Properties

Label 83.13.b.c.82.3
Level $83$
Weight $13$
Character 83.82
Analytic conductor $75.861$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [83,13,Mod(82,83)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(83, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("83.82");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 83.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(75.8614868339\)
Analytic rank: \(0\)
Dimension: \(80\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 82.3
Character \(\chi\) \(=\) 83.82
Dual form 83.13.b.c.82.78

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-118.839i q^{2} +844.819 q^{3} -10026.8 q^{4} -13361.5i q^{5} -100398. i q^{6} -171650. q^{7} +704815. i q^{8} +182277. q^{9} +O(q^{10})\) \(q-118.839i q^{2} +844.819 q^{3} -10026.8 q^{4} -13361.5i q^{5} -100398. i q^{6} -171650. q^{7} +704815. i q^{8} +182277. q^{9} -1.58787e6 q^{10} -2.15926e6 q^{11} -8.47084e6 q^{12} +1.84444e6i q^{13} +2.03988e7i q^{14} -1.12880e7i q^{15} +4.26900e7 q^{16} +3.93307e7 q^{17} -2.16618e7i q^{18} -3.06545e7i q^{19} +1.33973e8i q^{20} -1.45013e8 q^{21} +2.56605e8i q^{22} -1.60765e8 q^{23} +5.95441e8i q^{24} +6.56122e7 q^{25} +2.19193e8 q^{26} -2.94980e8 q^{27} +1.72110e9 q^{28} -1.30484e8 q^{29} -1.34146e9 q^{30} +7.79629e7 q^{31} -2.18634e9i q^{32} -1.82418e9 q^{33} -4.67403e9i q^{34} +2.29349e9i q^{35} -1.82766e9 q^{36} +2.07818e9 q^{37} -3.64296e9 q^{38} +1.55822e9i q^{39} +9.41736e9 q^{40} +1.11963e9 q^{41} +1.72333e10i q^{42} -8.39804e9i q^{43} +2.16505e10 q^{44} -2.43549e9i q^{45} +1.91052e10i q^{46} -5.29114e8i q^{47} +3.60653e10 q^{48} +1.56224e10 q^{49} -7.79732e9i q^{50} +3.32273e10 q^{51} -1.84939e10i q^{52} +4.10466e10i q^{53} +3.50552e10i q^{54} +2.88509e10i q^{55} -1.20982e11i q^{56} -2.58975e10i q^{57} +1.55066e10i q^{58} -1.85109e10 q^{59} +1.13183e11i q^{60} +5.15512e10 q^{61} -9.26507e9i q^{62} -3.12879e10 q^{63} -8.49648e10 q^{64} +2.46445e10 q^{65} +2.16785e11i q^{66} +3.45570e10i q^{67} -3.94361e11 q^{68} -1.35817e11 q^{69} +2.72557e11 q^{70} +2.62776e10i q^{71} +1.28472e11i q^{72} +6.18428e10i q^{73} -2.46970e11i q^{74} +5.54304e10 q^{75} +3.07367e11i q^{76} +3.70637e11 q^{77} +1.85178e11 q^{78} +1.40174e11i q^{79} -5.70401e11i q^{80} -3.46074e11 q^{81} -1.33057e11i q^{82} +(2.46839e11 + 2.14384e11i) q^{83} +1.45402e12 q^{84} -5.25515e11i q^{85} -9.98019e11 q^{86} -1.10235e11 q^{87} -1.52188e12i q^{88} -8.28753e11i q^{89} -2.89433e11 q^{90} -3.16599e11i q^{91} +1.61196e12 q^{92} +6.58645e10 q^{93} -6.28797e10 q^{94} -4.09589e11 q^{95} -1.84706e12i q^{96} +5.97811e11i q^{97} -1.85656e12i q^{98} -3.93584e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9} + 1177482 q^{10} + 510406 q^{11} - 9260402 q^{12} + 203791850 q^{16} + 5571718 q^{17} + 46565862 q^{21} + 804389464 q^{23} - 4263713272 q^{25} + 18061794 q^{26} - 2326165338 q^{27} - 204811652 q^{28} + 1486960270 q^{29} - 2621683648 q^{30} - 665743010 q^{31} + 135224502 q^{33} - 54936709824 q^{36} - 280627202 q^{37} + 6646145310 q^{38} - 7119722058 q^{40} + 51318109072 q^{41} - 11512674650 q^{44} + 85368259738 q^{48} + 148785395094 q^{49} - 100143389562 q^{51} - 63584241050 q^{59} - 29216180978 q^{61} - 332932206620 q^{63} - 323596534090 q^{64} + 362112989184 q^{65} - 86115426752 q^{68} + 272383417100 q^{69} + 105630718656 q^{70} + 785418808326 q^{75} - 663355117738 q^{77} + 1483841884620 q^{78} + 2430778545148 q^{81} + 837315119192 q^{83} - 3013574788354 q^{84} + 452180651958 q^{86} - 682263689498 q^{87} - 499714512022 q^{90} + 997428187414 q^{92} - 1487992716298 q^{93} + 3169817690580 q^{94} + 1542762610848 q^{95} + 2021347267420 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/83\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 118.839i 1.85687i −0.371499 0.928433i \(-0.621156\pi\)
0.371499 0.928433i \(-0.378844\pi\)
\(3\) 844.819 1.15887 0.579437 0.815017i \(-0.303273\pi\)
0.579437 + 0.815017i \(0.303273\pi\)
\(4\) −10026.8 −2.44795
\(5\) 13361.5i 0.855133i −0.903984 0.427567i \(-0.859371\pi\)
0.903984 0.427567i \(-0.140629\pi\)
\(6\) 100398.i 2.15187i
\(7\) −171650. −1.45900 −0.729500 0.683981i \(-0.760247\pi\)
−0.729500 + 0.683981i \(0.760247\pi\)
\(8\) 704815.i 2.68866i
\(9\) 182277. 0.342987
\(10\) −1.58787e6 −1.58787
\(11\) −2.15926e6 −1.21885 −0.609423 0.792845i \(-0.708599\pi\)
−0.609423 + 0.792845i \(0.708599\pi\)
\(12\) −8.47084e6 −2.83687
\(13\) 1.84444e6i 0.382125i 0.981578 + 0.191063i \(0.0611933\pi\)
−0.981578 + 0.191063i \(0.938807\pi\)
\(14\) 2.03988e7i 2.70917i
\(15\) 1.12880e7i 0.990991i
\(16\) 4.26900e7 2.54452
\(17\) 3.93307e7 1.62944 0.814719 0.579857i \(-0.196891\pi\)
0.814719 + 0.579857i \(0.196891\pi\)
\(18\) 2.16618e7i 0.636882i
\(19\) 3.06545e7i 0.651587i −0.945441 0.325794i \(-0.894369\pi\)
0.945441 0.325794i \(-0.105631\pi\)
\(20\) 1.33973e8i 2.09333i
\(21\) −1.45013e8 −1.69080
\(22\) 2.56605e8i 2.26323i
\(23\) −1.60765e8 −1.08598 −0.542992 0.839738i \(-0.682709\pi\)
−0.542992 + 0.839738i \(0.682709\pi\)
\(24\) 5.95441e8i 3.11581i
\(25\) 6.56122e7 0.268748
\(26\) 2.19193e8 0.709555
\(27\) −2.94980e8 −0.761395
\(28\) 1.72110e9 3.57157
\(29\) −1.30484e8 −0.219366 −0.109683 0.993967i \(-0.534984\pi\)
−0.109683 + 0.993967i \(0.534984\pi\)
\(30\) −1.34146e9 −1.84014
\(31\) 7.79629e7 0.0878452 0.0439226 0.999035i \(-0.486015\pi\)
0.0439226 + 0.999035i \(0.486015\pi\)
\(32\) 2.18634e9i 2.03619i
\(33\) −1.82418e9 −1.41249
\(34\) 4.67403e9i 3.02565i
\(35\) 2.29349e9i 1.24764i
\(36\) −1.82766e9 −0.839617
\(37\) 2.07818e9 0.809978 0.404989 0.914322i \(-0.367275\pi\)
0.404989 + 0.914322i \(0.367275\pi\)
\(38\) −3.64296e9 −1.20991
\(39\) 1.55822e9i 0.442835i
\(40\) 9.41736e9 2.29916
\(41\) 1.11963e9 0.235707 0.117853 0.993031i \(-0.462399\pi\)
