Properties

Label 128.12.e.b.33.14
Level $128$
Weight $12$
Character 128.33
Analytic conductor $98.348$
Analytic rank $0$
Dimension $42$
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] = [128,12,Mod(33,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.33");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 128.e (of order \(4\), degree \(2\), not 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: \(98.3479271116\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 33.14
Character \(\chi\) \(=\) 128.33
Dual form 128.12.e.b.97.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(179.701 + 179.701i) q^{3} +(-2961.29 + 2961.29i) q^{5} +5179.04i q^{7} -112562. i q^{9} +O(q^{10})\) \(q+(179.701 + 179.701i) q^{3} +(-2961.29 + 2961.29i) q^{5} +5179.04i q^{7} -112562. i q^{9} +(-694803. + 694803. i) q^{11} +(-524615. - 524615. i) q^{13} -1.06429e6 q^{15} +1.82572e6 q^{17} +(601185. + 601185. i) q^{19} +(-930679. + 930679. i) q^{21} -2.32829e7i q^{23} +3.12896e7i q^{25} +(5.20610e7 - 5.20610e7i) q^{27} +(-189495. - 189495. i) q^{29} -2.48770e8 q^{31} -2.49713e8 q^{33} +(-1.53366e7 - 1.53366e7i) q^{35} +(4.30934e8 - 4.30934e8i) q^{37} -1.88548e8i q^{39} +1.02672e8i q^{41} +(6.91645e7 - 6.91645e7i) q^{43} +(3.33329e8 + 3.33329e8i) q^{45} +2.50575e8 q^{47} +1.95050e9 q^{49} +(3.28084e8 + 3.28084e8i) q^{51} +(1.76937e9 - 1.76937e9i) q^{53} -4.11502e9i q^{55} +2.16067e8i q^{57} +(-3.82611e9 + 3.82611e9i) q^{59} +(-7.80093e9 - 7.80093e9i) q^{61} +5.82964e8 q^{63} +3.10707e9 q^{65} +(6.53658e9 + 6.53658e9i) q^{67} +(4.18397e9 - 4.18397e9i) q^{69} -1.53408e10i q^{71} +2.49847e10i q^{73} +(-5.62278e9 + 5.62278e9i) q^{75} +(-3.59841e9 - 3.59841e9i) q^{77} +2.46282e10 q^{79} -1.22922e9 q^{81} +(2.00793e10 + 2.00793e10i) q^{83} +(-5.40649e9 + 5.40649e9i) q^{85} -6.81047e7i q^{87} -3.72435e10i q^{89} +(2.71700e9 - 2.71700e9i) q^{91} +(-4.47042e10 - 4.47042e10i) q^{93} -3.56057e9 q^{95} -1.06041e11 q^{97} +(7.82084e10 + 7.82084e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 42 q + 2 q^{3} + 2 q^{5} + 540846 q^{11} + 2 q^{13} - 6075004 q^{15} - 4 q^{17} + 11291290 q^{19} - 354292 q^{21} + 66463304 q^{27} - 77673206 q^{29} + 343549808 q^{31} - 4 q^{33} + 434731684 q^{35} + 522762058 q^{37} - 3824193658 q^{43} - 97301954 q^{45} - 4586900144 q^{47} - 8474257474 q^{49} - 7074245796 q^{51} + 2100608058 q^{53} - 955824746 q^{59} - 2150827022 q^{61} + 27758037828 q^{63} - 1884965292 q^{65} + 3186519018 q^{67} + 16193060732 q^{69} - 28890034486 q^{75} + 22711870540 q^{77} + 48011833792 q^{79} - 90656394430 q^{81} - 55713221118 q^{83} + 84575506252 q^{85} + 147369662716 q^{91} + 69689773328 q^{93} + 375702304500 q^{95} - 4 q^{97} + 286271331106 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 179.701 + 179.701i 0.426956 + 0.426956i 0.887590 0.460634i \(-0.152378\pi\)
−0.460634 + 0.887590i \(0.652378\pi\)
\(4\) 0 0
\(5\) −2961.29 + 2961.29i −0.423785 + 0.423785i −0.886505 0.462719i \(-0.846874\pi\)
0.462719 + 0.886505i \(0.346874\pi\)
\(6\) 0 0
\(7\) 5179.04i 0.116469i 0.998303 + 0.0582345i \(0.0185471\pi\)
−0.998303 + 0.0582345i \(0.981453\pi\)
\(8\) 0 0
\(9\) 112562.i 0.635416i
\(10\) 0 0
\(11\) −694803. + 694803.i −1.30077 + 1.30077i −0.372903 + 0.927870i \(0.621638\pi\)
−0.927870 + 0.372903i \(0.878362\pi\)
\(12\) 0 0
\(13\) −524615. 524615.i −0.391879 0.391879i 0.483478 0.875357i \(-0.339373\pi\)
−0.875357 + 0.483478i \(0.839373\pi\)
\(14\) 0 0
\(15\) −1.06429e6 −0.361876
\(16\) 0 0
\(17\) 1.82572e6 0.311864 0.155932 0.987768i \(-0.450162\pi\)
0.155932 + 0.987768i \(0.450162\pi\)
\(18\) 0 0
\(19\) 601185. + 601185.i 0.0557011 + 0.0557011i 0.734409 0.678708i \(-0.237460\pi\)
−0.678708 + 0.734409i \(0.737460\pi\)
\(20\) 0 0
\(21\) −930679. + 930679.i −0.0497272 + 0.0497272i
\(22\) 0 0
\(23\) 2.32829e7i 0.754284i −0.926156 0.377142i \(-0.876907\pi\)
0.926156 0.377142i \(-0.123093\pi\)
\(24\) 0 0
\(25\) 3.12896e7i 0.640812i
\(26\) 0 0
\(27\) 5.20610e7 5.20610e7i 0.698252 0.698252i
\(28\) 0 0
\(29\) −189495. 189495.i −0.00171557 0.00171557i 0.706248 0.707964i \(-0.250386\pi\)
−0.707964 + 0.706248i \(0.750386\pi\)
\(30\) 0 0
\(31\) −2.48770e8 −1.56066 −0.780330 0.625367i \(-0.784949\pi\)
−0.780330 + 0.625367i \(0.784949\pi\)
\(32\) 0 0
\(33\) −2.49713e8 −1.11075
\(34\) 0 0
\(35\) −1.53366e7 1.53366e7i −0.0493578 0.0493578i
\(36\) 0 0
\(37\) 4.30934e8 4.30934e8i 1.02165 1.02165i 0.0218884 0.999760i \(-0.493032\pi\)
0.999760 0.0218884i \(-0.00696786\pi\)
\(38\) 0 0
\(39\) 1.88548e8i 0.334630i
\(40\) 0 0
\(41\) 1.02672e8i 0.138401i 0.997603 + 0.0692007i \(0.0220449\pi\)
−0.997603 + 0.0692007i \(0.977955\pi\)
\(42\) 0 0
\(43\) 6.91645e7 6.91645e7i 0.0717475 0.0717475i −0.670322 0.742070i \(-0.733844\pi\)
0.742070 + 0.670322i \(0.233844\pi\)
\(44\) 0 0
\(45\) 3.33329e8 + 3.33329e8i 0.269280 + 0.269280i
\(46\) 0 0
\(47\) 2.50575e8 0.159367 0.0796836 0.996820i \(-0.474609\pi\)
0.0796836 + 0.996820i \(0.474609\pi\)
\(48\) 0 0
\(49\) 1.95050e9 0.986435
\(50\) 0 0
\(51\) 3.28084e8 + 3.28084e8i 0.133152 + 0.133152i
\(52\) 0 0
\(53\) 1.76937e9 1.76937e9i 0.581169 0.581169i −0.354056 0.935224i \(-0.615198\pi\)
0.935224 + 0.354056i \(0.115198\pi\)
\(54\) 0 0
\(55\) 4.11502e9i 1.10250i
\(56\) 0 0
\(57\) 2.16067e8i 0.0475639i
\(58\) 0 0
\(59\) −3.82611e9 + 3.82611e9i −0.696742 + 0.696742i −0.963706 0.266964i \(-0.913979\pi\)
