Properties

Label 315.10.a.b.1.2
Level $315$
Weight $10$
Character 315.1
Self dual yes
Analytic conductor $162.236$
Analytic rank $1$
Dimension $2$
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] = [315,10,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("315.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(162.236288392\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+14.8284 q^{2} -292.118 q^{4} -625.000 q^{5} +2401.00 q^{7} -11923.8 q^{8} +O(q^{10})\) \(q+14.8284 q^{2} -292.118 q^{4} -625.000 q^{5} +2401.00 q^{7} -11923.8 q^{8} -9267.77 q^{10} +16523.6 q^{11} +26311.4 q^{13} +35603.1 q^{14} -27246.9 q^{16} +144003. q^{17} -159710. q^{19} +182574. q^{20} +245019. q^{22} -2.07393e6 q^{23} +390625. q^{25} +390156. q^{26} -701375. q^{28} +4.94938e6 q^{29} +4.22040e6 q^{31} +5.70096e6 q^{32} +2.13533e6 q^{34} -1.50062e6 q^{35} -1.29081e7 q^{37} -2.36824e6 q^{38} +7.45238e6 q^{40} +2.87518e7 q^{41} +3.54825e7 q^{43} -4.82683e6 q^{44} -3.07532e7 q^{46} -5.95633e7 q^{47} +5.76480e6 q^{49} +5.79235e6 q^{50} -7.68602e6 q^{52} -2.31161e6 q^{53} -1.03272e7 q^{55} -2.86290e7 q^{56} +7.33915e7 q^{58} +1.68651e8 q^{59} -6.70167e7 q^{61} +6.25819e7 q^{62} +9.84867e7 q^{64} -1.64446e7 q^{65} -1.56259e8 q^{67} -4.20657e7 q^{68} -2.22519e7 q^{70} -6.95067e7 q^{71} -7.83438e7 q^{73} -1.91407e8 q^{74} +4.66540e7 q^{76} +3.96731e7 q^{77} -4.26957e8 q^{79} +1.70293e7 q^{80} +4.26344e8 q^{82} +5.31242e8 q^{83} -9.00016e7 q^{85} +5.26149e8 q^{86} -1.97024e8 q^{88} +1.14168e8 q^{89} +6.31736e7 q^{91} +6.05833e8 q^{92} -8.83231e8 q^{94} +9.98185e7 q^{95} -1.46573e9 q^{97} +8.54829e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8} - 15000 q^{10} - 18566 q^{11} - 51090 q^{13} + 57624 q^{14} + 112768 q^{16} + 373910 q^{17} - 143276 q^{19} + 450000 q^{20} - 76808 q^{22} + 498908 q^{23} + 781250 q^{25} - 319736 q^{26} - 1728720 q^{28} + 11577554 q^{29} - 3953760 q^{31} + 11398656 q^{32} + 4243944 q^{34} - 3001250 q^{35} - 3205412 q^{37} - 2217520 q^{38} + 12840000 q^{40} - 1058992 q^{41} + 15948180 q^{43} + 10187376 q^{44} - 7156176 q^{46} - 65501290 q^{47} + 11529602 q^{49} + 9375000 q^{50} + 25432656 q^{52} + 25114688 q^{53} + 11603750 q^{55} - 49326144 q^{56} + 134182296 q^{58} + 116159208 q^{59} - 44688544 q^{61} - 12388000 q^{62} + 79055872 q^{64} + 31931250 q^{65} + 118092496 q^{67} - 140439024 q^{68} - 36015000 q^{70} + 294165824 q^{71} - 57419332 q^{73} - 102418064 q^{74} + 39622368 q^{76} - 44576966 q^{77} - 692852854 q^{79} - 70480000 q^{80} + 152932128 q^{82} + 540679928 q^{83} - 233693750 q^{85} + 346989008 q^{86} + 105455296 q^{88} + 779043704 q^{89} - 122667090 q^{91} - 495040608 q^{92} - 937691032 q^{94} + 89547500 q^{95} - 2673039406 q^{97} + 138355224 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.8284 0.655330 0.327665 0.944794i \(-0.393738\pi\)
0.327665 + 0.944794i \(0.393738\pi\)
\(3\) 0 0
\(4\) −292.118 −0.570542
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) −11923.8 −1.02922
\(9\) 0 0
\(10\) −9267.77 −0.293073
\(11\) 16523.6 0.340280 0.170140 0.985420i \(-0.445578\pi\)
0.170140 + 0.985420i \(0.445578\pi\)
\(12\) 0 0
\(13\) 26311.4 0.255505 0.127752 0.991806i \(-0.459224\pi\)
0.127752 + 0.991806i \(0.459224\pi\)
\(14\) 35603.1 0.247691
\(15\) 0 0
\(16\) −27246.9 −0.103939
\(17\) 144003. 0.418167 0.209084 0.977898i \(-0.432952\pi\)
0.209084 + 0.977898i \(0.432952\pi\)
\(18\) 0 0
\(19\) −159710. −0.281151 −0.140576 0.990070i \(-0.544895\pi\)
−0.140576 + 0.990070i \(0.544895\pi\)
\(20\) 182574. 0.255154
\(21\) 0 0
\(22\) 245019. 0.222996
\(23\) −2.07393e6 −1.54533 −0.772663 0.634817i \(-0.781075\pi\)
−0.772663 + 0.634817i \(0.781075\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 390156. 0.167440
\(27\) 0 0
\(28\) −701375. −0.215645
\(29\) 4.94938e6 1.29945 0.649725 0.760169i \(-0.274884\pi\)
0.649725 + 0.760169i \(0.274884\pi\)
\(30\) 0 0
\(31\) 4.22040e6 0.820779 0.410389 0.911910i \(-0.365393\pi\)
0.410389 + 0.911910i \(0.365393\pi\)
\(32\) 5.70096e6 0.961110
\(33\) 0 0
\(34\) 2.13533e6 0.274038
\(35\) −1.50062e6 −0.169031
\(36\) 0 0
\(37\) −1.29081e7 −1.13228 −0.566142 0.824308i \(-0.691565\pi\)
−0.566142 + 0.824308i \(0.691565\pi\)
\(38\) −2.36824e6 −0.184247
\(39\) 0 0
\(40\) 7.45238e6 0.460283
\(41\) 2.87518e7 1.58905 0.794525 0.607231i \(-0.207720\pi\)
0.794525 + 0.607231i \(0.207720\pi\)
\(42\) 0 0
\(43\) 3.54825e7 1.58273 0.791363 0.611347i \(-0.209372\pi\)
0.791363 + 0.611347i \(0.209372\pi\)
\(44\) −4.82683e6 −0.194144
\(45\) 0 0
\(46\) −3.07532e7 −1.01270
\(47\) −5.95633e7 −1.78049 −0.890243 0.455485i \(-0.849466\pi\)
−0.890243 + 0.455485i \(0.849466\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 5.79235e6 0.131066
\(51\) 0 0
\(52\) −7.68602e6 −0.145776
\(53\) −2.31161e6 −0.0402414 −0.0201207 0.999798i \(-0.506405\pi\)
−0.0201207 + 0.999798i \(0.506405\pi\)