0.117853 + 0.993031i \(0.462399\pi\)
\(42\) 1.72333e10i 3.13958i
\(43\) 8.39804e9i 1.32852i −0.747503 0.664259i \(-0.768747\pi\)
0.747503 0.664259i \(-0.231253\pi\)
\(44\) 2.16505e10 2.98368
\(45\) 2.43549e9i 0.293300i
\(46\) 1.91052e10i 2.01653i
\(47\) 5.29114e8i 0.0490865i −0.999699 0.0245433i \(-0.992187\pi\)
0.999699 0.0245433i \(-0.00781315\pi\)
\(48\) 3.60653e10 2.94878
\(49\) 1.56224e10 1.12868
\(50\) 7.79732e9i 0.499028i
\(51\) 3.32273e10 1.88831
\(52\) 1.84939e10i 0.935425i
\(53\) 4.10466e10i 1.85192i 0.377622 + 0.925960i \(0.376742\pi\)
−0.377622 + 0.925960i \(0.623258\pi\)
\(54\) 3.50552e10i 1.41381i
\(55\) 2.88509e10i 1.04228i
\(56\) 1.20982e11i 3.92275i
\(57\) 2.58975e10i 0.755107i
\(58\) 1.55066e10i 0.407333i
\(59\) −1.85109e10 −0.438850 −0.219425 0.975629i \(-0.570418\pi\)
−0.219425 + 0.975629i \(0.570418\pi\)
\(60\) 1.13183e11i 2.42590i
\(61\) 5.15512e10 1.00060 0.500299 0.865853i \(-0.333223\pi\)
0.500299 + 0.865853i \(0.333223\pi\)
\(62\) 9.26507e9i 0.163117i
\(63\) −3.12879e10 −0.500419
\(64\) −8.49648e10 −1.23640
\(65\) 2.46445e10 0.326768
\(66\) 2.16785e11i 2.62280i
\(67\) 3.45570e10i 0.382021i 0.981588 + 0.191010i \(0.0611764\pi\)
−0.981588 + 0.191010i \(0.938824\pi\)
\(68\) −3.94361e11 −3.98879
\(69\) −1.35817e11 −1.25852
\(70\) 2.72557e11 2.31670
\(71\) 2.62776e10i 0.205133i 0.994726 + 0.102567i \(0.0327055\pi\)
−0.994726 + 0.102567i \(0.967295\pi\)
\(72\) 1.28472e11i 0.922175i
\(73\) 6.18428e10i 0.408651i 0.978903 + 0.204325i \(0.0655000\pi\)
−0.978903 + 0.204325i \(0.934500\pi\)
\(74\) 2.46970e11i 1.50402i
\(75\) 5.54304e10 0.311444
\(76\) 3.07367e11i 1.59506i
\(77\) 3.70637e11 1.77830
\(78\) 1.85178e11 0.822285
\(79\) 1.40174e11i 0.576640i 0.957534 + 0.288320i \(0.0930966\pi\)
−0.957534 + 0.288320i \(0.906903\pi\)
\(80\) 5.70401e11i 2.17591i
\(81\) −3.46074e11 −1.22535
\(82\) 1.33057e11i 0.437676i
\(83\) 2.46839e11 + 2.14384e11i 0.754998 + 0.655727i
\(84\) 1.45402e12 4.13899
\(85\) 5.25515e11i 1.39339i
\(86\) −9.98019e11 −2.46688
\(87\) −1.10235e11 −0.254217
\(88\) 1.52188e12i 3.27706i
\(89\) 8.28753e11i 1.66757i −0.552087 0.833787i \(-0.686168\pi\)
0.552087 0.833787i \(-0.313832\pi\)
\(90\) −2.89433e11 −0.544618
\(91\) 3.16599e11i 0.557521i
\(92\) 1.61196e12 2.65844
\(93\) 6.58645e10 0.101801
\(94\) −6.28797e10 −0.0911472
\(95\) −4.09589e11 −0.557194
\(96\) 1.84706e12i 2.35968i
\(97\) 5.97811e11i 0.717685i 0.933398 + 0.358842i \(0.116828\pi\)
−0.933398 + 0.358842i \(0.883172\pi\)
\(98\) 1.85656e12i 2.09581i
\(99\) −3.93584e11 −0.418049
\(100\) −6.57882e11 −0.657882
\(101\) 1.21055e12i 1.14039i −0.821508 0.570197i \(-0.806867\pi\)
0.821508 0.570197i \(-0.193133\pi\)
\(102\) 3.94871e12i 3.50634i
\(103\) 1.62356e12i 1.35971i 0.733348 + 0.679853i \(0.237956\pi\)
−0.733348 + 0.679853i \(0.762044\pi\)
\(104\) −1.29999e12 −1.02740
\(105\) 1.93759e12i 1.44586i
\(106\) 4.87796e12 3.43877
\(107\) 2.53420e12i 1.68864i 0.535836 + 0.844322i \(0.319997\pi\)
−0.535836 + 0.844322i \(0.680003\pi\)
\(108\) 2.95771e12 1.86386
\(109\) −2.99841e12 −1.78785 −0.893927 0.448212i \(-0.852061\pi\)
−0.893927 + 0.448212i \(0.852061\pi\)
\(110\) 3.42862e12 1.93537
\(111\) 1.75569e12 0.938662
\(112\) −7.32774e12 −3.71246
\(113\) 2.58513e11 0.124169 0.0620843 0.998071i \(-0.480225\pi\)
0.0620843 + 0.998071i \(0.480225\pi\)
\(114\) −3.07764e12 −1.40213
\(115\) 2.14805e12i 0.928661i
\(116\) 1.30834e12 0.536997
\(117\) 3.36201e11i 0.131064i
\(118\) 2.19983e12i 0.814887i
\(119\) −6.75110e12 −2.37735
\(120\) 7.95596e12 2.66443
\(121\) 1.52398e12 0.485586
\(122\) 6.12632e12i 1.85798i
\(123\) 9.45886e11 0.273154
\(124\) −7.81720e11 −0.215041
\(125\) 4.13875e12i 1.08495i
\(126\) 3.71824e12i 0.929210i
\(127\) −6.54207e12 −1.55917 −0.779584 0.626298i \(-0.784569\pi\)
−0.779584 + 0.626298i \(0.784569\pi\)
\(128\) 1.14193e12i 0.259646i
\(129\) 7.09482e12i 1.53958i
\(130\) 2.92873e12i 0.606764i
\(131\) −4.37005e12 −0.864687 −0.432343 0.901709i \(-0.642313\pi\)
−0.432343 + 0.901709i \(0.642313\pi\)
\(132\) 1.82908e13 3.45771
\(133\) 5.26184e12i 0.950666i
\(134\) 4.10673e12 0.709361
\(135\) 3.94136e12i 0.651094i
\(136\) 2.77209e13i 4.38100i
\(137\) 8.58649e12i 1.29865i 0.760511 + 0.649325i \(0.224949\pi\)
−0.760511 + 0.649325i \(0.775051\pi\)
\(138\) 1.61404e13i 2.33690i
\(139\) 4.08815e12i 0.566811i −0.959000 0.283405i \(-0.908536\pi\)
0.959000 0.283405i \(-0.0914642\pi\)
\(140\) 2.29964e13i 3.05416i
\(141\) 4.47006e11i 0.0568851i
\(142\) 3.12282e12 0.380905
\(143\) 3.98264e12i 0.465752i
\(144\) 7.78143e12 0.872740
\(145\) 1.74345e12i 0.187587i
\(146\) 7.34937e12 0.758810
\(147\) 1.31981e13 1.30800
\(148\) −2.08376e13 −1.98279
\(149\) 3.85057e12i 0.351891i −0.984400 0.175945i \(-0.943702\pi\)
0.984400 0.175945i \(-0.0562982\pi\)
\(150\) 6.58732e12i 0.578311i
\(151\) 1.93265e13 1.63039 0.815196 0.579186i \(-0.196629\pi\)
0.815196 + 0.579186i \(0.196629\pi\)
\(152\) 2.16058e13 1.75190
\(153\) 7.16909e12 0.558876
\(154\) 4.40463e13i 3.30206i
\(155\) 1.04170e12i 0.0751193i
\(156\) 1.56240e13i 1.08404i
\(157\) 1.79822e13i 1.20073i −0.799726 0.600365i \(-0.795022\pi\)
0.799726 0.600365i \(-0.204978\pi\)
\(158\) 1.66582e13 1.07074
\(159\) 3.46769e13i 2.14614i
\(160\) −2.92127e13 −1.74121
\(161\) 2.75953e13 1.58445
\(162\) 4.11273e13i 2.27531i
\(163\) 3.63775e13i 1.93958i 0.243945 + 0.969789i \(0.421558\pi\)
−0.243945 + 0.969789i \(0.578442\pi\)
\(164\) −1.12264e13 −0.577000
\(165\) 2.43737e13i 1.20787i
\(166\) 2.54772e13 2.93343e13i 1.21760 1.40193i
\(167\) −3.86195e13 −1.78036 −0.890180 0.455608i \(-0.849422\pi\)
−0.890180 + 0.455608i \(0.849422\pi\)
\(168\) 1.02207e14i 4.54597i
\(169\) 1.98961e13 0.853980
\(170\) −6.24519e13 −2.58733
\(171\) 5.58763e12i 0.223486i
\(172\) 8.42057e13i 3.25215i