0.266964 + 0.963706i \(0.413979\pi\)
\(60\) 0 0
\(61\) −7.80093e9 7.80093e9i −1.18258 1.18258i −0.979072 0.203512i \(-0.934764\pi\)
−0.203512 0.979072i \(-0.565236\pi\)
\(62\) 0 0
\(63\) 5.82964e8 0.0740063
\(64\) 0 0
\(65\) 3.10707e9 0.332145
\(66\) 0 0
\(67\) 6.53658e9 + 6.53658e9i 0.591479 + 0.591479i 0.938031 0.346552i \(-0.112648\pi\)
−0.346552 + 0.938031i \(0.612648\pi\)
\(68\) 0 0
\(69\) 4.18397e9 4.18397e9i 0.322046 0.322046i
\(70\) 0 0
\(71\) 1.53408e10i 1.00908i −0.863388 0.504541i \(-0.831662\pi\)
0.863388 0.504541i \(-0.168338\pi\)
\(72\) 0 0
\(73\) 2.49847e10i 1.41058i 0.708917 + 0.705291i \(0.249184\pi\)
−0.708917 + 0.705291i \(0.750816\pi\)
\(74\) 0 0
\(75\) −5.62278e9 + 5.62278e9i −0.273599 + 0.273599i
\(76\) 0 0
\(77\) −3.59841e9 3.59841e9i −0.151500 0.151500i
\(78\) 0 0
\(79\) 2.46282e10 0.900499 0.450249 0.892903i \(-0.351335\pi\)
0.450249 + 0.892903i \(0.351335\pi\)
\(80\) 0 0
\(81\) −1.22922e9 −0.0391706
\(82\) 0 0
\(83\) 2.00793e10 + 2.00793e10i 0.559524 + 0.559524i 0.929172 0.369648i \(-0.120522\pi\)
−0.369648 + 0.929172i \(0.620522\pi\)
\(84\) 0 0
\(85\) −5.40649e9 + 5.40649e9i −0.132163 + 0.132163i
\(86\) 0 0
\(87\) 6.81047e7i 0.00146495i
\(88\) 0 0
\(89\) 3.72435e10i 0.706978i −0.935439 0.353489i \(-0.884995\pi\)
0.935439 0.353489i \(-0.115005\pi\)
\(90\) 0 0
\(91\) 2.71700e9 2.71700e9i 0.0456417 0.0456417i
\(92\) 0 0
\(93\) −4.47042e10 4.47042e10i −0.666334 0.666334i
\(94\) 0 0
\(95\) −3.56057e9 −0.0472106
\(96\) 0 0
\(97\) −1.06041e11 −1.25381 −0.626904 0.779097i \(-0.715678\pi\)
−0.626904 + 0.779097i \(0.715678\pi\)
\(98\) 0 0
\(99\) 7.82084e10 + 7.82084e10i 0.826533 + 0.826533i
\(100\) 0 0
\(101\) 1.45309e11 1.45309e11i 1.37571 1.37571i 0.523973 0.851735i \(-0.324449\pi\)
0.851735 0.523973i \(-0.175551\pi\)
\(102\) 0 0
\(103\) 9.17356e10i 0.779710i 0.920876 + 0.389855i \(0.127475\pi\)
−0.920876 + 0.389855i \(0.872525\pi\)
\(104\) 0 0
\(105\) 5.51202e9i 0.0421473i
\(106\) 0 0
\(107\) 1.50629e11 1.50629e11i 1.03824 1.03824i 0.0390029 0.999239i \(-0.487582\pi\)
0.999239 0.0390029i \(-0.0124182\pi\)
\(108\) 0 0
\(109\) 7.29906e10 + 7.29906e10i 0.454382 + 0.454382i 0.896806 0.442424i \(-0.145881\pi\)
−0.442424 + 0.896806i \(0.645881\pi\)
\(110\) 0 0
\(111\) 1.54879e11 0.872399
\(112\) 0 0
\(113\) 3.01998e11 1.54196 0.770979 0.636861i \(-0.219767\pi\)
0.770979 + 0.636861i \(0.219767\pi\)
\(114\) 0 0
\(115\) 6.89475e10 + 6.89475e10i 0.319654 + 0.319654i
\(116\) 0 0
\(117\) −5.90517e10 + 5.90517e10i −0.249006 + 0.249006i
\(118\) 0 0
\(119\) 9.45549e9i 0.0363225i
\(120\) 0 0
\(121\) 6.80189e11i 2.38402i
\(122\) 0 0
\(123\) −1.84502e10 + 1.84502e10i −0.0590913 + 0.0590913i
\(124\) 0 0
\(125\) −2.37252e11 2.37252e11i −0.695352 0.695352i
\(126\) 0 0
\(127\) 4.26018e11 1.14421 0.572107 0.820179i \(-0.306126\pi\)
0.572107 + 0.820179i \(0.306126\pi\)
\(128\) 0 0
\(129\) 2.48578e10 0.0612661
\(130\) 0 0
\(131\) −3.65762e11 3.65762e11i −0.828336 0.828336i 0.158951 0.987287i \(-0.449189\pi\)
−0.987287 + 0.158951i \(0.949189\pi\)
\(132\) 0 0
\(133\) −3.11356e9 + 3.11356e9i −0.00648744 + 0.00648744i
\(134\) 0 0
\(135\) 3.08335e11i 0.591817i
\(136\) 0 0
\(137\) 9.23893e11i 1.63553i −0.575552 0.817765i \(-0.695213\pi\)
0.575552 0.817765i \(-0.304787\pi\)
\(138\) 0 0
\(139\) 4.31354e11 4.31354e11i 0.705103 0.705103i −0.260398 0.965501i \(-0.583854\pi\)
0.965501 + 0.260398i \(0.0838538\pi\)
\(140\) 0 0
\(141\) 4.50285e10 + 4.50285e10i 0.0680428 + 0.0680428i
\(142\) 0 0
\(143\) 7.29007e11 1.01949
\(144\) 0 0
\(145\) 1.12230e9 0.00145406
\(146\) 0 0
\(147\) 3.50508e11 + 3.50508e11i 0.421165 + 0.421165i
\(148\) 0 0
\(149\) 6.92517e11 6.92517e11i 0.772513 0.772513i −0.206032 0.978545i \(-0.566055\pi\)
0.978545 + 0.206032i \(0.0660551\pi\)
\(150\) 0 0
\(151\) 1.06664e11i 0.110572i 0.998471 + 0.0552862i \(0.0176071\pi\)
−0.998471 + 0.0552862i \(0.982393\pi\)
\(152\) 0 0
\(153\) 2.05507e11i 0.198164i
\(154\) 0 0
\(155\) 7.36680e11 7.36680e11i 0.661385 0.661385i
\(156\) 0 0
\(157\) 2.38255e11 + 2.38255e11i 0.199339 + 0.199339i 0.799717 0.600377i \(-0.204983\pi\)
−0.600377 + 0.799717i \(0.704983\pi\)
\(158\) 0 0
\(159\) 6.35916e11 0.496267
\(160\) 0 0
\(161\) 1.20583e11 0.0878506
\(162\) 0 0
\(163\) 2.01643e12 + 2.01643e12i 1.37262 + 1.37262i 0.856539 + 0.516082i \(0.172610\pi\)
0.516082 + 0.856539i \(0.327390\pi\)
\(164\) 0 0
\(165\) 7.39474e11 7.39474e11i 0.470718 0.470718i
\(166\) 0 0
\(167\) 1.58210e12i 0.942525i 0.881993 + 0.471263i \(0.156202\pi\)
−0.881993 + 0.471263i \(0.843798\pi\)
\(168\) 0 0
\(169\) 1.24172e12i 0.692862i
\(170\) 0 0
\(171\) 6.76707e10 6.76707e10i 0.0353934 0.0353934i
\(172\) 0 0
\(173\) −2.50263e12 2.50263e12i −1.22784 1.22784i −0.964778 0.263066i \(-0.915266\pi\)
−0.263066 0.964778i \(-0.584734\pi\)
\(174\) 0 0
\(175\) −1.62050e11 −0.0746347
\(176\) 0 0
\(177\) −1.37511e12 −0.594957
\(178\) 0 0
\(179\) 1.34310e12 + 1.34310e12i 0.546283 + 0.546283i 0.925364 0.379081i \(-0.123760\pi\)
−0.379081 + 0.925364i \(0.623760\pi\)
\(180\) 0 0
\(181\) 1.48029e12 1.48029e12i 0.566390 0.566390i −0.364725 0.931115i \(-0.618837\pi\)
0.931115 + 0.364725i \(0.118837\pi\)
\(182\) 0 0
\(183\) 2.80367e12i 1.00982i
\(184\) 0 0
\(185\) 2.55224e12i 0.865920i
\(186\) 0 0
\(187\) −1.26852e12 + 1.26852e12i −0.405665 + 0.405665i
\(188\) 0 0
