Properties

Label 192.7.e.e.65.1
Level $192$
Weight $7$
Character 192.65
Analytic conductor $44.170$
Analytic rank $0$
Dimension $2$
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] = [192,7,Mod(65,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 192.e (of order \(2\), degree \(1\), 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: \(44.1703840550\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.1
Root \(-2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 192.65
Dual form 192.7.e.e.65.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.00000 - 26.8328i) q^{3} -160.997i q^{5} +242.000 q^{7} +(-711.000 - 160.997i) q^{9} +O(q^{10})\) \(q+(3.00000 - 26.8328i) q^{3} -160.997i q^{5} +242.000 q^{7} +(-711.000 - 160.997i) q^{9} +1770.97i q^{11} -2618.00 q^{13} +(-4320.00 - 482.991i) q^{15} +7083.86i q^{17} -5786.00 q^{19} +(726.000 - 6493.54i) q^{21} -9337.82i q^{23} -10295.0 q^{25} +(-6453.00 + 18595.1i) q^{27} +12396.8i q^{29} -20446.0 q^{31} +(47520.0 + 5312.90i) q^{33} -38961.2i q^{35} +46774.0 q^{37} +(-7854.00 + 70248.3i) q^{39} +3541.93i q^{41} -68618.0 q^{43} +(-25920.0 + 114469. i) q^{45} -21251.6i q^{47} -59085.0 q^{49} +(190080. + 21251.6i) q^{51} +171784. i q^{53} +285120. q^{55} +(-17358.0 + 155255. i) q^{57} -149566. i q^{59} -24794.0 q^{61} +(-172062. - 38961.2i) q^{63} +421490. i q^{65} +84358.0 q^{67} +(-250560. - 28013.5i) q^{69} -324248. i q^{71} -113806. q^{73} +(-30885.0 + 276244. i) q^{75} +428574. i q^{77} -159742. q^{79} +(479601. + 228938. i) q^{81} +515351. i q^{83} +1.14048e6 q^{85} +(332640. + 37190.3i) q^{87} +1.25610e6i q^{89} -633556. q^{91} +(-61338.0 + 548624. i) q^{93} +931528. i q^{95} +899522. q^{97} +(285120. - 1.25916e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 484 q^{7} - 1422 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 484 q^{7} - 1422 q^{9} - 5236 q^{13} - 8640 q^{15} - 11572 q^{19} + 1452 q^{21} - 20590 q^{25} - 12906 q^{27} - 40892 q^{31} + 95040 q^{33} + 93548 q^{37} - 15708 q^{39} - 137236 q^{43} - 51840 q^{45} - 118170 q^{49} + 380160 q^{51} + 570240 q^{55} - 34716 q^{57} - 49588 q^{61} - 344124 q^{63} + 168716 q^{67} - 501120 q^{69} - 227612 q^{73} - 61770 q^{75} - 319484 q^{79} + 959202 q^{81} + 2280960 q^{85} + 665280 q^{87} - 1267112 q^{91} - 122676 q^{93} + 1799044 q^{97} + 570240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 3.00000 26.8328i 0.111111 0.993808i
\(4\) 0 0
\(5\) 160.997i 1.28798i −0.765036 0.643988i \(-0.777279\pi\)
0.765036 0.643988i \(-0.222721\pi\)
\(6\) 0 0
\(7\) 242.000 0.705539 0.352770 0.935710i \(-0.385240\pi\)
0.352770 + 0.935710i \(0.385240\pi\)
\(8\) 0 0
\(9\) −711.000 160.997i −0.975309 0.220846i
\(10\) 0 0
\(11\) 1770.97i 1.33055i 0.746597 + 0.665276i \(0.231686\pi\)
−0.746597 + 0.665276i \(0.768314\pi\)
\(12\) 0 0
\(13\) −2618.00 −1.19162 −0.595812 0.803124i \(-0.703170\pi\)
−0.595812 + 0.803124i \(0.703170\pi\)
\(14\) 0 0
\(15\) −4320.00 482.991i −1.28000 0.143108i
\(16\) 0 0
\(17\) 7083.86i 1.44186i 0.693007 + 0.720931i \(0.256285\pi\)
−0.693007 + 0.720931i \(0.743715\pi\)
\(18\) 0 0
\(19\) −5786.00 −0.843563 −0.421782 0.906697i \(-0.638595\pi\)
−0.421782 + 0.906697i \(0.638595\pi\)
\(20\) 0 0
\(21\) 726.000 6493.54i 0.0783933 0.701171i
\(22\) 0 0
\(23\) 9337.82i 0.767471i −0.923443 0.383736i \(-0.874637\pi\)
0.923443 0.383736i \(-0.125363\pi\)
\(24\) 0 0
\(25\) −10295.0 −0.658880
\(26\) 0 0
\(27\) −6453.00 + 18595.1i −0.327846 + 0.944731i
\(28\) 0 0
\(29\) 12396.8i 0.508293i 0.967166 + 0.254147i \(0.0817946\pi\)
−0.967166 + 0.254147i \(0.918205\pi\)
\(30\) 0 0
\(31\) −20446.0 −0.686315 −0.343157 0.939278i \(-0.611496\pi\)
−0.343157 + 0.939278i \(0.611496\pi\)
\(32\) 0 0
\(33\) 47520.0 + 5312.90i 1.32231 + 0.147839i
\(34\) 0 0
\(35\) 38961.2i 0.908717i
\(36\) 0 0
\(37\) 46774.0 0.923420 0.461710 0.887031i \(-0.347236\pi\)
0.461710 + 0.887031i \(0.347236\pi\)
\(38\) 0 0
\(39\) −7854.00 + 70248.3i −0.132403 + 1.18425i
\(40\) 0 0
\(41\) 3541.93i 0.0513912i 0.999670 + 0.0256956i \(0.00818006\pi\)
−0.999670 + 0.0256956i \(0.991820\pi\)
\(42\) 0 0
\(43\) −68618.0 −0.863044 −0.431522 0.902103i \(-0.642023\pi\)
−0.431522 + 0.902103i \(0.642023\pi\)
\(44\) 0 0
\(45\) −25920.0 + 114469.i −0.284444 + 1.25617i
\(46\) 0 0
\(47\) 21251.6i 0.204691i −0.994749 0.102345i \(-0.967365\pi\)