\(173\) 1.21863e13 0.454565 0.227282 0.973829i \(-0.427016\pi\)
0.227282 + 0.973829i \(0.427016\pi\)
\(174\) 1.31003e13i 0.472047i
\(175\) −1.12623e13 −0.392103
\(176\) −9.21789e13 −3.10138
\(177\) −1.56384e13 −0.508572
\(178\) −9.84885e13 −3.09646
\(179\) 3.55533e13i 1.08084i 0.841395 + 0.540420i \(0.181735\pi\)
−0.841395 + 0.540420i \(0.818265\pi\)
\(180\) 2.44202e13i 0.717984i
\(181\) 3.32330e13i 0.945143i 0.881292 + 0.472572i \(0.156674\pi\)
−0.881292 + 0.472572i \(0.843326\pi\)
\(182\) −3.76244e13 −1.03524
\(183\) 4.35514e13 1.15957
\(184\) 1.13309e14i 2.91984i
\(185\) 2.77675e13i 0.692639i
\(186\) 7.82730e12i 0.189032i
\(187\) −8.49251e13 −1.98603
\(188\) 5.30533e12i 0.120162i
\(189\) 5.06333e13 1.11087
\(190\) 4.86753e13i 1.03463i
\(191\) 5.91788e13 1.21889 0.609447 0.792827i \(-0.291392\pi\)
0.609447 + 0.792827i \(0.291392\pi\)
\(192\) −7.17799e13 −1.43283
\(193\) 3.88589e12 0.0751877 0.0375939 0.999293i \(-0.488031\pi\)
0.0375939 + 0.999293i \(0.488031\pi\)
\(194\) 7.10436e13 1.33264
\(195\) 2.08201e13 0.378682
\(196\) −1.56643e14 −2.76296
\(197\) 8.53223e13 1.45971 0.729853 0.683605i \(-0.239589\pi\)
0.729853 + 0.683605i \(0.239589\pi\)
\(198\) 4.67734e13i 0.776261i
\(199\) −8.51346e13 −1.37084 −0.685421 0.728147i \(-0.740382\pi\)
−0.685421 + 0.728147i \(0.740382\pi\)
\(200\) 4.62445e13i 0.722570i
\(201\) 2.91944e13i 0.442713i
\(202\) −1.43861e14 −2.11756
\(203\) 2.23975e13 0.320055
\(204\) −3.33164e14 −4.62250
\(205\) 1.49599e13i 0.201561i
\(206\) 1.92943e14 2.52479
\(207\) −2.93038e13 −0.372479
\(208\) 7.87394e13i 0.972327i
\(209\) 6.61910e13i 0.794185i
\(210\) 2.30262e14 2.68476
\(211\) 3.73191e13i 0.422899i −0.977389 0.211449i \(-0.932182\pi\)
0.977389 0.211449i \(-0.0678184\pi\)
\(212\) 4.11567e14i 4.53341i
\(213\) 2.21998e13i 0.237723i
\(214\) 3.01163e14 3.13559
\(215\) −1.12210e14 −1.13606
\(216\) 2.07906e14i 2.04713i
\(217\) −1.33823e13 −0.128166
\(218\) 3.56330e14i 3.31981i
\(219\) 5.22460e13i 0.473574i
\(220\) 2.89282e14i 2.55144i
\(221\) 7.25432e13i 0.622649i
\(222\) 2.08645e14i 1.74297i
\(223\) 8.90915e13i 0.724448i 0.932091 + 0.362224i \(0.117982\pi\)
−0.932091 + 0.362224i \(0.882018\pi\)
\(224\) 3.75285e14i 2.97080i
\(225\) 1.19596e13 0.0921770
\(226\) 3.07216e13i 0.230565i
\(227\) 8.71919e13 0.637267 0.318633 0.947878i \(-0.396776\pi\)
0.318633 + 0.947878i \(0.396776\pi\)
\(228\) 2.59670e14i 1.84847i
\(229\) 6.84383e13 0.474555 0.237277 0.971442i \(-0.423745\pi\)
0.237277 + 0.971442i \(0.423745\pi\)
\(230\) 2.55273e14 1.72440
\(231\) 3.13121e14 2.06082
\(232\) 9.19670e13i 0.589799i
\(233\) 1.02919e14i 0.643218i 0.946873 + 0.321609i \(0.104224\pi\)
−0.946873 + 0.321609i \(0.895776\pi\)
\(234\) 3.99539e13 0.243368
\(235\) −7.06974e12 −0.0419755
\(236\) 1.85606e14 1.07429
\(237\) 1.18421e14i 0.668252i
\(238\) 8.02298e14i 4.41442i
\(239\) 1.78792e13i 0.0959312i −0.998849 0.0479656i \(-0.984726\pi\)
0.998849 0.0479656i \(-0.0152738\pi\)
\(240\) 4.81885e14i 2.52160i
\(241\) −1.67618e14 −0.855497 −0.427748 0.903898i \(-0.640693\pi\)
−0.427748 + 0.903898i \(0.640693\pi\)
\(242\) 1.81108e14i 0.901668i
\(243\) −1.35606e14 −0.658627
\(244\) −5.16895e14 −2.44942
\(245\) 2.08738e14i 0.965173i
\(246\) 1.12409e14i 0.507211i
\(247\) 5.65405e13 0.248988
\(248\) 5.49495e13i 0.236186i
\(249\) 2.08535e14 + 1.81115e14i 0.874947 + 0.759904i
\(250\) −4.91847e14 −2.01460
\(251\) 2.57841e13i 0.103112i 0.998670 + 0.0515560i \(0.0164181\pi\)
−0.998670 + 0.0515560i \(0.983582\pi\)
\(252\) 3.13718e14 1.22500
\(253\) 3.47133e14 1.32365
\(254\) 7.77456e14i 2.89517i
\(255\) 4.43965e14i 1.61476i
\(256\) −2.12309e14 −0.754273
\(257\) 3.59703e14i 1.24838i −0.781274 0.624189i \(-0.785430\pi\)
0.781274 0.624189i \(-0.214570\pi\)
\(258\) −8.43145e14 −2.85880
\(259\) −3.56720e14 −1.18176
\(260\) −2.47106e14 −0.799912
\(261\) −2.37843e13 −0.0752396
\(262\) 5.19335e14i 1.60561i
\(263\) 6.52934e14i 1.97303i 0.163659 + 0.986517i \(0.447670\pi\)
−0.163659 + 0.986517i \(0.552330\pi\)
\(264\) 1.28571e15i 3.79770i
\(265\) 5.48442e14 1.58364
\(266\) 6.25315e14 1.76526
\(267\) 7.00146e14i 1.93251i
\(268\) 3.46496e14i 0.935169i
\(269\) 4.72061e13i 0.124590i 0.998058 + 0.0622951i \(0.0198420\pi\)
−0.998058 + 0.0622951i \(0.980158\pi\)
\(270\) 4.68389e14 1.20899
\(271\) 4.78882e14i 1.20896i 0.796619 + 0.604481i \(0.206620\pi\)
−0.796619 + 0.604481i \(0.793380\pi\)
\(272\) 1.67903e15 4.14614
\(273\) 2.67469e14i 0.646096i
\(274\) 1.02041e15 2.41142
\(275\) −1.41674e14 −0.327562
\(276\) 1.36181e15 3.08080
\(277\) −4.75040e14 −1.05160 −0.525802 0.850607i \(-0.676235\pi\)
−0.525802 + 0.850607i \(0.676235\pi\)
\(278\) −4.85834e14 −1.05249
\(279\) 1.42109e13 0.0301298
\(280\) −1.61649e15 −3.35448
\(281\) 7.73695e14i 1.57156i −0.618504 0.785782i \(-0.712261\pi\)
0.618504 0.785782i \(-0.287739\pi\)
\(282\) −5.31219e13 −0.105628
\(283\) 9.80496e14i 1.90865i 0.298764 + 0.954327i \(0.403426\pi\)
−0.298764 + 0.954327i \(0.596574\pi\)
\(284\) 2.63481e14i 0.502157i
\(285\) −3.46028e14 −0.645717
\(286\) −4.73294e14 −0.864839
\(287\) −1.92185e14 −0.343896
\(288\) 3.98520e14i 0.698386i
\(289\) 9.64278e14 1.65507
\(290\) 2.07191e14 0.348324
\(291\) 5.05042e14i 0.831706i
\(292\) 6.20087e14i 1.00036i
\(293\) −9.86102e13 −0.155853 −0.0779267 0.996959i \(-0.524830\pi\)
−0.0779267 + 0.996959i \(0.524830\pi\)
\(294\) 1.56846e15i 2.42878i
\(295\) 2.47333e14i 0.375275i
\(296\) 1.46473e15i 2.17775i
\(297\) 6.36938e14 0.928023
\(298\) −4.57600e14 −0.653414
\(299\) 2.96522e14i 0.414982i
\(300\) −5.55791e14 −0.762401
\(301\) 1.44152e15i 1.93831i
\(302\) 2.29675e15i 3.02742i
\(303\) 1.02270e15i 1.32157i
\(304\) 1.30864e15i 1.65798i
\(305\) 6.88799e14i 0.855645i
\(306\) 8.51971e14i 1.03776i
\(307\) 6.04520e14i 0.722071i −0.932552 0.361036i \(-0.882423\pi\)
