Properties

Label 128.12.e.b.33.5
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 33.5
Character \(\chi\) \(=\) 128.33
Dual form 128.12.e.b.97.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-367.824 - 367.824i) q^{3} +(502.344 - 502.344i) q^{5} +33063.2i q^{7} +93442.1i q^{9} +O(q^{10})\) \(q+(-367.824 - 367.824i) q^{3} +(502.344 - 502.344i) q^{5} +33063.2i q^{7} +93442.1i q^{9} +(-335681. + 335681. i) q^{11} +(-568061. - 568061. i) q^{13} -369548. q^{15} -6.78457e6 q^{17} +(-1.33879e6 - 1.33879e6i) q^{19} +(1.21614e7 - 1.21614e7i) q^{21} -1.07133e6i q^{23} +4.83234e7i q^{25} +(-3.07887e7 + 3.07887e7i) q^{27} +(-5.01359e7 - 5.01359e7i) q^{29} +2.95969e8 q^{31} +2.46943e8 q^{33} +(1.66091e7 + 1.66091e7i) q^{35} +(-5.11883e8 + 5.11883e8i) q^{37} +4.17893e8i q^{39} +9.31803e8i q^{41} +(1.27362e9 - 1.27362e9i) q^{43} +(4.69401e7 + 4.69401e7i) q^{45} -1.80739e7 q^{47} +8.84154e8 q^{49} +(2.49553e9 + 2.49553e9i) q^{51} +(-2.29607e9 + 2.29607e9i) q^{53} +3.37255e8i q^{55} +9.84878e8i q^{57} +(4.81453e8 - 4.81453e8i) q^{59} +(-1.30887e9 - 1.30887e9i) q^{61} -3.08949e9 q^{63} -5.70724e8 q^{65} +(9.42251e9 + 9.42251e9i) q^{67} +(-3.94062e8 + 3.94062e8i) q^{69} -1.74203e10i q^{71} -8.91310e9i q^{73} +(1.77745e10 - 1.77745e10i) q^{75} +(-1.10987e10 - 1.10987e10i) q^{77} -4.07466e10 q^{79} +3.92026e10 q^{81} +(-1.16461e10 - 1.16461e10i) q^{83} +(-3.40819e9 + 3.40819e9i) q^{85} +3.68824e10i q^{87} -7.96706e10i q^{89} +(1.87819e10 - 1.87819e10i) q^{91} +(-1.08864e11 - 1.08864e11i) q^{93} -1.34507e9 q^{95} -1.03397e10 q^{97} +(-3.13667e10 - 3.13667e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) 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
\(3\) −367.824 367.824i −0.873923 0.873923i 0.118974 0.992897i \(-0.462039\pi\)
−0.992897 + 0.118974i \(0.962039\pi\)
\(4\) 0 0
\(5\) 502.344 502.344i 0.0718896 0.0718896i −0.670248 0.742137i \(-0.733812\pi\)
0.742137 + 0.670248i \(0.233812\pi\)
\(6\) 0 0
\(7\) 33063.2i 0.743542i 0.928325 + 0.371771i \(0.121249\pi\)
−0.928325 + 0.371771i \(0.878751\pi\)
\(8\) 0 0
\(9\) 93442.1i 0.527483i
\(10\) 0 0
\(11\) −335681. + 335681.i −0.628444 + 0.628444i −0.947677 0.319232i \(-0.896575\pi\)
0.319232 + 0.947677i \(0.396575\pi\)
\(12\) 0 0
\(13\) −568061. 568061.i −0.424333 0.424333i 0.462360 0.886692i \(-0.347003\pi\)
−0.886692 + 0.462360i \(0.847003\pi\)
\(14\) 0 0
\(15\) −369548. −0.125652
\(16\) 0 0
\(17\) −6.78457e6 −1.15892 −0.579460 0.815001i \(-0.696736\pi\)
−0.579460 + 0.815001i \(0.696736\pi\)
\(18\) 0 0
\(19\) −1.33879e6 1.33879e6i −0.124042 0.124042i 0.642361 0.766402i \(-0.277955\pi\)
−0.766402 + 0.642361i \(0.777955\pi\)
\(20\) 0 0
\(21\) 1.21614e7 1.21614e7i 0.649798 0.649798i
\(22\) 0 0
\(23\) 1.07133e6i 0.0347074i −0.999849 0.0173537i \(-0.994476\pi\)
0.999849 0.0173537i \(-0.00552412\pi\)
\(24\) 0 0
\(25\) 4.83234e7i 0.989664i
\(26\) 0 0
\(27\) −3.07887e7 + 3.07887e7i −0.412943 + 0.412943i
\(28\) 0 0
\(29\) −5.01359e7 5.01359e7i −0.453899 0.453899i 0.442747 0.896647i \(-0.354004\pi\)
−0.896647 + 0.442747i \(0.854004\pi\)
\(30\) 0 0
\(31\) 2.95969e8 1.85676 0.928382 0.371628i \(-0.121200\pi\)
0.928382 + 0.371628i \(0.121200\pi\)
\(32\) 0 0
\(33\) 2.46943e8 1.09842
\(34\) 0 0
\(35\) 1.66091e7 + 1.66091e7i 0.0534529 + 0.0534529i
\(36\) 0 0
\(37\) −5.11883e8 + 5.11883e8i −1.21356 + 1.21356i −0.243711 + 0.969848i \(0.578365\pi\)
−0.969848 + 0.243711i \(0.921635\pi\)
\(38\) 0 0
\(39\) 4.17893e8i 0.741669i
\(40\) 0 0
\(41\) 9.31803e8i 1.25607i 0.778186 + 0.628034i \(0.216140\pi\)
−0.778186 + 0.628034i \(0.783860\pi\)
\(42\) 0 0
\(43\) 1.27362e9 1.27362e9i 1.32118 1.32118i 0.408358 0.912822i \(-0.366101\pi\)
0.912822 0.408358i \(-0.133899\pi\)
\(44\) 0 0
\(45\) 4.69401e7 + 4.69401e7i 0.0379206 + 0.0379206i
\(46\) 0 0
\(47\) −1.80739e7 −0.0114951 −0.00574756 0.999983i \(-0.501830\pi\)
−0.00574756 + 0.999983i \(0.501830\pi\)
\(48\) 0 0
\(49\) 8.84154e8 0.447146
\(50\) 0 0
\(51\) 2.49553e9 + 2.49553e9i 1.01281 + 1.01281i
\(52\) 0 0
\(53\) −2.29607e9 + 2.29607e9i −0.754168 + 0.754168i −0.975254 0.221086i \(-0.929040\pi\)
0.221086 + 0.975254i \(0.429040\pi\)
\(54\) 0 0
\(55\) 3.37255e8i 0.0903572i
\(56\) 0 0
\(57\) 9.84878e8i 0.216806i
\(58\) 0 0
\(59\) 4.81453e8 4.81453e8i 0.0876734 0.0876734i −0.661910 0.749583i \(-0.730254\pi\)
0.749583 + 0.661910i \(0.230254\pi\)
\(60\) 0 0
\(61\) −1.30887e9 1.30887e9i −0.198419 0.198419i 0.600903 0.799322i \(-0.294808\pi\)
−0.799322 + 0.600903i \(0.794808\pi\)
\(62\) 0 0
\(63\) −3.08949e9 −0.392206
\(64\) 0 0
\(65\) −5.70724e8 −0.0610103
\(66\) 0 0
\(67\) 9.42251e9 + 9.42251e9i 0.852619 + 0.852619i 0.990455 0.137836i \(-0.0440147\pi\)
−0.137836 + 0.990455i \(0.544015\pi\)
\(68\) 0 0
\(69\) −3.94062e8 + 3.94062e8i −0.0303316 + 0.0303316i
\(70\) 0 0
\(71\) 1.74203e10i 1.14587i −0.819601 0.572934i \(-0.805805\pi\)
0.819601 0.572934i \(-0.194195\pi\)
\(72\) 0 0
\(73\) 8.91310e9i 0.503215i −0.967829 0.251607i \(-0.919041\pi\)
0.967829 0.251607i \(-0.0809591\pi\)
\(74\) 0 0
\(75\) 1.77745e10 1.77745e10i 0.864890 0.864890i
\(76\) 0 0
\(77\) −1.10987e10 1.10987e10i −0.467274 0.467274i
\(78\) 0 0
\(79\) −4.07466e10 −1.48985 −0.744924 0.667149i \(-0.767514\pi\)