0.994749 0.102345i \(-0.0326347\pi\)
\(48\) 0 0
\(49\) −59085.0 −0.502214
\(50\) 0 0
\(51\) 190080. + 21251.6i 1.43293 + 0.160207i
\(52\) 0 0
\(53\) 171784.i 1.15386i 0.816792 + 0.576932i \(0.195750\pi\)
−0.816792 + 0.576932i \(0.804250\pi\)
\(54\) 0 0
\(55\) 285120. 1.71372
\(56\) 0 0
\(57\) −17358.0 + 155255.i −0.0937292 + 0.838340i
\(58\) 0 0
\(59\) 149566.i 0.728244i −0.931351 0.364122i \(-0.881369\pi\)
0.931351 0.364122i \(-0.118631\pi\)
\(60\) 0 0
\(61\) −24794.0 −0.109234 −0.0546169 0.998507i \(-0.517394\pi\)
−0.0546169 + 0.998507i \(0.517394\pi\)
\(62\) 0 0
\(63\) −172062. 38961.2i −0.688119 0.155816i
\(64\) 0 0
\(65\) 421490.i 1.53478i
\(66\) 0 0
\(67\) 84358.0 0.280480 0.140240 0.990118i \(-0.455213\pi\)
0.140240 + 0.990118i \(0.455213\pi\)
\(68\) 0 0
\(69\) −250560. 28013.5i −0.762719 0.0852746i
\(70\) 0 0
\(71\) 324248.i 0.905945i −0.891524 0.452973i \(-0.850364\pi\)
0.891524 0.452973i \(-0.149636\pi\)
\(72\) 0 0
\(73\) −113806. −0.292548 −0.146274 0.989244i \(-0.546728\pi\)
−0.146274 + 0.989244i \(0.546728\pi\)
\(74\) 0 0
\(75\) −30885.0 + 276244.i −0.0732089 + 0.654800i
\(76\) 0 0
\(77\) 428574.i 0.938757i
\(78\) 0 0
\(79\) −159742. −0.323995 −0.161997 0.986791i \(-0.551794\pi\)
−0.161997 + 0.986791i \(0.551794\pi\)
\(80\) 0 0
\(81\) 479601. + 228938.i 0.902454 + 0.430786i
\(82\) 0 0
\(83\) 515351.i 0.901299i 0.892701 + 0.450650i \(0.148808\pi\)
−0.892701 + 0.450650i \(0.851192\pi\)
\(84\) 0 0
\(85\) 1.14048e6 1.85708
\(86\) 0 0
\(87\) 332640. + 37190.3i 0.505146 + 0.0564770i
\(88\) 0 0
\(89\) 1.25610e6i 1.78178i 0.454222 + 0.890889i \(0.349917\pi\)
−0.454222 + 0.890889i \(0.650083\pi\)
\(90\) 0 0
\(91\) −633556. −0.840738
\(92\) 0 0
\(93\) −61338.0 + 548624.i −0.0762572 + 0.682065i
\(94\) 0 0
\(95\) 931528.i 1.08649i
\(96\) 0 0
\(97\) 899522. 0.985591 0.492795 0.870145i \(-0.335975\pi\)
0.492795 + 0.870145i \(0.335975\pi\)
\(98\) 0 0
\(99\) 285120. 1.25916e6i 0.293848 1.29770i
\(100\) 0 0
\(101\) 1.19894e6i 1.16368i 0.813302 + 0.581842i \(0.197668\pi\)
−0.813302 + 0.581842i \(0.802332\pi\)
\(102\) 0 0
\(103\) −1.59701e6 −1.46149 −0.730743 0.682652i \(-0.760826\pi\)
−0.730743 + 0.682652i \(0.760826\pi\)
\(104\) 0 0
\(105\) −1.04544e6 116884.i −0.903090 0.100969i
\(106\) 0 0
\(107\) 214287.i 0.174922i 0.996168 + 0.0874610i \(0.0278753\pi\)
−0.996168 + 0.0874610i \(0.972125\pi\)
\(108\) 0 0
\(109\) −2.21811e6 −1.71278 −0.856392 0.516326i \(-0.827299\pi\)
−0.856392 + 0.516326i \(0.827299\pi\)
\(110\) 0 0
\(111\) 140322. 1.25508e6i 0.102602 0.917702i
\(112\) 0 0
\(113\) 2.01117e6i 1.39384i −0.717147 0.696922i \(-0.754552\pi\)
0.717147 0.696922i \(-0.245448\pi\)
\(114\) 0 0
\(115\) −1.50336e6 −0.988484
\(116\) 0 0
\(117\) 1.86140e6 + 421490.i 1.16220 + 0.263166i
\(118\) 0 0
\(119\) 1.71429e6i 1.01729i
\(120\) 0 0
\(121\) −1.36476e6 −0.770371
\(122\) 0 0
\(123\) 95040.0 + 10625.8i 0.0510730 + 0.00571013i
\(124\) 0 0
\(125\) 858113.i 0.439354i
\(126\) 0 0
\(127\) 709346. 0.346296 0.173148 0.984896i \(-0.444606\pi\)
0.173148 + 0.984896i \(0.444606\pi\)
\(128\) 0 0
\(129\) −205854. + 1.84121e6i −0.0958937 + 0.857700i
\(130\) 0 0
\(131\) 795164.i 0.353706i −0.984237 0.176853i \(-0.943408\pi\)
0.984237 0.176853i \(-0.0565917\pi\)
\(132\) 0 0
\(133\) −1.40021e6 −0.595167
\(134\) 0 0
\(135\) 2.99376e6 + 1.03891e6i 1.21679 + 0.422258i
\(136\) 0 0
\(137\) 1.46797e6i 0.570894i 0.958395 + 0.285447i \(0.0921420\pi\)
−0.958395 + 0.285447i \(0.907858\pi\)
\(138\) 0 0
\(139\) 5.07410e6 1.88936 0.944680 0.327993i \(-0.106372\pi\)
0.944680 + 0.327993i \(0.106372\pi\)
\(140\) 0 0
\(141\) −570240. 63754.8i −0.203423 0.0227434i
\(142\) 0 0
\(143\) 4.63639e6i 1.58552i
\(144\) 0 0
\(145\) 1.99584e6 0.654669
\(146\) 0 0
\(147\) −177255. + 1.58542e6i −0.0558016 + 0.499104i
\(148\) 0 0
\(149\) 3.32764e6i 1.00595i −0.864300 0.502977i \(-0.832238\pi\)
0.864300 0.502977i \(-0.167762\pi\)
\(150\) 0 0
\(151\) −3.93125e6 −1.14182 −0.570912 0.821011i \(-0.693410\pi\)
−0.570912 + 0.821011i \(0.693410\pi\)
\(152\) 0 0
\(153\) 1.14048e6 5.03663e6i 0.318430 1.40626i
\(154\) 0 0
\(155\) 3.29174e6i 0.883956i
\(156\) 0 0
\(157\) 3.61188e6 0.933328 0.466664 0.884435i \(-0.345456\pi\)
0.466664 + 0.884435i \(0.345456\pi\)
\(158\) 0 0
\(159\) 4.60944e6 + 515351.i 1.14672 + 0.128207i
\(160\) 0 0