\(189\) 2.69626e11 + 2.69626e11i 0.0813246 + 0.0813246i
\(190\) 0 0
\(191\) 3.79477e12 1.08020 0.540098 0.841602i \(-0.318387\pi\)
0.540098 + 0.841602i \(0.318387\pi\)
\(192\) 0 0
\(193\) 1.92445e12 0.517299 0.258650 0.965971i \(-0.416723\pi\)
0.258650 + 0.965971i \(0.416723\pi\)
\(194\) 0 0
\(195\) 5.58344e11 + 5.58344e11i 0.141811 + 0.141811i
\(196\) 0 0
\(197\) −1.87881e12 + 1.87881e12i −0.451147 + 0.451147i −0.895735 0.444588i \(-0.853350\pi\)
0.444588 + 0.895735i \(0.353350\pi\)
\(198\) 0 0
\(199\) 3.42680e12i 0.778390i −0.921155 0.389195i \(-0.872753\pi\)
0.921155 0.389195i \(-0.127247\pi\)
\(200\) 0 0
\(201\) 2.34926e12i 0.505071i
\(202\) 0 0
\(203\) 9.81400e8 9.81400e8i 0.000199810 0.000199810i
\(204\) 0 0
\(205\) −3.04041e11 3.04041e11i −0.0586525 0.0586525i
\(206\) 0 0
\(207\) −2.62078e12 −0.479284
\(208\) 0 0
\(209\) −8.35410e11 −0.144909
\(210\) 0 0
\(211\) −4.45545e12 4.45545e12i −0.733395 0.733395i 0.237896 0.971291i \(-0.423542\pi\)
−0.971291 + 0.237896i \(0.923542\pi\)
\(212\) 0 0
\(213\) 2.75675e12 2.75675e12i 0.430834 0.430834i
\(214\) 0 0
\(215\) 4.09632e11i 0.0608111i
\(216\) 0 0
\(217\) 1.28839e12i 0.181769i
\(218\) 0 0
\(219\) −4.48978e12 + 4.48978e12i −0.602257 + 0.602257i
\(220\) 0 0
\(221\) −9.57801e11 9.57801e11i −0.122213 0.122213i
\(222\) 0 0
\(223\) −5.95131e12 −0.722663 −0.361332 0.932437i \(-0.617678\pi\)
−0.361332 + 0.932437i \(0.617678\pi\)
\(224\) 0 0
\(225\) 3.52203e12 0.407183
\(226\) 0 0
\(227\) −1.36916e12 1.36916e12i −0.150769 0.150769i 0.627692 0.778462i \(-0.284000\pi\)
−0.778462 + 0.627692i \(0.784000\pi\)
\(228\) 0 0
\(229\) −5.82685e12 + 5.82685e12i −0.611419 + 0.611419i −0.943316 0.331897i \(-0.892311\pi\)
0.331897 + 0.943316i \(0.392311\pi\)
\(230\) 0 0
\(231\) 1.29328e12i 0.129368i
\(232\) 0 0
\(233\) 2.78470e12i 0.265657i 0.991139 + 0.132828i \(0.0424059\pi\)
−0.991139 + 0.132828i \(0.957594\pi\)
\(234\) 0 0
\(235\) −7.42024e11 + 7.42024e11i −0.0675375 + 0.0675375i
\(236\) 0 0
\(237\) 4.42571e12 + 4.42571e12i 0.384474 + 0.384474i
\(238\) 0 0
\(239\) −7.89713e12 −0.655059 −0.327530 0.944841i \(-0.606216\pi\)
−0.327530 + 0.944841i \(0.606216\pi\)
\(240\) 0 0
\(241\) 6.72861e12 0.533128 0.266564 0.963817i \(-0.414112\pi\)
0.266564 + 0.963817i \(0.414112\pi\)
\(242\) 0 0
\(243\) −9.44334e12 9.44334e12i −0.714976 0.714976i
\(244\) 0 0
\(245\) −5.77601e12 + 5.77601e12i −0.418037 + 0.418037i
\(246\) 0 0
\(247\) 6.30781e11i 0.0436562i
\(248\) 0 0
\(249\) 7.21653e12i 0.477784i
\(250\) 0 0
\(251\) 5.58474e12 5.58474e12i 0.353832 0.353832i −0.507701 0.861533i \(-0.669505\pi\)
0.861533 + 0.507701i \(0.169505\pi\)
\(252\) 0 0
\(253\) 1.61770e13 + 1.61770e13i 0.981152 + 0.981152i
\(254\) 0 0
\(255\) −1.94310e12 −0.112856
\(256\) 0 0
\(257\) −2.56783e13 −1.42868 −0.714340 0.699799i \(-0.753273\pi\)
−0.714340 + 0.699799i \(0.753273\pi\)
\(258\) 0 0
\(259\) 2.23183e12 + 2.23183e12i 0.118990 + 0.118990i
\(260\) 0 0
\(261\) −2.13299e10 + 2.13299e10i −0.00109010 + 0.00109010i
\(262\) 0 0
\(263\) 2.11569e13i 1.03680i 0.855138 + 0.518401i \(0.173473\pi\)
−0.855138 + 0.518401i \(0.826527\pi\)
\(264\) 0 0
\(265\) 1.04793e13i 0.492582i
\(266\) 0 0
\(267\) 6.69270e12 6.69270e12i 0.301849 0.301849i
\(268\) 0 0
\(269\) 1.13645e13 + 1.13645e13i 0.491940 + 0.491940i 0.908917 0.416977i \(-0.136911\pi\)
−0.416977 + 0.908917i \(0.636911\pi\)
\(270\) 0 0
\(271\) −4.20682e13 −1.74833 −0.874163 0.485633i \(-0.838589\pi\)
−0.874163 + 0.485633i \(0.838589\pi\)
\(272\) 0 0
\(273\) 9.76495e11 0.0389741
\(274\) 0 0
\(275\) −2.17401e13 2.17401e13i −0.833551 0.833551i
\(276\) 0 0
\(277\) −2.99399e13 + 2.99399e13i −1.10309 + 1.10309i −0.109056 + 0.994036i \(0.534783\pi\)
−0.994036 + 0.109056i \(0.965217\pi\)
\(278\) 0 0
\(279\) 2.80021e13i 0.991670i
\(280\) 0 0
\(281\) 1.93200e13i 0.657844i −0.944357 0.328922i \(-0.893315\pi\)
0.944357 0.328922i \(-0.106685\pi\)
\(282\) 0 0
\(283\) −2.01259e13 + 2.01259e13i −0.659068 + 0.659068i −0.955160 0.296092i \(-0.904317\pi\)
0.296092 + 0.955160i \(0.404317\pi\)
\(284\) 0 0
\(285\) −6.39837e11 6.39837e11i −0.0201569 0.0201569i
\(286\) 0 0
\(287\) −5.31742e11 −0.0161195
\(288\) 0 0
\(289\) −3.09386e13 −0.902741
\(290\) 0 0
\(291\) −1.90557e13 1.90557e13i −0.535321 0.535321i
\(292\) 0 0
\(293\) −1.16215e13 + 1.16215e13i −0.314406 + 0.314406i −0.846614 0.532207i \(-0.821363\pi\)
0.532207 + 0.846614i \(0.321363\pi\)
\(294\) 0 0
\(295\) 2.26605e13i 0.590538i
\(296\) 0 0
\(297\) 7.23442e13i 1.81653i
\(298\) 0 0
\(299\) −1.22146e13 + 1.22146e13i −0.295588 + 0.295588i
\(300\) 0 0
\(301\) 3.58206e11 + 3.58206e11i 0.00835635 + 0.00835635i
\(302\) 0 0
\(303\) 5.22245e13 1.17473
\(304\) 0 0
\(305\) 4.62016e13 1.00232
\(306\) 0 0
\(307\) 2.57889e13 + 2.57889e13i 0.539723 + 0.539723i 0.923448 0.383724i \(-0.125359\pi\)
−0.383724 + 0.923448i \(0.625359\pi\)
\(308\) 0 0
\(309\) −1.64850e13 + 1.64850e13i −0.332902 + 0.332902i
\(310\) 0 0
\(311\) 8.96609e13i 1.74752i −0.486361 0.873758i \(-0.661676\pi\)
0.486361 0.873758i \(-0.338324\pi\)
\(312\) 0 0
\(313\) 2.96628e13i 0.558108i −0.960275 0.279054i \(-0.909979\pi\)
0.960275 0.279054i \(-0.0900209\pi\)
\(314\) 0 0
\(315\) −1.72633e12 + 1.72633e12i −0.0313628 + 0.0313628i
\(316\) 0 0
\(317\) −2.58883e13 2.58883e13i −0.454233 0.454233i 0.442524 0.896757i \(-0.354083\pi\)
−0.896757 + 0.442524i \(0.854083\pi\)
\(318\) 0 0