0.932552 0.361036i \(-0.117577\pi\)
\(308\) −3.71631e15 −4.35319
\(309\) 1.37161e15i 1.57573i
\(310\) −1.23795e14 −0.139487
\(311\) 2.10845e14i 0.233024i 0.993189 + 0.116512i \(0.0371714\pi\)
−0.993189 + 0.116512i \(0.962829\pi\)
\(312\) −1.09826e15 −1.19063
\(313\) −1.23727e14 −0.131582 −0.0657912 0.997833i \(-0.520957\pi\)
−0.0657912 + 0.997833i \(0.520957\pi\)
\(314\) −2.13700e15 −2.22959
\(315\) 4.18052e14i 0.427924i
\(316\) 1.40550e15i 1.41159i
\(317\) −1.82391e15 −1.79742 −0.898708 0.438548i \(-0.855493\pi\)
−0.898708 + 0.438548i \(0.855493\pi\)
\(318\) 4.12099e15 3.98510
\(319\) 2.81749e14 0.267373
\(320\) 1.13525e15i 1.05729i
\(321\) 2.14094e15i 1.95692i
\(322\) 3.27941e15i 2.94212i
\(323\) 1.20566e15i 1.06172i
\(324\) 3.47002e15 2.99959
\(325\) 1.21018e14i 0.102695i
\(326\) 4.32308e15 3.60154
\(327\) −2.53311e15 −2.07190
\(328\) 7.89134e14i 0.633735i
\(329\) 9.08225e13i 0.0716173i
\(330\) 2.89656e15 2.24284
\(331\) 2.50773e15i 1.90684i 0.301654 + 0.953418i \(0.402461\pi\)
−0.301654 + 0.953418i \(0.597539\pi\)
\(332\) −2.47501e15 2.14959e15i −1.84820 1.60519i
\(333\) 3.78806e14 0.277812
\(334\) 4.58952e15i 3.30589i
\(335\) 4.61731e14 0.326678
\(336\) −6.19061e15 −4.30227
\(337\) 7.68904e14i 0.524920i −0.964943 0.262460i \(-0.915466\pi\)
0.964943 0.262460i \(-0.0845338\pi\)
\(338\) 2.36444e15i 1.58573i
\(339\) 2.18397e14 0.143896
\(340\) 5.26924e15i 3.41094i
\(341\) −1.68342e14 −0.107070
\(342\) −6.64030e14 −0.414984
\(343\) −3.05730e14 −0.187747
\(344\) 5.91907e15 3.57193
\(345\) 1.81471e15i 1.07620i
\(346\) 1.44822e15i 0.844066i
\(347\) 3.17159e14i 0.181677i −0.995866 0.0908386i \(-0.971045\pi\)
0.995866 0.0908386i \(-0.0289547\pi\)
\(348\) 1.10531e15 0.622312
\(349\) −9.46396e13 −0.0523746 −0.0261873 0.999657i \(-0.508337\pi\)
−0.0261873 + 0.999657i \(0.508337\pi\)
\(350\) 1.33841e15i 0.728083i
\(351\) 5.44074e14i 0.290948i
\(352\) 4.72087e15i 2.48180i
\(353\) 1.39545e15 0.721219 0.360610 0.932717i \(-0.382569\pi\)
0.360610 + 0.932717i \(0.382569\pi\)
\(354\) 1.85846e15i 0.944350i
\(355\) 3.51107e14 0.175416
\(356\) 8.30975e15i 4.08214i
\(357\) −5.70346e15 −2.75505
\(358\) 4.22513e15 2.00698
\(359\) 5.60158e14 0.261664 0.130832 0.991405i \(-0.458235\pi\)
0.130832 + 0.991405i \(0.458235\pi\)
\(360\) 1.71657e15 0.788583
\(361\) 1.27362e15 0.575434
\(362\) 3.94939e15 1.75500
\(363\) 1.28748e15 0.562732
\(364\) 3.17448e15i 1.36478i
\(365\) 8.26310e14 0.349451
\(366\) 5.17563e15i 2.15316i
\(367\) 8.60328e14i 0.352101i −0.984381 0.176051i \(-0.943668\pi\)
0.984381 0.176051i \(-0.0563322\pi\)
\(368\) −6.86305e15 −2.76332
\(369\) 2.04084e14 0.0808445
\(370\) −3.29988e15 −1.28614
\(371\) 7.04565e15i 2.70195i
\(372\) −6.60412e14 −0.249205
\(373\) −5.27027e15 −1.95695 −0.978475 0.206364i \(-0.933837\pi\)
−0.978475 + 0.206364i \(0.933837\pi\)
\(374\) 1.00925e16i 3.68780i
\(375\) 3.49649e15i 1.25732i
\(376\) 3.72928e14 0.131977
\(377\) 2.40670e14i 0.0838251i
\(378\) 6.01723e15i 2.06275i
\(379\) 3.72384e15i 1.25648i −0.778019 0.628241i \(-0.783775\pi\)
0.778019 0.628241i \(-0.216225\pi\)
\(380\) 4.10687e15 1.36399
\(381\) −5.52686e15 −1.80688
\(382\) 7.03277e15i 2.26332i
\(383\) −1.79701e15 −0.569322 −0.284661 0.958628i \(-0.591881\pi\)
−0.284661 + 0.958628i \(0.591881\pi\)
\(384\) 9.64728e14i 0.300897i
\(385\) 4.95225e15i 1.52068i
\(386\) 4.61798e14i 0.139614i
\(387\) 1.53077e15i 0.455665i
\(388\) 5.99415e15i 1.75686i
\(389\) 6.69572e15i 1.93241i 0.257774 + 0.966205i \(0.417011\pi\)
−0.257774 + 0.966205i \(0.582989\pi\)
\(390\) 2.47425e15i 0.703163i
\(391\) −6.32298e15 −1.76954
\(392\) 1.10109e16i 3.03464i
\(393\) −3.69190e15 −1.00206
\(394\) 1.01397e16i 2.71048i
\(395\) 1.87293e15 0.493104
\(396\) 3.94640e15 1.02336
\(397\) −2.15715e15 −0.550981 −0.275490 0.961304i \(-0.588840\pi\)
−0.275490 + 0.961304i \(0.588840\pi\)
\(398\) 1.01173e16i 2.54547i
\(399\) 4.44530e15i 1.10170i
\(400\) 2.80099e15 0.683835
\(401\) 8.21528e13 0.0197586 0.00987929 0.999951i \(-0.496855\pi\)
0.00987929 + 0.999951i \(0.496855\pi\)
\(402\) 3.46944e15 0.822060
\(403\) 1.43798e14i 0.0335678i
\(404\) 1.21380e16i 2.79163i
\(405\) 4.62405e15i 1.04783i
\(406\) 2.66171e15i 0.594299i
\(407\) −4.48734e15 −0.987239
\(408\) 2.34191e16i 5.07702i
\(409\) −2.04697e15 −0.437293 −0.218646 0.975804i \(-0.570164\pi\)
−0.218646 + 0.975804i \(0.570164\pi\)
\(410\) −1.77783e15 −0.374271
\(411\) 7.25402e15i 1.50497i
\(412\) 1.62791e16i 3.32850i
\(413\) 3.17740e15 0.640283
\(414\) 3.48245e15i 0.691644i
\(415\) 2.86448e15 3.29813e15i 0.560734 0.645624i
\(416\) 4.03258e15 0.778078
\(417\) 3.45375e15i 0.656862i
\(418\) 7.86611e15 1.47470
\(419\) −1.01743e15 −0.188027 −0.0940136 0.995571i \(-0.529970\pi\)
−0.0940136 + 0.995571i \(0.529970\pi\)
\(420\) 1.94278e16i 3.53939i
\(421\) 6.38091e15i 1.14601i 0.819550 + 0.573007i \(0.194223\pi\)
−0.819550 + 0.573007i \(0.805777\pi\)
\(422\) −4.43498e15 −0.785267
\(423\) 9.64456e13i 0.0168361i
\(424\) −2.89303e16 −4.97918
\(425\) 2.58057e15 0.437907
\(426\) 2.63821e15 0.441421
\(427\) −8.84876e15 −1.45987
\(428\) 2.54100e16i 4.13372i
\(429\) 3.36460e15i 0.539747i
\(430\) 1.33350e16i 2.10951i
\(431\) 5.76442e15 0.899274 0.449637 0.893211i \(-0.351553\pi\)
0.449637 + 0.893211i \(0.351553\pi\)
\(432\) −1.25927e16 −1.93739
\(433\) 5.86172e15i 0.889401i −0.895679 0.444701i \(-0.853310\pi\)
0.895679 0.444701i \(-0.146690\pi\)
\(434\) 1.59035e15i 0.237987i
\(435\) 1.47290e15i 0.217389i
\(436\) 3.00645e16 4.37659
\(437\) 4.92816e15i 0.707614i
\(438\) 6.20888e15 0.879364
\(439\) 3.25761e15i 0.455105i −0.973766 0.227553i \(-0.926928\pi\)
0.973766 0.227553i \(-0.0730724\pi\)
\(440\) −2.03345e16 −2.80232
\(441\) 2.84761e15 0.387124
\(442\) 8.62100e15 1.15618