\(161\) 2.25975e6i 0.541481i
\(162\) 0 0
\(163\) −3.76390e6 −0.869111 −0.434555 0.900645i \(-0.643094\pi\)
−0.434555 + 0.900645i \(0.643094\pi\)
\(164\) 0 0
\(165\) 855360. 7.65057e6i 0.190413 1.70311i
\(166\) 0 0
\(167\) 478161.i 0.102666i 0.998682 + 0.0513328i \(0.0163469\pi\)
−0.998682 + 0.0513328i \(0.983653\pi\)
\(168\) 0 0
\(169\) 2.02711e6 0.419970
\(170\) 0 0
\(171\) 4.11385e6 + 931528.i 0.822734 + 0.186298i
\(172\) 0 0
\(173\) 5.01006e6i 0.967620i −0.875173 0.483810i \(-0.839253\pi\)
0.875173 0.483810i \(-0.160747\pi\)
\(174\) 0 0
\(175\) −2.49139e6 −0.464866
\(176\) 0 0
\(177\) −4.01328e6 448698.i −0.723735 0.0809160i
\(178\) 0 0
\(179\) 9.60685e6i 1.67503i −0.546417 0.837513i \(-0.684009\pi\)
0.546417 0.837513i \(-0.315991\pi\)
\(180\) 0 0
\(181\) −3.85591e6 −0.650267 −0.325133 0.945668i \(-0.605409\pi\)
−0.325133 + 0.945668i \(0.605409\pi\)
\(182\) 0 0
\(183\) −74382.0 + 665293.i −0.0121371 + 0.108557i
\(184\) 0 0
\(185\) 7.53047e6i 1.18934i
\(186\) 0 0
\(187\) −1.25453e7 −1.91847
\(188\) 0 0
\(189\) −1.56163e6 + 4.50002e6i −0.231309 + 0.666545i
\(190\) 0 0
\(191\) 1.36268e6i 0.195566i −0.995208 0.0977829i \(-0.968825\pi\)
0.995208 0.0977829i \(-0.0311751\pi\)
\(192\) 0 0
\(193\) −4.53158e6 −0.630344 −0.315172 0.949034i \(-0.602062\pi\)
−0.315172 + 0.949034i \(0.602062\pi\)
\(194\) 0 0
\(195\) 1.13098e7 + 1.26447e6i 1.52528 + 0.170531i
\(196\) 0 0
\(197\) 6.08681e6i 0.796143i 0.917354 + 0.398071i \(0.130320\pi\)
−0.917354 + 0.398071i \(0.869680\pi\)
\(198\) 0 0
\(199\) −2.91777e6 −0.370248 −0.185124 0.982715i \(-0.559269\pi\)
−0.185124 + 0.982715i \(0.559269\pi\)
\(200\) 0 0
\(201\) 253074. 2.26356e6i 0.0311644 0.278743i
\(202\) 0 0
\(203\) 3.00002e6i 0.358621i
\(204\) 0 0
\(205\) 570240. 0.0661906
\(206\) 0 0
\(207\) −1.50336e6 + 6.63919e6i −0.169493 + 0.748521i
\(208\) 0 0
\(209\) 1.02468e7i 1.12241i
\(210\) 0 0
\(211\) −1.12078e7 −1.19309 −0.596547 0.802578i \(-0.703461\pi\)
−0.596547 + 0.802578i \(0.703461\pi\)
\(212\) 0 0
\(213\) −8.70048e6 972743.i −0.900336 0.100661i
\(214\) 0 0
\(215\) 1.10473e7i 1.11158i
\(216\) 0 0
\(217\) −4.94793e6 −0.484222
\(218\) 0 0
\(219\) −341418. + 3.05374e6i −0.0325053 + 0.290736i
\(220\) 0 0
\(221\) 1.85456e7i 1.71816i
\(222\) 0 0
\(223\) −1.48732e7 −1.34119 −0.670593 0.741825i \(-0.733960\pi\)
−0.670593 + 0.741825i \(0.733960\pi\)
\(224\) 0 0
\(225\) 7.31974e6 + 1.65746e6i 0.642611 + 0.145511i
\(226\) 0 0
\(227\) 2.73614e6i 0.233917i −0.993137 0.116958i \(-0.962686\pi\)
0.993137 0.116958i \(-0.0373144\pi\)
\(228\) 0 0
\(229\) 8.96133e6 0.746219 0.373109 0.927787i \(-0.378292\pi\)
0.373109 + 0.927787i \(0.378292\pi\)
\(230\) 0 0
\(231\) 1.14998e7 + 1.28572e6i 0.932945 + 0.104306i
\(232\) 0 0
\(233\) 1.06293e7i 0.840308i 0.907453 + 0.420154i \(0.138024\pi\)
−0.907453 + 0.420154i \(0.861976\pi\)
\(234\) 0 0
\(235\) −3.42144e6 −0.263636
\(236\) 0 0
\(237\) −479226. + 4.28633e6i −0.0359994 + 0.321988i
\(238\) 0 0
\(239\) 2.39080e7i 1.75126i −0.482984 0.875629i \(-0.660447\pi\)
0.482984 0.875629i \(-0.339553\pi\)
\(240\) 0 0
\(241\) −1.00217e7 −0.715964 −0.357982 0.933728i \(-0.616535\pi\)
−0.357982 + 0.933728i \(0.616535\pi\)
\(242\) 0 0
\(243\) 7.58184e6 1.21822e7i 0.528392 0.849001i
\(244\) 0 0
\(245\) 9.51250e6i 0.646839i
\(246\) 0 0
\(247\) 1.51477e7 1.00521
\(248\) 0 0
\(249\) 1.38283e7 + 1.54605e6i 0.895718 + 0.100144i
\(250\) 0 0
\(251\) 2.58186e7i 1.63272i 0.577544 + 0.816359i \(0.304011\pi\)
−0.577544 + 0.816359i \(0.695989\pi\)
\(252\) 0 0
\(253\) 1.65370e7 1.02116
\(254\) 0 0
\(255\) 3.42144e6 3.06023e7i 0.206342 1.84558i
\(256\) 0 0
\(257\) 1.39410e7i 0.821289i 0.911796 + 0.410644i \(0.134696\pi\)
−0.911796 + 0.410644i \(0.865304\pi\)
\(258\) 0 0
\(259\) 1.13193e7 0.651509
\(260\) 0 0
\(261\) 1.99584e6 8.81410e6i 0.112255 0.495743i
\(262\) 0 0
\(263\) 2.87711e7i 1.58157i 0.612092 + 0.790787i \(0.290328\pi\)
−0.612092 + 0.790787i \(0.709672\pi\)
\(264\) 0 0
\(265\) 2.76566e7 1.48615
\(266\) 0 0
\(267\) 3.37046e7 + 3.76829e6i 1.77074 + 0.197975i
\(268\) 0 0
\(269\) 2.03927e7i 1.04765i 0.851825 + 0.523826i \(0.175496\pi\)
−0.851825 + 0.523826i \(0.824504\pi\)
\(270\) 0 0
\(271\) −9.57504e6 −0.481097 −0.240548 0.970637i \(-0.577327\pi\)
−0.240548 + 0.970637i \(0.577327\pi\)
\(272\) 0 0
\(273\) −1.90067e6 + 1.70001e7i −0.0934154 + 0.835532i