−0.744924 + 0.667149i \(0.767514\pi\)
\(80\) 0 0
\(81\) 3.92026e10 1.24924
\(82\) 0 0
\(83\) −1.16461e10 1.16461e10i −0.324527 0.324527i 0.525974 0.850501i \(-0.323701\pi\)
−0.850501 + 0.525974i \(0.823701\pi\)
\(84\) 0 0
\(85\) −3.40819e9 + 3.40819e9i −0.0833143 + 0.0833143i
\(86\) 0 0
\(87\) 3.68824e10i 0.793346i
\(88\) 0 0
\(89\) 7.96706e10i 1.51235i −0.654368 0.756176i \(-0.727065\pi\)
0.654368 0.756176i \(-0.272935\pi\)
\(90\) 0 0
\(91\) 1.87819e10 1.87819e10i 0.315509 0.315509i
\(92\) 0 0
\(93\) −1.08864e11 1.08864e11i −1.62267 1.62267i
\(94\) 0 0
\(95\) −1.34507e9 −0.0178346
\(96\) 0 0
\(97\) −1.03397e10 −0.122254 −0.0611269 0.998130i \(-0.519469\pi\)
−0.0611269 + 0.998130i \(0.519469\pi\)
\(98\) 0 0
\(99\) −3.13667e10 3.13667e10i −0.331494 0.331494i
\(100\) 0 0
\(101\) 3.36018e10 3.36018e10i 0.318123 0.318123i −0.529923 0.848046i \(-0.677779\pi\)
0.848046 + 0.529923i \(0.177779\pi\)
\(102\) 0 0
\(103\) 9.27041e10i 0.787942i −0.919123 0.393971i \(-0.871101\pi\)
0.919123 0.393971i \(-0.128899\pi\)
\(104\) 0 0
\(105\) 1.22184e10i 0.0934275i
\(106\) 0 0
\(107\) −2.88146e10 + 2.88146e10i −0.198610 + 0.198610i −0.799404 0.600794i \(-0.794851\pi\)
0.600794 + 0.799404i \(0.294851\pi\)
\(108\) 0 0
\(109\) 2.21468e10 + 2.21468e10i 0.137868 + 0.137868i 0.772673 0.634804i \(-0.218919\pi\)
−0.634804 + 0.772673i \(0.718919\pi\)
\(110\) 0 0
\(111\) 3.76565e11 2.12111
\(112\) 0 0
\(113\) −1.71538e11 −0.875849 −0.437924 0.899012i \(-0.644286\pi\)
−0.437924 + 0.899012i \(0.644286\pi\)
\(114\) 0 0
\(115\) −5.38178e8 5.38178e8i −0.00249510 0.00249510i
\(116\) 0 0
\(117\) 5.30808e10 5.30808e10i 0.223829 0.223829i
\(118\) 0 0
\(119\) 2.24319e11i 0.861705i
\(120\) 0 0
\(121\) 5.99484e10i 0.210116i
\(122\) 0 0
\(123\) 3.42740e11 3.42740e11i 1.09771 1.09771i
\(124\) 0 0
\(125\) 4.88035e10 + 4.88035e10i 0.143036 + 0.143036i
\(126\) 0 0
\(127\) 5.54635e11 1.48966 0.744830 0.667254i \(-0.232531\pi\)
0.744830 + 0.667254i \(0.232531\pi\)
\(128\) 0 0
\(129\) −9.36933e11 −2.30922
\(130\) 0 0
\(131\) 4.50085e11 + 4.50085e11i 1.01930 + 1.01930i 0.999810 + 0.0194915i \(0.00620473\pi\)
0.0194915 + 0.999810i \(0.493795\pi\)
\(132\) 0 0
\(133\) 4.42646e10 4.42646e10i 0.0922301 0.0922301i
\(134\) 0 0
\(135\) 3.09330e10i 0.0593727i
\(136\) 0 0
\(137\) 7.75745e11i 1.37327i −0.727002 0.686635i \(-0.759087\pi\)
0.727002 0.686635i \(-0.240913\pi\)
\(138\) 0 0
\(139\) 3.80649e11 3.80649e11i 0.622219 0.622219i −0.323880 0.946098i \(-0.604987\pi\)
0.946098 + 0.323880i \(0.104987\pi\)
\(140\) 0 0
\(141\) 6.64802e9 + 6.64802e9i 0.0100459 + 0.0100459i
\(142\) 0 0
\(143\) 3.81374e11 0.533339
\(144\) 0 0
\(145\) −5.03709e10 −0.0652613
\(146\) 0 0
\(147\) −3.25213e11 3.25213e11i −0.390771 0.390771i
\(148\) 0 0
\(149\) 1.21482e11 1.21482e11i 0.135515 0.135515i −0.636095 0.771610i \(-0.719451\pi\)
0.771610 + 0.636095i \(0.219451\pi\)
\(150\) 0 0
\(151\) 4.18981e11i 0.434331i −0.976135 0.217166i \(-0.930319\pi\)
0.976135 0.217166i \(-0.0696812\pi\)
\(152\) 0 0
\(153\) 6.33965e11i 0.611311i
\(154\) 0 0
\(155\) 1.48678e11 1.48678e11i 0.133482 0.133482i
\(156\) 0 0
\(157\) −1.64950e12 1.64950e12i −1.38008 1.38008i −0.844456 0.535626i \(-0.820076\pi\)
−0.535626 0.844456i \(-0.679924\pi\)
\(158\) 0 0
\(159\) 1.68910e12 1.31817
\(160\) 0 0
\(161\) 3.54217e10 0.0258064
\(162\) 0 0
\(163\) −1.07698e12 1.07698e12i −0.733121 0.733121i 0.238116 0.971237i \(-0.423470\pi\)
−0.971237 + 0.238116i \(0.923470\pi\)
\(164\) 0 0
\(165\) 1.24050e11 1.24050e11i 0.0789653 0.0789653i
\(166\) 0 0
\(167\) 1.59845e12i 0.952268i −0.879373 0.476134i \(-0.842038\pi\)
0.879373 0.476134i \(-0.157962\pi\)
\(168\) 0 0
\(169\) 1.14677e12i 0.639883i
\(170\) 0 0
\(171\) 1.25099e11 1.25099e11i 0.0654299 0.0654299i
\(172\) 0 0
\(173\) −1.45486e12 1.45486e12i −0.713785 0.713785i 0.253540 0.967325i \(-0.418405\pi\)
−0.967325 + 0.253540i \(0.918405\pi\)
\(174\) 0 0
\(175\) −1.59773e12 −0.735856
\(176\) 0 0
\(177\) −3.54180e11 −0.153240
\(178\) 0 0
\(179\) 1.30891e12 + 1.30891e12i 0.532376 + 0.532376i 0.921279 0.388903i \(-0.127146\pi\)
−0.388903 + 0.921279i \(0.627146\pi\)
\(180\) 0 0
\(181\) 2.38152e12 2.38152e12i 0.911219 0.911219i −0.0851488 0.996368i \(-0.527137\pi\)
0.996368 + 0.0851488i \(0.0271366\pi\)
\(182\) 0 0
\(183\) 9.62869e11i 0.346806i
\(184\) 0 0
\(185\) 5.14282e11i 0.174485i
\(186\) 0 0
\(187\) 2.27745e12 2.27745e12i 0.728316 0.728316i
\(188\) 0 0
\(189\) −1.01797e12 1.01797e12i −0.307040 0.307040i
\(190\) 0 0
\(191\) −3.52280e12 −1.00278 −0.501388 0.865223i \(-0.667177\pi\)
−0.501388 + 0.865223i \(0.667177\pi\)
\(192\) 0 0
\(193\) 4.01739e12 1.07989 0.539944 0.841701i \(-0.318445\pi\)
0.539944 + 0.841701i \(0.318445\pi\)
\(194\) 0 0
\(195\) 2.09926e11 + 2.09926e11i 0.0533183 + 0.0533183i
\(196\) 0 0
\(197\) −3.98851e12 + 3.98851e12i −0.957737 + 0.957737i −0.999142 0.0414057i \(-0.986816\pi\)
0.0414057 + 0.999142i \(0.486816\pi\)
\(198\) 0 0
\(199\) 1.41250e11i 0.0320846i −0.999871 0.0160423i \(-0.994893\pi\)
0.999871 0.0160423i \(-0.00510665\pi\)
\(200\) 0 0
\(201\) 6.93165e12i 1.49025i
\(202\) 0 0
\(203\) 1.65765e12 1.65765e12i 0.337493 0.337493i
\(204\) 0 0
\(205\) 4.68086e11 + 4.68086e11i 0.0902982 + 0.0902982i
\(206\) 0 0
