0.122749
\(333\) 0 0
\(334\) −2.28761e8 −0.335946
\(335\) 5.36144e8 0.779156
\(336\) 0 0
\(337\) −4.77653e8 −0.679842 −0.339921 0.940454i \(-0.610400\pi\)
−0.339921 + 0.940454i \(0.610400\pi\)
\(338\) 1.18330e8 0.166682
\(339\) 0 0
\(340\) 3.20611e8 0.442387
\(341\) −2.24547e8 −0.306666
\(342\) 0 0
\(343\) 7.47963e8 1.00081
\(344\) −1.90208e8 −0.251927
\(345\) 0 0
\(346\) −4.75543e8 −0.617196
\(347\) 9.37582e8 1.20464 0.602318 0.798256i \(-0.294244\pi\)
0.602318 + 0.798256i \(0.294244\pi\)
\(348\) 0 0
\(349\) 9.69080e8 1.22031 0.610155 0.792282i \(-0.291107\pi\)
0.610155 + 0.792282i \(0.291107\pi\)
\(350\) 2.53775e8 0.316382
\(351\) 0 0
\(352\) −3.59202e7 −0.0438975
\(353\) 4.61929e8 0.558938 0.279469 0.960155i \(-0.409842\pi\)
0.279469 + 0.960155i \(0.409842\pi\)
\(354\) 0 0
\(355\) −6.69872e8 −0.794680
\(356\) −2.48616e8 −0.292047
\(357\) 0 0
\(358\) −8.73019e8 −1.00562
\(359\) 7.75397e8 0.884491 0.442245 0.896894i \(-0.354182\pi\)
0.442245 + 0.896894i \(0.354182\pi\)
\(360\) 0 0
\(361\) −2.12115e7 −0.0237299
\(362\) 7.91545e8 0.876992
\(363\) 0 0
\(364\) −2.48959e8 −0.270566
\(365\) 1.48042e7 0.0159353
\(366\) 0 0
\(367\) −2.89183e8 −0.305381 −0.152690 0.988274i \(-0.548794\pi\)
−0.152690 + 0.988274i \(0.548794\pi\)
\(368\) 4.98360e7 0.0521286
\(369\) 0 0
\(370\) −1.44450e9 −1.48256
\(371\) 5.69124e8 0.578626
\(372\) 0 0
\(373\) 6.15666e8 0.614277 0.307138 0.951665i \(-0.400629\pi\)
0.307138 + 0.951665i \(0.400629\pi\)
\(374\) 1.19745e8 0.118361
\(375\) 0 0
\(376\) 4.76498e8 0.462278
\(377\) −1.04061e9 −1.00021
\(378\) 0 0
\(379\) 9.95325e8 0.939134 0.469567 0.882897i \(-0.344410\pi\)
0.469567 + 0.882897i \(0.344410\pi\)
\(380\) −6.93620e8 −0.648453
\(381\) 0 0
\(382\) −4.95292e8 −0.454610
\(383\) 2.09760e8 0.190777 0.0953887 0.995440i \(-0.469591\pi\)
0.0953887 + 0.995440i \(0.469591\pi\)
\(384\) 0 0
\(385\) 2.25906e8 0.201751
\(386\) −8.27874e8 −0.732671
\(387\) 0 0
\(388\) −5.41761e8 −0.470865
\(389\) −2.53008e8 −0.217927 −0.108963 0.994046i \(-0.534753\pi\)
−0.108963 + 0.994046i \(0.534753\pi\)
\(390\) 0 0
\(391\) −1.66136e8 −0.140554
\(392\) 2.60102e8 0.218094
\(393\) 0 0
\(394\) −8.95889e8 −0.737934
\(395\) 8.66666e8 0.707558
\(396\) 0 0
\(397\) 1.23414e9 0.989914 0.494957 0.868917i \(-0.335184\pi\)
0.494957 + 0.868917i \(0.335184\pi\)
\(398\) 8.78887e7 0.0698783
\(399\) 0 0
\(400\) 2.31312e8 0.180713
\(401\) −1.41825e9 −1.09837 −0.549185 0.835701i \(-0.685062\pi\)
−0.549185 + 0.835701i \(0.685062\pi\)
\(402\) 0 0
\(403\) −1.41855e9 −1.07964
\(404\) −3.70921e8 −0.279863
\(405\) 0 0
\(406\) −6.75260e8 −0.500760
\(407\) −5.39508e8 −0.396659
\(408\) 0 0
\(409\) 8.33868e8 0.602651 0.301325 0.953521i \(-0.402571\pi\)
0.301325 + 0.953521i \(0.402571\pi\)
\(410\) −1.22249e9 −0.875996
\(411\) 0 0
\(412\) 8.57810e7 0.0604298
\(413\) −1.28610e9 −0.898360
\(414\) 0 0
\(415\) 4.69180e8 0.322234