\(274\) 0 0
\(275\) 1.82321e7i 0.876675i
\(276\) 0 0
\(277\) −3.28669e7 −1.54639 −0.773196 0.634167i \(-0.781343\pi\)
−0.773196 + 0.634167i \(0.781343\pi\)
\(278\) 0 0
\(279\) 1.45371e7 + 3.29174e6i 0.669369 + 0.151570i
\(280\) 0 0
\(281\) 2.37416e7i 1.07002i 0.844847 + 0.535008i \(0.179691\pi\)
−0.844847 + 0.535008i \(0.820309\pi\)
\(282\) 0 0
\(283\) −3.93043e7 −1.73413 −0.867063 0.498198i \(-0.833995\pi\)
−0.867063 + 0.498198i \(0.833995\pi\)
\(284\) 0 0
\(285\) 2.49955e7 + 2.79458e6i 1.07976 + 0.120721i
\(286\) 0 0
\(287\) 857147.i 0.0362585i
\(288\) 0 0
\(289\) −2.60436e7 −1.07896
\(290\) 0 0
\(291\) 2.69857e6 2.41367e7i 0.109510 0.979488i
\(292\) 0 0
\(293\) 2.32156e7i 0.922947i 0.887154 + 0.461474i \(0.152679\pi\)
−0.887154 + 0.461474i \(0.847321\pi\)
\(294\) 0 0
\(295\) −2.40797e7 −0.937961
\(296\) 0 0
\(297\) −3.29314e7 1.14280e7i −1.25701 0.436217i
\(298\) 0 0
\(299\) 2.44464e7i 0.914538i
\(300\) 0 0
\(301\) −1.66056e7 −0.608911
\(302\) 0 0
\(303\) 3.21710e7 + 3.59683e6i 1.15648 + 0.129298i
\(304\) 0 0
\(305\) 3.99176e6i 0.140690i
\(306\) 0 0
\(307\) −9.05148e6 −0.312827 −0.156414 0.987692i \(-0.549993\pi\)
−0.156414 + 0.987692i \(0.549993\pi\)
\(308\) 0 0
\(309\) −4.79102e6 + 4.28522e7i −0.162387 + 1.45244i
\(310\) 0 0
\(311\) 2.17120e6i 0.0721804i 0.999349 + 0.0360902i \(0.0114904\pi\)
−0.999349 + 0.0360902i \(0.988510\pi\)
\(312\) 0 0
\(313\) 1.08512e7 0.353871 0.176936 0.984222i \(-0.443382\pi\)
0.176936 + 0.984222i \(0.443382\pi\)
\(314\) 0 0
\(315\) −6.27264e6 + 2.77014e7i −0.200687 + 0.886280i
\(316\) 0 0
\(317\) 3.19143e7i 1.00186i 0.865488 + 0.500930i \(0.167009\pi\)
−0.865488 + 0.500930i \(0.832991\pi\)
\(318\) 0 0
\(319\) −2.19542e7 −0.676311
\(320\) 0 0
\(321\) 5.74992e6 + 642861.i 0.173839 + 0.0194358i
\(322\) 0 0
\(323\) 4.09872e7i 1.21630i
\(324\) 0 0
\(325\) 2.69523e7 0.785138
\(326\) 0 0
\(327\) −6.65432e6 + 5.95180e7i −0.190309 + 1.70218i
\(328\) 0 0
\(329\) 5.14288e6i 0.144417i
\(330\) 0 0
\(331\) 5.68600e7 1.56792 0.783958 0.620814i \(-0.213198\pi\)
0.783958 + 0.620814i \(0.213198\pi\)
\(332\) 0 0
\(333\) −3.32563e7 7.53047e6i −0.900620 0.203934i
\(334\) 0 0
\(335\) 1.35814e7i 0.361251i
\(336\) 0 0
\(337\) 4.50571e7 1.17726 0.588632 0.808401i \(-0.299667\pi\)
0.588632 + 0.808401i \(0.299667\pi\)
\(338\) 0 0
\(339\) −5.39654e7 6.03352e6i −1.38521 0.154872i
\(340\) 0 0
\(341\) 3.62092e7i 0.913178i
\(342\) 0 0
\(343\) −4.27696e7 −1.05987
\(344\) 0 0
\(345\) −4.51008e6 + 4.03394e7i −0.109832 + 0.982363i
\(346\) 0 0
\(347\) 2.11294e7i 0.505707i −0.967505 0.252853i \(-0.918631\pi\)
0.967505 0.252853i \(-0.0813690\pi\)
\(348\) 0 0
\(349\) 1.41108e7 0.331953 0.165976 0.986130i \(-0.446922\pi\)
0.165976 + 0.986130i \(0.446922\pi\)
\(350\) 0 0
\(351\) 1.68940e7 4.86821e7i 0.390670 1.12577i
\(352\) 0 0
\(353\) 1.60958e7i 0.365923i 0.983120 + 0.182961i \(0.0585683\pi\)
−0.983120 + 0.182961i \(0.941432\pi\)
\(354\) 0 0
\(355\) −5.22029e7 −1.16683
\(356\) 0 0
\(357\) 4.59994e7 + 5.14288e6i 1.01099 + 0.113032i
\(358\) 0 0
\(359\) 1.11677e7i 0.241369i −0.992691 0.120684i \(-0.961491\pi\)
0.992691 0.120684i \(-0.0385089\pi\)
\(360\) 0 0
\(361\) −1.35681e7 −0.288401
\(362\) 0 0
\(363\) −4.09428e6 + 3.66203e7i −0.0855968 + 0.765601i
\(364\) 0 0
\(365\) 1.83224e7i 0.376794i
\(366\) 0 0
\(367\) −5.39616e6 −0.109166 −0.0545829 0.998509i \(-0.517383\pi\)
−0.0545829 + 0.998509i \(0.517383\pi\)
\(368\) 0 0
\(369\) 570240. 2.51831e6i 0.0113495 0.0501223i
\(370\) 0 0
\(371\) 4.15717e7i 0.814096i
\(372\) 0 0
\(373\) 2.12151e7 0.408807 0.204403 0.978887i \(-0.434475\pi\)
0.204403 + 0.978887i \(0.434475\pi\)
\(374\) 0 0
\(375\) −2.30256e7 2.57434e6i −0.436634 0.0488171i
\(376\) 0 0
\(377\) 3.24547e7i 0.605695i
\(378\) 0 0
\(379\) 6.90261e6 0.126793 0.0633966 0.997988i \(-0.479807\pi\)
0.0633966 + 0.997988i \(0.479807\pi\)
\(380\) 0 0
\(381\) 2.12804e6 1.90338e7i 0.0384773 0.344151i
\(382\) 0 0
\(383\) 6.12716e7i 1.09059i −0.838243 0.545296i \(-0.816417\pi\)
0.838243 0.545296i \(-0.183583\pi\)
\(384\) 0 0
\(385\) 6.89990e7 1.20910
\(386\) 0 0
\(387\) 4.87874e7 + 1.10473e7i 0.841734 + 0.190600i
\(388\) 0 0
\(389\) 7.49552e7i 1.27336i −0.771126 0.636682i \(-0.780306\pi\)
0.771126 0.636682i \(-0.219694\pi\)
\(390\) 0 0
\(391\) 6.61478e7 1.10659
\(392\) 0 0
\(393\) −2.13365e7 2.38549e6i −0.351516 0.0393007i