\(54\) 0 0
\(55\) −1.03272e7 −0.152178
\(56\) −2.86290e7 −0.389010
\(57\) 0 0
\(58\) 7.33915e7 0.851569
\(59\) 1.68651e8 1.81198 0.905992 0.423296i \(-0.139127\pi\)
0.905992 + 0.423296i \(0.139127\pi\)
\(60\) 0 0
\(61\) −6.70167e7 −0.619725 −0.309863 0.950781i \(-0.600283\pi\)
−0.309863 + 0.950781i \(0.600283\pi\)
\(62\) 6.25819e7 0.537881
\(63\) 0 0
\(64\) 9.84867e7 0.733783
\(65\) −1.64446e7 −0.114265
\(66\) 0 0
\(67\) −1.56259e8 −0.947343 −0.473671 0.880702i \(-0.657072\pi\)
−0.473671 + 0.880702i \(0.657072\pi\)
\(68\) −4.20657e7 −0.238582
\(69\) 0 0
\(70\) −2.22519e7 −0.110771
\(71\) −6.95067e7 −0.324612 −0.162306 0.986740i \(-0.551893\pi\)
−0.162306 + 0.986740i \(0.551893\pi\)
\(72\) 0 0
\(73\) −7.83438e7 −0.322888 −0.161444 0.986882i \(-0.551615\pi\)
−0.161444 + 0.986882i \(0.551615\pi\)
\(74\) −1.91407e8 −0.742020
\(75\) 0 0
\(76\) 4.66540e7 0.160409
\(77\) 3.96731e7 0.128614
\(78\) 0 0
\(79\) −4.26957e8 −1.23328 −0.616641 0.787245i \(-0.711507\pi\)
−0.616641 + 0.787245i \(0.711507\pi\)
\(80\) 1.70293e7 0.0464828
\(81\) 0 0
\(82\) 4.26344e8 1.04135
\(83\) 5.31242e8 1.22869 0.614343 0.789039i \(-0.289421\pi\)
0.614343 + 0.789039i \(0.289421\pi\)
\(84\) 0 0
\(85\) −9.00016e7 −0.187010
\(86\) 5.26149e8 1.03721
\(87\) 0 0
\(88\) −1.97024e8 −0.350225
\(89\) 1.14168e8 0.192881 0.0964404 0.995339i \(-0.469254\pi\)
0.0964404 + 0.995339i \(0.469254\pi\)
\(90\) 0 0
\(91\) 6.31736e7 0.0965716
\(92\) 6.05833e8 0.881674
\(93\) 0 0
\(94\) −8.83231e8 −1.16681
\(95\) 9.98185e7 0.125735
\(96\) 0 0
\(97\) −1.46573e9 −1.68105 −0.840524 0.541774i \(-0.817753\pi\)
−0.840524 + 0.541774i \(0.817753\pi\)
\(98\) 8.54829e7 0.0936186
\(99\) 0 0
\(100\) −1.14108e8 −0.114108
\(101\) −7.53733e8 −0.720728 −0.360364 0.932812i \(-0.617348\pi\)
−0.360364 + 0.932812i \(0.617348\pi\)
\(102\) 0 0
\(103\) −1.32143e9 −1.15685 −0.578425 0.815736i \(-0.696332\pi\)
−0.578425 + 0.815736i \(0.696332\pi\)
\(104\) −3.13732e8 −0.262971
\(105\) 0 0
\(106\) −3.42775e7 −0.0263714
\(107\) −1.07364e9 −0.791831 −0.395915 0.918287i \(-0.629573\pi\)
−0.395915 + 0.918287i \(0.629573\pi\)
\(108\) 0 0
\(109\) 1.08650e9 0.737245 0.368623 0.929579i \(-0.379830\pi\)
0.368623 + 0.929579i \(0.379830\pi\)
\(110\) −1.53137e8 −0.0997268
\(111\) 0 0
\(112\) −6.54199e7 −0.0392852
\(113\) −2.85559e9 −1.64757 −0.823783 0.566906i \(-0.808140\pi\)
−0.823783 + 0.566906i \(0.808140\pi\)
\(114\) 0 0
\(115\) 1.29621e9 0.691090
\(116\) −1.44580e9 −0.741392
\(117\) 0 0
\(118\) 2.50083e9 1.18745
\(119\) 3.45750e8 0.158052
\(120\) 0 0
\(121\) −2.08492e9 −0.884209
\(122\) −9.93753e8 −0.406125
\(123\) 0 0
\(124\) −1.23285e9 −0.468289
\(125\) −2.44141e8 −0.0894427
\(126\) 0 0
\(127\) −3.86082e9 −1.31693 −0.658465 0.752611i \(-0.728794\pi\)
−0.658465 + 0.752611i \(0.728794\pi\)
\(128\) −1.45849e9 −0.480240
\(129\) 0 0
\(130\) −2.43848e8 −0.0748814
\(131\) −4.01900e8 −0.119233 −0.0596166 0.998221i \(-0.518988\pi\)
−0.0596166 + 0.998221i \(0.518988\pi\)
\(132\) 0 0
\(133\) −3.83463e8 −0.106265
\(134\) −2.31707e9 −0.620822
\(135\) 0 0
\(136\) −1.71706e9 −0.430388
\(137\) −3.04172e9 −0.737695 −0.368847 0.929490i \(-0.620247\pi\)
−0.368847 + 0.929490i \(0.620247\pi\)
\(138\) 0 0
\(139\) 8.36421e8 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(140\) 4.38359e8 0.0964393
\(141\) 0 0
\(142\) −1.03068e9 −0.212728
\(143\) 4.34758e8 0.0869431
\(144\) 0 0
\(145\) −3.09336e9 −0.581132
\(146\) −1.16172e9 −0.211598
\(147\) 0 0
\(148\) 3.77069e9 0.646016
\(149\) 7.76217e9 1.29016 0.645082 0.764114i \(-0.276823\pi\)
0.645082 + 0.764114i \(0.276823\pi\)
\(150\) 0 0
\(151\) 8.38042e9 1.31181 0.655903 0.754846i \(-0.272288\pi\)
0.655903 + 0.754846i \(0.272288\pi\)
\(152\) 1.90435e9 0.289367
\(153\) 0 0
\(154\) 5.88290e8 0.0842845
\(155\) −2.63775e9 −0.367063
\(156\) 0 0
\(157\) −8.49699e9 −1.11613 −0.558067 0.829796i \(-0.688457\pi\)
−0.558067 + 0.829796i \(0.688457\pi\)
\(158\) −6.33110e9 −0.808206
\(159\) 0 0
\(160\) −3.56310e9 −0.429821
\(161\) −4.97952e9 −0.584078
\(162\) 0 0
\(163\) 4.25863e9 0.472526 0.236263 0.971689i \(-0.424077\pi\)
0.236263 + 0.971689i \(0.424077\pi\)
\(164\) −8.39891e9 −0.906621
\(165\) 0 0
\(166\) 7.87749e9 0.805195
\(167\) −9.23536e8 −0.0918818 −0.0459409 0.998944i \(-0.514629\pi\)
−0.0459409 + 0.998944i \(0.514629\pi\)
\(168\) 0 0
\(169\) −9.91221e9 −0.934717
\(170\) −1.33458e9 −0.122553
\(171\) 0 0
\(172\) −1.03651e10 −0.903012
\(173\) 4.76006e9 0.404022 0.202011 0.979383i \(-0.435252\pi\)
0.202011 + 0.979383i \(0.435252\pi\)
\(174\) 0 0
\(175\) 9.37891e8 0.0755929
\(176\) −4.50217e8 −0.0353683
\(177\) 0 0
\(178\) 1.69293e9 0.126401
\(179\) 9.47187e9 0.689599 0.344800 0.938676i \(-0.387947\pi\)
0.344800 + 0.938676i \(0.387947\pi\)
\(180\) 0 0
\(181\) −7.85395e9 −0.543919 −0.271960 0.962309i \(-0.587672\pi\)
−0.271960 + 0.962309i \(0.587672\pi\)
\(182\) 9.36766e8 0.0632863