\(416\) −2.26922e8 −0.154543
\(417\) 0 0
\(418\) −2.59061e8 −0.173494
\(419\) 5.56088e8 0.369313 0.184656 0.982803i \(-0.440883\pi\)
0.184656 + 0.982803i \(0.440883\pi\)
\(420\) 0 0
\(421\) 1.76819e9 1.15489 0.577447 0.816428i \(-0.304049\pi\)
0.577447 + 0.816428i \(0.304049\pi\)
\(422\) −5.71242e8 −0.370021
\(423\) 0 0
\(424\) 5.18747e8 0.330503
\(425\) −7.71113e8 −0.487256
\(426\) 0 0
\(427\) −1.66037e9 −1.03207
\(428\) 8.82014e8 0.543779
\(429\) 0 0
\(430\) 1.09036e9 0.661346
\(431\) −1.86575e9 −1.12249 −0.561244 0.827650i \(-0.689677\pi\)
−0.561244 + 0.827650i \(0.689677\pi\)
\(432\) 0 0
\(433\) −8.86037e8 −0.524498 −0.262249 0.965000i \(-0.584464\pi\)
−0.262249 + 0.965000i \(0.584464\pi\)
\(434\) −9.20510e8 −0.540523
\(435\) 0 0
\(436\) 6.02043e8 0.347876
\(437\) 3.59423e8 0.206026
\(438\) 0 0
\(439\) −1.11162e9 −0.627094 −0.313547 0.949573i \(-0.601517\pi\)
−0.313547 + 0.949573i \(0.601517\pi\)
\(440\) 2.05910e8 0.115237
\(441\) 0 0
\(442\) 7.56478e8 0.416695
\(443\) −3.79851e8 −0.207587 −0.103794 0.994599i \(-0.533098\pi\)
−0.103794 + 0.994599i \(0.533098\pi\)
\(444\) 0 0
\(445\) 1.42517e9 0.766668
\(446\) −9.68993e8 −0.517188
\(447\) 0 0
\(448\) −1.47252e8 −0.0773726
\(449\) −1.04853e9 −0.546664 −0.273332 0.961920i \(-0.588126\pi\)
−0.273332 + 0.961920i \(0.588126\pi\)
\(450\) 0 0
\(451\) −4.56589e8 −0.234373
\(452\) 4.97114e7 0.0253204
\(453\) 0 0
\(454\) 2.58955e8 0.129876
\(455\) 1.42714e9 0.710275
\(456\) 0 0
\(457\) −1.49813e9 −0.734250 −0.367125 0.930172i \(-0.619658\pi\)
−0.367125 + 0.930172i \(0.619658\pi\)
\(458\) −1.71739e9 −0.835293
\(459\) 0 0
\(460\) −2.85682e8 −0.136845
\(461\) −1.49236e9 −0.709447 −0.354723 0.934971i \(-0.615425\pi\)
−0.354723 + 0.934971i \(0.615425\pi\)
\(462\) 0 0
\(463\) −3.06867e9 −1.43687 −0.718433 0.695596i \(-0.755140\pi\)
−0.718433 + 0.695596i \(0.755140\pi\)
\(464\) −6.15489e8 −0.286027
\(465\) 0 0
\(466\) −1.87664e9 −0.859074
\(467\) −1.68526e9 −0.765698 −0.382849 0.923811i \(-0.625057\pi\)
−0.382849 + 0.923811i \(0.625057\pi\)
\(468\) 0 0
\(469\) 8.20886e8 0.367433
\(470\) −2.73149e9 −1.21355
\(471\) 0 0
\(472\) −1.17226e9 −0.513131
\(473\) 4.07238e8 0.176943
\(474\) 0 0
\(475\) 1.66825e9 0.714223
\(476\) 4.90885e8 0.208620
\(477\) 0 0
\(478\) −1.38821e9 −0.581376
\(479\) 2.69509e9 1.12047 0.560233 0.828335i \(-0.310712\pi\)
0.560233 + 0.828335i \(0.310712\pi\)
\(480\) 0 0
\(481\) −3.40829e9 −1.39646
\(482\) −1.51719e9 −0.617129
\(483\) 0 0
\(484\) −1.17027e9 −0.469168
\(485\) 3.10561e9 1.23609
\(486\) 0 0
\(487\) −2.88969e9 −1.13371 −0.566853 0.823819i \(-0.691839\pi\)
−0.566853 + 0.823819i \(0.691839\pi\)
\(488\) −1.51340e9 −0.589502
\(489\) 0 0
\(490\) −1.49102e9 −0.572528
\(491\) 2.94886e9 1.12427 0.562133 0.827047i \(-0.309981\pi\)
0.562133 + 0.827047i \(0.309981\pi\)
\(492\) 0 0
\(493\) 2.05182e9 0.771215
\(494\) −1.63659e9 −0.610795
\(495\) 0 0
\(496\) −8.39030e8 −0.308739