\(207\) 1.00108e11 0.0183076
\(208\) 0 0
\(209\) 8.98812e11 0.155907
\(210\) 0 0
\(211\) 3.64182e12 + 3.64182e12i 0.599466 + 0.599466i 0.940170 0.340704i \(-0.110666\pi\)
−0.340704 + 0.940170i \(0.610666\pi\)
\(212\) 0 0
\(213\) −6.40761e12 + 6.40761e12i −1.00140 + 1.00140i
\(214\) 0 0
\(215\) 1.27959e12i 0.189958i
\(216\) 0 0
\(217\) 9.78567e12i 1.38058i
\(218\) 0 0
\(219\) −3.27845e12 + 3.27845e12i −0.439771 + 0.439771i
\(220\) 0 0
\(221\) 3.85405e12 + 3.85405e12i 0.491768 + 0.491768i
\(222\) 0 0
\(223\) −5.31451e12 −0.645337 −0.322668 0.946512i \(-0.604580\pi\)
−0.322668 + 0.946512i \(0.604580\pi\)
\(224\) 0 0
\(225\) −4.51544e12 −0.522031
\(226\) 0 0
\(227\) 4.88964e12 + 4.88964e12i 0.538437 + 0.538437i 0.923070 0.384633i \(-0.125672\pi\)
−0.384633 + 0.923070i \(0.625672\pi\)
\(228\) 0 0
\(229\) 7.07133e12 7.07133e12i 0.742003 0.742003i −0.230960 0.972963i \(-0.574187\pi\)
0.972963 + 0.230960i \(0.0741867\pi\)
\(230\) 0 0
\(231\) 8.16472e12i 0.816724i
\(232\) 0 0
\(233\) 1.54810e13i 1.47687i 0.674324 + 0.738435i \(0.264435\pi\)
−0.674324 + 0.738435i \(0.735565\pi\)
\(234\) 0 0
\(235\) −9.07932e9 + 9.07932e9i −0.000826380 + 0.000826380i
\(236\) 0 0
\(237\) 1.49876e13 + 1.49876e13i 1.30201 + 1.30201i
\(238\) 0 0
\(239\) 1.47869e13 1.22656 0.613280 0.789866i \(-0.289850\pi\)
0.613280 + 0.789866i \(0.289850\pi\)
\(240\) 0 0
\(241\) −3.70990e12 −0.293946 −0.146973 0.989140i \(-0.546953\pi\)
−0.146973 + 0.989140i \(0.546953\pi\)
\(242\) 0 0
\(243\) −8.96555e12 8.96555e12i −0.678801 0.678801i
\(244\) 0 0
\(245\) 4.44149e11 4.44149e11i 0.0321452 0.0321452i
\(246\) 0 0
\(247\) 1.52103e12i 0.105270i
\(248\) 0 0
\(249\) 8.56743e12i 0.567223i
\(250\) 0 0
\(251\) −1.64619e13 + 1.64619e13i −1.04297 + 1.04297i −0.0439391 + 0.999034i \(0.513991\pi\)
−0.999034 + 0.0439391i \(0.986009\pi\)
\(252\) 0 0
\(253\) 3.59626e11 + 3.59626e11i 0.0218116 + 0.0218116i
\(254\) 0 0
\(255\) 2.50723e12 0.145621
\(256\) 0 0
\(257\) 2.20187e13 1.22506 0.612532 0.790446i \(-0.290151\pi\)
0.612532 + 0.790446i \(0.290151\pi\)
\(258\) 0 0
\(259\) −1.69245e13 1.69245e13i −0.902331 0.902331i
\(260\) 0 0
\(261\) 4.68480e12 4.68480e12i 0.239424 0.239424i
\(262\) 0 0
\(263\) 4.64085e12i 0.227426i 0.993514 + 0.113713i \(0.0362745\pi\)
−0.993514 + 0.113713i \(0.963726\pi\)
\(264\) 0 0
\(265\) 2.30684e12i 0.108434i
\(266\) 0 0
\(267\) −2.93047e13 + 2.93047e13i −1.32168 + 1.32168i
\(268\) 0 0
\(269\) 3.23177e12 + 3.23177e12i 0.139895 + 0.139895i 0.773586 0.633691i \(-0.218461\pi\)
−0.633691 + 0.773586i \(0.718461\pi\)
\(270\) 0 0
\(271\) −3.04462e13 −1.26533 −0.632663 0.774427i \(-0.718038\pi\)
−0.632663 + 0.774427i \(0.718038\pi\)
\(272\) 0 0
\(273\) −1.38169e13 −0.551461
\(274\) 0 0
\(275\) −1.62212e13 1.62212e13i −0.621949 0.621949i
\(276\) 0 0
\(277\) −1.12987e13 + 1.12987e13i −0.416282 + 0.416282i −0.883920 0.467638i \(-0.845105\pi\)
0.467638 + 0.883920i \(0.345105\pi\)
\(278\) 0 0
\(279\) 2.76559e13i 0.979412i
\(280\) 0 0
\(281\) 2.01696e13i 0.686772i −0.939194 0.343386i \(-0.888426\pi\)
0.939194 0.343386i \(-0.111574\pi\)
\(282\) 0 0
\(283\) 2.67176e12 2.67176e12i 0.0874929 0.0874929i −0.662006 0.749499i \(-0.730295\pi\)
0.749499 + 0.662006i \(0.230295\pi\)
\(284\) 0 0
\(285\) 4.94748e11 + 4.94748e11i 0.0155861 + 0.0155861i
\(286\) 0 0
\(287\) −3.08084e13 −0.933938
\(288\) 0 0
\(289\) 1.17585e13 0.343095
\(290\) 0 0
\(291\) 3.80318e12 + 3.80318e12i 0.106840 + 0.106840i
\(292\) 0 0
\(293\) 3.85320e13 3.85320e13i 1.04244 1.04244i 0.0433791 0.999059i \(-0.486188\pi\)
0.999059 0.0433791i \(-0.0138123\pi\)
\(294\) 0 0
\(295\) 4.83710e11i 0.0126056i
\(296\) 0 0
\(297\) 2.06703e13i 0.519024i
\(298\) 0 0
\(299\) −6.08583e11 + 6.08583e11i −0.0147275 + 0.0147275i
\(300\) 0 0
\(301\) 4.21098e13 + 4.21098e13i 0.982352 + 0.982352i
\(302\) 0 0
\(303\) −2.47191e13 −0.556031
\(304\) 0 0
\(305\) −1.31501e12 −0.0285285
\(306\) 0 0
\(307\) 3.24055e13 + 3.24055e13i 0.678199 + 0.678199i 0.959593 0.281393i \(-0.0907966\pi\)
−0.281393 + 0.959593i \(0.590797\pi\)
\(308\) 0 0
\(309\) −3.40988e13 + 3.40988e13i −0.688601 + 0.688601i
\(310\) 0 0
\(311\) 5.32038e13i 1.03696i −0.855091 0.518478i \(-0.826499\pi\)
0.855091 0.518478i \(-0.173501\pi\)
\(312\) 0 0
\(313\) 7.41435e13i 1.39502i 0.716577 + 0.697508i \(0.245708\pi\)
−0.716577 + 0.697508i \(0.754292\pi\)
\(314\) 0 0
\(315\) −1.55199e12 + 1.55199e12i −0.0281955 + 0.0281955i
\(316\) 0 0
\(317\) −6.45432e12 6.45432e12i −0.113246 0.113246i 0.648213 0.761459i \(-0.275517\pi\)
−0.761459 + 0.648213i \(0.775517\pi\)
\(318\) 0 0
\(319\) 3.36593e13 0.570501
\(320\) 0 0
\(321\) 2.11974e13 0.347140
\(322\) 0 0
\(323\) 9.08312e12 + 9.08312e12i 0.143754 + 0.143754i
\(324\) 0 0
\(325\) 2.74507e13 2.74507e13i 0.419947 0.419947i
\(326\) 0 0
\(327\) 1.62922e13i 0.240973i
\(328\) 0 0
\(329\) 5.97580e11i 0.00854710i
\(330\) 0 0
\(331\) 5.80148e13 5.80148e13i 0.802574 0.802574i −0.180923 0.983497i \(-0.557909\pi\)
0.983497 + 0.180923i \(0.0579085\pi\)
\(332\) 0 0
\(333\) −4.78314e13 4.78314e13i −0.640132 0.640132i
\(334\) 0 0
\(335\) 9.46668e12 0.122589
\(336\) 0 0
\(337\) −7.30225e13 −0.915150 −0.457575 0.889171i \(-0.651282\pi\)
−0.457575 + 0.889171i \(0.651282\pi\)