\(183\) 0 0
\(184\) 2.47292e10 1.59049
\(185\) 8.06758e9 0.506373
\(186\) 0 0
\(187\) 2.37944e9 0.142294
\(188\) 1.73995e10 1.01584
\(189\) 0 0
\(190\) 1.48015e9 0.0823976
\(191\) −1.78210e10 −0.968909 −0.484454 0.874817i \(-0.660982\pi\)
−0.484454 + 0.874817i \(0.660982\pi\)
\(192\) 0 0
\(193\) −1.93101e9 −0.100179 −0.0500894 0.998745i \(-0.515951\pi\)
−0.0500894 + 0.998745i \(0.515951\pi\)
\(194\) −2.17344e10 −1.10164
\(195\) 0 0
\(196\) −1.68400e9 −0.0815061
\(197\) −3.24598e10 −1.53549 −0.767746 0.640755i \(-0.778622\pi\)
−0.767746 + 0.640755i \(0.778622\pi\)
\(198\) 0 0
\(199\) −2.75463e10 −1.24516 −0.622578 0.782558i \(-0.713915\pi\)
−0.622578 + 0.782558i \(0.713915\pi\)
\(200\) −4.65773e9 −0.205845
\(201\) 0 0
\(202\) −1.11767e10 −0.472315
\(203\) 1.18835e10 0.491146
\(204\) 0 0
\(205\) −1.79699e10 −0.710645
\(206\) −1.95947e10 −0.758118
\(207\) 0 0
\(208\) −7.16904e8 −0.0265568
\(209\) −2.63897e9 −0.0956702
\(210\) 0 0
\(211\) 1.94450e10 0.675362 0.337681 0.941261i \(-0.390357\pi\)
0.337681 + 0.941261i \(0.390357\pi\)
\(212\) 6.75261e8 0.0229594
\(213\) 0 0
\(214\) −1.59204e10 −0.518911
\(215\) −2.21765e10 −0.707817
\(216\) 0 0
\(217\) 1.01332e10 0.310225
\(218\) 1.61111e10 0.483139
\(219\) 0 0
\(220\) 3.01677e9 0.0868240
\(221\) 3.78891e9 0.106844
\(222\) 0 0
\(223\) −4.90459e10 −1.32810 −0.664050 0.747688i \(-0.731164\pi\)
−0.664050 + 0.747688i \(0.731164\pi\)
\(224\) 1.36880e10 0.363265
\(225\) 0 0
\(226\) −4.23439e10 −1.07970
\(227\) 6.87990e10 1.71975 0.859876 0.510503i \(-0.170541\pi\)
0.859876 + 0.510503i \(0.170541\pi\)
\(228\) 0 0
\(229\) −3.08456e9 −0.0741198 −0.0370599 0.999313i \(-0.511799\pi\)
−0.0370599 + 0.999313i \(0.511799\pi\)
\(230\) 1.92207e10 0.452892
\(231\) 0 0
\(232\) −5.90154e10 −1.33743
\(233\) 1.48643e10 0.330402 0.165201 0.986260i \(-0.447173\pi\)
0.165201 + 0.986260i \(0.447173\pi\)
\(234\) 0 0
\(235\) 3.72271e10 0.796258
\(236\) −4.92659e10 −1.03381
\(237\) 0 0
\(238\) 5.12693e9 0.103576
\(239\) −4.06416e10 −0.805713 −0.402857 0.915263i \(-0.631983\pi\)
−0.402857 + 0.915263i \(0.631983\pi\)
\(240\) 0 0
\(241\) −2.09799e10 −0.400615 −0.200308 0.979733i \(-0.564194\pi\)
−0.200308 + 0.979733i \(0.564194\pi\)
\(242\) −3.09161e10 −0.579449
\(243\) 0 0
\(244\) 1.95768e10 0.353580
\(245\) −3.60300e9 −0.0638877
\(246\) 0 0
\(247\) −4.20218e9 −0.0718354
\(248\) −5.03232e10 −0.844765
\(249\) 0 0
\(250\) −3.62022e9 −0.0586145
\(251\) 9.95566e10 1.58321 0.791605 0.611034i \(-0.209246\pi\)
0.791605 + 0.611034i \(0.209246\pi\)
\(252\) 0 0
\(253\) −3.42688e10 −0.525844
\(254\) −5.72499e10 −0.863024
\(255\) 0 0
\(256\) −7.20522e10 −1.04850
\(257\) 1.30781e11 1.87002 0.935009 0.354624i \(-0.115391\pi\)
0.935009 + 0.354624i \(0.115391\pi\)
\(258\) 0 0
\(259\) −3.09924e10 −0.427963
\(260\) 4.80376e9 0.0651931
\(261\) 0 0
\(262\) −5.95954e9 −0.0781371
\(263\) 3.27938e10 0.422660 0.211330 0.977415i \(-0.432221\pi\)
0.211330 + 0.977415i \(0.432221\pi\)
\(264\) 0 0
\(265\) 1.44475e9 0.0179965
\(266\) −5.68615e9 −0.0696387
\(267\) 0 0
\(268\) 4.56459e10 0.540499
\(269\) −7.47690e9 −0.0870636 −0.0435318 0.999052i \(-0.513861\pi\)
−0.0435318 + 0.999052i \(0.513861\pi\)
\(270\) 0 0
\(271\) 6.72735e9 0.0757674 0.0378837 0.999282i \(-0.487938\pi\)
0.0378837 + 0.999282i \(0.487938\pi\)
\(272\) −3.92363e9 −0.0434638
\(273\) 0 0
\(274\) −4.51039e10 −0.483434
\(275\) 6.45452e9 0.0680561
\(276\) 0 0
\(277\) −1.52676e11 −1.55815 −0.779077 0.626928i \(-0.784312\pi\)
−0.779077 + 0.626928i \(0.784312\pi\)
\(278\) 1.24028e10 0.124543
\(279\) 0 0
\(280\) 1.78932e10 0.173971
\(281\) −3.09796e10 −0.296413 −0.148206 0.988956i \(-0.547350\pi\)
−0.148206 + 0.988956i \(0.547350\pi\)
\(282\) 0 0
\(283\) −8.92078e10 −0.826731 −0.413365 0.910565i \(-0.635647\pi\)
−0.413365 + 0.910565i \(0.635647\pi\)
\(284\) 2.03041e10 0.185205
\(285\) 0 0
\(286\) 6.44678e9 0.0569765
\(287\) 6.90331e10 0.600605
\(288\) 0 0
\(289\) −9.78512e10 −0.825136
\(290\) −4.58697e10 −0.380833
\(291\) 0 0
\(292\) 2.28856e10 0.184221
\(293\) −6.30090e10 −0.499457 −0.249728 0.968316i \(-0.580341\pi\)
−0.249728 + 0.968316i \(0.580341\pi\)
\(294\) 0 0
\(295\) −1.05407e11 −0.810344
\(296\) 1.53914e11 1.16537
\(297\) 0 0
\(298\) 1.15101e11 0.845483
\(299\) −5.45681e10 −0.394838
\(300\) 0 0
\(301\) 8.51934e10 0.598214
\(302\) 1.24268e11 0.859665
\(303\) 0 0
\(304\) 4.35160e9 0.0292225
\(305\) 4.18855e10 0.277150
\(306\) 0 0
\(307\) −1.22461e11 −0.786821 −0.393411 0.919363i \(-0.628705\pi\)
−0.393411 + 0.919363i \(0.628705\pi\)
\(308\) −1.15892e10 −0.0733797
\(309\) 0 0
\(310\) −3.91137e10 −0.240548
\(311\) −2.77926e11 −1.68464 −0.842321 0.538975i \(-0.818812\pi\)
−0.842321 + 0.538975i \(0.818812\pi\)
\(312\) 0 0
\(313\) 1.22493e11 0.721374 0.360687 0.932687i \(-0.382542\pi\)
0.360687 + 0.932687i \(0.382542\pi\)
\(314\) −1.25997e11 −0.731436
\(315\) 0 0
\(316\) 1.24722e11 0.703639