\(497\) −1.02564e9 −0.374754
\(498\) 0 0
\(499\) 3.10068e9 1.11713 0.558567 0.829459i \(-0.311351\pi\)
0.558567 + 0.829459i \(0.311351\pi\)
\(500\) 5.08396e8 0.181889
\(501\) 0 0
\(502\) 2.16884e9 0.765181
\(503\) 6.25596e8 0.219183 0.109591 0.993977i \(-0.465046\pi\)
0.109591 + 0.993977i \(0.465046\pi\)
\(504\) 0 0
\(505\) 2.12628e9 0.734683
\(506\) −1.06699e8 −0.0366130
\(507\) 0 0
\(508\) −1.29778e9 −0.439217
\(509\) −4.35887e9 −1.46508 −0.732541 0.680723i \(-0.761666\pi\)
−0.732541 + 0.680723i \(0.761666\pi\)
\(510\) 0 0
\(511\) 2.26666e7 0.00751472
\(512\) −1.34218e8 −0.0441942
\(513\) 0 0
\(514\) 3.49008e9 1.13361
\(515\) −4.91734e8 −0.158637
\(516\) 0 0
\(517\) −1.02019e9 −0.324685
\(518\) −2.21167e9 −0.699142
\(519\) 0 0
\(520\) 1.30082e9 0.405699
\(521\) −1.86006e9 −0.576229 −0.288115 0.957596i \(-0.593028\pi\)
−0.288115 + 0.957596i \(0.593028\pi\)
\(522\) 0 0
\(523\) 1.89981e9 0.580703 0.290352 0.956920i \(-0.406228\pi\)
0.290352 + 0.956920i \(0.406228\pi\)
\(524\) 3.66946e8 0.111415
\(525\) 0 0
\(526\) 1.01865e9 0.305194
\(527\) 2.79703e9 0.832454
\(528\) 0 0
\(529\) 1.48036e8 0.0434783
\(530\) −2.97368e9 −0.867620
\(531\) 0 0
\(532\) −1.06200e9 −0.305796
\(533\) −2.88446e9 −0.825123
\(534\) 0 0
\(535\) −5.05608e9 −1.42750
\(536\) 7.48225e8 0.209872
\(537\) 0 0
\(538\) 2.00792e9 0.555914
\(539\) −5.56882e8 −0.153180
\(540\) 0 0
\(541\) 6.17189e9 1.67582 0.837910 0.545808i \(-0.183777\pi\)
0.837910 + 0.545808i \(0.183777\pi\)
\(542\) −9.53670e8 −0.257277
\(543\) 0 0
\(544\) 4.47435e8 0.119161
\(545\) −3.45117e9 −0.913226
\(546\) 0 0
\(547\) 4.97696e9 1.30020 0.650098 0.759851i \(-0.274728\pi\)
0.650098 + 0.759851i \(0.274728\pi\)
\(548\) 1.90061e9 0.493358
\(549\) 0 0
\(550\) −4.95242e8 −0.126925
\(551\) −4.43898e9 −1.13045
\(552\) 0 0
\(553\) 1.32695e9 0.333669
\(554\) 2.16146e9 0.540087
\(555\) 0 0
\(556\) 1.72928e9 0.426681
\(557\) 2.16771e9 0.531506 0.265753 0.964041i \(-0.414379\pi\)
0.265753 + 0.964041i \(0.414379\pi\)
\(558\) 0 0
\(559\) 2.57268e9 0.622938
\(560\) 8.44111e8 0.203115
\(561\) 0 0
\(562\) −5.06408e9 −1.20344
\(563\) −4.33727e9 −1.02432 −0.512162 0.858889i \(-0.671155\pi\)
−0.512162 + 0.858889i \(0.671155\pi\)
\(564\) 0 0
\(565\) −2.84967e8 −0.0664699
\(566\) −5.34644e9 −1.23939
\(567\) 0 0
\(568\) −9.34852e8 −0.214054
\(569\) 1.76759e9 0.402243 0.201122 0.979566i \(-0.435541\pi\)
0.201122 + 0.979566i \(0.435541\pi\)
\(570\) 0 0
\(571\) 4.35096e9 0.978046 0.489023 0.872271i \(-0.337353\pi\)
0.489023 + 0.872271i \(0.337353\pi\)
\(572\) 4.85843e8 0.108545
\(573\) 0 0
\(574\) −1.87175e9 −0.413101
\(575\) 6.87104e8 0.150725
\(576\) 0 0
\(577\) 7.46172e9 1.61705 0.808525 0.588462i \(-0.200266\pi\)
0.808525 + 0.588462i \(0.200266\pi\)
\(578\) 1.79112e9 0.385813
\(579\) 0 0
\(580\) 3.52825e9 0.750864
\(581\) 7.18358e8 0.151958
\(582\) 0 0
\(583\) −1.11064e9 −0.232132