\(338\) 0 0
\(339\) 6.30958e13 + 6.30958e13i 0.765425 + 0.765425i
\(340\) 0 0
\(341\) −9.93510e13 + 9.93510e13i −1.16687 + 1.16687i
\(342\) 0 0
\(343\) 9.46096e13i 1.07601i
\(344\) 0 0
\(345\) 3.95910e11i 0.00436105i
\(346\) 0 0
\(347\) 4.07174e13 4.07174e13i 0.434478 0.434478i −0.455670 0.890149i \(-0.650600\pi\)
0.890149 + 0.455670i \(0.150600\pi\)
\(348\) 0 0
\(349\) −7.50423e13 7.50423e13i −0.775829 0.775829i 0.203290 0.979119i \(-0.434837\pi\)
−0.979119 + 0.203290i \(0.934837\pi\)
\(350\) 0 0
\(351\) 3.49797e13 0.350451
\(352\) 0 0
\(353\) 1.92000e14 1.86441 0.932205 0.361931i \(-0.117882\pi\)
0.932205 + 0.361931i \(0.117882\pi\)
\(354\) 0 0
\(355\) −8.75099e12 8.75099e12i −0.0823761 0.0823761i
\(356\) 0 0
\(357\) −8.25101e13 + 8.25101e13i −0.753064 + 0.753064i
\(358\) 0 0
\(359\) 6.04811e13i 0.535304i −0.963516 0.267652i \(-0.913752\pi\)
0.963516 0.267652i \(-0.0862477\pi\)
\(360\) 0 0
\(361\) 1.12906e14i 0.969227i
\(362\) 0 0
\(363\) 2.20505e13 2.20505e13i 0.183625 0.183625i
\(364\) 0 0
\(365\) −4.47744e12 4.47744e12i −0.0361759 0.0361759i
\(366\) 0 0
\(367\) 7.73489e13 0.606444 0.303222 0.952920i \(-0.401938\pi\)
0.303222 + 0.952920i \(0.401938\pi\)
\(368\) 0 0
\(369\) −8.70696e13 −0.662555
\(370\) 0 0
\(371\) −7.59154e13 7.59154e13i −0.560755 0.560755i
\(372\) 0 0
\(373\) 7.33207e13 7.33207e13i 0.525809 0.525809i −0.393511 0.919320i \(-0.628740\pi\)
0.919320 + 0.393511i \(0.128740\pi\)
\(374\) 0 0
\(375\) 3.59022e13i 0.250005i
\(376\) 0 0
\(377\) 5.69605e13i 0.385209i
\(378\) 0 0
\(379\) 1.92106e14 1.92106e14i 1.26190 1.26190i 0.311734 0.950169i \(-0.399090\pi\)
0.950169 0.311734i \(-0.100910\pi\)
\(380\) 0 0
\(381\) −2.04008e14 2.04008e14i −1.30185 1.30185i
\(382\) 0 0
\(383\) 1.28679e14 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(384\) 0 0
\(385\) −1.11507e13 −0.0671844
\(386\) 0 0
\(387\) 1.19009e14 + 1.19009e14i 0.696900 + 0.696900i
\(388\) 0 0
\(389\) 2.00352e14 2.00352e14i 1.14044 1.14044i 0.152068 0.988370i \(-0.451407\pi\)
0.988370 0.152068i \(-0.0485934\pi\)
\(390\) 0 0
\(391\) 7.26854e12i 0.0402230i
\(392\) 0 0
\(393\) 3.31104e14i 1.78158i
\(394\) 0 0
\(395\) −2.04688e13 + 2.04688e13i −0.107105 + 0.107105i
\(396\) 0 0
\(397\) 1.54966e14 + 1.54966e14i 0.788657 + 0.788657i 0.981274 0.192617i \(-0.0616974\pi\)
−0.192617 + 0.981274i \(0.561697\pi\)
\(398\) 0 0
\(399\) −3.25632e13 −0.161204
\(400\) 0 0
\(401\) −3.58680e14 −1.72748 −0.863740 0.503938i \(-0.831884\pi\)
−0.863740 + 0.503938i \(0.831884\pi\)
\(402\) 0 0
\(403\) −1.68128e14 1.68128e14i −0.787886 0.787886i
\(404\) 0 0
\(405\) 1.96932e13 1.96932e13i 0.0898077 0.0898077i
\(406\) 0 0
\(407\) 3.43658e14i 1.52531i
\(408\) 0 0
\(409\) 4.56728e13i 0.197324i 0.995121 + 0.0986618i \(0.0314562\pi\)
−0.995121 + 0.0986618i \(0.968544\pi\)
\(410\) 0 0
\(411\) −2.85338e14 + 2.85338e14i −1.20013 + 1.20013i
\(412\) 0 0
\(413\) 1.59184e13 + 1.59184e13i 0.0651888 + 0.0651888i
\(414\) 0 0
\(415\) −1.17007e13 −0.0466602
\(416\) 0 0
\(417\) −2.80024e14 −1.08754
\(418\) 0 0
\(419\) −2.11630e13 2.11630e13i −0.0800572 0.0800572i 0.665944 0.746001i \(-0.268029\pi\)
−0.746001 + 0.665944i \(0.768029\pi\)
\(420\) 0 0
\(421\) 1.92706e14 1.92706e14i 0.710138 0.710138i −0.256426 0.966564i \(-0.582545\pi\)
0.966564 + 0.256426i \(0.0825450\pi\)
\(422\) 0 0
\(423\) 1.68886e12i 0.00606349i
\(424\) 0 0
\(425\) 3.27854e14i 1.14694i
\(426\) 0 0
\(427\) 4.32754e13 4.32754e13i 0.147533 0.147533i
\(428\) 0 0
\(429\) −1.40279e14 1.40279e14i −0.466097 0.466097i
\(430\) 0 0
\(431\) 3.15533e14 1.02193 0.510964 0.859602i \(-0.329289\pi\)
0.510964 + 0.859602i \(0.329289\pi\)
\(432\) 0 0
\(433\) 9.43423e13 0.297867 0.148934 0.988847i \(-0.452416\pi\)
0.148934 + 0.988847i \(0.452416\pi\)
\(434\) 0 0
\(435\) 1.85276e13 + 1.85276e13i 0.0570334 + 0.0570334i
\(436\) 0 0
\(437\) −1.43429e12 + 1.43429e12i −0.00430516 + 0.00430516i
\(438\) 0 0
\(439\) 4.26118e12i 0.0124731i −0.999981 0.00623655i \(-0.998015\pi\)
0.999981 0.00623655i \(-0.00198517\pi\)
\(440\) 0 0
\(441\) 8.26172e13i 0.235862i
\(442\) 0 0
\(443\) −1.07675e13 + 1.07675e13i −0.0299843 + 0.0299843i −0.721940 0.691956i \(-0.756749\pi\)
0.691956 + 0.721940i \(0.256749\pi\)
\(444\) 0 0
\(445\) −4.00220e13 4.00220e13i −0.108722 0.108722i
\(446\) 0 0
\(447\) −8.93681e13 −0.236860
\(448\) 0 0
\(449\) 1.49573e14 0.386811 0.193405 0.981119i \(-0.438047\pi\)
0.193405 + 0.981119i \(0.438047\pi\)
\(450\) 0 0
\(451\) −3.12788e14 3.12788e14i −0.789368 0.789368i
\(452\) 0 0
\(453\) −1.54111e14 + 1.54111e14i −0.379572 + 0.379572i
\(454\) 0 0
\(455\) 1.88700e13i 0.0453637i
\(456\) 0 0
\(457\) 5.73073e13i 0.134484i −0.997737 0.0672420i \(-0.978580\pi\)
0.997737 0.0672420i \(-0.0214200\pi\)
\(458\) 0 0
\(459\) 2.08888e14 2.08888e14i 0.478568 0.478568i
\(460\) 0 0
\(461\) 5.04689e14 + 5.04689e14i 1.12893 + 1.12893i 0.990351 + 0.138583i \(0.0442548\pi\)
0.138583 + 0.990351i \(0.455745\pi\)
\(462\) 0 0
\(463\) −1.04606e14 −0.228487 −0.114243 0.993453i \(-0.536444\pi\)
−0.114243 + 0.993453i \(0.536444\pi\)
\(464\) 0 0
\(465\) −1.09375e14 −0.233306
\(466\) 0 0
\(467\) 5.83143e14 + 5.83143e14i 1.21488 + 1.21488i 0.969404 + 0.245472i \(0.0789430\pi\)
0.245472 + 0.969404i \(0.421057\pi\)