\(443\) −1.27860e16 −1.69165 −0.845826 0.533459i \(-0.820892\pi\)
−0.845826 + 0.533459i \(0.820892\pi\)
\(444\) −1.76040e16 −2.29780
\(445\) −1.10733e16 −1.42600
\(446\) 1.05876e16 1.34520
\(447\) 3.25304e15i 0.407797i
\(448\) 1.45842e16 1.80391
\(449\) 8.57615e15i 1.04668i −0.852124 0.523340i \(-0.824686\pi\)
0.852124 0.523340i \(-0.175314\pi\)
\(450\) 1.42128e15i 0.171160i
\(451\) −2.41758e15 −0.287290
\(452\) −2.59206e15 −0.303959
\(453\) 1.63274e16 1.88942
\(454\) 1.03618e16i 1.18332i
\(455\) −4.23022e15 −0.476754
\(456\) 1.82530e16 2.03022
\(457\) 1.72254e15i 0.189092i 0.995521 + 0.0945459i \(0.0301399\pi\)
−0.995521 + 0.0945459i \(0.969860\pi\)
\(458\) 8.13317e15i 0.881185i
\(459\) −1.16017e16 −1.24064
\(460\) 2.15381e16i 2.27332i
\(461\) 6.71326e15i 0.699404i 0.936861 + 0.349702i \(0.113717\pi\)
−0.936861 + 0.349702i \(0.886283\pi\)
\(462\) 3.72111e16i 3.82667i
\(463\) 1.42783e16 1.44941 0.724705 0.689059i \(-0.241976\pi\)
0.724705 + 0.689059i \(0.241976\pi\)
\(464\) −5.57036e15 −0.558182
\(465\) 8.80046e14i 0.0870537i
\(466\) 1.22308e16 1.19437
\(467\) 7.28270e15i 0.702087i 0.936359 + 0.351044i \(0.114173\pi\)
−0.936359 + 0.351044i \(0.885827\pi\)
\(468\) 3.37102e15i 0.320839i
\(469\) 5.93170e15i 0.557368i
\(470\) 8.40164e14i 0.0779430i
\(471\) 1.51917e16i 1.39149i
\(472\) 1.30468e16i 1.17992i
\(473\) 1.81336e16i 1.61926i
\(474\) 1.40731e16 1.24086
\(475\) 2.01131e15i 0.175112i
\(476\) 6.76921e16 5.81964
\(477\) 7.48187e15i 0.635185i
\(478\) −2.12475e15 −0.178131
\(479\) 1.64745e16 1.36395 0.681975 0.731376i \(-0.261121\pi\)
0.681975 + 0.731376i \(0.261121\pi\)
\(480\) −2.46794e16 −2.01784
\(481\) 3.83309e15i 0.309513i
\(482\) 1.99196e16i 1.58854i
\(483\) 2.33130e16 1.83618
\(484\) −1.52806e16 −1.18869
\(485\) 7.98763e15 0.613716
\(486\) 1.61153e16i 1.22298i
\(487\) 2.13245e16i 1.59847i 0.601019 + 0.799235i \(0.294762\pi\)
−0.601019 + 0.799235i \(0.705238\pi\)
\(488\) 3.63341e16i 2.69027i
\(489\) 3.07324e16i 2.24773i
\(490\) −2.48063e16 −1.79220
\(491\) 7.18491e15i 0.512782i −0.966573 0.256391i \(-0.917467\pi\)
0.966573 0.256391i \(-0.0825334\pi\)
\(492\) −9.48423e15 −0.668669
\(493\) −5.13201e15 −0.357443
\(494\) 6.71925e15i 0.462337i
\(495\) 5.25886e15i 0.357487i
\(496\) 3.32824e15 0.223524
\(497\) 4.51055e15i 0.299289i
\(498\) 2.15236e16 2.47821e16i 1.41104 1.62466i
\(499\) 5.21234e15 0.337621 0.168810 0.985649i \(-0.446007\pi\)
0.168810 + 0.985649i \(0.446007\pi\)
\(500\) 4.14985e16i 2.65590i
\(501\) −3.26265e16 −2.06321
\(502\) 3.06417e15 0.191465
\(503\) 1.18370e16i 0.730858i 0.930839 + 0.365429i \(0.119078\pi\)
−0.930839 + 0.365429i \(0.880922\pi\)
\(504\) 2.20522e16i 1.34545i
\(505\) −1.61747e16 −0.975189
\(506\) 4.12531e16i 2.45784i
\(507\) 1.68086e16 0.989655
\(508\) 6.55961e16 3.81677
\(509\) −6.07670e15 −0.349431 −0.174715 0.984619i \(-0.555901\pi\)
−0.174715 + 0.984619i \(0.555901\pi\)
\(510\) −5.27605e16 −2.99839
\(511\) 1.06153e16i 0.596221i
\(512\) 2.99081e16i 1.66023i
\(513\) 9.04246e15i 0.496115i
\(514\) −4.27470e16 −2.31807
\(515\) 2.16931e16 1.16273
\(516\) 7.11385e16i 3.76883i
\(517\) 1.14250e15i 0.0598289i
\(518\) 4.23924e16i 2.19437i
\(519\) 1.02952e16 0.526783
\(520\) 1.73698e16i 0.878567i
\(521\) −1.73245e16 −0.866231 −0.433116 0.901338i \(-0.642586\pi\)
−0.433116 + 0.901338i \(0.642586\pi\)
\(522\) 2.82651e15i 0.139710i
\(523\) −7.23137e15 −0.353354 −0.176677 0.984269i \(-0.556535\pi\)
−0.176677 + 0.984269i \(0.556535\pi\)
\(524\) 4.38177e16 2.11671
\(525\) −9.51462e15 −0.454397
\(526\) 7.75943e16 3.66366
\(527\) 3.06633e15 0.143138
\(528\) −7.78745e16 −3.59411
\(529\) 3.93067e15 0.179363
\(530\) 6.51766e16i 2.94060i
\(531\) −3.37413e15 −0.150520
\(532\) 5.27596e16i 2.32719i
\(533\) 2.06510e15i 0.0900695i
\(534\) −8.32049e16 −3.58841
\(535\) 3.38606e16 1.44402
\(536\) −2.43563e16 −1.02712
\(537\) 3.00361e16i 1.25256i
\(538\) 5.60994e15 0.231347
\(539\) −3.37329e16 −1.37569
\(540\) 3.95193e16i 1.59385i
\(541\) 2.71530e16i 1.08302i −0.840696 0.541508i \(-0.817854\pi\)
0.840696 0.541508i \(-0.182146\pi\)
\(542\) 5.69101e16 2.24488
\(543\) 2.80758e16i 1.09530i
\(544\) 8.59901e16i 3.31784i
\(545\) 4.00631e16i 1.52885i
\(546\) −3.17858e16 −1.19971
\(547\) 1.88442e16 0.703484 0.351742 0.936097i \(-0.385589\pi\)
0.351742 + 0.936097i \(0.385589\pi\)
\(548\) 8.60952e16i 3.17904i
\(549\) 9.39662e15 0.343192
\(550\) 1.68364e16i 0.608239i
\(551\) 3.99992e15i 0.142936i
\(552\) 9.57259e16i 3.38373i
\(553\) 2.40608e16i 0.841317i
\(554\) 5.64536e16i 1.95269i
\(555\) 2.34585e16i 0.802681i
\(556\) 4.09912e16i 1.38753i
\(557\) 2.01451e16 0.674588 0.337294 0.941399i \(-0.390488\pi\)
0.337294 + 0.941399i \(0.390488\pi\)
\(558\) 1.68881e15i 0.0559470i
\(559\) 1.54897e16 0.507660
\(560\) 9.79093e16i 3.17465i
\(561\) −7.17463e16 −2.30156
\(562\) −9.19455e16 −2.91818
\(563\) −4.95890e16 −1.55716 −0.778582 0.627542i \(-0.784061\pi\)
−0.778582 + 0.627542i \(0.784061\pi\)
\(564\) 4.48205e15i 0.139252i
\(565\) 3.45411e15i 0.106181i
\(566\) 1.16522e17 3.54412
\(567\) 5.94036e16 1.78778
\(568\) −1.85209e16 −0.551533
\(569\) 3.06515e16i 0.903187i 0.892224 + 0.451594i \(0.149144\pi\)
−0.892224 + 0.451594i \(0.850856\pi\)
\(570\) 4.11218e16i 1.19901i
\(571\) 3.02745e16i 0.873495i −0.899584 0.436748i \(-0.856130\pi\)
0.899584 0.436748i \(-0.143870\pi\)
\(572\) 3.99332e16i 1.14014i
\(573\) 4.99953e16 1.41254
\(574\) 2.28391e16i 0.638570i
\(575\) −1.05481e16 −0.291856
\(576\) −1.54872e16 −0.424070
\(577\) 1.73130e16i 0.469156i −0.972097 0.234578i \(-0.924629\pi\)
0.972097 0.234578i \(-0.0753708\pi\)
\(578\) 1.14594e17i 3.07324i
\(579\) 3.28288e15 0.0871331
\(580\) 1.74813e16i 0.459204i
\(581\) −4.23700e16 3.67989e16i −1.10154 0.956706i
\(582\) 6.00189e16 1.54437
\(583\) 8.86303e16i 2.25721i
\(584\) −4.35878e16 −1.09872