\(584\) 2.06603e7 0.00429231
\(585\) 0 0
\(586\) 4.29122e9 0.880925
\(587\) 3.27341e9 0.667985 0.333992 0.942576i \(-0.391604\pi\)
0.333992 + 0.942576i \(0.391604\pi\)
\(588\) 0 0
\(589\) −6.05118e9 −1.22022
\(590\) 6.71991e9 1.34704
\(591\) 0 0
\(592\) −2.01590e9 −0.399340
\(593\) 5.07631e9 0.999670 0.499835 0.866121i \(-0.333394\pi\)
0.499835 + 0.866121i \(0.333394\pi\)
\(594\) 0 0
\(595\) −2.81397e9 −0.547658
\(596\) 7.82513e8 0.151401
\(597\) 0 0
\(598\) −6.74063e8 −0.128898
\(599\) 9.57665e9 1.82062 0.910311 0.413926i \(-0.135843\pi\)
0.910311 + 0.413926i \(0.135843\pi\)
\(600\) 0 0
\(601\) −1.02812e9 −0.193189 −0.0965945 0.995324i \(-0.530795\pi\)
−0.0965945 + 0.995324i \(0.530795\pi\)
\(602\) 1.66944e9 0.311876
\(603\) 0 0
\(604\) −5.06523e9 −0.935341
\(605\) 6.70851e9 1.23164
\(606\) 0 0
\(607\) 2.45045e9 0.444719 0.222359 0.974965i \(-0.428624\pi\)
0.222359 + 0.974965i \(0.428624\pi\)
\(608\) −9.67994e8 −0.174667
\(609\) 0 0
\(610\) 8.67548e9 1.54753
\(611\) −6.44492e9 −1.14307
\(612\) 0 0
\(613\) 8.90460e8 0.156136 0.0780679 0.996948i \(-0.475125\pi\)
0.0780679 + 0.996948i \(0.475125\pi\)
\(614\) 8.57767e8 0.149548
\(615\) 0 0
\(616\) 3.15268e8 0.0543434
\(617\) −3.73214e9 −0.639675 −0.319838 0.947472i \(-0.603628\pi\)
−0.319838 + 0.947472i \(0.603628\pi\)
\(618\) 0 0
\(619\) −9.73916e9 −1.65046 −0.825228 0.564799i \(-0.808954\pi\)
−0.825228 + 0.564799i \(0.808954\pi\)
\(620\) 4.80968e9 0.810486
\(621\) 0 0
\(622\) −7.79689e9 −1.29914
\(623\) 2.18207e9 0.361544
\(624\) 0 0
\(625\) −7.32628e9 −1.20034
\(626\) 5.17675e9 0.843426
\(627\) 0 0
\(628\) −4.26147e9 −0.686595
\(629\) 6.72030e9 1.07674
\(630\) 0 0
\(631\) −8.10602e8 −0.128441 −0.0642207 0.997936i \(-0.520456\pi\)
−0.0642207 + 0.997936i \(0.520456\pi\)
\(632\) 1.20949e9 0.190587
\(633\) 0 0
\(634\) 4.18370e9 0.652001
\(635\) 7.43946e9 1.15301
\(636\) 0 0
\(637\) −3.51805e9 −0.539279
\(638\) 1.31777e9 0.200894
\(639\) 0 0
\(640\) 7.69394e8 0.116016
\(641\) 1.53566e9 0.230299 0.115150 0.993348i \(-0.463265\pi\)
0.115150 + 0.993348i \(0.463265\pi\)
\(642\) 0 0
\(643\) 9.03556e9 1.34035 0.670173 0.742205i \(-0.266220\pi\)
0.670173 + 0.742205i \(0.266220\pi\)
\(644\) −4.37405e8 −0.0645332
\(645\) 0 0
\(646\) 3.22695e9 0.470954
\(647\) 6.44225e8 0.0935132 0.0467566 0.998906i \(-0.485111\pi\)
0.0467566 + 0.998906i \(0.485111\pi\)
\(648\) 0 0
\(649\) 2.50983e9 0.360402
\(650\) −3.12864e9 −0.446847
\(651\) 0 0
\(652\) 3.57055e9 0.504508
\(653\) 3.14760e9 0.442368 0.221184 0.975232i \(-0.429008\pi\)
0.221184 + 0.975232i \(0.429008\pi\)
\(654\) 0 0
\(655\) −2.10349e9 −0.292480
\(656\) −1.70607e9 −0.235957
\(657\) 0 0
\(658\) −4.18217e9 −0.572283
\(659\) −1.35636e10 −1.84619 −0.923093 0.384577i \(-0.874347\pi\)
−0.923093 + 0.384577i \(0.874347\pi\)
\(660\) 0 0
\(661\) 1.38238e10 1.86176 0.930880 0.365325i \(-0.119042\pi\)
0.930880 + 0.365325i \(0.119042\pi\)