\(468\) 0 0
\(469\) −3.11538e14 + 3.11538e14i −0.633958 + 0.633958i
\(470\) 0 0
\(471\) 1.21345e15i 2.41217i
\(472\) 0 0
\(473\) 8.55057e14i 1.66058i
\(474\) 0 0
\(475\) 6.46949e13 6.46949e13i 0.122760 0.122760i
\(476\) 0 0
\(477\) −2.14550e14 2.14550e14i −0.397811 0.397811i
\(478\) 0 0
\(479\) −9.29337e12 −0.0168394 −0.00841972 0.999965i \(-0.502680\pi\)
−0.00841972 + 0.999965i \(0.502680\pi\)
\(480\) 0 0
\(481\) 5.81561e14 1.02991
\(482\) 0 0
\(483\) −1.30289e13 1.30289e13i −0.0225528 0.0225528i
\(484\) 0 0
\(485\) −5.19407e12 + 5.19407e12i −0.00878878 + 0.00878878i
\(486\) 0 0
\(487\) 2.67485e14i 0.442477i −0.975220 0.221238i \(-0.928990\pi\)
0.975220 0.221238i \(-0.0710099\pi\)
\(488\) 0 0
\(489\) 7.92278e14i 1.28138i
\(490\) 0 0
\(491\) −3.39834e13 + 3.39834e13i −0.0537425 + 0.0537425i −0.733467 0.679725i \(-0.762099\pi\)
0.679725 + 0.733467i \(0.262099\pi\)
\(492\) 0 0
\(493\) 3.40150e14 + 3.40150e14i 0.526033 + 0.526033i
\(494\) 0 0
\(495\) −3.15138e13 −0.0476619
\(496\) 0 0
\(497\) 5.75970e14 0.852001
\(498\) 0 0
\(499\) −1.03300e14 1.03300e14i −0.149468 0.149468i 0.628412 0.777880i \(-0.283705\pi\)
−0.777880 + 0.628412i \(0.783705\pi\)
\(500\) 0 0
\(501\) −5.87949e14 + 5.87949e14i −0.832209 + 0.832209i
\(502\) 0 0
\(503\) 1.03966e15i 1.43969i −0.694136 0.719843i \(-0.744214\pi\)
0.694136 0.719843i \(-0.255786\pi\)
\(504\) 0 0
\(505\) 3.37594e13i 0.0457395i
\(506\) 0 0
\(507\) −4.21811e14 + 4.21811e14i −0.559209 + 0.559209i
\(508\) 0 0
\(509\) −5.30191e14 5.30191e14i −0.687835 0.687835i 0.273918 0.961753i \(-0.411680\pi\)
−0.961753 + 0.273918i \(0.911680\pi\)
\(510\) 0 0
\(511\) 2.94695e14 0.374161
\(512\) 0 0
\(513\) 8.24391e13 0.102444
\(514\) 0 0
\(515\) −4.65693e13 4.65693e13i −0.0566448 0.0566448i
\(516\) 0 0
\(517\) 6.06706e12 6.06706e12i 0.00722405 0.00722405i
\(518\) 0 0
\(519\) 1.07026e15i 1.24759i
\(520\) 0 0
\(521\) 7.25048e14i 0.827484i 0.910394 + 0.413742i \(0.135778\pi\)
−0.910394 + 0.413742i \(0.864222\pi\)
\(522\) 0 0
\(523\) −6.35477e14 + 6.35477e14i −0.710134 + 0.710134i −0.966563 0.256429i \(-0.917454\pi\)
0.256429 + 0.966563i \(0.417454\pi\)
\(524\) 0 0
\(525\) 5.87682e14 + 5.87682e14i 0.643082 + 0.643082i
\(526\) 0 0
\(527\) −2.00802e15 −2.15184
\(528\) 0 0
\(529\) 9.51662e14 0.998795
\(530\) 0 0
\(531\) 4.49880e13 + 4.49880e13i 0.0462463 + 0.0462463i
\(532\) 0 0
\(533\) 5.29321e14 5.29321e14i 0.532991 0.532991i
\(534\) 0 0
\(535\) 2.89497e13i 0.0285560i
\(536\) 0 0
\(537\) 9.62898e14i 0.930511i
\(538\) 0 0
\(539\) −2.96793e14 + 2.96793e14i −0.281006 + 0.281006i
\(540\) 0 0
\(541\) 1.14149e15 + 1.14149e15i 1.05898 + 1.05898i 0.998148 + 0.0608311i \(0.0193751\pi\)
0.0608311 + 0.998148i \(0.480625\pi\)
\(542\) 0 0
\(543\) −1.75196e15 −1.59267
\(544\) 0 0
\(545\) 2.22506e13 0.0198226
\(546\) 0 0
\(547\) 7.92097e14 + 7.92097e14i 0.691589 + 0.691589i 0.962581 0.270993i \(-0.0873519\pi\)
−0.270993 + 0.962581i \(0.587352\pi\)
\(548\) 0 0
\(549\) 1.22304e14 1.22304e14i 0.104663 0.104663i
\(550\) 0 0
\(551\) 1.34243e14i 0.112605i
\(552\) 0 0
\(553\) 1.34721e15i 1.10776i
\(554\) 0 0
\(555\) 1.89165e14 1.89165e14i 0.152486 0.152486i
\(556\) 0 0
\(557\) −1.32985e15 1.32985e15i −1.05099 1.05099i −0.998628 0.0523642i \(-0.983324\pi\)
−0.0523642 0.998628i \(-0.516676\pi\)
\(558\) 0 0
\(559\) −1.44698e15 −1.12124
\(560\) 0 0
\(561\) −1.67540e15 −1.27299
\(562\) 0 0
\(563\) −6.09588e14 6.09588e14i −0.454193 0.454193i 0.442551 0.896743i \(-0.354074\pi\)
−0.896743 + 0.442551i \(0.854074\pi\)
\(564\) 0 0
\(565\) −8.61711e13 + 8.61711e13i −0.0629644 + 0.0629644i
\(566\) 0 0
\(567\) 1.29616e15i 0.928865i
\(568\) 0 0
\(569\) 1.37610e15i 0.967234i 0.875280 + 0.483617i \(0.160677\pi\)
−0.875280 + 0.483617i \(0.839323\pi\)
\(570\) 0 0
\(571\) 4.57011e14 4.57011e14i 0.315085 0.315085i −0.531790 0.846876i \(-0.678481\pi\)
0.846876 + 0.531790i \(0.178481\pi\)
\(572\) 0 0
\(573\) 1.29577e15 + 1.29577e15i 0.876349 + 0.876349i
\(574\) 0 0
\(575\) 5.17705e13 0.0343486
\(576\) 0 0
\(577\) 1.10753e15 0.720921 0.360460 0.932775i \(-0.382620\pi\)
0.360460 + 0.932775i \(0.382620\pi\)
\(578\) 0 0
\(579\) −1.47769e15 1.47769e15i −0.943740 0.943740i
\(580\) 0 0
\(581\) 3.85057e14 3.85057e14i 0.241299 0.241299i
\(582\) 0 0
\(583\) 1.54149e15i 0.947905i
\(584\) 0 0
\(585\) 5.33297e13i 0.0321819i
\(586\) 0 0
\(587\) −7.20142e14 + 7.20142e14i −0.426490 + 0.426490i −0.887431 0.460941i \(-0.847512\pi\)
0.460941 + 0.887431i \(0.347512\pi\)
\(588\) 0 0
\(589\) −3.96240e14 3.96240e14i −0.230316 0.230316i
\(590\) 0 0
\(591\) 2.93414e15 1.67398
\(592\) 0 0
\(593\) 1.05889e15 0.592996 0.296498 0.955034i \(-0.404181\pi\)
0.296498 + 0.955034i \(0.404181\pi\)
\(594\) 0 0
\(595\) −1.12686e14 1.12686e14i −0.0619476 0.0619476i
\(596\) 0 0
\(597\) −5.19552e13 + 5.19552e13i −0.0280395 + 0.0280395i
\(598\) 0 0
\(599\) 8.03403e14i 0.425682i −0.977087 0.212841i \(-0.931728\pi\)
0.977087 0.212841i \(-0.0682717\pi\)
\(600\) 0 0
\(601\) 1.38624e15i 0.721155i −0.932729 0.360578i \(-0.882580\pi\)
0.932729 0.360578i \(-0.117420\pi\)
\(602\) 0 0
\(603\) −8.80459e14 + 8.80459e14i −0.449742 + 0.449742i
\(604\) 0 0
\(605\) 3.01147e13 + 3.01147e13i 0.0151051 + 0.0151051i