\(585\) 4.49213e15 0.112077
\(586\) 1.17188e16i 0.289399i
\(587\) 6.26769e16i 1.53207i −0.642799 0.766035i \(-0.722227\pi\)
0.642799 0.766035i \(-0.277773\pi\)
\(588\) −1.32335e17 −3.20192
\(589\) 2.38991e15i 0.0572388i
\(590\) 2.93929e16 0.696836
\(591\) 7.20819e16 1.69161
\(592\) 8.87177e16 2.06101
\(593\) 9.06460e15 0.208459 0.104229 0.994553i \(-0.466762\pi\)
0.104229 + 0.994553i \(0.466762\pi\)
\(594\) 7.56934e16i 1.72321i
\(595\) 9.02046e16i 2.03295i
\(596\) 3.86090e16i 0.861412i
\(597\) −7.19233e16 −1.58863
\(598\) −3.52385e16 −0.770566
\(599\) 2.72962e16i 0.590938i −0.955352 0.295469i \(-0.904524\pi\)
0.955352 0.295469i \(-0.0954759\pi\)
\(600\) 3.90682e16i 0.837367i
\(601\) 3.44890e16i 0.731869i 0.930641 + 0.365935i \(0.119251\pi\)
−0.930641 + 0.365935i \(0.880749\pi\)
\(602\) 1.71310e17 3.59918
\(603\) 6.29896e15i 0.131028i
\(604\) −1.93783e17 −3.99112
\(605\) 2.03625e16i 0.415240i
\(606\) −1.21537e17 −2.45398
\(607\) −5.10224e16 −1.02007 −0.510034 0.860154i \(-0.670367\pi\)
−0.510034 + 0.860154i \(0.670367\pi\)
\(608\) −6.70211e16 −1.32675
\(609\) 1.89219e16 0.370903
\(610\) −8.18565e16 −1.58882
\(611\) 9.75922e14 0.0187572
\(612\) −7.18832e16 −1.36810
\(613\) 5.53492e16i 1.04315i 0.853204 + 0.521577i \(0.174656\pi\)
−0.853204 + 0.521577i \(0.825344\pi\)
\(614\) −7.18408e16 −1.34079
\(615\) 1.26384e16i 0.233583i
\(616\) 2.61231e17i 4.78123i
\(617\) −1.21775e16 −0.220723 −0.110362 0.993891i \(-0.535201\pi\)
−0.110362 + 0.993891i \(0.535201\pi\)
\(618\) 1.63002e17 2.92591
\(619\) 5.58277e16 0.992444 0.496222 0.868196i \(-0.334720\pi\)
0.496222 + 0.868196i \(0.334720\pi\)
\(620\) 1.04449e16i 0.183889i
\(621\) 4.74223e16 0.826863
\(622\) 2.50567e16 0.432695
\(623\) 1.42255e17i 2.43299i
\(624\) 6.65205e16i 1.12680i
\(625\) −3.92811e16 −0.659027
\(626\) 1.47036e16i 0.244331i
\(627\) 5.59194e16i 0.920359i
\(628\) 1.80304e17i 2.93933i
\(629\) 8.17363e16 1.31981
\(630\) 4.96811e16 0.794599
\(631\) 3.85729e16i 0.611092i −0.952177 0.305546i \(-0.901161\pi\)
0.952177 0.305546i \(-0.0988389\pi\)
\(632\) −9.87967e16 −1.55039
\(633\) 3.15279e16i 0.490086i
\(634\) 2.16753e17i 3.33756i
\(635\) 8.74115e16i 1.33330i
\(636\) 3.47700e17i 5.25365i
\(637\) 2.88147e16i 0.431298i
\(638\) 3.34829e16i 0.496476i
\(639\) 4.78982e15i 0.0703581i
\(640\) 1.52579e16 0.222032
\(641\) 7.93407e16i 1.14379i −0.820326 0.571897i \(-0.806208\pi\)
0.820326 0.571897i \(-0.193792\pi\)
\(642\) 2.54428e17 3.63375
\(643\) 4.46023e16i 0.631091i −0.948910 0.315545i \(-0.897813\pi\)
0.948910 0.315545i \(-0.102187\pi\)
\(644\) −2.76693e17 −3.87867
\(645\) −9.47972e16 −1.31655
\(646\) −1.43280e17 −1.97147
\(647\) 6.95026e16i 0.947491i 0.880662 + 0.473746i \(0.157098\pi\)
−0.880662 + 0.473746i \(0.842902\pi\)
\(648\) 2.43918e17i 3.29454i
\(649\) 3.99699e16 0.534891
\(650\) 1.43817e16 0.190691
\(651\) −1.13056e16 −0.148528
\(652\) 3.64751e17i 4.74800i
\(653\) 9.60289e16i 1.23858i −0.785164 0.619288i \(-0.787421\pi\)
0.785164 0.619288i \(-0.212579\pi\)
\(654\) 3.01034e17i 3.84724i
\(655\) 5.83902e16i 0.739422i
\(656\) 4.77972e16 0.599762
\(657\) 1.12726e16i 0.140162i
\(658\) 1.07933e16 0.132984
\(659\) −5.88008e16 −0.717911 −0.358956 0.933355i \(-0.616867\pi\)
−0.358956 + 0.933355i \(0.616867\pi\)
\(660\) 2.44391e17i 2.95680i
\(661\) 9.31680e16i 1.11701i 0.829500 + 0.558506i \(0.188625\pi\)
−0.829500 + 0.558506i \(0.811375\pi\)
\(662\) 2.98018e17 3.54074
\(663\) 6.12859e16i 0.721571i
\(664\) −1.51101e17 + 1.73976e17i −1.76303 + 2.02993i
\(665\) 7.03059e16 0.812946
\(666\) 4.50171e16i 0.515860i
\(667\) 2.09772e16 0.238228
\(668\) 3.87231e17 4.35824
\(669\) 7.52661e16i 0.839543i
\(670\) 5.48719e16i 0.606598i
\(671\) −1.11312e17 −1.21958
\(672\) 3.17048e17i 3.44278i
\(673\) −8.41391e15 −0.0905539 −0.0452770 0.998974i \(-0.514417\pi\)
−0.0452770 + 0.998974i \(0.514417\pi\)
\(674\) −9.13762e16 −0.974706
\(675\) −1.93543e16 −0.204623
\(676\) −1.99495e17 −2.09050
\(677\) 4.74659e15i 0.0493003i 0.999696 + 0.0246502i \(0.00784719\pi\)
−0.999696 + 0.0246502i \(0.992153\pi\)
\(678\) 2.59541e16i 0.267195i
\(679\) 1.02614e17i 1.04710i
\(680\) 3.70391e17 3.74634
\(681\) 7.36614e16 0.738511
\(682\) 2.00057e16i 0.198814i
\(683\) 3.02791e16i 0.298276i −0.988816 0.149138i \(-0.952350\pi\)
0.988816 0.149138i \(-0.0476498\pi\)
\(684\) 5.60261e16i 0.547084i
\(685\) 1.14728e17 1.11052
\(686\) 3.63328e16i 0.348622i
\(687\) 5.78180e16 0.549949
\(688\) 3.58513e17i 3.38045i
\(689\) −7.57082e16 −0.707665
\(690\) 2.15660e17 1.99836
\(691\) 1.30429e17 1.19813 0.599066 0.800699i \(-0.295539\pi\)
0.599066 + 0.800699i \(0.295539\pi\)
\(692\) −1.22190e17 −1.11275
\(693\) 6.75588e16 0.609933
\(694\) −3.76910e16 −0.337350
\(695\) −5.46237e16 −0.484699
\(696\) 7.76955e16i 0.683503i
\(697\) 4.40359e16 0.384070
\(698\) 1.12469e16i 0.0972526i
\(699\) 8.69476e16i 0.745409i
\(700\) 1.12925e17 0.959849
\(701\) −1.24581e17 −1.04989 −0.524947 0.851135i \(-0.675915\pi\)
−0.524947 + 0.851135i \(0.675915\pi\)
\(702\) −6.46575e16 −0.540252
\(703\) 6.37056e16i 0.527772i
\(704\) 1.83461e17 1.50698
\(705\) −5.97265e15 −0.0486443
\(706\) 1.65835e17i 1.33921i
\(707\) 2.07791e17i 1.66384i
\(708\) 1.56803e17 1.24496
\(709\) 3.77878e16i 0.297491i −0.988875 0.148746i \(-0.952476\pi\)
0.988875 0.148746i \(-0.0475236\pi\)
\(710\) 4.17254e16i 0.325724i
\(711\) 2.55505e16i 0.197780i
\(712\) 5.84118e17 4.48353
\(713\) −1.25337e16 −0.0953985
\(714\) 6.77796e17i 5.11575i
\(715\) −5.32138e16 −0.398280
\(716\) 3.56486e17i 2.64585i
\(717\) 1.51046e16i 0.111172i
\(718\) 6.65689e16i 0.485875i
\(719\) 1.62645e17i 1.17724i −0.808409 0.588622i \(-0.799671\pi\)
0.808409 0.588622i \(-0.200329\pi\)
\(720\) 1.03971e17i 0.746308i
\(721\) 2.78684e17i 1.98381i
\(722\) 1.51356e17i 1.06850i