\(662\) 8.04261e8 0.107744
\(663\) 0 0
\(664\) 6.54772e8 0.0867964
\(665\) 6.08782e9 0.802761
\(666\) 0 0
\(667\) −1.82828e9 −0.238563
\(668\) 1.83009e9 0.237550
\(669\) 0 0
\(670\) −4.28915e9 −0.550946
\(671\) 3.24021e9 0.414042
\(672\) 0 0
\(673\) −3.45394e9 −0.436780 −0.218390 0.975862i \(-0.570080\pi\)
−0.218390 + 0.975862i \(0.570080\pi\)
\(674\) 3.82123e9 0.480721
\(675\) 0 0
\(676\) −9.46644e8 −0.117862
\(677\) −6.53708e9 −0.809699 −0.404849 0.914383i \(-0.632676\pi\)
−0.404849 + 0.914383i \(0.632676\pi\)
\(678\) 0 0
\(679\) 4.75498e9 0.582914
\(680\) −2.56489e9 −0.312815
\(681\) 0 0
\(682\) 1.79637e9 0.216846
\(683\) 1.53942e10 1.84877 0.924386 0.381457i \(-0.124578\pi\)
0.924386 + 0.381457i \(0.124578\pi\)
\(684\) 0 0
\(685\) −1.08951e10 −1.29514
\(686\) −5.98370e9 −0.707678
\(687\) 0 0
\(688\) 1.52167e9 0.178139
\(689\) −7.01638e9 −0.817233
\(690\) 0 0
\(691\) −6.43606e9 −0.742074 −0.371037 0.928618i \(-0.620998\pi\)
−0.371037 + 0.928618i \(0.620998\pi\)
\(692\) 3.80434e9 0.436424
\(693\) 0 0
\(694\) −7.50066e9 −0.851807
\(695\) −9.91297e9 −1.12010
\(696\) 0 0
\(697\) 5.68744e9 0.636212
\(698\) −7.75264e9 −0.862890
\(699\) 0 0
\(700\) −2.03020e9 −0.223716
\(701\) −3.29232e9 −0.360985 −0.180492 0.983576i \(-0.557769\pi\)
−0.180492 + 0.983576i \(0.557769\pi\)
\(702\) 0 0
\(703\) −1.45389e10 −1.57829
\(704\) 2.87362e8 0.0310402
\(705\) 0 0
\(706\) −3.69543e9 −0.395229
\(707\) 3.25553e9 0.346460
\(708\) 0 0
\(709\) −1.33790e10 −1.40981 −0.704907 0.709300i \(-0.749011\pi\)
−0.704907 + 0.709300i \(0.749011\pi\)
\(710\) 5.35898e9 0.561924
\(711\) 0 0
\(712\) 1.98892e9 0.206509
\(713\) −2.49231e9 −0.257506
\(714\) 0 0
\(715\) −2.78506e9 −0.284947
\(716\) 6.98415e9 0.711079
\(717\) 0 0
\(718\) −6.20317e9 −0.625430
\(719\) −9.99931e9 −1.00327 −0.501636 0.865079i \(-0.667268\pi\)
−0.501636 + 0.865079i \(0.667268\pi\)
\(720\) 0 0
\(721\) −7.52890e8 −0.0748098
\(722\) 1.69692e8 0.0167796
\(723\) 0 0
\(724\) −6.33236e9 −0.620127
\(725\) −8.48592e9 −0.827020
\(726\) 0 0
\(727\) 3.87514e9 0.374039 0.187019 0.982356i \(-0.440117\pi\)
0.187019 + 0.982356i \(0.440117\pi\)
\(728\) 1.99167e9 0.191319
\(729\) 0 0
\(730\) −1.18433e8 −0.0112679
\(731\) −5.07270e9 −0.480317
\(732\) 0 0
\(733\) 2.03185e10 1.90559 0.952793 0.303620i \(-0.0981954\pi\)
0.952793 + 0.303620i \(0.0981954\pi\)
\(734\) 2.31346e9 0.215937
\(735\) 0 0
\(736\) −3.98688e8 −0.0368605
\(737\) −1.60196e9 −0.147406
\(738\) 0 0
\(739\) 6.01270e9 0.548042 0.274021 0.961724i \(-0.411646\pi\)
0.274021 + 0.961724i \(0.411646\pi\)
\(740\) 1.15560e10 1.04833
\(741\) 0 0
\(742\) −4.55299e9 −0.409150
\(743\) −7.10581e9 −0.635554 −0.317777 0.948165i \(-0.602936\pi\)
−0.317777 + 0.948165i \(0.602936\pi\)
\(744\) 0 0
\(745\) −4.48570e9 −0.397451
\(746\) −4.92533e9 −0.434359
\(747\) 0 0
\(748\) −9.57962e8 −0.0836937
\(749\) −7.74134e9 −0.673178
\(750\) 0 0