\(394\) 0 0
\(395\) 2.57180e7i 0.417297i
\(396\) 0 0
\(397\) −9.39352e7 −1.50126 −0.750631 0.660721i \(-0.770251\pi\)
−0.750631 + 0.660721i \(0.770251\pi\)
\(398\) 0 0
\(399\) −4.20064e6 + 3.75716e7i −0.0661297 + 0.591482i
\(400\) 0 0
\(401\) 2.85125e7i 0.442184i 0.975253 + 0.221092i \(0.0709621\pi\)
−0.975253 + 0.221092i \(0.929038\pi\)
\(402\) 0 0
\(403\) 5.35276e7 0.817830
\(404\) 0 0
\(405\) 3.68582e7 7.72143e7i 0.554842 1.16234i
\(406\) 0 0
\(407\) 8.28352e7i 1.22866i
\(408\) 0 0
\(409\) −1.13436e8 −1.65799 −0.828996 0.559255i \(-0.811087\pi\)
−0.828996 + 0.559255i \(0.811087\pi\)
\(410\) 0 0
\(411\) 3.93898e7 + 4.40391e6i 0.567359 + 0.0634327i
\(412\) 0 0
\(413\) 3.61950e7i 0.513805i
\(414\) 0 0
\(415\) 8.29699e7 1.16085
\(416\) 0 0
\(417\) 1.52223e7 1.36152e8i 0.209929 1.87766i
\(418\) 0 0
\(419\) 1.15880e8i 1.57531i −0.616119 0.787653i \(-0.711296\pi\)
0.616119 0.787653i \(-0.288704\pi\)
\(420\) 0 0
\(421\) −6.58338e7 −0.882272 −0.441136 0.897440i \(-0.645424\pi\)
−0.441136 + 0.897440i \(0.645424\pi\)
\(422\) 0 0
\(423\) −3.42144e6 + 1.51099e7i −0.0452051 + 0.199636i
\(424\) 0 0
\(425\) 7.29284e7i 0.950013i
\(426\) 0 0
\(427\) −6.00015e6 −0.0770688
\(428\) 0 0
\(429\) −1.24407e8 1.39092e7i −1.57570 0.176169i
\(430\) 0 0
\(431\) 1.15347e8i 1.44070i −0.693612 0.720349i \(-0.743982\pi\)
0.693612 0.720349i \(-0.256018\pi\)
\(432\) 0 0
\(433\) −9.17231e7 −1.12984 −0.564918 0.825147i \(-0.691092\pi\)
−0.564918 + 0.825147i \(0.691092\pi\)
\(434\) 0 0
\(435\) 5.98752e6 5.35540e7i 0.0727410 0.650615i
\(436\) 0 0
\(437\) 5.40286e7i 0.647410i
\(438\) 0 0
\(439\) 1.70641e7 0.201693 0.100847 0.994902i \(-0.467845\pi\)
0.100847 + 0.994902i \(0.467845\pi\)
\(440\) 0 0
\(441\) 4.20094e7 + 9.51250e6i 0.489814 + 0.110912i
\(442\) 0 0
\(443\) 1.02080e8i 1.17416i 0.809528 + 0.587081i \(0.199723\pi\)
−0.809528 + 0.587081i \(0.800277\pi\)
\(444\) 0 0
\(445\) 2.02228e8 2.29488
\(446\) 0 0
\(447\) −8.92901e7 9.98293e6i −0.999725 0.111773i
\(448\) 0 0
\(449\) 6.91565e7i 0.764002i −0.924162 0.382001i \(-0.875235\pi\)
0.924162 0.382001i \(-0.124765\pi\)
\(450\) 0 0
\(451\) −6.27264e6 −0.0683787
\(452\) 0 0
\(453\) −1.17937e7 + 1.05486e8i −0.126869 + 1.13475i
\(454\) 0 0
\(455\) 1.02001e8i 1.08285i
\(456\) 0 0
\(457\) 4.98054e7 0.521829 0.260914 0.965362i \(-0.415976\pi\)
0.260914 + 0.965362i \(0.415976\pi\)
\(458\) 0 0
\(459\) −1.31725e8 4.57122e7i −1.36217 0.472709i
\(460\) 0 0
\(461\) 5.00776e7i 0.511141i −0.966790 0.255571i \(-0.917737\pi\)
0.966790 0.255571i \(-0.0822633\pi\)
\(462\) 0 0
\(463\) −4.15949e6 −0.0419080 −0.0209540 0.999780i \(-0.506670\pi\)
−0.0209540 + 0.999780i \(0.506670\pi\)
\(464\) 0 0
\(465\) 8.83267e7 + 9.87523e6i 0.878483 + 0.0982174i
\(466\) 0 0
\(467\) 1.00127e8i 0.983107i −0.870847 0.491553i \(-0.836429\pi\)
0.870847 0.491553i \(-0.163571\pi\)
\(468\) 0 0
\(469\) 2.04146e7 0.197890
\(470\) 0 0
\(471\) 1.08356e7 9.69169e7i 0.103703 0.927548i
\(472\) 0 0
\(473\) 1.21520e8i 1.14832i
\(474\) 0 0
\(475\) 5.95669e7 0.555807
\(476\) 0 0
\(477\) 2.76566e7 1.22138e8i 0.254826 1.12537i
\(478\) 0 0
\(479\) 2.04823e8i 1.86368i 0.362867 + 0.931841i \(0.381798\pi\)
−0.362867 + 0.931841i \(0.618202\pi\)
\(480\) 0 0
\(481\) −1.22454e8 −1.10037
\(482\) 0 0
\(483\) −6.06355e7 6.77926e6i −0.538128 0.0601646i
\(484\) 0 0
\(485\) 1.44820e8i 1.26942i
\(486\) 0 0
\(487\) 1.45737e8 1.26178 0.630888 0.775874i \(-0.282691\pi\)
0.630888 + 0.775874i \(0.282691\pi\)
\(488\) 0 0
\(489\) −1.12917e7 + 1.00996e8i −0.0965678 + 0.863729i
\(490\) 0 0
\(491\) 1.01485e8i 0.857350i 0.903459 + 0.428675i \(0.141019\pi\)
−0.903459 + 0.428675i \(0.858981\pi\)
\(492\) 0 0
\(493\) −8.78170e7 −0.732888
\(494\) 0 0
\(495\) −2.02720e8 4.59034e7i −1.67140 0.378468i
\(496\) 0 0
\(497\) 7.84680e7i 0.639180i
\(498\) 0 0
\(499\) 9.42057e7 0.758186 0.379093 0.925359i \(-0.376236\pi\)
0.379093 + 0.925359i \(0.376236\pi\)
\(500\) 0 0
\(501\) 1.28304e7 + 1.43448e6i 0.102030 + 0.0114073i
\(502\) 0 0
\(503\) 6.54655e7i 0.514409i −0.966357 0.257205i \(-0.917199\pi\)
0.966357 0.257205i \(-0.0828014\pi\)
\(504\) 0 0
\(505\) 1.93026e8 1.49879
\(506\) 0 0
\(507\) 6.08134e6 5.43932e7i 0.0466633 0.417370i
\(508\) 0 0
\(509\) 8.77803e6i 0.0665647i −0.999446 0.0332823i \(-0.989404\pi\)
0.999446 0.0332823i \(-0.0105961\pi\)