\(319\) 2.63323e11 0.00446313
\(320\) 0 0
\(321\) 5.41365e13 0.886568
\(322\) 0 0
\(323\) 1.09760e12 + 1.09760e12i 0.0173712 + 0.0173712i
\(324\) 0 0
\(325\) 1.64150e13 1.64150e13i 0.251121 0.251121i
\(326\) 0 0
\(327\) 2.62330e13i 0.388003i
\(328\) 0 0
\(329\) 1.29774e12i 0.0185613i
\(330\) 0 0
\(331\) −2.92511e13 + 2.92511e13i −0.404658 + 0.404658i −0.879871 0.475213i \(-0.842371\pi\)
0.475213 + 0.879871i \(0.342371\pi\)
\(332\) 0 0
\(333\) −4.85069e13 4.85069e13i −0.649173 0.649173i
\(334\) 0 0
\(335\) −3.87134e13 −0.501320
\(336\) 0 0
\(337\) −1.74548e13 −0.218751 −0.109376 0.994000i \(-0.534885\pi\)
−0.109376 + 0.994000i \(0.534885\pi\)
\(338\) 0 0
\(339\) 5.42693e13 + 5.42693e13i 0.658349 + 0.658349i
\(340\) 0 0
\(341\) 1.72846e14 1.72846e14i 2.03007 2.03007i
\(342\) 0 0
\(343\) 2.03424e13i 0.231358i
\(344\) 0 0
\(345\) 2.47799e13i 0.272957i
\(346\) 0 0
\(347\) 1.65389e13 1.65389e13i 0.176480 0.176480i −0.613340 0.789819i \(-0.710174\pi\)
0.789819 + 0.613340i \(0.210174\pi\)
\(348\) 0 0
\(349\) 4.01630e13 + 4.01630e13i 0.415228 + 0.415228i 0.883555 0.468327i \(-0.155143\pi\)
−0.468327 + 0.883555i \(0.655143\pi\)
\(350\) 0 0
\(351\) −5.46239e13 −0.547260
\(352\) 0 0
\(353\) −6.14061e13 −0.596281 −0.298140 0.954522i \(-0.596366\pi\)
−0.298140 + 0.954522i \(0.596366\pi\)
\(354\) 0 0
\(355\) 4.54285e13 + 4.54285e13i 0.427634 + 0.427634i
\(356\) 0 0
\(357\) −1.69916e12 + 1.69916e12i −0.0155081 + 0.0155081i
\(358\) 0 0
\(359\) 2.12990e13i 0.188512i −0.995548 0.0942560i \(-0.969953\pi\)
0.995548 0.0942560i \(-0.0300472\pi\)
\(360\) 0 0
\(361\) 1.15767e14i 0.993795i
\(362\) 0 0
\(363\) 1.22231e14 1.22231e14i 1.01787 1.01787i
\(364\) 0 0
\(365\) −7.39870e13 7.39870e13i −0.597784 0.597784i
\(366\) 0 0
\(367\) 1.60814e14 1.26084 0.630420 0.776254i \(-0.282883\pi\)
0.630420 + 0.776254i \(0.282883\pi\)
\(368\) 0 0
\(369\) 1.15570e13 0.0879425
\(370\) 0 0
\(371\) 9.16366e12 + 9.16366e12i 0.0676881 + 0.0676881i
\(372\) 0 0
\(373\) 5.26476e13 5.26476e13i 0.377555 0.377555i −0.492665 0.870219i \(-0.663977\pi\)
0.870219 + 0.492665i \(0.163977\pi\)
\(374\) 0 0
\(375\) 8.52688e13i 0.593770i
\(376\) 0 0
\(377\) 1.98823e11i 0.00134459i
\(378\) 0 0
\(379\) 1.01266e13 1.01266e13i 0.0665196 0.0665196i −0.673064 0.739584i \(-0.735022\pi\)
0.739584 + 0.673064i \(0.235022\pi\)
\(380\) 0 0
\(381\) 7.65558e13 + 7.65558e13i 0.488529 + 0.488529i
\(382\) 0 0
\(383\) 2.19772e14 1.36263 0.681315 0.731990i \(-0.261408\pi\)
0.681315 + 0.731990i \(0.261408\pi\)
\(384\) 0 0
\(385\) 2.13119e13 0.128407
\(386\) 0 0
\(387\) −7.78530e12 7.78530e12i −0.0455895 0.0455895i
\(388\) 0 0
\(389\) −1.11456e14 + 1.11456e14i −0.634428 + 0.634428i −0.949175 0.314748i \(-0.898080\pi\)
0.314748 + 0.949175i \(0.398080\pi\)
\(390\) 0 0
\(391\) 4.25082e13i 0.235234i
\(392\) 0 0
\(393\) 1.31456e14i 0.707327i
\(394\) 0 0
\(395\) −7.29312e13 + 7.29312e13i −0.381618 + 0.381618i
\(396\) 0 0
\(397\) −1.72989e14 1.72989e14i −0.880379 0.880379i 0.113194 0.993573i \(-0.463892\pi\)
−0.993573 + 0.113194i \(0.963892\pi\)
\(398\) 0 0
\(399\) −1.11902e12 −0.00553971
\(400\) 0 0
\(401\) −6.99669e13 −0.336976 −0.168488 0.985704i \(-0.553888\pi\)
−0.168488 + 0.985704i \(0.553888\pi\)
\(402\) 0 0
\(403\) 1.30508e14 + 1.30508e14i 0.611590 + 0.611590i
\(404\) 0 0
\(405\) 3.64006e12 3.64006e12i 0.0165999 0.0165999i
\(406\) 0 0
\(407\) 5.98829e14i 2.65787i
\(408\) 0 0
\(409\) 3.35300e14i 1.44862i −0.689472 0.724312i \(-0.742157\pi\)
0.689472 0.724312i \(-0.257843\pi\)
\(410\) 0 0
\(411\) 1.66025e14 1.66025e14i 0.698300 0.698300i
\(412\) 0 0
\(413\) −1.98156e13 1.98156e13i −0.0811488 0.0811488i
\(414\) 0 0
\(415\) −1.18921e14 −0.474236
\(416\) 0 0
\(417\) 1.55030e14 0.602097
\(418\) 0 0
\(419\) −2.19319e13 2.19319e13i −0.0829657 0.0829657i 0.664406 0.747372i \(-0.268685\pi\)
−0.747372 + 0.664406i \(0.768685\pi\)
\(420\) 0 0
\(421\) −2.94928e13 + 2.94928e13i −0.108684 + 0.108684i −0.759357 0.650674i \(-0.774487\pi\)
0.650674 + 0.759357i \(0.274487\pi\)
\(422\) 0 0
\(423\) 2.82052e13i 0.101265i
\(424\) 0 0
\(425\) 5.71262e13i 0.199846i
\(426\) 0 0
\(427\) 4.04013e13 4.04013e13i 0.137734 0.137734i
\(428\) 0 0
\(429\) 1.31003e14 + 1.31003e14i 0.435278 + 0.435278i
\(430\) 0 0
\(431\) 4.93229e14 1.59744 0.798718 0.601705i \(-0.205512\pi\)
0.798718 + 0.601705i \(0.205512\pi\)
\(432\) 0 0
\(433\) 2.29557e14 0.724781 0.362391 0.932026i \(-0.381961\pi\)
0.362391 + 0.932026i \(0.381961\pi\)
\(434\) 0 0
\(435\) 2.01678e11 + 2.01678e11i 0.000620822 + 0.000620822i
\(436\) 0 0
\(437\) 1.39974e13 1.39974e13i 0.0420144 0.0420144i
\(438\) 0 0
\(439\) 1.66738e14i 0.488067i −0.969767 0.244033i \(-0.921529\pi\)
0.969767 0.244033i \(-0.0784707\pi\)
\(440\) 0 0
\(441\) 2.19553e14i 0.626797i
\(442\) 0 0
\(443\) −4.92405e13 + 4.92405e13i −0.137120 + 0.137120i −0.772335 0.635215i \(-0.780911\pi\)
0.635215 + 0.772335i \(0.280911\pi\)
\(444\) 0 0
\(445\) 1.10289e14 + 1.10289e14i 0.299607 + 0.299607i
\(446\) 0 0
\(447\) 2.48892e14 0.659659
\(448\) 0 0
\(449\) −4.21793e14 −1.09080 −0.545399 0.838177i \(-0.683622\pi\)
−0.545399 + 0.838177i \(0.683622\pi\)
\(450\) 0 0
\(451\) −7.13367e13 7.13367e13i −0.180029 0.180029i
\(452\) 0 0
\(453\) −1.91677e13 + 1.91677e13i −0.0472096 + 0.0472096i
\(454\) 0 0
\(455\) 1.60917e13i 0.0386846i
\(456\) 0 0
\(457\) 6.74966e13i 0.158396i 0.996859 + 0.0791978i \(0.0252359\pi\)