\(751\) −2.66674e9 −0.229743 −0.114871 0.993380i \(-0.536646\pi\)
−0.114871 + 0.993380i \(0.536646\pi\)
\(752\) −3.81198e9 −0.326880
\(753\) 0 0
\(754\) 8.32486e9 0.707257
\(755\) 2.90361e10 2.45541
\(756\) 0 0
\(757\) 1.21053e10 1.01424 0.507119 0.861876i \(-0.330711\pi\)
0.507119 + 0.861876i \(0.330711\pi\)
\(758\) −7.96260e9 −0.664068
\(759\) 0 0
\(760\) 5.54896e9 0.458526
\(761\) 2.97000e9 0.244293 0.122146 0.992512i \(-0.461022\pi\)
0.122146 + 0.992512i \(0.461022\pi\)
\(762\) 0 0
\(763\) −5.28406e9 −0.430657
\(764\) 3.96233e9 0.321458
\(765\) 0 0
\(766\) −1.67808e9 −0.134900
\(767\) 1.58556e10 1.26881
\(768\) 0 0
\(769\) 1.95706e10 1.55190 0.775948 0.630797i \(-0.217272\pi\)
0.775948 + 0.630797i \(0.217272\pi\)
\(770\) −1.80725e9 −0.142659
\(771\) 0 0
\(772\) 6.62299e9 0.518076
\(773\) 1.54223e9 0.120094 0.0600468 0.998196i \(-0.480875\pi\)
0.0600468 + 0.998196i \(0.480875\pi\)
\(774\) 0 0
\(775\) −1.15680e10 −0.892690
\(776\) 4.33409e9 0.332952
\(777\) 0 0
\(778\) 2.02406e9 0.154098
\(779\) −1.23044e10 −0.932564
\(780\) 0 0
\(781\) 2.00153e9 0.150343
\(782\) 1.32909e9 0.0993870
\(783\) 0 0
\(784\) −2.08082e9 −0.154216
\(785\) 2.44286e10 1.80241
\(786\) 0 0
\(787\) −1.76348e10 −1.28961 −0.644805 0.764347i \(-0.723061\pi\)
−0.644805 + 0.764347i \(0.723061\pi\)
\(788\) 7.16712e9 0.521798
\(789\) 0 0
\(790\) −6.93333e9 −0.500319
\(791\) −4.36311e8 −0.0313458
\(792\) 0 0
\(793\) 2.04697e10 1.45766
\(794\) −9.87312e9 −0.699975
\(795\) 0 0
\(796\) −7.03110e8 −0.0494114
\(797\) −2.10641e10 −1.47380 −0.736901 0.676000i \(-0.763712\pi\)
−0.736901 + 0.676000i \(0.763712\pi\)
\(798\) 0 0
\(799\) 1.27078e10 0.881367
\(800\) −1.85050e9 −0.127783
\(801\) 0 0
\(802\) 1.13460e10 0.776664
\(803\) −4.42338e7 −0.00301474
\(804\) 0 0
\(805\) 2.50740e9 0.169409
\(806\) 1.13484e10 0.763417
\(807\) 0 0
\(808\) 2.96736e9 0.197893
\(809\) 2.12308e10 1.40977 0.704884 0.709323i \(-0.250999\pi\)
0.704884 + 0.709323i \(0.250999\pi\)
\(810\) 0 0
\(811\) 9.31477e9 0.613196 0.306598 0.951839i \(-0.400809\pi\)
0.306598 + 0.951839i \(0.400809\pi\)
\(812\) 5.40208e9 0.354091
\(813\) 0 0
\(814\) 4.31606e9 0.280480
\(815\) −2.04679e10 −1.32441
\(816\) 0 0
\(817\) 1.09744e10 0.704052
\(818\) −6.67094e9 −0.426138
\(819\) 0 0
\(820\) 9.77993e9 0.619423
\(821\) 1.51846e10 0.957642 0.478821 0.877913i \(-0.341064\pi\)
0.478821 + 0.877913i \(0.341064\pi\)
\(822\) 0 0
\(823\) −9.38917e9 −0.587121 −0.293561 0.955940i \(-0.594840\pi\)
−0.293561 + 0.955940i \(0.594840\pi\)
\(824\) −6.86248e8 −0.0427303
\(825\) 0 0
\(826\) 1.02888e10 0.635236
\(827\) −1.40878e10 −0.866114 −0.433057 0.901366i \(-0.642565\pi\)
−0.433057 + 0.901366i \(0.642565\pi\)
\(828\) 0 0
\(829\) 1.99910e10 1.21869 0.609345 0.792905i \(-0.291432\pi\)
0.609345 + 0.792905i \(0.291432\pi\)
\(830\) −3.75344e9 −0.227854
\(831\) 0 0
\(832\) 1.81538e9 0.109279
\(833\) 6.93672e9 0.415812