\(317\) −2.32000e11 −1.29039 −0.645197 0.764016i \(-0.723225\pi\)
−0.645197 + 0.764016i \(0.723225\pi\)
\(318\) 0 0
\(319\) 8.17814e10 0.442177
\(320\) −6.15542e10 −0.328158
\(321\) 0 0
\(322\) −7.38384e10 −0.382764
\(323\) −2.29986e10 −0.117568
\(324\) 0 0
\(325\) 1.02779e10 0.0511009
\(326\) 6.31487e10 0.309660
\(327\) 0 0
\(328\) −3.42831e11 −1.63549
\(329\) −1.43012e11 −0.672961
\(330\) 0 0
\(331\) −3.74253e11 −1.71372 −0.856859 0.515551i \(-0.827587\pi\)
−0.856859 + 0.515551i \(0.827587\pi\)
\(332\) −1.55185e11 −0.701018
\(333\) 0 0
\(334\) −1.36946e10 −0.0602129
\(335\) 9.76616e10 0.423665
\(336\) 0 0
\(337\) 3.35878e11 1.41856 0.709280 0.704927i \(-0.249020\pi\)
0.709280 + 0.704927i \(0.249020\pi\)
\(338\) −1.46982e11 −0.612548
\(339\) 0 0
\(340\) 2.62911e10 0.106697
\(341\) 6.97361e10 0.279295
\(342\) 0 0
\(343\) 1.38413e10 0.0539949
\(344\) −4.23086e11 −1.62898
\(345\) 0 0
\(346\) 7.05842e10 0.264768
\(347\) 3.63698e11 1.34666 0.673331 0.739341i \(-0.264863\pi\)
0.673331 + 0.739341i \(0.264863\pi\)
\(348\) 0 0
\(349\) 1.08740e11 0.392350 0.196175 0.980569i \(-0.437148\pi\)
0.196175 + 0.980569i \(0.437148\pi\)
\(350\) 1.39074e10 0.0495383
\(351\) 0 0
\(352\) 9.42002e10 0.327047
\(353\) 3.36249e11 1.15259 0.576296 0.817241i \(-0.304498\pi\)
0.576296 + 0.817241i \(0.304498\pi\)
\(354\) 0 0
\(355\) 4.34417e10 0.145171
\(356\) −3.33505e10 −0.110047
\(357\) 0 0
\(358\) 1.40453e11 0.451915
\(359\) −5.49944e11 −1.74741 −0.873703 0.486460i \(-0.838288\pi\)
−0.873703 + 0.486460i \(0.838288\pi\)
\(360\) 0 0
\(361\) −2.97181e11 −0.920954
\(362\) −1.16462e11 −0.356447
\(363\) 0 0
\(364\) −1.84541e10 −0.0550982
\(365\) 4.89649e10 0.144400
\(366\) 0 0
\(367\) −1.32204e11 −0.380405 −0.190203 0.981745i \(-0.560914\pi\)
−0.190203 + 0.981745i \(0.560914\pi\)
\(368\) 5.65084e10 0.160619
\(369\) 0 0
\(370\) 1.19630e11 0.331841
\(371\) −5.55017e9 −0.0152098
\(372\) 0 0
\(373\) 4.88064e11 1.30553 0.652765 0.757560i \(-0.273609\pi\)
0.652765 + 0.757560i \(0.273609\pi\)
\(374\) 3.52833e10 0.0932496
\(375\) 0 0
\(376\) 7.10222e11 1.83252
\(377\) 1.30225e11 0.332015
\(378\) 0 0
\(379\) 3.00162e11 0.747274 0.373637 0.927575i \(-0.378111\pi\)
0.373637 + 0.927575i \(0.378111\pi\)
\(380\) −2.91587e10 −0.0717369
\(381\) 0 0
\(382\) −2.64258e11 −0.634955
\(383\) −7.74202e11 −1.83848 −0.919241 0.393694i \(-0.871197\pi\)
−0.919241 + 0.393694i \(0.871197\pi\)
\(384\) 0 0
\(385\) −2.47957e10 −0.0575179
\(386\) −2.86338e10 −0.0656501
\(387\) 0 0
\(388\) 4.28165e11 0.959109
\(389\) 1.70127e11 0.376703 0.188352 0.982102i \(-0.439686\pi\)
0.188352 + 0.982102i \(0.439686\pi\)
\(390\) 0 0
\(391\) −2.98652e11 −0.646204
\(392\) −6.87383e10 −0.147032
\(393\) 0 0
\(394\) −4.81327e11 −1.00625
\(395\) 2.66848e11 0.551540
\(396\) 0 0
\(397\) −5.47242e11 −1.10566 −0.552831 0.833293i \(-0.686452\pi\)
−0.552831 + 0.833293i \(0.686452\pi\)
\(398\) −4.08468e11 −0.815988
\(399\) 0 0
\(400\) −1.06433e10 −0.0207878
\(401\) 4.57174e11 0.882941 0.441471 0.897276i \(-0.354457\pi\)
0.441471 + 0.897276i \(0.354457\pi\)
\(402\) 0 0
\(403\) 1.11045e11 0.209713
\(404\) 2.20179e11 0.411206
\(405\) 0 0
\(406\) 1.76213e11 0.321863
\(407\) −2.13288e11 −0.385294
\(408\) 0 0
\(409\) 9.42996e10 0.166631 0.0833153 0.996523i \(-0.473449\pi\)
0.0833153 + 0.996523i \(0.473449\pi\)
\(410\) −2.66465e11 −0.465707
\(411\) 0 0
\(412\) 3.86013e11 0.660032
\(413\) 4.04930e11 0.684865
\(414\) 0 0
\(415\) −3.32026e11 −0.549485
\(416\) 1.50000e11 0.245568
\(417\) 0 0
\(418\) −3.91318e10 −0.0626955
\(419\) −1.66874e11 −0.264500 −0.132250 0.991216i \(-0.542220\pi\)
−0.132250 + 0.991216i \(0.542220\pi\)
\(420\) 0 0
\(421\) −5.52779e11 −0.857594 −0.428797 0.903401i \(-0.641062\pi\)
−0.428797 + 0.903401i \(0.641062\pi\)
\(422\) 2.88339e11 0.442585
\(423\) 0 0
\(424\) 2.75631e10 0.0414174
\(425\) 5.62510e10 0.0836334
\(426\) 0 0
\(427\) −1.60907e11 −0.234234
\(428\) 3.13630e11 0.451773
\(429\) 0 0
\(430\) −3.28843e11 −0.463853
\(431\) 4.37408e11 0.610575 0.305288 0.952260i \(-0.401247\pi\)
0.305288 + 0.952260i \(0.401247\pi\)
\(432\) 0 0
\(433\) 2.79837e11 0.382569 0.191285 0.981535i \(-0.438735\pi\)
0.191285 + 0.981535i \(0.438735\pi\)
\(434\) 1.50259e11 0.203300
\(435\) 0 0
\(436\) −3.17387e11 −0.420630
\(437\) 3.31227e11 0.434470
\(438\) 0 0
\(439\) 8.87150e11 1.14000 0.570002 0.821643i \(-0.306942\pi\)
0.570002 + 0.821643i \(0.306942\pi\)
\(440\) 1.23140e11 0.156625
\(441\) 0 0
\(442\) 5.61835e10 0.0700178
\(443\) 5.87852e11 0.725189 0.362594 0.931947i \(-0.381891\pi\)
0.362594 + 0.931947i \(0.381891\pi\)
\(444\) 0 0
\(445\) −7.13549e10 −0.0862589
\(446\) −7.27274e11 −0.870344
\(447\) 0 0
\(448\) 2.36466e11 0.277344
\(449\) 5.19918e11 0.603707 0.301854 0.953354i \(-0.402395\pi\)
0.301854 + 0.953354i \(0.402395\pi\)
\(450\) 0 0
\(451\) 4.75082e11 0.540723
\(452\) 8.34168e11 0.940006