\(723\) −1.41607e17 −0.991412
\(724\) 3.33221e17i 2.31367i
\(725\) −8.56133e15 −0.0589540
\(726\) 1.53004e17i 1.04492i
\(727\) −1.39320e17 −0.943638 −0.471819 0.881695i \(-0.656402\pi\)
−0.471819 + 0.881695i \(0.656402\pi\)
\(728\) 2.23144e17 1.49898
\(729\) 6.93559e16 0.462081
\(730\) 9.81982e16i 0.648883i
\(731\) 3.30301e17i 2.16474i
\(732\) −4.36682e17 −2.83857
\(733\) −8.26984e16 −0.533179 −0.266590 0.963810i \(-0.585897\pi\)
−0.266590 + 0.963810i \(0.585897\pi\)
\(734\) −1.02241e17 −0.653805
\(735\) 1.76346e17i 1.11851i
\(736\) 3.51486e17i 2.21127i
\(737\) 7.46175e16i 0.465624i
\(738\) 2.42532e16i 0.150117i
\(739\) −2.61294e17 −1.60422 −0.802109 0.597178i \(-0.796289\pi\)
−0.802109 + 0.597178i \(0.796289\pi\)
\(740\) 2.78420e17i 1.69555i
\(741\) 4.77665e16 0.288545
\(742\) −8.37301e17 −5.01716
\(743\) 8.11764e16i 0.482499i −0.970463 0.241250i \(-0.922443\pi\)
0.970463 0.241250i \(-0.0775573\pi\)
\(744\) 4.64223e16i 0.273709i
\(745\) −5.14492e16 −0.300913
\(746\) 6.26316e17i 3.63380i
\(747\) 4.49933e16 + 3.90773e16i 0.258955 + 0.224906i
\(748\) 8.51529e17 4.86172
\(749\) 4.34995e17i 2.46373i
\(750\) −4.15521e17 −2.33467
\(751\) −6.90851e16 −0.385074 −0.192537 0.981290i \(-0.561672\pi\)
−0.192537 + 0.981290i \(0.561672\pi\)
\(752\) 2.25879e16i 0.124902i
\(753\) 2.17829e16i 0.119494i
\(754\) −2.86011e16 −0.155652
\(755\) 2.58230e17i 1.39420i
\(756\) −5.07691e17 −2.71937
\(757\) 8.38383e16 0.445520 0.222760 0.974873i \(-0.428493\pi\)
0.222760 + 0.974873i \(0.428493\pi\)
\(758\) −4.42540e17 −2.33312
\(759\) 2.93264e17 1.53394
\(760\) 2.88684e17i 1.49810i
\(761\) 5.84528e16i 0.300952i 0.988614 + 0.150476i \(0.0480807\pi\)
−0.988614 + 0.150476i \(0.951919\pi\)
\(762\) 6.56809e17i 3.35513i
\(763\) 5.14677e17 2.60848
\(764\) −5.93375e17 −2.98379
\(765\) 9.57895e16i 0.477913i
\(766\) 2.13556e17i 1.05716i
\(767\) 3.41424e16i 0.167696i
\(768\) −1.79363e17 −0.874107
\(769\) 1.73611e16i 0.0839498i −0.999119 0.0419749i \(-0.986635\pi\)
0.999119 0.0419749i \(-0.0133649\pi\)
\(770\) −5.88522e17 −2.82370
\(771\) 3.03884e17i 1.44671i
\(772\) −3.89632e16 −0.184056
\(773\) −3.54239e16 −0.166043 −0.0830213 0.996548i \(-0.526457\pi\)
−0.0830213 + 0.996548i \(0.526457\pi\)
\(774\) −1.81916e17 −0.846109
\(775\) 5.11532e15 0.0236082
\(776\) −4.21347e17 −1.92961
\(777\) −3.01364e17 −1.36951
\(778\) 7.95715e17 3.58823
\(779\) 3.43218e16i 0.153584i
\(780\) −2.08759e17 −0.926997
\(781\) 5.67402e16i 0.250026i
\(782\) 7.51420e17i 3.28581i
\(783\) 3.84901e16 0.167024
\(784\) 6.66921e17 2.87196
\(785\) −2.40268e17 −1.02678
\(786\) 4.38744e17i 1.86070i
\(787\) 1.73853e17 0.731702 0.365851 0.930673i \(-0.380778\pi\)
0.365851 + 0.930673i \(0.380778\pi\)
\(788\) −8.55511e17 −3.57329
\(789\) 5.51610e17i 2.28650i
\(790\) 2.22578e17i 0.915628i
\(791\) −4.43737e16 −0.181162
\(792\) 2.77404e17i 1.12399i
\(793\) 9.50833e16i 0.382354i
\(794\) 2.56354e17i 1.02310i
\(795\) 4.63334e17 1.83524
\(796\) 8.53629e17 3.35576
\(797\) 1.42743e17i 0.556935i −0.960446 0.278467i \(-0.910174\pi\)
0.960446 0.278467i \(-0.0898264\pi\)
\(798\) 5.28277e17 2.04571
\(799\) 2.08104e16i 0.0799834i
\(800\) 1.43450e17i 0.547220i
\(801\) 1.51063e17i 0.571956i
\(802\) 9.76299e15i 0.0366891i
\(803\) 1.33535e17i 0.498082i
\(804\) 2.92727e17i 1.08374i
\(805\) 3.68713e17i 1.35492i
\(806\) 1.70889e16 0.0623310
\(807\) 3.98806e16i 0.144384i
\(808\) 8.53215e17 3.06613
\(809\) 8.77739e16i 0.313094i −0.987671 0.156547i \(-0.949964\pi\)
0.987671 0.156547i \(-0.0500362\pi\)
\(810\) 5.49520e17 1.94569
\(811\) 1.86891e17 0.656845 0.328423 0.944531i \(-0.393483\pi\)
0.328423 + 0.944531i \(0.393483\pi\)
\(812\) −2.24576e17 −0.783479
\(813\) 4.04568e17i 1.40103i
\(814\) 5.33273e17i 1.83317i
\(815\) 4.86056e17 1.65860
\(816\) 1.41847e18 4.80485
\(817\) −2.57438e17 −0.865645
\(818\) 2.43261e17i 0.811994i
\(819\) 5.77088e16i 0.191222i
\(820\) 1.50000e17i 0.493411i
\(821\) 2.33519e17i 0.762540i −0.924464 0.381270i \(-0.875487\pi\)
0.924464 0.381270i \(-0.124513\pi\)
\(822\) 8.62064e17 2.79453
\(823\) 5.39826e17i 1.73722i −0.495495 0.868611i \(-0.665013\pi\)
0.495495 0.868611i \(-0.334987\pi\)
\(824\) −1.14431e18 −3.65578
\(825\) −1.19689e17 −0.379603
\(826\) 3.77601e17i 1.18892i
\(827\) 9.87890e16i 0.308799i 0.988009 + 0.154399i \(0.0493443\pi\)
−0.988009 + 0.154399i \(0.950656\pi\)
\(828\) 2.93824e17 0.911811
\(829\) 5.70217e17i 1.75676i 0.477961 + 0.878381i \(0.341376\pi\)
−0.477961 + 0.878381i \(0.658624\pi\)
\(830\) −3.91948e17 3.40413e17i −1.19884 1.04121i
\(831\) −4.01323e17 −1.21868
\(832\) 1.56713e17i 0.472460i
\(833\) 6.14440e17 1.83912
\(834\) −4.10441e17 −1.21971
\(835\) 5.16013e17i 1.52245i
\(836\) 6.63686e17i 1.94413i
\(837\) −2.29975e16 −0.0668848
\(838\) 1.20911e17i 0.349142i
\(839\) −4.61102e17 −1.32198 −0.660990 0.750394i \(-0.729864\pi\)
−0.660990 + 0.750394i \(0.729864\pi\)
\(840\) −1.36564e18 −3.88741
\(841\) −3.36789e17 −0.951879
\(842\) 7.58304e17 2.12800
\(843\) 6.53632e17i 1.82124i
\(844\) 3.74192e17i 1.03524i
\(845\) 2.65841e17i 0.730267i
\(846\) −1.14615e16 −0.0312623
\(847\) −2.61590e17 −0.708470
\(848\) 1.75228e18i 4.71226i
\(849\) 8.28341e17i 2.21189i
\(850\) 3.06674e17i 0.813135i
\(851\) −3.34098e17 −0.879624
\(852\) 2.22594e17i 0.581936i
\(853\) 3.76065e17 0.976268 0.488134 0.872769i \(-0.337678\pi\)
0.488134 + 0.872769i \(0.337678\pi\)
\(854\) 1.05158e18i 2.71079i
\(855\) −7.46588e16 −0.191110
\(856\) −1.78614e18 −4.54019
\(857\) 1.20139e17 0.303250 0.151625 0.988438i \(-0.451549\pi\)
0.151625 + 0.988438i \(0.451549\pi\)
\(858\) −3.99848e17 −1.00224
\(859\) −3.66071e17 −0.911184 −0.455592 0.890189i \(-0.650572\pi\)
−0.455592 + 0.890189i \(0.650572\pi\)
\(860\) 1.12511e18 2.78102
\(861\) −1.62361e17 −0.398532
\(862\) 6.85040e17i 1.66983i