\(606\) 0 0
\(607\) 3.18636e15 1.56948 0.784742 0.619823i \(-0.212796\pi\)
0.784742 + 0.619823i \(0.212796\pi\)
\(608\) 0 0
\(609\) −1.21945e15 −0.589886
\(610\) 0 0
\(611\) 1.02671e13 + 1.02671e13i 0.00487776 + 0.00487776i
\(612\) 0 0
\(613\) −1.13972e15 + 1.13972e15i −0.531820 + 0.531820i −0.921114 0.389294i \(-0.872719\pi\)
0.389294 + 0.921114i \(0.372719\pi\)
\(614\) 0 0
\(615\) 3.44346e14i 0.157827i
\(616\) 0 0
\(617\) 4.02609e15i 1.81265i 0.422578 + 0.906326i \(0.361125\pi\)
−0.422578 + 0.906326i \(0.638875\pi\)
\(618\) 0 0
\(619\) −1.12455e14 + 1.12455e14i −0.0497370 + 0.0497370i −0.731538 0.681801i \(-0.761197\pi\)
0.681801 + 0.731538i \(0.261197\pi\)
\(620\) 0 0
\(621\) 3.29849e13 + 3.29849e13i 0.0143322 + 0.0143322i
\(622\) 0 0
\(623\) 2.63416e15 1.12450
\(624\) 0 0
\(625\) −2.31051e15 −0.969098
\(626\) 0 0
\(627\) −3.30605e14 3.30605e14i −0.136250 0.136250i
\(628\) 0 0
\(629\) 3.47290e15 3.47290e15i 1.40642 1.40642i
\(630\) 0 0
\(631\) 3.54561e15i 1.41101i 0.708706 + 0.705504i \(0.249279\pi\)
−0.708706 + 0.705504i \(0.750721\pi\)
\(632\) 0 0
\(633\) 2.67910e15i 1.04777i
\(634\) 0 0
\(635\) 2.78618e14 2.78618e14i 0.107091 0.107091i
\(636\) 0 0
\(637\) −5.02253e14 5.02253e14i −0.189739 0.189739i
\(638\) 0 0
\(639\) 1.62779e15 0.604427
\(640\) 0 0
\(641\) −3.99466e15 −1.45801 −0.729005 0.684509i \(-0.760017\pi\)
−0.729005 + 0.684509i \(0.760017\pi\)
\(642\) 0 0
\(643\) 7.20860e14 + 7.20860e14i 0.258637 + 0.258637i 0.824500 0.565863i \(-0.191457\pi\)
−0.565863 + 0.824500i \(0.691457\pi\)
\(644\) 0 0
\(645\) −4.70663e14 + 4.70663e14i −0.166009 + 0.166009i
\(646\) 0 0
\(647\) 2.38050e15i 0.825458i −0.910854 0.412729i \(-0.864576\pi\)
0.910854 0.412729i \(-0.135424\pi\)
\(648\) 0 0
\(649\) 3.23229e14i 0.110196i
\(650\) 0 0
\(651\) 3.59940e15 3.59940e15i 1.20652 1.20652i
\(652\) 0 0
\(653\) −3.05366e15 3.05366e15i −1.00646 1.00646i −0.999979 0.00648374i \(-0.997936\pi\)
−0.00648374 0.999979i \(-0.502064\pi\)
\(654\) 0 0
\(655\) 4.52195e14 0.146554
\(656\) 0 0
\(657\) 8.32859e14 0.265437
\(658\) 0 0
\(659\) 3.38786e15 + 3.38786e15i 1.06183 + 1.06183i 0.997958 + 0.0638750i \(0.0203459\pi\)
0.0638750 + 0.997958i \(0.479654\pi\)
\(660\) 0 0
\(661\) −2.17225e15 + 2.17225e15i −0.669580 + 0.669580i −0.957619 0.288039i \(-0.906997\pi\)
0.288039 + 0.957619i \(0.406997\pi\)
\(662\) 0 0
\(663\) 2.83523e15i 0.859534i
\(664\) 0 0
\(665\) 4.44722e13i 0.0132608i
\(666\) 0 0
\(667\) −5.37122e13 + 5.37122e13i −0.0157536 + 0.0157536i
\(668\) 0 0
\(669\) 1.95480e15 + 1.95480e15i 0.563975 + 0.563975i
\(670\) 0 0
\(671\) 8.78726e14 0.249390
\(672\) 0 0
\(673\) −2.38223e14 −0.0665121 −0.0332561 0.999447i \(-0.510588\pi\)
−0.0332561 + 0.999447i \(0.510588\pi\)
\(674\) 0 0
\(675\) −1.48781e15 1.48781e15i −0.408675 0.408675i
\(676\) 0 0
\(677\) −1.90659e15 + 1.90659e15i −0.515252 + 0.515252i −0.916131 0.400879i \(-0.868705\pi\)
0.400879 + 0.916131i \(0.368705\pi\)
\(678\) 0 0
\(679\) 3.41862e14i 0.0909007i
\(680\) 0 0
\(681\) 3.59706e15i 0.941106i
\(682\) 0 0
\(683\) 2.55329e15 2.55329e15i 0.657334 0.657334i −0.297415 0.954748i \(-0.596124\pi\)
0.954748 + 0.297415i \(0.0961243\pi\)
\(684\) 0 0
\(685\) −3.89691e14 3.89691e14i −0.0987239 0.0987239i
\(686\) 0 0
\(687\) −5.20201e15 −1.29691
\(688\) 0 0
\(689\) 2.60862e15 0.640037
\(690\) 0 0
\(691\) 4.73213e14 + 4.73213e14i 0.114269 + 0.114269i 0.761929 0.647660i \(-0.224252\pi\)
−0.647660 + 0.761929i \(0.724252\pi\)
\(692\) 0 0
\(693\) 1.03708e15 1.03708e15i 0.246479 0.246479i
\(694\) 0 0
\(695\) 3.82433e14i 0.0894621i
\(696\) 0 0
\(697\) 6.32188e15i 1.45568i
\(698\) 0 0
\(699\) 5.69430e15 5.69430e15i 1.29067 1.29067i
\(700\) 0 0
\(701\) −4.29380e14 4.29380e14i −0.0958061 0.0958061i 0.657579 0.753385i \(-0.271580\pi\)
−0.753385 + 0.657579i \(0.771580\pi\)
\(702\) 0 0
\(703\) 1.37061e15 0.301064
\(704\) 0 0
\(705\) 6.67918e12 0.00144439
\(706\) 0 0
\(707\) 1.11098e15 + 1.11098e15i 0.236538 + 0.236538i
\(708\) 0 0
\(709\) 1.83229e14 1.83229e14i 0.0384097 0.0384097i −0.687641 0.726051i \(-0.741354\pi\)
0.726051 + 0.687641i \(0.241354\pi\)
\(710\) 0 0
\(711\) 3.80744e15i 0.785870i
\(712\) 0 0
\(713\) 3.17081e14i 0.0644433i
\(714\) 0 0
\(715\) 1.91581e14 1.91581e14i 0.0383415 0.0383415i
\(716\) 0 0
\(717\) −5.43898e15 5.43898e15i −1.07192 1.07192i
\(718\) 0 0
\(719\) −4.63852e15 −0.900265 −0.450133 0.892962i \(-0.648623\pi\)
−0.450133 + 0.892962i \(0.648623\pi\)
\(720\) 0 0
\(721\) 3.06509e15 0.585867
\(722\) 0 0
\(723\) 1.36459e15 + 1.36459e15i 0.256887 + 0.256887i
\(724\) 0 0
\(725\) 2.42274e15 2.42274e15i 0.449208 0.449208i
\(726\) 0 0
\(727\) 3.43021e15i 0.626442i 0.949680 + 0.313221i \(0.101408\pi\)
−0.949680 + 0.313221i \(0.898592\pi\)
\(728\) 0 0
\(729\) 3.49140e14i 0.0628055i
\(730\) 0 0
\(731\) −8.64094e15 + 8.64094e15i −1.53114 + 1.53114i
\(732\) 0 0
\(733\) −4.05712e15 4.05712e15i −0.708184 0.708184i 0.257969 0.966153i \(-0.416947\pi\)
−0.966153 + 0.257969i \(0.916947\pi\)
\(734\) 0 0
\(735\) −3.26738e14 −0.0561848
\(736\) 0 0
\(737\) −6.32591e15 −1.07165
\(738\) 0 0
\(739\) −1.67030e15 1.67030e15i −0.278772 0.278772i 0.553846 0.832619i \(-0.313159\pi\)
−0.832619 + 0.553846i \(0.813159\pi\)
\(740\) 0 0