\(453\) 0 0
\(454\) 1.02018e12 1.12701
\(455\) −3.94835e10 −0.0431881
\(456\) 0 0
\(457\) −1.03055e12 −1.10521 −0.552607 0.833442i \(-0.686367\pi\)
−0.552607 + 0.833442i \(0.686367\pi\)
\(458\) −4.57392e10 −0.0485729
\(459\) 0 0
\(460\) −3.78646e11 −0.394296
\(461\) −9.37018e11 −0.966259 −0.483130 0.875549i \(-0.660500\pi\)
−0.483130 + 0.875549i \(0.660500\pi\)
\(462\) 0 0
\(463\) 1.08281e12 1.09506 0.547529 0.836787i \(-0.315569\pi\)
0.547529 + 0.836787i \(0.315569\pi\)
\(464\) −1.34855e11 −0.135063
\(465\) 0 0
\(466\) 2.20414e11 0.216522
\(467\) −1.52419e11 −0.148290 −0.0741451 0.997247i \(-0.523623\pi\)
−0.0741451 + 0.997247i \(0.523623\pi\)
\(468\) 0 0
\(469\) −3.75177e11 −0.358062
\(470\) 5.52019e11 0.521812
\(471\) 0 0
\(472\) −2.01096e12 −1.86494
\(473\) 5.86297e11 0.538570
\(474\) 0 0
\(475\) −6.23865e10 −0.0562302
\(476\) −1.01000e11 −0.0901756
\(477\) 0 0
\(478\) −6.02652e11 −0.528008
\(479\) 2.02244e11 0.175535 0.0877677 0.996141i \(-0.472027\pi\)
0.0877677 + 0.996141i \(0.472027\pi\)
\(480\) 0 0
\(481\) −3.39631e11 −0.289304
\(482\) −3.11099e11 −0.262535
\(483\) 0 0
\(484\) 6.09042e11 0.504479
\(485\) 9.16079e11 0.751788
\(486\) 0 0
\(487\) 8.35572e11 0.673137 0.336568 0.941659i \(-0.390734\pi\)
0.336568 + 0.941659i \(0.390734\pi\)
\(488\) 7.99094e11 0.637836
\(489\) 0 0
\(490\) −5.34268e10 −0.0418675
\(491\) 1.06242e12 0.824953 0.412476 0.910968i \(-0.364664\pi\)
0.412476 + 0.910968i \(0.364664\pi\)
\(492\) 0 0
\(493\) 7.12723e11 0.543387
\(494\) −6.23117e10 −0.0470759
\(495\) 0 0
\(496\) −1.14993e11 −0.0853108
\(497\) −1.66886e11 −0.122692
\(498\) 0 0
\(499\) −3.33938e11 −0.241109 −0.120554 0.992707i \(-0.538467\pi\)
−0.120554 + 0.992707i \(0.538467\pi\)
\(500\) 7.13178e10 0.0510309
\(501\) 0 0
\(502\) 1.47627e12 1.03752
\(503\) 3.60934e11 0.251404 0.125702 0.992068i \(-0.459882\pi\)
0.125702 + 0.992068i \(0.459882\pi\)
\(504\) 0 0
\(505\) 4.71083e11 0.322319
\(506\) −5.08152e11 −0.344601
\(507\) 0 0
\(508\) 1.12781e12 0.751365
\(509\) −4.09926e11 −0.270692 −0.135346 0.990798i \(-0.543215\pi\)
−0.135346 + 0.990798i \(0.543215\pi\)
\(510\) 0 0
\(511\) −1.88104e11 −0.122040
\(512\) −3.21676e11 −0.206873
\(513\) 0 0
\(514\) 1.93928e12 1.22548
\(515\) 8.25894e11 0.517359
\(516\) 0 0
\(517\) −9.84199e11 −0.605865
\(518\) −4.59569e11 −0.280457
\(519\) 0 0
\(520\) 1.96082e11 0.117604
\(521\) 1.09884e12 0.653377 0.326688 0.945132i \(-0.394067\pi\)
0.326688 + 0.945132i \(0.394067\pi\)
\(522\) 0 0
\(523\) 2.97552e12 1.73902 0.869511 0.493914i \(-0.164434\pi\)
0.869511 + 0.493914i \(0.164434\pi\)
\(524\) 1.17402e11 0.0680276
\(525\) 0 0
\(526\) 4.86280e11 0.276982
\(527\) 6.07748e11 0.343223
\(528\) 0 0
\(529\) 2.50005e12 1.38803
\(530\) 2.14234e10 0.0117936
\(531\) 0 0
\(532\) 1.12016e11 0.0606288
\(533\) 7.56500e11 0.406010
\(534\) 0 0
\(535\) 6.71026e11 0.354118
\(536\) 1.86320e12 0.975028
\(537\) 0 0
\(538\) −1.10871e11 −0.0570554
\(539\) 9.52551e10 0.0486115
\(540\) 0 0
\(541\) 2.97095e12 1.49110 0.745551 0.666449i \(-0.232187\pi\)
0.745551 + 0.666449i \(0.232187\pi\)
\(542\) 9.97561e10 0.0496527
\(543\) 0 0
\(544\) 8.20952e11 0.401904
\(545\) −6.79065e11 −0.329706
\(546\) 0 0
\(547\) −1.50514e12 −0.718842 −0.359421 0.933176i \(-0.617026\pi\)
−0.359421 + 0.933176i \(0.617026\pi\)
\(548\) 8.88540e11 0.420886
\(549\) 0 0
\(550\) 9.57104e10 0.0445992
\(551\) −7.90463e11 −0.365342
\(552\) 0 0
\(553\) −1.02512e12 −0.466136
\(554\) −2.26394e12 −1.02111
\(555\) 0 0
\(556\) −2.44333e11 −0.108429
\(557\) 2.17354e12 0.956797 0.478399 0.878143i \(-0.341217\pi\)
0.478399 + 0.878143i \(0.341217\pi\)
\(558\) 0 0
\(559\) 9.33593e11 0.404394
\(560\) 4.08874e10 0.0175689
\(561\) 0 0
\(562\) −4.59379e11 −0.194248
\(563\) −1.40104e12 −0.587710 −0.293855 0.955850i \(-0.594938\pi\)
−0.293855 + 0.955850i \(0.594938\pi\)
\(564\) 0 0
\(565\) 1.78474e12 0.736814
\(566\) −1.32281e12 −0.541781
\(567\) 0 0
\(568\) 8.28784e11 0.334098
\(569\) 1.81226e12 0.724795 0.362397 0.932024i \(-0.381958\pi\)
0.362397 + 0.932024i \(0.381958\pi\)
\(570\) 0 0
\(571\) −3.38885e12 −1.33410 −0.667052 0.745012i \(-0.732444\pi\)
−0.667052 + 0.745012i \(0.732444\pi\)
\(572\) −1.27001e11 −0.0496048
\(573\) 0 0
\(574\) 1.02365e12 0.393594
\(575\) −8.10131e11 −0.309065
\(576\) 0 0
\(577\) 2.86129e12 1.07466 0.537329 0.843373i \(-0.319433\pi\)
0.537329 + 0.843373i \(0.319433\pi\)
\(578\) −1.45098e12 −0.540737
\(579\) 0 0
\(580\) 9.03626e11 0.331560
\(581\) 1.27551e12 0.464400
\(582\) 0 0
\(583\) −3.81960e10 −0.0136933
\(584\) 9.34156e11 0.332324
\(585\) 0 0
\(586\) −9.34324e11 −0.327309
\(587\) 7.85729e11 0.273150 0.136575 0.990630i \(-0.456391\pi\)
0.136575 + 0.990630i \(0.456391\pi\)
\(588\) 0 0
\(589\) −6.74039e11 −0.230763
\(590\) −1.56302e12 −0.531043
\(591\) 0 0
\(592\) 3.51707e11 0.117688
\(593\) −8.24271e11 −0.273731 −0.136865 0.990590i \(-0.543703\pi\)