\(863\) −5.51779e17 −1.33567 −0.667837 0.744307i \(-0.732780\pi\)
−0.667837 + 0.744307i \(0.732780\pi\)
\(864\) 6.44926e17i 1.55034i
\(865\) 1.62827e17i 0.388713i
\(866\) −6.96604e17 −1.65150
\(867\) 8.14640e17 1.91801
\(868\) 1.34182e17 0.313745
\(869\) 3.02672e17i 0.702835i
\(870\) 1.75039e17 0.403663
\(871\) −6.37384e16 −0.145980
\(872\) 2.11333e18i 4.80693i
\(873\) 1.08968e17i 0.246157i
\(874\) 5.85660e17 1.31394
\(875\) 7.10416e17i 1.58294i
\(876\) 5.23861e17i 1.15929i
\(877\) 3.89314e17i 0.855661i −0.903859 0.427831i \(-0.859278\pi\)
0.903859 0.427831i \(-0.140722\pi\)
\(878\) −3.87133e17 −0.845070
\(879\) −8.33077e16 −0.180614
\(880\) 1.23164e18i 2.65210i
\(881\) 1.94636e17 0.416264 0.208132 0.978101i \(-0.433262\pi\)
0.208132 + 0.978101i \(0.433262\pi\)
\(882\) 3.38409e17i 0.718837i
\(883\) 4.09135e17i 0.863181i −0.902070 0.431591i \(-0.857953\pi\)
0.902070 0.431591i \(-0.142047\pi\)
\(884\) 7.27378e17i 1.52422i
\(885\) 2.08952e17i 0.434897i
\(886\) 1.51948e18i 3.14117i
\(887\) 7.71233e17i 1.58359i 0.610785 + 0.791797i \(0.290854\pi\)
−0.610785 + 0.791797i \(0.709146\pi\)
\(888\) 1.23744e18i 2.52374i
\(889\) 1.12295e18 2.27483
\(890\) 1.31595e18i 2.64789i
\(891\) 7.47264e17 1.49351
\(892\) 8.93304e17i 1.77341i
\(893\) −1.62197e16 −0.0319842
\(894\) −3.86589e17 −0.757224
\(895\) 4.75043e17 0.924262
\(896\) 1.96013e17i 0.378824i
\(897\) 2.50507e17i 0.480912i
\(898\) −1.01918e18 −1.94355
\(899\) −1.01729e16 −0.0192702
\(900\) −1.19917e17 −0.225645
\(901\) 1.61439e18i 3.01759i
\(902\) 2.87304e17i 0.533460i
\(903\) 1.21783e18i 2.24625i
\(904\) 1.82204e17i 0.333847i
\(905\) 4.44041e17 0.808223
\(906\) 1.94034e18i 3.50839i
\(907\) 1.74730e17 0.313852 0.156926 0.987610i \(-0.449842\pi\)
0.156926 + 0.987610i \(0.449842\pi\)
\(908\) −8.74258e17 −1.56000
\(909\) 2.20656e17i 0.391141i
\(910\) 5.02717e17i 0.885269i
\(911\) 7.85468e14 0.00137410 0.000687050 1.00000i \(-0.499781\pi\)
0.000687050 1.00000i \(0.499781\pi\)
\(912\) 1.10557e18i 1.92139i
\(913\) −5.32990e17 4.62910e17i −0.920227 0.799230i
\(914\) 2.04706e17 0.351118
\(915\) 5.81910e17i 0.991584i
\(916\) −6.86219e17 −1.16169
\(917\) 7.50119e17 1.26158
\(918\) 1.37875e18i 2.30371i
\(919\) 3.44048e17i 0.571119i 0.958361 + 0.285559i \(0.0921794\pi\)
−0.958361 + 0.285559i \(0.907821\pi\)
\(920\) −1.51398e18 −2.49685
\(921\) 5.10709e17i 0.836789i
\(922\) 7.97800e17 1.29870
\(923\) −4.84676e16 −0.0783865
\(924\) −3.13961e18 −5.04479
\(925\) 1.36354e17 0.217680
\(926\) 1.69683e18i 2.69136i
\(927\) 2.95938e17i 0.466362i
\(928\) 2.85282e17i 0.446669i
\(929\) 1.06336e18 1.65419 0.827095 0.562063i \(-0.189992\pi\)
0.827095 + 0.562063i \(0.189992\pi\)
\(930\) −1.04584e17 −0.161647
\(931\) 4.78897e17i 0.735435i
\(932\) 1.03195e18i 1.57457i
\(933\) 1.78126e17i 0.270046i
\(934\) 8.65472e17 1.30368
\(935\) 1.13472e18i 1.69832i
\(936\) −2.36959e17 −0.352386
\(937\) 5.99788e17i 0.886258i 0.896458 + 0.443129i \(0.146132\pi\)
−0.896458 + 0.443129i \(0.853868\pi\)
\(938\) −7.04920e17 −1.03496
\(939\) −1.04527e17 −0.152487
\(940\) 7.08870e16 0.102754
\(941\) −5.91926e16 −0.0852569 −0.0426284 0.999091i \(-0.513573\pi\)
−0.0426284 + 0.999091i \(0.513573\pi\)
\(942\) −1.80537e18 −2.58382
\(943\) −1.79997e17 −0.255974
\(944\) −7.90233e17 −1.11667
\(945\) 6.76534e17i 0.949946i
\(946\) 2.15498e18 3.00675
\(947\) 7.98630e17i 1.10725i −0.832766 0.553625i \(-0.813244\pi\)
0.832766 0.553625i \(-0.186756\pi\)
\(948\) 1.18739e18i 1.63585i
\(949\) −1.14066e17 −0.156156
\(950\) −2.39023e17 −0.325161
\(951\) −1.54088e18 −2.08298
\(952\) 4.75828e18i 6.39188i
\(953\) 9.91381e17 1.32338 0.661688 0.749779i \(-0.269840\pi\)
0.661688 + 0.749779i \(0.269840\pi\)
\(954\) 8.89142e17 1.17945
\(955\) 7.90714e17i 1.04232i
\(956\) 1.79271e17i 0.234835i
\(957\) 2.38026e17 0.309851
\(958\) 1.95782e18i 2.53267i
\(959\) 1.47387e18i 1.89473i
\(960\) 9.59083e17i 1.22526i
\(961\) −7.81585e17 −0.992283
\(962\) 4.55523e17 0.574724
\(963\) 4.61928e17i 0.579183i
\(964\) 1.68068e18 2.09422
\(965\) 5.19212e16i 0.0642955i
\(966\) 2.77050e18i 3.40954i
\(967\) 1.09903e18i 1.34415i 0.740481 + 0.672077i \(0.234598\pi\)
−0.740481 + 0.672077i \(0.765402\pi\)
\(968\) 1.07412e18i 1.30557i
\(969\) 1.01857e18i 1.23040i
\(970\) 9.49245e17i 1.13959i
\(971\) 4.00487e17i 0.477830i −0.971040 0.238915i \(-0.923208\pi\)
0.971040 0.238915i \(-0.0767917\pi\)
\(972\) 1.35969e18 1.61229
\(973\) 7.01731e17i 0.826977i
\(974\) 2.53419e18 2.96815
\(975\) 1.02238e17i 0.119011i
\(976\) 2.20072e18 2.54605
\(977\) 3.72887e17 0.428756 0.214378 0.976751i \(-0.431228\pi\)
0.214378 + 0.976751i \(0.431228\pi\)
\(978\) 3.65222e18 4.17373
\(979\) 1.78949e18i 2.03252i
\(980\) 2.09298e18i 2.36270i
\(981\) −5.46543e17 −0.613211
\(982\) −8.53851e17 −0.952168
\(983\) −1.57247e18 −1.74286 −0.871428 0.490524i \(-0.836806\pi\)
−0.871428 + 0.490524i \(0.836806\pi\)
\(984\) 6.66675e17i 0.734419i
\(985\) 1.14003e18i 1.24824i
\(986\) 6.09886e17i 0.663723i
\(987\) 7.67285e16i 0.0829954i
\(988\) −5.66922e17 −0.609511
\(989\) 1.35011e18i 1.44275i
\(990\) 6.24960e17 0.663806
\(991\) −7.20299e17 −0.760451 −0.380225 0.924894i \(-0.624154\pi\)
−0.380225 + 0.924894i \(0.624154\pi\)
\(992\) 1.70453e17i 0.178869i
\(993\) 2.11858e18i 2.20978i
\(994\) −5.36031e17 −0.555740
\(995\) 1.13752e18i 1.17225i
\(996\) −2.09094e18 1.81601e18i −2.14183 1.86021i
\(997\) −1.28675e18 −1.31015 −0.655077 0.755562i \(-0.727364\pi\)
−0.655077 + 0.755562i \(0.727364\pi\)
\(998\) 6.19432e17i 0.626917i
\(999\) −6.13022e17 −0.616713
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 83.13.b.c.82.3 80
83.82 odd 2 inner 83.13.b.c.82.78 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
83.13.b.c.82.3 80 1.1 even 1 trivial
83.13.b.c.82.78 yes 80 83.82 odd 2 inner