\(741\) 5.59471e14 5.59471e14i 0.0919978 0.0919978i
\(742\) 0 0
\(743\) 4.61265e15i 0.747330i −0.927564 0.373665i \(-0.878101\pi\)
0.927564 0.373665i \(-0.121899\pi\)
\(744\) 0 0
\(745\) 1.22052e14i 0.0194843i
\(746\) 0 0
\(747\) 1.08824e15 1.08824e15i 0.171183 0.171183i
\(748\) 0 0
\(749\) −9.52701e14 9.52701e14i −0.147675 0.147675i
\(750\) 0 0
\(751\) −6.38439e14 −0.0975214 −0.0487607 0.998810i \(-0.515527\pi\)
−0.0487607 + 0.998810i \(0.515527\pi\)
\(752\) 0 0
\(753\) 1.21101e16 1.82296
\(754\) 0 0
\(755\) −2.10473e14 2.10473e14i −0.0312239 0.0312239i
\(756\) 0 0
\(757\) 3.03835e15 3.03835e15i 0.444232 0.444232i −0.449199 0.893432i \(-0.648291\pi\)
0.893432 + 0.449199i \(0.148291\pi\)
\(758\) 0 0
\(759\) 2.64558e14i 0.0381234i
\(760\) 0 0
\(761\) 2.97530e15i 0.422586i −0.977423 0.211293i \(-0.932233\pi\)
0.977423 0.211293i \(-0.0677674\pi\)
\(762\) 0 0
\(763\) −7.32242e14 + 7.32242e14i −0.102511 + 0.102511i
\(764\) 0 0
\(765\) −3.18468e14 3.18468e14i −0.0439469 0.0439469i
\(766\) 0 0
\(767\) −5.46989e14 −0.0744054
\(768\) 0 0
\(769\) −4.50036e15 −0.603465 −0.301733 0.953393i \(-0.597565\pi\)
−0.301733 + 0.953393i \(0.597565\pi\)
\(770\) 0 0
\(771\) −8.09899e15 8.09899e15i −1.07061 1.07061i
\(772\) 0 0
\(773\) −6.33340e15 + 6.33340e15i −0.825372 + 0.825372i −0.986873 0.161500i \(-0.948367\pi\)
0.161500 + 0.986873i \(0.448367\pi\)
\(774\) 0 0
\(775\) 1.43022e16i 1.83757i
\(776\) 0 0
\(777\) 1.24504e16i 1.57714i
\(778\) 0 0
\(779\) 1.24749e15 1.24749e15i 0.155805 0.155805i
\(780\) 0 0
\(781\) 5.84766e15 + 5.84766e15i 0.720115 + 0.720115i
\(782\) 0 0
\(783\) 3.08723e15 0.374869
\(784\) 0 0
\(785\) −1.65723e15 −0.198427
\(786\) 0 0
\(787\) 3.25291e15 + 3.25291e15i 0.384071 + 0.384071i 0.872566 0.488496i \(-0.162454\pi\)
−0.488496 + 0.872566i \(0.662454\pi\)
\(788\) 0 0
\(789\) 1.70702e15 1.70702e15i 0.198753 0.198753i
\(790\) 0 0
\(791\) 5.67159e15i 0.651230i
\(792\) 0 0
\(793\) 1.48704e15i 0.168391i
\(794\) 0 0
\(795\) 8.48510e14 8.48510e14i 0.0947628 0.0947628i
\(796\) 0 0
\(797\) 2.11952e15 + 2.11952e15i 0.233462 + 0.233462i 0.814136 0.580674i \(-0.197211\pi\)
−0.580674 + 0.814136i \(0.697211\pi\)
\(798\) 0 0
\(799\) 1.22624e14 0.0133219
\(800\) 0 0
\(801\) 7.44458e15 0.797740
\(802\) 0 0
\(803\) 2.99196e15 + 2.99196e15i 0.316242 + 0.316242i
\(804\) 0 0
\(805\) 1.77939e13 1.77939e13i 0.00185521 0.00185521i
\(806\) 0 0
\(807\) 2.37745e15i 0.244515i
\(808\) 0 0
\(809\) 6.90043e15i 0.700098i 0.936731 + 0.350049i \(0.113835\pi\)
−0.936731 + 0.350049i \(0.886165\pi\)
\(810\) 0 0
\(811\) 1.16275e15 1.16275e15i 0.116378 0.116378i −0.646519 0.762898i \(-0.723776\pi\)
0.762898 + 0.646519i \(0.223776\pi\)
\(812\) 0 0
\(813\) 1.11989e16 + 1.11989e16i 1.10580 + 1.10580i
\(814\) 0 0
\(815\) −1.08203e15 −0.105408
\(816\) 0 0
\(817\) −3.41021e15 −0.327763
\(818\) 0 0
\(819\) 1.75502e15 + 1.75502e15i 0.166426 + 0.166426i
\(820\) 0 0
\(821\) −8.05348e15 + 8.05348e15i −0.753523 + 0.753523i −0.975135 0.221612i \(-0.928868\pi\)
0.221612 + 0.975135i \(0.428868\pi\)
\(822\) 0 0
\(823\) 1.72899e16i 1.59622i 0.602510 + 0.798111i \(0.294167\pi\)
−0.602510 + 0.798111i \(0.705833\pi\)
\(824\) 0 0
\(825\) 1.19331e16i 1.08707i
\(826\) 0 0
\(827\) −1.23292e16 + 1.23292e16i −1.10829 + 1.10829i −0.114913 + 0.993376i \(0.536659\pi\)
−0.993376 + 0.114913i \(0.963341\pi\)
\(828\) 0 0
\(829\) 1.23351e16 + 1.23351e16i 1.09419 + 1.09419i 0.995076 + 0.0991110i \(0.0315999\pi\)
0.0991110 + 0.995076i \(0.468400\pi\)
\(830\) 0 0
\(831\) 8.31183e15 0.727598
\(832\) 0 0
\(833\) −5.99860e15 −0.518206
\(834\) 0 0
\(835\) −8.02973e14 8.02973e14i −0.0684582 0.0684582i
\(836\) 0 0
\(837\) −9.11249e15 + 9.11249e15i −0.766738 + 0.766738i
\(838\) 0 0
\(839\) 5.85813e15i 0.486484i −0.969966 0.243242i \(-0.921789\pi\)
0.969966 0.243242i \(-0.0782109\pi\)
\(840\) 0 0
\(841\) 7.17330e15i 0.587951i
\(842\) 0 0
\(843\) −7.41886e15 + 7.41886e15i −0.600186 + 0.600186i
\(844\) 0 0
\(845\) −5.76075e14 5.76075e14i −0.0460010 0.0460010i
\(846\) 0 0
\(847\) −1.98209e15 −0.156230
\(848\) 0 0
\(849\) −1.96548e15 −0.152924
\(850\) 0 0
\(851\) 5.48397e14 + 5.48397e14i 0.0421194 + 0.0421194i
\(852\) 0 0
\(853\) 1.14683e16 1.14683e16i 0.869522 0.869522i −0.122898 0.992419i \(-0.539219\pi\)
0.992419 + 0.122898i \(0.0392187\pi\)
\(854\) 0 0
\(855\) 1.25686e14i 0.00940747i
\(856\) 0 0
\(857\) 3.71190e15i 0.274285i 0.990551 + 0.137142i \(0.0437918\pi\)
−0.990551 + 0.137142i \(0.956208\pi\)
\(858\) 0 0
\(859\) −1.20180e15 + 1.20180e15i −0.0876735 + 0.0876735i −0.749583 0.661910i \(-0.769746\pi\)
0.661910 + 0.749583i \(0.269746\pi\)
\(860\) 0 0
\(861\) 1.13321e16 + 1.13321e16i 0.816190 + 0.816190i
\(862\) 0 0
\(863\) 4.89836e15 0.348331 0.174165 0.984716i \(-0.444277\pi\)
0.174165 + 0.984716i \(0.444277\pi\)
\(864\) 0 0
\(865\) −1.46168e15 −0.102627
\(866\) 0 0
\(867\) −4.32506e15 4.32506e15i −0.299839 0.299839i
\(868\) 0 0
\(869\) 1.36778e16 1.36778e16i 0.936287 0.936287i
\(870\) 0 0
\(871\) 1.07051e16i 0.723589i
\(872\) 0 0
\(873\) 9.66161e14i 0.0644868i
\(874\) 0 0
\(875\) −1.61360e15 + 1.61360e15i −0.106353 + 0.106353i
\(876\) 0 0
\(877\) −3.94432e15 3.94432e15i −0.256729 0.256729i 0.566993 0.823722i \(-0.308106\pi\)