−0.136865 + 0.990590i \(0.543703\pi\)
\(594\) 0 0
\(595\) −2.16094e11 −0.0706831
\(596\) −2.26747e12 −0.736093
\(597\) 0 0
\(598\) −8.09159e11 −0.258749
\(599\) 2.60594e10 0.00827073 0.00413537 0.999991i \(-0.498684\pi\)
0.00413537 + 0.999991i \(0.498684\pi\)
\(600\) 0 0
\(601\) 1.90372e11 0.0595207 0.0297603 0.999557i \(-0.490526\pi\)
0.0297603 + 0.999557i \(0.490526\pi\)
\(602\) 1.26328e12 0.392028
\(603\) 0 0
\(604\) −2.44807e12 −0.748441
\(605\) 1.30307e12 0.395430
\(606\) 0 0
\(607\) 1.77675e12 0.531222 0.265611 0.964080i \(-0.414426\pi\)
0.265611 + 0.964080i \(0.414426\pi\)
\(608\) −9.10497e11 −0.270217
\(609\) 0 0
\(610\) 6.21096e11 0.181624
\(611\) −1.56719e12 −0.454922
\(612\) 0 0
\(613\) −2.48518e11 −0.0710863 −0.0355431 0.999368i \(-0.511316\pi\)
−0.0355431 + 0.999368i \(0.511316\pi\)
\(614\) −1.81591e12 −0.515628
\(615\) 0 0
\(616\) −4.73054e11 −0.132372
\(617\) −3.52347e12 −0.978786 −0.489393 0.872063i \(-0.662782\pi\)
−0.489393 + 0.872063i \(0.662782\pi\)
\(618\) 0 0
\(619\) −6.69712e12 −1.83350 −0.916748 0.399466i \(-0.869196\pi\)
−0.916748 + 0.399466i \(0.869196\pi\)
\(620\) 7.70534e11 0.209425
\(621\) 0 0
\(622\) −4.12121e12 −1.10400
\(623\) 2.74117e11 0.0729021
\(624\) 0 0
\(625\) 1.52588e11 0.0400000
\(626\) 1.81637e12 0.472738
\(627\) 0 0
\(628\) 2.48212e12 0.636802
\(629\) −1.85880e12 −0.473484
\(630\) 0 0
\(631\) 6.29917e11 0.158180 0.0790900 0.996867i \(-0.474799\pi\)
0.0790900 + 0.996867i \(0.474799\pi\)
\(632\) 5.09095e12 1.26932
\(633\) 0 0
\(634\) −3.44020e12 −0.845634
\(635\) 2.41301e12 0.588949
\(636\) 0 0
\(637\) 1.51680e11 0.0365006
\(638\) 1.21269e12 0.289772
\(639\) 0 0
\(640\) 9.11555e11 0.214770
\(641\) −2.82049e12 −0.659877 −0.329938 0.944002i \(-0.607028\pi\)
−0.329938 + 0.944002i \(0.607028\pi\)
\(642\) 0 0
\(643\) 7.19804e12 1.66060 0.830300 0.557317i \(-0.188169\pi\)
0.830300 + 0.557317i \(0.188169\pi\)
\(644\) 1.45461e12 0.333241
\(645\) 0 0
\(646\) −3.41033e11 −0.0770459
\(647\) −7.17803e12 −1.61041 −0.805204 0.592998i \(-0.797944\pi\)
−0.805204 + 0.592998i \(0.797944\pi\)
\(648\) 0 0
\(649\) 2.78671e12 0.616582
\(650\) 1.52405e11 0.0334880
\(651\) 0 0
\(652\) −1.24402e12 −0.269596
\(653\) 1.54599e12 0.332734 0.166367 0.986064i \(-0.446796\pi\)
0.166367 + 0.986064i \(0.446796\pi\)
\(654\) 0 0
\(655\) 2.51187e11 0.0533227
\(656\) −7.83398e11 −0.165164
\(657\) 0 0
\(658\) −2.12064e12 −0.441011
\(659\) −2.38624e12 −0.492868 −0.246434 0.969160i \(-0.579259\pi\)
−0.246434 + 0.969160i \(0.579259\pi\)
\(660\) 0 0
\(661\) −2.35488e12 −0.479802 −0.239901 0.970797i \(-0.577115\pi\)
−0.239901 + 0.970797i \(0.577115\pi\)
\(662\) −5.54958e12 −1.12305
\(663\) 0 0
\(664\) −6.33443e12 −1.26459
\(665\) 2.39664e11 0.0475232
\(666\) 0 0
\(667\) −1.02647e13 −2.00807
\(668\) 2.69781e11 0.0524225
\(669\) 0 0
\(670\) 1.44817e12 0.277640
\(671\) −1.10736e12 −0.210880
\(672\) 0 0
\(673\) −6.32709e12 −1.18887 −0.594437 0.804142i \(-0.702625\pi\)
−0.594437 + 0.804142i \(0.702625\pi\)
\(674\) 4.98055e12 0.929625
\(675\) 0 0
\(676\) 2.89553e12 0.533296
\(677\) −3.16670e12 −0.579372 −0.289686 0.957122i \(-0.593551\pi\)
−0.289686 + 0.957122i \(0.593551\pi\)
\(678\) 0 0
\(679\) −3.51921e12 −0.635376
\(680\) 1.07316e12 0.192475
\(681\) 0 0
\(682\) 1.03408e12 0.183030
\(683\) 6.06916e12 1.06717 0.533587 0.845745i \(-0.320844\pi\)
0.533587 + 0.845745i \(0.320844\pi\)
\(684\) 0 0
\(685\) 1.90107e12 0.329907
\(686\) 2.05245e11 0.0353845
\(687\) 0 0
\(688\) −9.66788e11 −0.164507
\(689\) −6.08216e10 −0.0102818
\(690\) 0 0
\(691\) −1.36410e12 −0.227611 −0.113806 0.993503i \(-0.536304\pi\)
−0.113806 + 0.993503i \(0.536304\pi\)
\(692\) −1.39050e12 −0.230512
\(693\) 0 0
\(694\) 5.39307e12 0.882508
\(695\) −5.22763e11 −0.0849911
\(696\) 0 0
\(697\) 4.14033e12 0.664489
\(698\) 1.61244e12 0.257119
\(699\) 0 0
\(700\) −2.73974e11 −0.0431290
\(701\) −4.27032e12 −0.667927 −0.333964 0.942586i \(-0.608386\pi\)
−0.333964 + 0.942586i \(0.608386\pi\)
\(702\) 0 0
\(703\) 2.06155e12 0.318343
\(704\) 1.62735e12 0.249692
\(705\) 0 0
\(706\) 4.98605e12 0.755328
\(707\) −1.80971e12 −0.272410
\(708\) 0 0
\(709\) 4.76449e12 0.708122 0.354061 0.935222i \(-0.384800\pi\)
0.354061 + 0.935222i \(0.384800\pi\)
\(710\) 6.44172e11 0.0951348
\(711\) 0 0
\(712\) −1.36132e12 −0.198517
\(713\) −8.75284e12 −1.26837
\(714\) 0 0
\(715\) −2.71724e11 −0.0388822
\(716\) −2.76690e12 −0.393446
\(717\) 0 0
\(718\) −8.15481e12 −1.14513
\(719\) 2.64192e12 0.368671 0.184336 0.982863i \(-0.440987\pi\)
0.184336 + 0.982863i \(0.440987\pi\)
\(720\) 0 0
\(721\) −3.17276e12 −0.437248
\(722\) −4.40672e12 −0.603529
\(723\) 0 0
\(724\) 2.29428e12 0.310329
\(725\) 1.93335e12 0.259890
\(726\) 0 0
\(727\) −3.54861e12 −0.471144 −0.235572 0.971857i \(-0.575696\pi\)
−0.235572 + 0.971857i \(0.575696\pi\)
\(728\) −7.53270e11 −0.0993938
\(729\) 0 0
\(730\) 7.26072e11 0.0946296