\(510\) 0 0
\(511\) −2.75411e7 −0.206404
\(512\) 0 0
\(513\) 3.73371e7 1.07591e8i 0.276559 0.796940i
\(514\) 0 0
\(515\) 2.57113e8i 1.88236i
\(516\) 0 0
\(517\) 3.76358e7 0.272352
\(518\) 0 0
\(519\) −1.34434e8 1.50302e7i −0.961628 0.107513i
\(520\) 0 0
\(521\) 1.39544e8i 0.986730i 0.869822 + 0.493365i \(0.164233\pi\)
−0.869822 + 0.493365i \(0.835767\pi\)
\(522\) 0 0
\(523\) −1.66441e8 −1.16347 −0.581737 0.813377i \(-0.697627\pi\)
−0.581737 + 0.813377i \(0.697627\pi\)
\(524\) 0 0
\(525\) −7.47417e6 + 6.68510e7i −0.0516518 + 0.461987i
\(526\) 0 0
\(527\) 1.44837e8i 0.989570i
\(528\) 0 0
\(529\) 6.08410e7 0.410988
\(530\) 0 0
\(531\) −2.40797e7 + 1.06342e8i −0.160830 + 0.710263i
\(532\) 0 0
\(533\) 9.27278e6i 0.0612390i
\(534\) 0 0
\(535\) 3.44995e7 0.225295
\(536\) 0 0
\(537\) −2.57779e8 2.88205e7i −1.66465 0.186114i
\(538\) 0 0
\(539\) 1.04638e8i 0.668223i
\(540\) 0 0
\(541\) 8.26283e7 0.521840 0.260920 0.965360i \(-0.415974\pi\)
0.260920 + 0.965360i \(0.415974\pi\)
\(542\) 0 0
\(543\) −1.15677e7 + 1.03465e8i −0.0722519 + 0.646240i
\(544\) 0 0
\(545\) 3.57108e8i 2.20602i
\(546\) 0 0
\(547\) 4.42353e7 0.270276 0.135138 0.990827i \(-0.456852\pi\)
0.135138 + 0.990827i \(0.456852\pi\)
\(548\) 0 0
\(549\) 1.76285e7 + 3.99176e6i 0.106537 + 0.0241239i
\(550\) 0 0
\(551\) 7.17277e7i 0.428777i
\(552\) 0 0
\(553\) −3.86576e7 −0.228591
\(554\) 0 0
\(555\) −2.02064e8 2.25914e7i −1.18198 0.132149i
\(556\) 0 0
\(557\) 2.33744e8i 1.35262i 0.736617 + 0.676310i \(0.236422\pi\)
−0.736617 + 0.676310i \(0.763578\pi\)
\(558\) 0 0
\(559\) 1.79642e8 1.02842
\(560\) 0 0
\(561\) −3.76358e7 + 3.36625e8i −0.213164 + 1.90659i
\(562\) 0 0
\(563\) 8.34284e7i 0.467508i −0.972296 0.233754i \(-0.924899\pi\)
0.972296 0.233754i \(-0.0751010\pi\)
\(564\) 0 0
\(565\) −3.23793e8 −1.79524
\(566\) 0 0
\(567\) 1.16063e8 + 5.54029e7i 0.636717 + 0.303937i
\(568\) 0 0
\(569\) 5.02777e7i 0.272922i 0.990645 + 0.136461i \(0.0435729\pi\)
−0.990645 + 0.136461i \(0.956427\pi\)
\(570\) 0 0
\(571\) 3.44757e8 1.85184 0.925922 0.377716i \(-0.123290\pi\)
0.925922 + 0.377716i \(0.123290\pi\)
\(572\) 0 0
\(573\) −3.65645e7 4.08803e6i −0.194355 0.0217295i
\(574\) 0 0
\(575\) 9.61329e7i 0.505671i
\(576\) 0 0
\(577\) −1.46692e8 −0.763622 −0.381811 0.924240i \(-0.624699\pi\)
−0.381811 + 0.924240i \(0.624699\pi\)
\(578\) 0 0
\(579\) −1.35947e7 + 1.21595e8i −0.0700383 + 0.626441i
\(580\) 0 0
\(581\) 1.24715e8i 0.635902i
\(582\) 0 0
\(583\) −3.04223e8 −1.53528
\(584\) 0 0
\(585\) 6.78586e7 2.99679e8i 0.338951 1.49689i
\(586\) 0 0
\(587\) 3.70598e8i 1.83227i −0.400875 0.916133i \(-0.631294\pi\)
0.400875 0.916133i \(-0.368706\pi\)
\(588\) 0 0
\(589\) 1.18301e8 0.578950
\(590\) 0 0
\(591\) 1.63326e8 + 1.82604e7i 0.791213 + 0.0884603i
\(592\) 0 0
\(593\) 2.20414e8i 1.05700i −0.848933 0.528501i \(-0.822754\pi\)
0.848933 0.528501i \(-0.177246\pi\)
\(594\) 0 0
\(595\) 2.75996e8 1.31024
\(596\) 0 0
\(597\) −8.75332e6 + 7.82921e7i −0.0411386 + 0.367955i
\(598\) 0 0
\(599\) 2.96272e8i 1.37851i −0.724519 0.689255i \(-0.757938\pi\)
0.724519 0.689255i \(-0.242062\pi\)
\(600\) 0 0
\(601\) 3.78140e8 1.74192 0.870962 0.491350i \(-0.163496\pi\)
0.870962 + 0.491350i \(0.163496\pi\)
\(602\) 0 0
\(603\) −5.99785e7 1.35814e7i −0.273555 0.0619429i
\(604\) 0 0
\(605\) 2.19722e8i 0.992219i
\(606\) 0 0
\(607\) 2.71119e8 1.21226 0.606128 0.795367i \(-0.292722\pi\)
0.606128 + 0.795367i \(0.292722\pi\)
\(608\) 0 0
\(609\) 8.04989e7 + 9.00005e6i 0.356400 + 0.0398468i
\(610\) 0 0
\(611\) 5.56367e7i 0.243914i
\(612\) 0 0
\(613\) 8.95583e7 0.388798 0.194399 0.980922i \(-0.437724\pi\)
0.194399 + 0.980922i \(0.437724\pi\)
\(614\) 0 0
\(615\) 1.71072e6 1.53011e7i 0.00735451 0.0657807i
\(616\) 0 0
\(617\) 8.35648e7i 0.355769i 0.984051 + 0.177884i \(0.0569253\pi\)
−0.984051 + 0.177884i \(0.943075\pi\)
\(618\) 0 0
\(619\) 3.01493e8 1.27118 0.635588 0.772028i \(-0.280758\pi\)
0.635588 + 0.772028i \(0.280758\pi\)
\(620\) 0 0
\(621\) 1.73638e8 + 6.02570e7i 0.725054 + 0.251613i
\(622\) 0 0
\(623\) 3.03976e8i 1.25711i
\(624\) 0 0
\(625\) −2.99013e8 −1.22476
\(626\) 0 0
\(627\) −2.74951e8 3.07404e7i −1.11546 0.124712i
\(628\) 0 0
\(629\) 3.31341e8i 1.33144i
\(630\) 0 0
\(631\) 1.78992e8 0.712435 0.356218 0.934403i \(-0.384066\pi\)
0.356218 + 0.934403i \(0.384066\pi\)
\(632\) 0 0