−0.996859 + 0.0791978i \(0.974764\pi\)
\(458\) 0 0
\(459\) 9.50489e13 9.50489e13i 0.217760 0.217760i
\(460\) 0 0
\(461\) 1.27861e14 + 1.27861e14i 0.286011 + 0.286011i 0.835501 0.549489i \(-0.185178\pi\)
−0.549489 + 0.835501i \(0.685178\pi\)
\(462\) 0 0
\(463\) −7.04481e14 −1.53877 −0.769385 0.638785i \(-0.779437\pi\)
−0.769385 + 0.638785i \(0.779437\pi\)
\(464\) 0 0
\(465\) 2.64764e14 0.564765
\(466\) 0 0
\(467\) −3.44528e14 3.44528e14i −0.717764 0.717764i 0.250383 0.968147i \(-0.419443\pi\)
−0.968147 + 0.250383i \(0.919443\pi\)
\(468\) 0 0
\(469\) −3.38532e13 + 3.38532e13i −0.0688889 + 0.0688889i
\(470\) 0 0
\(471\) 8.56292e13i 0.170219i
\(472\) 0 0
\(473\) 9.61113e13i 0.186654i
\(474\) 0 0
\(475\) −1.88109e13 + 1.88109e13i −0.0356939 + 0.0356939i
\(476\) 0 0
\(477\) −1.99164e14 1.99164e14i −0.369284 0.369284i
\(478\) 0 0
\(479\) −6.93135e14 −1.25595 −0.627975 0.778234i \(-0.716116\pi\)
−0.627975 + 0.778234i \(0.716116\pi\)
\(480\) 0 0
\(481\) −4.52149e14 −0.800726
\(482\) 0 0
\(483\) 2.16689e13 + 2.16689e13i 0.0375084 + 0.0375084i
\(484\) 0 0
\(485\) 3.14019e14 3.14019e14i 0.531345 0.531345i
\(486\) 0 0
\(487\) 2.22683e14i 0.368365i −0.982892 0.184182i \(-0.941036\pi\)
0.982892 0.184182i \(-0.0589637\pi\)
\(488\) 0 0
\(489\) 7.24708e14i 1.17210i
\(490\) 0 0
\(491\) −5.87841e14 + 5.87841e14i −0.929634 + 0.929634i −0.997682 0.0680481i \(-0.978323\pi\)
0.0680481 + 0.997682i \(0.478323\pi\)
\(492\) 0 0
\(493\) −3.45964e11 3.45964e11i −0.000535024 0.000535024i
\(494\) 0 0
\(495\) −4.63196e14 −0.700545
\(496\) 0 0
\(497\) 7.94506e13 0.117527
\(498\) 0 0
\(499\) 9.60683e13 + 9.60683e13i 0.139004 + 0.139004i 0.773185 0.634181i \(-0.218663\pi\)
−0.634181 + 0.773185i \(0.718663\pi\)
\(500\) 0 0
\(501\) −2.84305e14 + 2.84305e14i −0.402417 + 0.402417i
\(502\) 0 0
\(503\) 9.12018e14i 1.26293i −0.775404 0.631466i \(-0.782454\pi\)
0.775404 0.631466i \(-0.217546\pi\)
\(504\) 0 0
\(505\) 8.60607e14i 1.16601i
\(506\) 0 0
\(507\) 2.23138e14 2.23138e14i 0.295822 0.295822i
\(508\) 0 0
\(509\) 4.32018e14 + 4.32018e14i 0.560472 + 0.560472i 0.929441 0.368970i \(-0.120289\pi\)
−0.368970 + 0.929441i \(0.620289\pi\)
\(510\) 0 0
\(511\) −1.29397e14 −0.164289
\(512\) 0 0
\(513\) 6.25966e13 0.0777867
\(514\) 0 0
\(515\) −2.71656e14 2.71656e14i −0.330430 0.330430i
\(516\) 0 0
\(517\) −1.74100e14 + 1.74100e14i −0.207301 + 0.207301i
\(518\) 0 0
\(519\) 8.99450e14i 1.04847i
\(520\) 0 0
\(521\) 1.04084e15i 1.18789i 0.804504 + 0.593947i \(0.202431\pi\)
−0.804504 + 0.593947i \(0.797569\pi\)
\(522\) 0 0
\(523\) 2.46703e12 2.46703e12i 0.00275686 0.00275686i −0.705727 0.708484i \(-0.749379\pi\)
0.708484 + 0.705727i \(0.249379\pi\)
\(524\) 0 0
\(525\) −2.91206e13 2.91206e13i −0.0318658 0.0318658i
\(526\) 0 0
\(527\) −4.54185e14 −0.486714
\(528\) 0 0
\(529\) 4.10715e14 0.431056
\(530\) 0 0
\(531\) 4.30676e14 + 4.30676e14i 0.442721 + 0.442721i
\(532\) 0 0
\(533\) 5.38632e13 5.38632e13i 0.0542366 0.0542366i
\(534\) 0 0
\(535\) 8.92114e14i 0.879983i
\(536\) 0 0
\(537\) 4.82714e14i 0.466478i
\(538\) 0 0
\(539\) −1.35522e15 + 1.35522e15i −1.28313 + 1.28313i
\(540\) 0 0
\(541\) −1.03487e15 1.03487e15i −0.960065 0.960065i 0.0391680 0.999233i \(-0.487529\pi\)
−0.999233 + 0.0391680i \(0.987529\pi\)
\(542\) 0 0
\(543\) 5.32020e14 0.483648
\(544\) 0 0
\(545\) −4.32293e14 −0.385121
\(546\) 0 0
\(547\) −5.79056e14 5.79056e14i −0.505580 0.505580i 0.407586 0.913167i \(-0.366371\pi\)
−0.913167 + 0.407586i \(0.866371\pi\)
\(548\) 0 0
\(549\) −8.78089e14 + 8.78089e14i −0.751434 + 0.751434i
\(550\) 0 0
\(551\) 2.27843e11i 0.000191118i
\(552\) 0 0
\(553\) 1.27550e14i 0.104880i
\(554\) 0 0
\(555\) −4.58641e14 + 4.58641e14i −0.369710 + 0.369710i
\(556\) 0 0
\(557\) 7.19074e14 + 7.19074e14i 0.568290 + 0.568290i 0.931649 0.363359i \(-0.118370\pi\)
−0.363359 + 0.931649i \(0.618370\pi\)
\(558\) 0 0
\(559\) −7.25694e13 −0.0562327
\(560\) 0 0
\(561\) −4.55907e14 −0.346402
\(562\) 0 0
\(563\) −5.57706e14 5.57706e14i −0.415536 0.415536i 0.468126 0.883662i \(-0.344929\pi\)
−0.883662 + 0.468126i \(0.844929\pi\)
\(564\) 0 0
\(565\) −8.94303e14 + 8.94303e14i −0.653459 + 0.653459i
\(566\) 0 0
\(567\) 6.36616e12i 0.00456216i
\(568\) 0 0
\(569\) 1.80466e15i 1.26846i −0.773143 0.634231i \(-0.781317\pi\)
0.773143 0.634231i \(-0.218683\pi\)
\(570\) 0 0
\(571\) −1.82563e14 + 1.82563e14i −0.125868 + 0.125868i −0.767235 0.641367i \(-0.778368\pi\)
0.641367 + 0.767235i \(0.278368\pi\)
\(572\) 0 0
\(573\) 6.81924e14 + 6.81924e14i 0.461196 + 0.461196i
\(574\) 0 0
\(575\) 7.28515e14 0.483354
\(576\) 0 0
\(577\) 2.15369e15 1.40190 0.700950 0.713211i \(-0.252760\pi\)
0.700950 + 0.713211i \(0.252760\pi\)
\(578\) 0 0
\(579\) 3.45826e14 + 3.45826e14i 0.220864 + 0.220864i
\(580\) 0 0
\(581\) −1.03991e14 + 1.03991e14i −0.0651671 + 0.0651671i
\(582\) 0 0
\(583\) 2.45873e15i 1.51194i
\(584\) 0 0
\(585\) 3.49739e14i 0.211051i
\(586\) 0 0
\(587\) 1.54304e15 1.54304e15i 0.913833 0.913833i −0.0827383 0.996571i \(-0.526367\pi\)
0.996571 + 0.0827383i \(0.0263666\pi\)
\(588\) 0 0
\(589\) −1.49557e14 1.49557e14i −0.0869305 0.0869305i
\(590\) 0 0
\(591\) −6.75247e14 −0.385240
\(592\) 0 0
\(593\) −2.91327e15 −1.63147 −0.815737 0.578424i \(-0.803668\pi\)
−0.815737 + 0.578424i \(0.803668\pi\)
\(594\) 0 0
\(595\) −2.80004e13 2.80004e13i −0.0153929 0.0153929i
\(596\) 0 0