\(834\) 0 0
\(835\) −1.04909e10 −0.623604
\(836\) 2.07248e9 0.122679
\(837\) 0 0
\(838\) −4.44870e9 −0.261143
\(839\) −2.49729e10 −1.45983 −0.729915 0.683538i \(-0.760440\pi\)
−0.729915 + 0.683538i \(0.760440\pi\)
\(840\) 0 0
\(841\) 5.32994e9 0.308984
\(842\) −1.41455e10 −0.816633
\(843\) 0 0
\(844\) 4.56993e9 0.261644
\(845\) 5.42657e9 0.309405
\(846\) 0 0
\(847\) 1.02714e10 0.580813
\(848\) −4.14998e9 −0.233701
\(849\) 0 0
\(850\) 6.16891e9 0.344542
\(851\) −5.98815e9 −0.333073
\(852\) 0 0
\(853\) 2.15537e10 1.18905 0.594526 0.804077i \(-0.297340\pi\)
0.594526 + 0.804077i \(0.297340\pi\)
\(854\) 1.32830e10 0.729781
\(855\) 0 0
\(856\) −7.05611e9 −0.384510
\(857\) 1.83252e9 0.0994524 0.0497262 0.998763i \(-0.484165\pi\)
0.0497262 + 0.998763i \(0.484165\pi\)
\(858\) 0 0
\(859\) 2.88323e10 1.55204 0.776020 0.630708i \(-0.217235\pi\)
0.776020 + 0.630708i \(0.217235\pi\)
\(860\) −8.72284e9 −0.467642
\(861\) 0 0
\(862\) 1.49260e10 0.793719
\(863\) 2.80605e10 1.48613 0.743067 0.669217i \(-0.233370\pi\)
0.743067 + 0.669217i \(0.233370\pi\)
\(864\) 0 0
\(865\) −2.18081e10 −1.14568
\(866\) 7.08829e9 0.370876
\(867\) 0 0
\(868\) 7.36408e9 0.382208
\(869\) −2.58953e9 −0.133860
\(870\) 0 0
\(871\) −1.01202e10 −0.518950
\(872\) −4.81634e9 −0.245986
\(873\) 0 0
\(874\) −2.87539e9 −0.145682
\(875\) −4.46214e9 −0.225172
\(876\) 0 0
\(877\) −1.07928e10 −0.540299 −0.270149 0.962818i \(-0.587073\pi\)
−0.270149 + 0.962818i \(0.587073\pi\)
\(878\) 8.89300e9 0.443422
\(879\) 0 0
\(880\) −1.64728e9 −0.0814851
\(881\) 2.86597e10 1.41207 0.706034 0.708178i \(-0.250483\pi\)
0.706034 + 0.708178i \(0.250483\pi\)
\(882\) 0 0
\(883\) −3.47774e10 −1.69994 −0.849972 0.526828i \(-0.823381\pi\)
−0.849972 + 0.526828i \(0.823381\pi\)
\(884\) −6.05183e9 −0.294648
\(885\) 0 0
\(886\) 3.03881e9 0.146786
\(887\) −6.42202e9 −0.308986 −0.154493 0.987994i \(-0.549374\pi\)
−0.154493 + 0.987994i \(0.549374\pi\)
\(888\) 0 0
\(889\) 1.13905e10 0.543734
\(890\) −1.14014e10 −0.542116
\(891\) 0 0
\(892\) 7.75195e9 0.365707
\(893\) −2.74925e10 −1.29191
\(894\) 0 0
\(895\) −4.00362e10 −1.86669
\(896\) 1.17801e9 0.0547107
\(897\) 0 0
\(898\) 8.38827e9 0.386550
\(899\) 3.07807e10 1.41292
\(900\) 0 0
\(901\) 1.38346e10 0.630128
\(902\) 3.65271e9 0.165727
\(903\) 0 0
\(904\) −3.97691e8 −0.0179043
\(905\) 3.62998e10 1.62793
\(906\) 0 0
\(907\) −3.96724e10 −1.76548 −0.882740 0.469861i \(-0.844304\pi\)
−0.882740 + 0.469861i \(0.844304\pi\)
\(908\) −2.07164e9 −0.0918360
\(909\) 0 0
\(910\) −1.14171e10 −0.502240
\(911\) −3.39607e10 −1.48820 −0.744102 0.668066i \(-0.767122\pi\)
−0.744102 + 0.668066i \(0.767122\pi\)
\(912\) 0 0
\(913\) −1.40187e9 −0.0609623
\(914\) 1.19851e10 0.519193
\(915\) 0 0
\(916\) 1.37391e10 0.590642
\(917\) −3.22064e9 −0.137927
\(918\) 0 0
\(919\) 1.98186e10 0.842304 0.421152 0.906990i \(-0.361626\pi\)
0.421152 + 0.906990i \(0.361626\pi\)