\(633\) −3.36235e7 + 3.00738e8i −0.132566 + 1.18571i
\(634\) 0 0
\(635\) 1.14203e8i 0.446020i
\(636\) 0 0
\(637\) 1.54685e8 0.598451
\(638\) 0 0
\(639\) −5.22029e7 + 2.30540e8i −0.200075 + 0.883576i
\(640\) 0 0
\(641\) 2.55168e8i 0.968842i 0.874835 + 0.484421i \(0.160970\pi\)
−0.874835 + 0.484421i \(0.839030\pi\)
\(642\) 0 0
\(643\) −4.20260e8 −1.58083 −0.790416 0.612571i \(-0.790135\pi\)
−0.790416 + 0.612571i \(0.790135\pi\)
\(644\) 0 0
\(645\) 2.96430e8 + 3.31419e7i 1.10470 + 0.123509i
\(646\) 0 0
\(647\) 1.20078e8i 0.443355i 0.975120 + 0.221677i \(0.0711532\pi\)
−0.975120 + 0.221677i \(0.928847\pi\)
\(648\) 0 0
\(649\) 2.64876e8 0.968968
\(650\) 0 0
\(651\) −1.48438e7 + 1.32767e8i −0.0538024 + 0.481224i
\(652\) 0 0
\(653\) 4.03305e8i 1.44842i −0.689581 0.724209i \(-0.742205\pi\)
0.689581 0.724209i \(-0.257795\pi\)
\(654\) 0 0
\(655\) −1.28019e8 −0.455565
\(656\) 0 0
\(657\) 8.09161e7 + 1.83224e7i 0.285324 + 0.0646080i
\(658\) 0 0
\(659\) 3.33542e8i 1.16545i 0.812669 + 0.582726i \(0.198014\pi\)
−0.812669 + 0.582726i \(0.801986\pi\)
\(660\) 0 0
\(661\) −4.90583e8 −1.69867 −0.849334 0.527856i \(-0.822996\pi\)
−0.849334 + 0.527856i \(0.822996\pi\)
\(662\) 0 0
\(663\) −4.97629e8 5.56367e7i −1.70752 0.190906i
\(664\) 0 0
\(665\) 2.25430e8i 0.766560i
\(666\) 0 0
\(667\) 1.15759e8 0.390100
\(668\) 0 0
\(669\) −4.46195e7 + 3.99089e8i −0.149021 + 1.33288i
\(670\) 0 0
\(671\) 4.39093e7i 0.145341i
\(672\) 0 0
\(673\) 7.80962e7 0.256203 0.128102 0.991761i \(-0.459112\pi\)
0.128102 + 0.991761i \(0.459112\pi\)
\(674\) 0 0
\(675\) 6.64336e7 1.91437e8i 0.216011 0.622464i
\(676\) 0 0
\(677\) 1.31174e8i 0.422747i 0.977405 + 0.211374i \(0.0677937\pi\)
−0.977405 + 0.211374i \(0.932206\pi\)
\(678\) 0 0
\(679\) 2.17684e8 0.695373
\(680\) 0 0
\(681\) −7.34184e7 8.20843e6i −0.232468 0.0259907i
\(682\) 0 0
\(683\) 1.37604e6i 0.00431886i 0.999998 + 0.00215943i \(0.000687368\pi\)
−0.999998 + 0.00215943i \(0.999313\pi\)
\(684\) 0 0
\(685\) 2.36339e8 0.735297
\(686\) 0 0
\(687\) 2.68840e7 2.40458e8i 0.0829132 0.741598i
\(688\) 0 0
\(689\) 4.49730e8i 1.37497i
\(690\) 0 0
\(691\) 5.59620e6 0.0169613 0.00848065 0.999964i \(-0.497300\pi\)
0.00848065 + 0.999964i \(0.497300\pi\)
\(692\) 0 0
\(693\) 6.89990e7 3.04716e8i 0.207321 0.915578i
\(694\) 0 0
\(695\) 8.16915e8i 2.43345i
\(696\) 0 0
\(697\) −2.50906e7 −0.0740989
\(698\) 0 0
\(699\) 2.85215e8 + 3.18880e7i 0.835105 + 0.0933675i
\(700\) 0 0
\(701\) 4.43508e8i 1.28750i −0.765235 0.643751i \(-0.777377\pi\)
0.765235 0.643751i \(-0.222623\pi\)
\(702\) 0 0
\(703\) −2.70634e8 −0.778963
\(704\) 0 0
\(705\) −1.02643e7 + 9.18069e7i −0.0292929 + 0.262004i
\(706\) 0 0
\(707\) 2.90144e8i 0.821024i
\(708\) 0 0
\(709\) −2.12264e8 −0.595576 −0.297788 0.954632i \(-0.596249\pi\)
−0.297788 + 0.954632i \(0.596249\pi\)
\(710\) 0 0
\(711\) 1.13577e8 + 2.57180e7i 0.315995 + 0.0715530i
\(712\) 0 0
\(713\) 1.90921e8i 0.526727i
\(714\) 0 0
\(715\) −7.46444e8 −2.04211
\(716\) 0 0
\(717\) −6.41520e8 7.17241e7i −1.74041 0.194584i
\(718\) 0 0
\(719\) 4.22024e8i 1.13540i 0.823234 + 0.567702i \(0.192167\pi\)
−0.823234 + 0.567702i \(0.807833\pi\)
\(720\) 0 0
\(721\) −3.86475e8 −1.03114
\(722\) 0 0
\(723\) −3.00652e7 + 2.68911e8i −0.0795516 + 0.711531i
\(724\) 0 0
\(725\) 1.27625e8i 0.334904i
\(726\) 0 0
\(727\) −4.59541e8 −1.19597 −0.597986 0.801507i \(-0.704032\pi\)
−0.597986 + 0.801507i \(0.704032\pi\)
\(728\) 0 0
\(729\) −3.04138e8 2.39989e8i −0.785034 0.619453i
\(730\) 0 0
\(731\) 4.86081e8i 1.24439i
\(732\) 0 0
\(733\) 7.63121e6 0.0193768 0.00968838 0.999953i \(-0.496916\pi\)
0.00968838 + 0.999953i \(0.496916\pi\)
\(734\) 0 0
\(735\) 2.55247e8 + 2.85375e7i 0.642834 + 0.0718710i
\(736\) 0 0
\(737\) 1.49395e8i 0.373193i
\(738\) 0 0
\(739\) 1.62220e8 0.401949 0.200975 0.979596i \(-0.435589\pi\)
0.200975 + 0.979596i \(0.435589\pi\)
\(740\) 0 0
\(741\) 4.54432e7 4.06457e8i 0.111690 0.998987i
\(742\) 0 0
\(743\) 5.95948e8i 1.45292i 0.687209 + 0.726460i \(0.258836\pi\)
−0.687209 + 0.726460i \(0.741164\pi\)
\(744\) 0 0
\(745\) −5.35740e8 −1.29564
\(746\) 0 0
\(747\) 8.29699e7 3.66415e8i 0.199048 0.879045i
\(748\) 0 0
\(749\) 5.18574e7i 0.123414i
\(750\) 0 0
\(751\) 4.72042e7 0.111445 0.0557225 0.998446i \(-0.482254\pi\)
0.0557225 + 0.998446i \(0.482254\pi\)
\(752\) 0 0
\(753\) 6.92785e8 + 7.74558e7i 1.62261 + 0.181413i