\(597\) 6.15800e14 6.15800e14i 0.332339 0.332339i
\(598\) 0 0
\(599\) 1.31209e15i 0.695210i −0.937641 0.347605i \(-0.886995\pi\)
0.937641 0.347605i \(-0.113005\pi\)
\(600\) 0 0
\(601\) 2.98203e15i 1.55132i 0.631148 + 0.775662i \(0.282584\pi\)
−0.631148 + 0.775662i \(0.717416\pi\)
\(602\) 0 0
\(603\) 7.35772e14 7.35772e14i 0.375836 0.375836i
\(604\) 0 0
\(605\) 2.01424e15 + 2.01424e15i 1.01031 + 1.01031i
\(606\) 0 0
\(607\) −8.61957e14 −0.424569 −0.212284 0.977208i \(-0.568090\pi\)
−0.212284 + 0.977208i \(0.568090\pi\)
\(608\) 0 0
\(609\) 3.52717e11 0.000170621
\(610\) 0 0
\(611\) −1.31455e14 1.31455e14i −0.0624526 0.0624526i
\(612\) 0 0
\(613\) 8.86904e14 8.86904e14i 0.413851 0.413851i −0.469227 0.883078i \(-0.655467\pi\)
0.883078 + 0.469227i \(0.155467\pi\)
\(614\) 0 0
\(615\) 1.09273e14i 0.0500841i
\(616\) 0 0
\(617\) 3.31474e15i 1.49239i −0.665730 0.746193i \(-0.731880\pi\)
0.665730 0.746193i \(-0.268120\pi\)
\(618\) 0 0
\(619\) 2.60524e15 2.60524e15i 1.15226 1.15226i 0.166156 0.986099i \(-0.446864\pi\)
0.986099 0.166156i \(-0.0531356\pi\)
\(620\) 0 0
\(621\) −1.21213e15 1.21213e15i −0.526680 0.526680i
\(622\) 0 0
\(623\) 1.92886e14 0.0823409
\(624\) 0 0
\(625\) −1.22671e14 −0.0514520
\(626\) 0 0
\(627\) −1.50124e14 1.50124e14i −0.0618698 0.0618698i
\(628\) 0 0
\(629\) 7.86766e14 7.86766e14i 0.318616 0.318616i
\(630\) 0 0
\(631\) 2.45185e15i 0.975737i 0.872917 + 0.487868i \(0.162225\pi\)
−0.872917 + 0.487868i \(0.837775\pi\)
\(632\) 0 0
\(633\) 1.60130e15i 0.626255i
\(634\) 0 0
\(635\) −1.26156e15 + 1.26156e15i −0.484901 + 0.484901i
\(636\) 0 0
\(637\) −1.02326e15 1.02326e15i −0.386563 0.386563i
\(638\) 0 0
\(639\) −1.72679e15 −0.641188
\(640\) 0 0
\(641\) −5.41404e14 −0.197607 −0.0988035 0.995107i \(-0.531502\pi\)
−0.0988035 + 0.995107i \(0.531502\pi\)
\(642\) 0 0
\(643\) −6.59814e14 6.59814e14i −0.236734 0.236734i 0.578762 0.815496i \(-0.303536\pi\)
−0.815496 + 0.578762i \(0.803536\pi\)
\(644\) 0 0
\(645\) −7.36113e13 + 7.36113e13i −0.0259637 + 0.0259637i
\(646\) 0 0
\(647\) 1.35482e15i 0.469795i −0.972020 0.234898i \(-0.924524\pi\)
0.972020 0.234898i \(-0.0754755\pi\)
\(648\) 0 0
\(649\) 5.31679e15i 1.81261i
\(650\) 0 0
\(651\) 2.31525e14 2.31525e14i 0.0776072 0.0776072i
\(652\) 0 0
\(653\) −9.93736e14 9.93736e14i −0.327528 0.327528i 0.524118 0.851646i \(-0.324395\pi\)
−0.851646 + 0.524118i \(0.824395\pi\)
\(654\) 0 0
\(655\) 2.16625e15 0.702073
\(656\) 0 0
\(657\) 2.81233e15 0.896308
\(658\) 0 0
\(659\) 1.25994e15 + 1.25994e15i 0.394894 + 0.394894i 0.876428 0.481534i \(-0.159920\pi\)
−0.481534 + 0.876428i \(0.659920\pi\)
\(660\) 0 0
\(661\) 7.61831e14 7.61831e14i 0.234828 0.234828i −0.579876 0.814705i \(-0.696899\pi\)
0.814705 + 0.579876i \(0.196899\pi\)
\(662\) 0 0
\(663\) 3.44235e14i 0.104359i
\(664\) 0 0
\(665\) 1.84403e13i 0.00549857i
\(666\) 0 0
\(667\) −4.41199e12 + 4.41199e12i −0.00129402 + 0.00129402i
\(668\) 0 0
\(669\) −1.06946e15 1.06946e15i −0.308546 0.308546i
\(670\) 0 0
\(671\) 1.08402e16 3.07655
\(672\) 0 0
\(673\) 4.75717e15 1.32821 0.664103 0.747641i \(-0.268814\pi\)
0.664103 + 0.747641i \(0.268814\pi\)
\(674\) 0 0
\(675\) 1.62897e15 + 1.62897e15i 0.447448 + 0.447448i
\(676\) 0 0
\(677\) 1.85656e15 1.85656e15i 0.501733 0.501733i −0.410243 0.911976i \(-0.634556\pi\)
0.911976 + 0.410243i \(0.134556\pi\)
\(678\) 0 0
\(679\) 5.49193e14i 0.146030i
\(680\) 0 0
\(681\) 4.92080e14i 0.128744i
\(682\) 0 0
\(683\) 4.05012e14 4.05012e14i 0.104269 0.104269i −0.653048 0.757317i \(-0.726510\pi\)
0.757317 + 0.653048i \(0.226510\pi\)
\(684\) 0 0
\(685\) 2.73592e15 + 2.73592e15i 0.693114 + 0.693114i
\(686\) 0 0
\(687\) −2.09418e15 −0.522098
\(688\) 0 0
\(689\) −1.85648e15 −0.455496
\(690\) 0 0
\(691\) 2.79390e15 + 2.79390e15i 0.674656 + 0.674656i 0.958786 0.284130i \(-0.0917047\pi\)
−0.284130 + 0.958786i \(0.591705\pi\)
\(692\) 0 0
\(693\) −4.05045e14 + 4.05045e14i −0.0962654 + 0.0962654i
\(694\) 0 0
\(695\) 2.55473e15i 0.597625i
\(696\) 0 0
\(697\) 1.87450e14i 0.0431624i
\(698\) 0 0
\(699\) −5.00413e14 + 5.00413e14i −0.113424 + 0.113424i
\(700\) 0 0
\(701\) −3.76553e15 3.76553e15i −0.840189 0.840189i 0.148694 0.988883i \(-0.452493\pi\)
−0.988883 + 0.148694i \(0.952493\pi\)
\(702\) 0 0
\(703\) 5.18143e14 0.113814
\(704\) 0 0
\(705\) −2.66685e14 −0.0576711
\(706\) 0 0
\(707\) 7.52564e14 + 7.52564e14i 0.160227 + 0.160227i
\(708\) 0 0
\(709\) 3.41294e15 3.41294e15i 0.715442 0.715442i −0.252226 0.967668i \(-0.581163\pi\)
0.967668 + 0.252226i \(0.0811628\pi\)
\(710\) 0 0
\(711\) 2.77220e15i 0.572192i
\(712\) 0 0
\(713\) 5.79209e15i 1.17718i
\(714\) 0 0
\(715\) −2.15880e15 + 2.15880e15i −0.432046 + 0.432046i
\(716\) 0 0
\(717\) −1.41912e15 1.41912e15i −0.279682 0.279682i
\(718\) 0 0
\(719\) 4.05222e15 0.786474 0.393237 0.919437i \(-0.371355\pi\)
0.393237 + 0.919437i \(0.371355\pi\)
\(720\) 0 0
\(721\) −4.75102e14 −0.0908120
\(722\) 0 0
\(723\) 1.20914e15 + 1.20914e15i 0.227623 + 0.227623i
\(724\) 0 0
\(725\) 5.92922e12 5.92922e12i 0.00109936 0.00109936i
\(726\) 0 0
\(727\) 3.88621e15i 0.709720i −0.934920 0.354860i \(-0.884529\pi\)
0.934920 0.354860i \(-0.115471\pi\)
\(728\) 0 0
\(729\) 3.17620e15i 0.571356i
\(730\) 0 0
\(731\) 1.26275e14 1.26275e14i 0.0223755 0.0223755i
\(732\) 0 0
\(733\) −5.83093e15 5.83093e15i −1.01781 1.01781i −0.999839 0.0179691i \(-0.994280\pi\)