−0.823722 + 0.566993i \(0.808106\pi\)
\(878\) 0 0
\(879\) −2.83460e16 −1.82202
\(880\) 0 0
\(881\) −8.41637e15 −0.534266 −0.267133 0.963660i \(-0.586076\pi\)
−0.267133 + 0.963660i \(0.586076\pi\)
\(882\) 0 0
\(883\) 4.32184e15 + 4.32184e15i 0.270947 + 0.270947i 0.829482 0.558534i \(-0.188636\pi\)
−0.558534 + 0.829482i \(0.688636\pi\)
\(884\) 0 0
\(885\) −1.77920e14 + 1.77920e14i −0.0110163 + 0.0110163i
\(886\) 0 0
\(887\) 2.87494e16i 1.75812i 0.476707 + 0.879062i \(0.341830\pi\)
−0.476707 + 0.879062i \(0.658170\pi\)
\(888\) 0 0
\(889\) 1.83380e16i 1.10762i
\(890\) 0 0
\(891\) −1.31596e16 + 1.31596e16i −0.785081 + 0.785081i
\(892\) 0 0
\(893\) 2.41972e13 + 2.41972e13i 0.00142587 + 0.00142587i
\(894\) 0 0
\(895\) 1.31505e15 0.0765446
\(896\) 0 0
\(897\) 4.47703e14 0.0257414
\(898\) 0 0
\(899\) −1.48387e16 1.48387e16i −0.842784 0.842784i
\(900\) 0 0
\(901\) 1.55779e16 1.55779e16i 0.874020 0.874020i
\(902\) 0 0
\(903\) 3.09780e16i 1.71700i
\(904\) 0 0
\(905\) 2.39269e15i 0.131014i
\(906\) 0 0
\(907\) 5.17663e15 5.17663e15i 0.280031 0.280031i −0.553090 0.833121i \(-0.686551\pi\)
0.833121 + 0.553090i \(0.186551\pi\)
\(908\) 0 0
\(909\) 3.13983e15 + 3.13983e15i 0.167805 + 0.167805i
\(910\) 0 0
\(911\) −2.22340e16 −1.17400 −0.586998 0.809588i \(-0.699690\pi\)
−0.586998 + 0.809588i \(0.699690\pi\)
\(912\) 0 0
\(913\) 7.81874e15 0.407894
\(914\) 0 0
\(915\) 4.83691e14 + 4.83691e14i 0.0249317 + 0.0249317i
\(916\) 0 0
\(917\) −1.48812e16 + 1.48812e16i −0.757893 + 0.757893i
\(918\) 0 0
\(919\) 1.58068e16i 0.795444i 0.917506 + 0.397722i \(0.130199\pi\)
−0.917506 + 0.397722i \(0.869801\pi\)
\(920\) 0 0
\(921\) 2.38390e16i 1.18539i
\(922\) 0 0
\(923\) −9.89580e15 + 9.89580e15i −0.486230 + 0.486230i
\(924\) 0 0
\(925\) −2.47359e16 2.47359e16i −1.20102 1.20102i
\(926\) 0 0
\(927\) 8.66246e15 0.415626
\(928\) 0 0
\(929\) 3.46893e16 1.64479 0.822393 0.568920i \(-0.192639\pi\)
0.822393 + 0.568920i \(0.192639\pi\)
\(930\) 0 0
\(931\) −1.18370e15 1.18370e15i −0.0554647 0.0554647i
\(932\) 0 0
\(933\) −1.95696e16 + 1.95696e16i −0.906220 + 0.906220i
\(934\) 0 0
\(935\) 2.28813e15i 0.104717i
\(936\) 0 0
\(937\) 2.05502e16i 0.929496i −0.885443 0.464748i \(-0.846145\pi\)
0.885443 0.464748i \(-0.153855\pi\)
\(938\) 0 0
\(939\) 2.72718e16 2.72718e16i 1.21914 1.21914i
\(940\) 0 0
\(941\) −1.82034e16 1.82034e16i −0.804286 0.804286i 0.179476 0.983762i \(-0.442560\pi\)
−0.983762 + 0.179476i \(0.942560\pi\)
\(942\) 0 0
\(943\) 9.98271e14 0.0435948
\(944\) 0 0
\(945\) −1.02274e15 −0.0441460
\(946\) 0 0
\(947\) −2.56323e15 2.56323e15i −0.109361 0.109361i 0.650309 0.759670i \(-0.274640\pi\)
−0.759670 + 0.650309i \(0.774640\pi\)
\(948\) 0 0
\(949\) −5.06319e15 + 5.06319e15i −0.213530 + 0.213530i
\(950\) 0 0
\(951\) 4.74811e15i 0.197937i
\(952\) 0 0
\(953\) 7.68907e15i 0.316857i 0.987370 + 0.158428i \(0.0506427\pi\)
−0.987370 + 0.158428i \(0.949357\pi\)
\(954\) 0 0
\(955\) −1.76966e15 + 1.76966e15i −0.0720892 + 0.0720892i
\(956\) 0 0
\(957\) −1.23807e16 1.23807e16i −0.498574 0.498574i
\(958\) 0 0
\(959\) 2.56486e16 1.02108
\(960\) 0 0
\(961\) 6.21890e16 2.44757
\(962\) 0 0
\(963\) −2.69249e15 2.69249e15i −0.104764 0.104764i
\(964\) 0 0
\(965\) 2.01811e15 2.01811e15i 0.0776328 0.0776328i
\(966\) 0 0
\(967\) 2.35131e16i 0.894260i 0.894469 + 0.447130i \(0.147554\pi\)
−0.894469 + 0.447130i \(0.852446\pi\)
\(968\) 0 0
\(969\) 6.68198e15i 0.251261i
\(970\) 0 0
\(971\) 2.96572e16 2.96572e16i 1.10262 1.10262i 0.108522 0.994094i \(-0.465388\pi\)
0.994094 0.108522i \(-0.0346117\pi\)
\(972\) 0 0
\(973\) 1.25855e16 + 1.25855e16i 0.462645 + 0.462645i
\(974\) 0 0
\(975\) −2.01940e16 −0.734003
\(976\) 0 0
\(977\) −3.19671e16 −1.14890 −0.574450 0.818539i \(-0.694784\pi\)
−0.574450 + 0.818539i \(0.694784\pi\)
\(978\) 0 0
\(979\) 2.67439e16 + 2.67439e16i 0.950429 + 0.950429i
\(980\) 0 0
\(981\) −2.06944e15 + 2.06944e15i −0.0727233 + 0.0727233i
\(982\) 0 0
\(983\) 3.59807e16i 1.25033i −0.780492 0.625166i \(-0.785031\pi\)
0.780492 0.625166i \(-0.214969\pi\)
\(984\) 0 0
\(985\) 4.00721e15i 0.137703i
\(986\) 0 0
\(987\) −2.19804e14 + 2.19804e14i −0.00746951 + 0.00746951i
\(988\) 0 0
\(989\) −1.36447e15 1.36447e15i −0.0458547 0.0458547i
\(990\) 0 0
\(991\) −2.28338e16 −0.758881 −0.379441 0.925216i \(-0.623884\pi\)
−0.379441 + 0.925216i \(0.623884\pi\)
\(992\) 0 0
\(993\) −4.26785e16 −1.40278
\(994\) 0 0
\(995\) −7.09562e13 7.09562e13i −0.00230655 0.00230655i
\(996\) 0 0
\(997\) 2.26548e16 2.26548e16i 0.728346 0.728346i −0.241945 0.970290i \(-0.577785\pi\)
0.970290 + 0.241945i \(0.0777852\pi\)
\(998\) 0 0
\(999\) 3.15204e16i 1.00226i
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.33.5 42
4.3 odd 2 128.12.e.a.33.17 42
8.3 odd 2 64.12.e.a.17.5 42
8.5 even 2 16.12.e.a.13.14 yes 42
16.3 odd 4 64.12.e.a.49.5 42
16.5 even 4 inner 128.12.e.b.97.5 42
16.11 odd 4 128.12.e.a.97.17 42
16.13 even 4 16.12.e.a.5.14 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.12.e.a.5.14 42 16.13 even 4
16.12.e.a.13.14 yes 42 8.5 even 2
64.12.e.a.17.5 42 8.3 odd 2
64.12.e.a.49.5 42 16.3 odd 4
128.12.e.a.33.17 42 4.3 odd 2
128.12.e.a.97.17 42 16.11 odd 4
128.12.e.b.33.5 42 1.1 even 1 trivial
128.12.e.b.97.5 42 16.5 even 4 inner