\(731\) 5.10956e12 0.661844
\(732\) 0 0
\(733\) −1.34909e13 −1.72613 −0.863063 0.505095i \(-0.831457\pi\)
−0.863063 + 0.505095i \(0.831457\pi\)
\(734\) −1.96037e12 −0.249291
\(735\) 0 0
\(736\) −1.18234e13 −1.48523
\(737\) −2.58195e12 −0.322362
\(738\) 0 0
\(739\) −1.33151e13 −1.64227 −0.821133 0.570736i \(-0.806658\pi\)
−0.821133 + 0.570736i \(0.806658\pi\)
\(740\) −2.35668e12 −0.288907
\(741\) 0 0
\(742\) −8.23003e10 −0.00996744
\(743\) −1.37811e13 −1.65896 −0.829479 0.558538i \(-0.811363\pi\)
−0.829479 + 0.558538i \(0.811363\pi\)
\(744\) 0 0
\(745\) −4.85135e12 −0.576979
\(746\) 7.23722e12 0.855553
\(747\) 0 0
\(748\) −6.95075e11 −0.0811848
\(749\) −2.57781e12 −0.299284
\(750\) 0 0
\(751\) −1.31166e12 −0.150467 −0.0752334 0.997166i \(-0.523970\pi\)
−0.0752334 + 0.997166i \(0.523970\pi\)
\(752\) 1.62292e12 0.185062
\(753\) 0 0
\(754\) 1.93103e12 0.217580
\(755\) −5.23776e12 −0.586657
\(756\) 0 0
\(757\) 1.31405e13 1.45439 0.727196 0.686430i \(-0.240823\pi\)
0.727196 + 0.686430i \(0.240823\pi\)
\(758\) 4.45093e12 0.489711
\(759\) 0 0
\(760\) −1.19022e12 −0.129409
\(761\) −4.85816e12 −0.525099 −0.262549 0.964919i \(-0.584563\pi\)
−0.262549 + 0.964919i \(0.584563\pi\)
\(762\) 0 0
\(763\) 2.60869e12 0.278652
\(764\) 5.20584e12 0.552804
\(765\) 0 0
\(766\) −1.14802e13 −1.20481
\(767\) 4.43743e12 0.462970
\(768\) 0 0
\(769\) −1.57384e13 −1.62290 −0.811452 0.584420i \(-0.801322\pi\)
−0.811452 + 0.584420i \(0.801322\pi\)
\(770\) −3.67681e11 −0.0376932
\(771\) 0 0
\(772\) 5.64081e11 0.0571562
\(773\) −1.97088e12 −0.198542 −0.0992710 0.995060i \(-0.531651\pi\)
−0.0992710 + 0.995060i \(0.531651\pi\)
\(774\) 0 0
\(775\) 1.64859e12 0.164156
\(776\) 1.74770e13 1.73017
\(777\) 0 0
\(778\) 2.52271e12 0.246865
\(779\) −4.59194e12 −0.446763
\(780\) 0 0
\(781\) −1.14850e12 −0.110459
\(782\) −4.42854e12 −0.423477
\(783\) 0 0
\(784\) −1.57073e11 −0.0148484
\(785\) 5.31062e12 0.499150
\(786\) 0 0
\(787\) −1.29627e13 −1.20451 −0.602253 0.798306i \(-0.705730\pi\)
−0.602253 + 0.798306i \(0.705730\pi\)
\(788\) 9.48207e12 0.876063
\(789\) 0 0
\(790\) 3.95694e12 0.361441
\(791\) −6.85627e12 −0.622721
\(792\) 0 0
\(793\) −1.76330e12 −0.158343
\(794\) −8.11474e12 −0.724573
\(795\) 0 0
\(796\) 8.04675e12 0.710414
\(797\) −1.04429e13 −0.916762 −0.458381 0.888756i \(-0.651571\pi\)
−0.458381 + 0.888756i \(0.651571\pi\)
\(798\) 0 0
\(799\) −8.57727e12 −0.744541
\(800\) 2.22694e12 0.192222
\(801\) 0 0
\(802\) 6.77917e12 0.578618
\(803\) −1.29452e12 −0.109872
\(804\) 0 0
\(805\) 3.11220e12 0.261208
\(806\) 1.64662e12 0.137431
\(807\) 0 0
\(808\) 8.98737e12 0.741791
\(809\) 6.99142e12 0.573848 0.286924 0.957953i \(-0.407367\pi\)
0.286924 + 0.957953i \(0.407367\pi\)
\(810\) 0 0
\(811\) −1.14380e13 −0.928442 −0.464221 0.885719i \(-0.653666\pi\)
−0.464221 + 0.885719i \(0.653666\pi\)
\(812\) −3.47137e12 −0.280220
\(813\) 0 0
\(814\) −3.16273e12 −0.252495
\(815\) −2.66164e12 −0.211320
\(816\) 0 0
\(817\) −5.66689e12 −0.444985
\(818\) 1.39831e12 0.109198
\(819\) 0 0
\(820\) 5.24932e12 0.405453
\(821\) 4.42435e12 0.339864 0.169932 0.985456i \(-0.445645\pi\)
0.169932 + 0.985456i \(0.445645\pi\)
\(822\) 0 0
\(823\) 1.71948e13 1.30647 0.653234 0.757156i \(-0.273412\pi\)
0.653234 + 0.757156i \(0.273412\pi\)
\(824\) 1.57565e13 1.19066
\(825\) 0 0
\(826\) 6.00448e12 0.448813
\(827\) −3.04498e12 −0.226365 −0.113182 0.993574i \(-0.536104\pi\)
−0.113182 + 0.993574i \(0.536104\pi\)
\(828\) 0 0
\(829\) −8.03416e12 −0.590806 −0.295403 0.955373i \(-0.595454\pi\)
−0.295403 + 0.955373i \(0.595454\pi\)
\(830\) −4.92343e12 −0.360094
\(831\) 0 0
\(832\) 2.59132e12 0.187485
\(833\) 8.30146e11 0.0597382
\(834\) 0 0
\(835\) 5.77210e11 0.0410908
\(836\) 7.70891e11 0.0545839
\(837\) 0 0
\(838\) −2.47448e12 −0.173335
\(839\) −1.36209e13 −0.949020 −0.474510 0.880250i \(-0.657375\pi\)
−0.474510 + 0.880250i \(0.657375\pi\)
\(840\) 0 0
\(841\) 9.98921e12 0.688571
\(842\) −8.19684e12 −0.562007
\(843\) 0 0
\(844\) −5.68023e12 −0.385323
\(845\) 6.19513e12 0.418018
\(846\) 0 0
\(847\) −5.00589e12 −0.334200
\(848\) 6.29842e10 0.00418264
\(849\) 0 0
\(850\) 8.34114e11 0.0548075
\(851\) 2.67706e13 1.74975
\(852\) 0 0
\(853\) 3.64211e12 0.235549 0.117775 0.993040i \(-0.462424\pi\)
0.117775 + 0.993040i \(0.462424\pi\)
\(854\) −2.38600e12 −0.153501
\(855\) 0 0
\(856\) 1.28019e13 0.814971
\(857\) −1.62518e13 −1.02917 −0.514587 0.857438i \(-0.672055\pi\)
−0.514587 + 0.857438i \(0.672055\pi\)
\(858\) 0 0
\(859\) 1.69063e13 1.05944 0.529722 0.848171i \(-0.322296\pi\)
0.529722 + 0.848171i \(0.322296\pi\)
\(860\) 6.47816e12 0.403839
\(861\) 0 0
\(862\) 6.48608e12 0.400128
\(863\) 1.56857e13 0.962619 0.481310 0.876551i \(-0.340161\pi\)
0.481310 + 0.876551i \(0.340161\pi\)
\(864\) 0 0
\(865\) −2.97504e12 −0.180684
\(866\) 4.14955e12 0.250709
\(867\) 0 0
\(868\) −2.96008e12 −0.176997
\(869\) −7.05485e12 −0.419661