−0.0179691 0.999839i \(-0.505720\pi\)
\(734\) 0 0
\(735\) −2.07591e15 −0.356967
\(736\) 0 0
\(737\) −9.08327e15 −1.53876
\(738\) 0 0
\(739\) 3.69206e15 + 3.69206e15i 0.616203 + 0.616203i 0.944555 0.328352i \(-0.106493\pi\)
−0.328352 + 0.944555i \(0.606493\pi\)
\(740\) 0 0
\(741\) 1.13352e14 1.13352e14i 0.0186393 0.0186393i
\(742\) 0 0
\(743\) 1.42369e15i 0.230663i −0.993327 0.115331i \(-0.963207\pi\)
0.993327 0.115331i \(-0.0367930\pi\)
\(744\) 0 0
\(745\) 4.10149e15i 0.654759i
\(746\) 0 0
\(747\) 2.26017e15 2.26017e15i 0.355531 0.355531i
\(748\) 0 0
\(749\) 7.80115e14 + 7.80115e14i 0.120923 + 0.120923i
\(750\) 0 0
\(751\) −1.14591e16 −1.75037 −0.875185 0.483788i \(-0.839261\pi\)
−0.875185 + 0.483788i \(0.839261\pi\)
\(752\) 0 0
\(753\) 2.00716e15 0.302142
\(754\) 0 0
\(755\) −3.15865e14 3.15865e14i −0.0468590 0.0468590i
\(756\) 0 0
\(757\) −5.28545e15 + 5.28545e15i −0.772778 + 0.772778i −0.978591 0.205813i \(-0.934016\pi\)
0.205813 + 0.978591i \(0.434016\pi\)
\(758\) 0 0
\(759\) 5.81406e15i 0.837818i
\(760\) 0 0
\(761\) 1.00264e15i 0.142407i 0.997462 + 0.0712034i \(0.0226839\pi\)
−0.997462 + 0.0712034i \(0.977316\pi\)
\(762\) 0 0
\(763\) −3.78021e14 + 3.78021e14i −0.0529214 + 0.0529214i
\(764\) 0 0
\(765\) 6.08566e14 + 6.08566e14i 0.0839789 + 0.0839789i
\(766\) 0 0
\(767\) 4.01447e15 0.546077
\(768\) 0 0
\(769\) 6.50821e15 0.872703 0.436352 0.899776i \(-0.356270\pi\)
0.436352 + 0.899776i \(0.356270\pi\)
\(770\) 0 0
\(771\) −4.61442e15 4.61442e15i −0.609984 0.609984i
\(772\) 0 0
\(773\) 4.27938e15 4.27938e15i 0.557691 0.557691i −0.370958 0.928650i \(-0.620971\pi\)
0.928650 + 0.370958i \(0.120971\pi\)
\(774\) 0 0
\(775\) 7.78392e15i 1.00009i
\(776\) 0 0
\(777\) 8.02123e14i 0.101607i
\(778\) 0 0
\(779\) −6.17248e13 + 6.17248e13i −0.00770910 + 0.00770910i
\(780\) 0 0
\(781\) 1.06588e16 + 1.06588e16i 1.31259 + 1.31259i
\(782\) 0 0
\(783\) −1.97306e13 −0.00239580
\(784\) 0 0
\(785\) −1.41108e15 −0.168954
\(786\) 0 0
\(787\) 5.86881e15 + 5.86881e15i 0.692929 + 0.692929i 0.962875 0.269946i \(-0.0870059\pi\)
−0.269946 + 0.962875i \(0.587006\pi\)
\(788\) 0 0
\(789\) −3.80192e15 + 3.80192e15i −0.442669 + 0.442669i
\(790\) 0 0
\(791\) 1.56406e15i 0.179590i
\(792\) 0 0
\(793\) 8.18496e15i 0.926860i
\(794\) 0 0
\(795\) −1.88313e15 + 1.88313e15i −0.210311 + 0.210311i
\(796\) 0 0
\(797\) 6.58799e15 + 6.58799e15i 0.725658 + 0.725658i 0.969752 0.244093i \(-0.0784903\pi\)
−0.244093 + 0.969752i \(0.578490\pi\)
\(798\) 0 0
\(799\) 4.57479e14 0.0497009
\(800\) 0 0
\(801\) −4.19221e15 −0.449225
\(802\) 0 0
\(803\) −1.73594e16 1.73594e16i −1.83485 1.83485i
\(804\) 0 0
\(805\) −3.57082e14 + 3.57082e14i −0.0372298 + 0.0372298i
\(806\) 0 0
\(807\) 4.08441e15i 0.420074i
\(808\) 0 0
\(809\) 4.37501e15i 0.443877i −0.975061 0.221938i \(-0.928762\pi\)
0.975061 0.221938i \(-0.0712384\pi\)
\(810\) 0 0
\(811\) 4.70532e14 4.70532e14i 0.0470949 0.0470949i −0.683167 0.730262i \(-0.739398\pi\)
0.730262 + 0.683167i \(0.239398\pi\)
\(812\) 0 0
\(813\) −7.55969e15 7.55969e15i −0.746459 0.746459i
\(814\) 0 0
\(815\) −1.19424e16 −1.16339
\(816\) 0 0
\(817\) 8.31613e13 0.00799282
\(818\) 0 0
\(819\) −3.05831e14 3.05831e14i −0.0290015 0.0290015i
\(820\) 0 0
\(821\) 5.92788e15 5.92788e15i 0.554641 0.554641i −0.373136 0.927777i \(-0.621717\pi\)
0.927777 + 0.373136i \(0.121717\pi\)
\(822\) 0 0
\(823\) 1.89761e16i 1.75189i 0.482408 + 0.875947i \(0.339762\pi\)
−0.482408 + 0.875947i \(0.660238\pi\)
\(824\) 0 0
\(825\) 7.81344e15i 0.711780i
\(826\) 0 0
\(827\) −4.01913e14 + 4.01913e14i −0.0361286 + 0.0361286i −0.724940 0.688812i \(-0.758133\pi\)
0.688812 + 0.724940i \(0.258133\pi\)
\(828\) 0 0
\(829\) 5.59288e15 + 5.59288e15i 0.496119 + 0.496119i 0.910228 0.414109i \(-0.135907\pi\)
−0.414109 + 0.910228i \(0.635907\pi\)
\(830\) 0 0
\(831\) −1.07605e16 −0.941944
\(832\) 0 0
\(833\) 3.56108e15 0.307634
\(834\) 0 0
\(835\) −4.68505e15 4.68505e15i −0.399428 0.399428i
\(836\) 0 0
\(837\) −1.29512e16 + 1.29512e16i −1.08973 + 1.08973i
\(838\) 0 0
\(839\) 1.32289e16i 1.09858i −0.835631 0.549291i \(-0.814898\pi\)
0.835631 0.549291i \(-0.185102\pi\)
\(840\) 0 0
\(841\) 1.22004e16i 0.999994i
\(842\) 0 0
\(843\) 3.47182e15 3.47182e15i 0.280871 0.280871i
\(844\) 0 0
\(845\) 3.67709e15 + 3.67709e15i 0.293625 + 0.293625i
\(846\) 0 0
\(847\) 3.52273e15 0.277665
\(848\) 0 0
\(849\) −7.23329e15 −0.562787
\(850\) 0 0
\(851\) −1.00334e16 1.00334e16i −0.770613 0.770613i
\(852\) 0 0
\(853\) −8.41615e15 + 8.41615e15i −0.638107 + 0.638107i −0.950088 0.311981i \(-0.899007\pi\)
0.311981 + 0.950088i \(0.399007\pi\)
\(854\) 0 0
\(855\) 4.00785e14i 0.0299984i
\(856\) 0 0
\(857\) 1.65720e16i 1.22456i 0.790641 + 0.612280i \(0.209748\pi\)
−0.790641 + 0.612280i \(0.790252\pi\)
\(858\) 0 0
\(859\) −1.37657e16 + 1.37657e16i −1.00424 + 1.00424i −0.00424510 + 0.999991i \(0.501351\pi\)
−0.999991 + 0.00424510i \(0.998649\pi\)
\(860\) 0 0
\(861\) −9.55545e13 9.55545e13i −0.00688231 0.00688231i
\(862\) 0 0
\(863\) 8.13724e15 0.578652 0.289326 0.957231i \(-0.406569\pi\)
0.289326 + 0.957231i \(0.406569\pi\)
\(864\) 0 0
\(865\) 1.48220e16 1.04068
\(866\) 0 0
\(867\) −5.55970e15 5.55970e15i −0.385431 0.385431i
\(868\) 0 0
\(869\) −1.71117e16 + 1.71117e16i −1.17134 + 1.17134i
\(870\) 0 0
\(871\) 6.85837e15i 0.463576i
\(872\) 0 0
\(873\) 1.19362e16i 0.796690i
\(874\) 0 0
\(875\) 1.22874e15 1.22874e15i 0.0809869 0.0809869i