\(920\) 2.28545e9 0.0967642
\(921\) 0 0
\(922\) 1.19389e10 0.501655
\(923\) 1.26444e10 0.529290
\(924\) 0 0
\(925\) −2.77938e10 −1.15465
\(926\) 2.45493e10 1.01602
\(927\) 0 0
\(928\) 4.92391e9 0.202252
\(929\) 2.08980e10 0.855164 0.427582 0.903976i \(-0.359365\pi\)
0.427582 + 0.903976i \(0.359365\pi\)
\(930\) 0 0
\(931\) −1.50071e10 −0.609499
\(932\) 1.50131e10 0.607457
\(933\) 0 0
\(934\) 1.34821e10 0.541430
\(935\) 5.49145e9 0.219708
\(936\) 0 0
\(937\) −4.61325e9 −0.183197 −0.0915986 0.995796i \(-0.529198\pi\)
−0.0915986 + 0.995796i \(0.529198\pi\)
\(938\) −6.56709e9 −0.259814
\(939\) 0 0
\(940\) 2.18519e10 0.858109
\(941\) −2.61949e10 −1.02483 −0.512417 0.858737i \(-0.671250\pi\)
−0.512417 + 0.858737i \(0.671250\pi\)
\(942\) 0 0
\(943\) −5.06781e9 −0.196802
\(944\) 9.37810e9 0.362838
\(945\) 0 0
\(946\) −3.25790e9 −0.125118
\(947\) 1.01216e10 0.387278 0.193639 0.981073i \(-0.437971\pi\)
0.193639 + 0.981073i \(0.437971\pi\)
\(948\) 0 0
\(949\) −2.79443e8 −0.0106136
\(950\) −1.33460e10 −0.505032
\(951\) 0 0
\(952\) −3.92708e9 −0.147517
\(953\) 5.16517e9 0.193312 0.0966562 0.995318i \(-0.469185\pi\)
0.0966562 + 0.995318i \(0.469185\pi\)
\(954\) 0 0
\(955\) −2.27138e10 −0.843875
\(956\) 1.11057e10 0.411095
\(957\) 0 0
\(958\) −2.15607e10 −0.792289
\(959\) −1.66815e10 −0.610759
\(960\) 0 0
\(961\) 1.44474e10 0.525119
\(962\) 2.72663e10 0.987446
\(963\) 0 0
\(964\) 1.21375e10 0.436376
\(965\) −3.79659e10 −1.36003
\(966\) 0 0
\(967\) 2.33871e10 0.831734 0.415867 0.909425i \(-0.363478\pi\)
0.415867 + 0.909425i \(0.363478\pi\)
\(968\) 9.36219e9 0.331752
\(969\) 0 0
\(970\) −2.48449e10 −0.874049
\(971\) 4.05447e9 0.142124 0.0710619 0.997472i \(-0.477361\pi\)
0.0710619 + 0.997472i \(0.477361\pi\)
\(972\) 0 0
\(973\) −1.51777e10 −0.528215
\(974\) 2.31176e10 0.801651
\(975\) 0 0
\(976\) 1.21072e10 0.416841
\(977\) −5.68903e10 −1.95167 −0.975837 0.218499i \(-0.929884\pi\)
−0.975837 + 0.218499i \(0.929884\pi\)
\(978\) 0 0
\(979\) −4.25831e9 −0.145043
\(980\) 1.19282e10 0.404839
\(981\) 0 0
\(982\) −2.35909e10 −0.794977
\(983\) −3.74559e10 −1.25772 −0.628859 0.777519i \(-0.716478\pi\)
−0.628859 + 0.777519i \(0.716478\pi\)
\(984\) 0 0
\(985\) −4.10850e10 −1.36980
\(986\) −1.64146e10 −0.545331
\(987\) 0 0
\(988\) 1.30927e10 0.431897
\(989\) 4.52004e9 0.148578
\(990\) 0 0
\(991\) 1.00112e8 0.00326759 0.00163379 0.999999i \(-0.499480\pi\)
0.00163379 + 0.999999i \(0.499480\pi\)
\(992\) 6.71224e9 0.218312
\(993\) 0 0
\(994\) 8.20509e9 0.264991
\(995\) 4.03053e9 0.129712
\(996\) 0 0
\(997\) −3.72236e10 −1.18956 −0.594779 0.803889i \(-0.702760\pi\)
−0.594779 + 0.803889i \(0.702760\pi\)
\(998\) −2.48055e10 −0.789934
\(999\) 0 0
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 414.8.a.i.1.1 4
3.2 odd 2 138.8.a.h.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.8.a.h.1.4 4 3.2 odd 2
414.8.a.i.1.1 4 1.1 even 1 trivial