\(870\) 0 0
\(871\) −4.11138e12 −0.242050
\(872\) −1.29553e13 −0.758790
\(873\) 0 0
\(874\) 4.91158e12 0.284721
\(875\) −5.86182e11 −0.0338062
\(876\) 0 0
\(877\) 2.37521e13 1.35583 0.677914 0.735141i \(-0.262884\pi\)
0.677914 + 0.735141i \(0.262884\pi\)
\(878\) 1.31550e13 0.747079
\(879\) 0 0
\(880\) 2.81385e11 0.0158172
\(881\) −1.13698e13 −0.635858 −0.317929 0.948115i \(-0.602987\pi\)
−0.317929 + 0.948115i \(0.602987\pi\)
\(882\) 0 0
\(883\) −1.21711e12 −0.0673762 −0.0336881 0.999432i \(-0.510725\pi\)
−0.0336881 + 0.999432i \(0.510725\pi\)
\(884\) −1.10681e12 −0.0609588
\(885\) 0 0
\(886\) 8.71692e12 0.475238
\(887\) −3.43743e13 −1.86457 −0.932283 0.361730i \(-0.882186\pi\)
−0.932283 + 0.361730i \(0.882186\pi\)
\(888\) 0 0
\(889\) −9.26983e12 −0.497753
\(890\) −1.05808e12 −0.0565281
\(891\) 0 0
\(892\) 1.43272e13 0.757738
\(893\) 9.51284e12 0.500586
\(894\) 0 0
\(895\) −5.91992e12 −0.308398
\(896\) −3.50183e12 −0.181513
\(897\) 0 0
\(898\) 7.70957e12 0.395627
\(899\) 2.08884e13 1.06656
\(900\) 0 0
\(901\) −3.32877e11 −0.0168276
\(902\) 7.04472e12 0.354352
\(903\) 0 0
\(904\) 3.40495e13 1.69571
\(905\) 4.90872e12 0.243248
\(906\) 0 0
\(907\) −3.08586e13 −1.51406 −0.757030 0.653381i \(-0.773350\pi\)
−0.757030 + 0.653381i \(0.773350\pi\)
\(908\) −2.00974e13 −0.981192
\(909\) 0 0
\(910\) −5.85478e11 −0.0283025
\(911\) 3.36006e13 1.61627 0.808136 0.588996i \(-0.200477\pi\)
0.808136 + 0.588996i \(0.200477\pi\)
\(912\) 0 0
\(913\) 8.77802e12 0.418098
\(914\) −1.52814e13 −0.724280
\(915\) 0 0
\(916\) 9.01056e11 0.0422885
\(917\) −9.64961e11 −0.0450659
\(918\) 0 0
\(919\) −1.46200e13 −0.676127 −0.338063 0.941123i \(-0.609772\pi\)
−0.338063 + 0.941123i \(0.609772\pi\)
\(920\) −1.54557e13 −0.711287
\(921\) 0 0
\(922\) −1.38945e13 −0.633219
\(923\) −1.82882e12 −0.0829398
\(924\) 0 0
\(925\) −5.04224e12 −0.226457
\(926\) 1.60563e13 0.717625
\(927\) 0 0
\(928\) 2.82162e13 1.24891
\(929\) 1.36244e13 0.600130 0.300065 0.953919i \(-0.402992\pi\)
0.300065 + 0.953919i \(0.402992\pi\)
\(930\) 0 0
\(931\) −9.20694e11 −0.0401644
\(932\) −4.34213e12 −0.188508
\(933\) 0 0
\(934\) −2.26013e12 −0.0971790
\(935\) −1.48715e12 −0.0636358
\(936\) 0 0
\(937\) −1.06580e13 −0.451697 −0.225848 0.974162i \(-0.572515\pi\)
−0.225848 + 0.974162i \(0.572515\pi\)
\(938\) −5.56328e12 −0.234649
\(939\) 0 0
\(940\) −1.08747e13 −0.454299
\(941\) −1.54253e13 −0.641328 −0.320664 0.947193i \(-0.603906\pi\)
−0.320664 + 0.947193i \(0.603906\pi\)
\(942\) 0 0
\(943\) −5.96294e13 −2.45560
\(944\) −4.59522e12 −0.188335
\(945\) 0 0
\(946\) 8.69386e12 0.352941
\(947\) −1.97267e13 −0.797039 −0.398519 0.917160i \(-0.630476\pi\)
−0.398519 + 0.917160i \(0.630476\pi\)
\(948\) 0 0
\(949\) −2.06133e12 −0.0824993
\(950\) −9.25094e11 −0.0368493
\(951\) 0 0
\(952\) −4.12265e12 −0.162671
\(953\) 1.39214e12 0.0546721 0.0273361 0.999626i \(-0.491298\pi\)
0.0273361 + 0.999626i \(0.491298\pi\)
\(954\) 0 0
\(955\) 1.11381e13 0.433309
\(956\) 1.18721e13 0.459694
\(957\) 0 0
\(958\) 2.99895e12 0.115034
\(959\) −7.30317e12 −0.278822
\(960\) 0 0
\(961\) −8.62783e12 −0.326322
\(962\) −5.03619e12 −0.189589
\(963\) 0 0
\(964\) 6.12861e12 0.228568
\(965\) 1.20688e12 0.0448013
\(966\) 0 0
\(967\) 5.29760e12 0.194832 0.0974160 0.995244i \(-0.468942\pi\)
0.0974160 + 0.995244i \(0.468942\pi\)
\(968\) 2.48602e13 0.910049
\(969\) 0 0
\(970\) 1.35840e13 0.492669
\(971\) 7.97890e12 0.288042 0.144021 0.989575i \(-0.453997\pi\)
0.144021 + 0.989575i \(0.453997\pi\)
\(972\) 0 0
\(973\) 2.00825e12 0.0718306
\(974\) 1.23902e13 0.441127
\(975\) 0 0
\(976\) 1.82600e12 0.0644135
\(977\) 6.27831e12 0.220454 0.110227 0.993906i \(-0.464842\pi\)
0.110227 + 0.993906i \(0.464842\pi\)
\(978\) 0 0
\(979\) 1.88646e12 0.0656335
\(980\) 1.05250e12 0.0364506
\(981\) 0 0
\(982\) 1.57540e13 0.540616
\(983\) −1.25053e12 −0.0427173 −0.0213586 0.999772i \(-0.506799\pi\)
−0.0213586 + 0.999772i \(0.506799\pi\)
\(984\) 0 0
\(985\) 2.02874e13 0.686692
\(986\) 1.05686e13 0.356098
\(987\) 0 0
\(988\) 1.22753e12 0.0409851
\(989\) −7.35883e13 −2.44583
\(990\) 0 0
\(991\) 1.89095e13 0.622801 0.311400 0.950279i \(-0.399202\pi\)
0.311400 + 0.950279i \(0.399202\pi\)
\(992\) 2.40603e13 0.788858
\(993\) 0 0
\(994\) −2.47465e12 −0.0804036
\(995\) 1.72164e13 0.556851
\(996\) 0 0
\(997\) −7.17482e12 −0.229976 −0.114988 0.993367i \(-0.536683\pi\)
−0.114988 + 0.993367i \(0.536683\pi\)
\(998\) −4.95178e12 −0.158006
\(999\) 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 315.10.a.b.1.2 2
3.2 odd 2 35.10.a.b.1.1 2
15.2 even 4 175.10.b.c.99.1 4
15.8 even 4 175.10.b.c.99.4 4
15.14 odd 2 175.10.a.c.1.2 2
21.20 even 2 245.10.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.1 2 3.2 odd 2
175.10.a.c.1.2 2 15.14 odd 2
175.10.b.c.99.1 4 15.2 even 4
175.10.b.c.99.4 4 15.8 even 4
245.10.a.c.1.1 2 21.20 even 2
315.10.a.b.1.2 2 1.1 even 1 trivial