\(876\) 0 0
\(877\) 8.99391e15 + 8.99391e15i 0.585397 + 0.585397i 0.936381 0.350984i \(-0.114153\pi\)
−0.350984 + 0.936381i \(0.614153\pi\)
\(878\) 0 0
\(879\) −4.17680e15 −0.268476
\(880\) 0 0
\(881\) 1.35360e16 0.859255 0.429628 0.903006i \(-0.358645\pi\)
0.429628 + 0.903006i \(0.358645\pi\)
\(882\) 0 0
\(883\) 5.86996e15 + 5.86996e15i 0.368003 + 0.368003i 0.866749 0.498745i \(-0.166206\pi\)
−0.498745 + 0.866749i \(0.666206\pi\)
\(884\) 0 0
\(885\) 4.07211e15 4.07211e15i 0.252134 0.252134i
\(886\) 0 0
\(887\) 1.36497e15i 0.0834723i −0.999129 0.0417362i \(-0.986711\pi\)
0.999129 0.0417362i \(-0.0132889\pi\)
\(888\) 0 0
\(889\) 2.20636e15i 0.133265i
\(890\) 0 0
\(891\) 8.54062e14 8.54062e14i 0.0509521 0.0509521i
\(892\) 0 0
\(893\) 1.50642e14 + 1.50642e14i 0.00887692 + 0.00887692i
\(894\) 0 0
\(895\) −7.95463e15 −0.463013
\(896\) 0 0
\(897\) −4.38994e15 −0.252406
\(898\) 0 0
\(899\) 4.71406e13 + 4.71406e13i 0.00267742 + 0.00267742i
\(900\) 0 0
\(901\) 3.23038e15 3.23038e15i 0.181246 0.181246i
\(902\) 0 0
\(903\) 1.28740e14i 0.00713560i
\(904\) 0 0
\(905\) 8.76715e15i 0.480056i
\(906\) 0 0
\(907\) 4.23091e15 4.23091e15i 0.228873 0.228873i −0.583349 0.812222i \(-0.698258\pi\)
0.812222 + 0.583349i \(0.198258\pi\)
\(908\) 0 0
\(909\) −1.63563e16 1.63563e16i −0.874148 0.874148i
\(910\) 0 0
\(911\) −1.76552e16 −0.932229 −0.466114 0.884724i \(-0.654346\pi\)
−0.466114 + 0.884724i \(0.654346\pi\)
\(912\) 0 0
\(913\) −2.79023e16 −1.45563
\(914\) 0 0
\(915\) 8.30247e15 + 8.30247e15i 0.427949 + 0.427949i
\(916\) 0 0
\(917\) 1.89430e15 1.89430e15i 0.0964754 0.0964754i
\(918\) 0 0
\(919\) 1.51316e16i 0.761464i 0.924685 + 0.380732i \(0.124328\pi\)
−0.924685 + 0.380732i \(0.875672\pi\)
\(920\) 0 0
\(921\) 9.26856e15i 0.460876i
\(922\) 0 0
\(923\) −8.04800e15 + 8.04800e15i −0.395438 + 0.395438i
\(924\) 0 0
\(925\) 1.34838e16 + 1.34838e16i 0.654685 + 0.654685i
\(926\) 0 0
\(927\) 1.03260e16 0.495441
\(928\) 0 0
\(929\) −3.73662e16 −1.77171 −0.885856 0.463961i \(-0.846428\pi\)
−0.885856 + 0.463961i \(0.846428\pi\)
\(930\) 0 0
\(931\) 1.17261e15 + 1.17261e15i 0.0549455 + 0.0549455i
\(932\) 0 0
\(933\) 1.61122e16 1.61122e16i 0.746113 0.746113i
\(934\) 0 0
\(935\) 7.51289e15i 0.343829i
\(936\) 0 0
\(937\) 1.60131e15i 0.0724284i 0.999344 + 0.0362142i \(0.0115299\pi\)
−0.999344 + 0.0362142i \(0.988470\pi\)
\(938\) 0 0
\(939\) 5.33044e15 5.33044e15i 0.238288 0.238288i
\(940\) 0 0
\(941\) −1.89984e16 1.89984e16i −0.839411 0.839411i 0.149370 0.988781i \(-0.452275\pi\)
−0.988781 + 0.149370i \(0.952275\pi\)
\(942\) 0 0
\(943\) 2.39050e15 0.104394
\(944\) 0 0
\(945\) −1.59688e15 −0.0689284
\(946\) 0 0
\(947\) 2.90069e16 + 2.90069e16i 1.23759 + 1.23759i 0.960984 + 0.276603i \(0.0892088\pi\)
0.276603 + 0.960984i \(0.410791\pi\)
\(948\) 0 0
\(949\) 1.31073e16 1.31073e16i 0.552778 0.552778i
\(950\) 0 0
\(951\) 9.30432e15i 0.387875i
\(952\) 0 0
\(953\) 9.02554e15i 0.371931i −0.982556 0.185965i \(-0.940459\pi\)
0.982556 0.185965i \(-0.0595412\pi\)
\(954\) 0 0
\(955\) −1.12374e16 + 1.12374e16i −0.457771 + 0.457771i
\(956\) 0 0
\(957\) 4.73193e13 + 4.73193e13i 0.00190556 + 0.00190556i
\(958\) 0 0
\(959\) 4.78488e15 0.190489
\(960\) 0 0
\(961\) 3.64780e16 1.43566
\(962\) 0 0
\(963\) −1.69552e16 1.69552e16i −0.659716 0.659716i
\(964\) 0 0
\(965\) −5.69886e15 + 5.69886e15i −0.219224 + 0.219224i
\(966\) 0 0
\(967\) 1.17299e16i 0.446118i −0.974805 0.223059i \(-0.928396\pi\)
0.974805 0.223059i \(-0.0716042\pi\)
\(968\) 0 0
\(969\) 3.94479e14i 0.0148335i
\(970\) 0 0
\(971\) 2.94649e16 2.94649e16i 1.09547 1.09547i 0.100534 0.994934i \(-0.467945\pi\)
0.994934 0.100534i \(-0.0320552\pi\)
\(972\) 0 0
\(973\) 2.23400e15 + 2.23400e15i 0.0821227 + 0.0821227i
\(974\) 0 0
\(975\) 5.89959e15 0.214435
\(976\) 0 0
\(977\) 2.36971e16 0.851677 0.425838 0.904799i \(-0.359979\pi\)
0.425838 + 0.904799i \(0.359979\pi\)
\(978\) 0 0
\(979\) 2.58769e16 + 2.58769e16i 0.919618 + 0.919618i
\(980\) 0 0
\(981\) 8.21598e15 8.21598e15i 0.288722 0.288722i
\(982\) 0 0
\(983\) 2.60613e16i 0.905634i 0.891604 + 0.452817i \(0.149581\pi\)
−0.891604 + 0.452817i \(0.850419\pi\)
\(984\) 0 0
\(985\) 1.11274e16i 0.382379i
\(986\) 0 0
\(987\) −2.33204e14 + 2.33204e14i −0.00792487 + 0.00792487i
\(988\) 0 0
\(989\) −1.61035e15 1.61035e15i −0.0541179 0.0541179i
\(990\) 0 0
\(991\) 4.19445e16 1.39402 0.697011 0.717060i \(-0.254513\pi\)
0.697011 + 0.717060i \(0.254513\pi\)
\(992\) 0 0
\(993\) −1.05129e16 −0.345542
\(994\) 0 0
\(995\) 1.01478e16 + 1.01478e16i 0.329870 + 0.329870i
\(996\) 0 0
\(997\) 2.65082e16 2.65082e16i 0.852231 0.852231i −0.138177 0.990408i \(-0.544124\pi\)
0.990408 + 0.138177i \(0.0441242\pi\)
\(998\) 0 0
\(999\) 4.48698e16i 1.42674i
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 128.12.e.b.33.14 42
4.3 odd 2 128.12.e.a.33.8 42
8.3 odd 2 64.12.e.a.17.14 42
8.5 even 2 16.12.e.a.13.20 yes 42
16.3 odd 4 64.12.e.a.49.14 42
16.5 even 4 inner 128.12.e.b.97.14 42
16.11 odd 4 128.12.e.a.97.8 42
16.13 even 4 16.12.e.a.5.20 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.12.e.a.5.20 42 16.13 even 4
16.12.e.a.13.20 yes 42 8.5 even 2
64.12.e.a.17.14 42 8.3 odd 2
64.12.e.a.49.14 42 16.3 odd 4
128.12.e.a.33.8 42 4.3 odd 2
128.12.e.a.97.8 42 16.11 odd 4
128.12.e.b.33.14 42 1.1 even 1 trivial
128.12.e.b.97.14 42 16.5 even 4 inner