\(754\) 0 0
\(755\) 6.32918e8i 1.47064i
\(756\) 0 0
\(757\) 6.50504e8 1.49956 0.749778 0.661690i \(-0.230160\pi\)
0.749778 + 0.661690i \(0.230160\pi\)
\(758\) 0 0
\(759\) 4.96109e7 4.43733e8i 0.113462 1.01484i
\(760\) 0 0
\(761\) 6.49863e8i 1.47458i −0.675577 0.737289i \(-0.736106\pi\)
0.675577 0.737289i \(-0.263894\pi\)
\(762\) 0 0
\(763\) −5.36782e8 −1.20844
\(764\) 0 0
\(765\) −8.10881e8 1.83614e8i −1.81123 0.410129i
\(766\) 0 0
\(767\) 3.91564e8i 0.867794i
\(768\) 0 0
\(769\) 2.41673e8 0.531433 0.265717 0.964051i \(-0.414391\pi\)
0.265717 + 0.964051i \(0.414391\pi\)
\(770\) 0 0
\(771\) 3.74077e8 + 4.18231e7i 0.816203 + 0.0912543i
\(772\) 0 0
\(773\) 2.63004e8i 0.569409i 0.958615 + 0.284705i \(0.0918955\pi\)
−0.958615 + 0.284705i \(0.908104\pi\)
\(774\) 0 0
\(775\) 2.10492e8 0.452199
\(776\) 0 0
\(777\) 3.39579e7 3.03729e8i 0.0723899 0.647475i
\(778\) 0 0
\(779\) 2.04936e7i 0.0433517i
\(780\) 0 0
\(781\) 5.74232e8 1.20541
\(782\) 0 0
\(783\) −2.30520e8 7.99963e7i −0.480200 0.166642i
\(784\) 0 0
\(785\) 5.81501e8i 1.20210i
\(786\) 0 0
\(787\) 3.26577e8 0.669979 0.334990 0.942222i \(-0.391267\pi\)
0.334990 + 0.942222i \(0.391267\pi\)
\(788\) 0 0
\(789\) 7.72010e8 + 8.63133e7i 1.57178 + 0.175730i
\(790\) 0 0
\(791\) 4.86704e8i 0.983412i
\(792\) 0 0
\(793\) 6.49107e7 0.130166
\(794\) 0 0
\(795\) 8.29699e7 7.42106e8i 0.165127 1.47694i
\(796\) 0 0
\(797\) 9.53699e7i 0.188381i −0.995554 0.0941903i \(-0.969974\pi\)
0.995554 0.0941903i \(-0.0300262\pi\)
\(798\) 0 0
\(799\) 1.50543e8 0.295135
\(800\) 0 0
\(801\) 2.02228e8 8.93086e8i 0.393499 1.73778i
\(802\) 0 0
\(803\) 2.01547e8i 0.389250i
\(804\) 0 0
\(805\) −3.63813e8 −0.697414
\(806\) 0 0
\(807\) 5.47193e8 + 6.11780e7i 1.04117 + 0.116406i
\(808\) 0 0
\(809\) 2.82791e8i 0.534098i 0.963683 + 0.267049i \(0.0860485\pi\)
−0.963683 + 0.267049i \(0.913952\pi\)
\(810\) 0 0
\(811\) 2.15001e8 0.403067 0.201533 0.979482i \(-0.435408\pi\)
0.201533 + 0.979482i \(0.435408\pi\)
\(812\) 0 0
\(813\) −2.87251e7 + 2.56925e8i −0.0534552 + 0.478118i
\(814\) 0 0
\(815\) 6.05976e8i 1.11939i
\(816\) 0 0
\(817\) 3.97024e8 0.728032
\(818\) 0 0
\(819\) 4.50458e8 + 1.02001e8i 0.819979 + 0.185674i
\(820\) 0 0
\(821\) 7.97415e8i 1.44097i −0.693471 0.720485i \(-0.743919\pi\)
0.693471 0.720485i \(-0.256081\pi\)
\(822\) 0 0
\(823\) 4.95961e8 0.889710 0.444855 0.895603i \(-0.353255\pi\)
0.444855 + 0.895603i \(0.353255\pi\)
\(824\) 0 0
\(825\) −4.89218e8 5.46963e7i −0.871246 0.0974083i
\(826\) 0 0
\(827\) 2.30031e7i 0.0406696i 0.999793 + 0.0203348i \(0.00647321\pi\)
−0.999793 + 0.0203348i \(0.993527\pi\)
\(828\) 0 0
\(829\) −5.03474e8 −0.883718 −0.441859 0.897085i \(-0.645681\pi\)
−0.441859 + 0.897085i \(0.645681\pi\)
\(830\) 0 0
\(831\) −9.86008e7 + 8.81912e8i −0.171821 + 1.53682i
\(832\) 0 0
\(833\) 4.18550e8i 0.724123i
\(834\) 0 0
\(835\) 7.69824e7 0.132231
\(836\) 0 0
\(837\) 1.31938e8 3.80196e8i 0.225006 0.648383i
\(838\) 0 0
\(839\) 3.06275e8i 0.518592i −0.965798 0.259296i \(-0.916510\pi\)
0.965798 0.259296i \(-0.0834905\pi\)
\(840\) 0 0
\(841\) 4.41144e8 0.741638
\(842\) 0 0
\(843\) 6.37053e8 + 7.12247e7i 1.06339 + 0.118891i
\(844\) 0 0
\(845\) 3.26359e8i 0.540911i
\(846\) 0 0
\(847\) −3.30272e8 −0.543527
\(848\) 0 0
\(849\) −1.17913e8 + 1.05464e9i −0.192681 + 1.72339i
\(850\) 0 0
\(851\) 4.36767e8i 0.708698i
\(852\) 0 0
\(853\) 2.74562e6 0.00442378 0.00221189 0.999998i \(-0.499296\pi\)
0.00221189 + 0.999998i \(0.499296\pi\)
\(854\) 0 0
\(855\) 1.49973e8 6.62316e8i 0.239947 1.05966i
\(856\) 0 0
\(857\) 3.08435e8i 0.490028i −0.969520 0.245014i \(-0.921207\pi\)
0.969520 0.245014i \(-0.0787926\pi\)
\(858\) 0 0
\(859\) −1.04043e9 −1.64147 −0.820736 0.571307i \(-0.806437\pi\)
−0.820736 + 0.571307i \(0.806437\pi\)
\(860\) 0 0
\(861\) 2.29997e7 + 2.57144e6i 0.0360340 + 0.00402872i
\(862\) 0 0
\(863\) 7.72127e8i 1.20131i −0.799507 0.600657i \(-0.794906\pi\)
0.799507 0.600657i \(-0.205094\pi\)
\(864\) 0 0
\(865\) −8.06604e8 −1.24627
\(866\) 0 0
\(867\) −7.81307e7 + 6.98822e8i −0.119885 + 1.07228i
\(868\) 0 0
\(869\) 2.82898e8i 0.431092i
\(870\) 0 0
\(871\) −2.20849e8 −0.334227
\(872\) 0 0
\(873\) −6.39560e8 1.44820e8i −0.961255 0.217664i
\(874\) 0 0
\(875\) 2.07663e8i 0.309982i
\(876\) 0 0
\(877\) 3.17817e8 0.471171 0.235586 0.971854i \(-0.424299\pi\)
0.235586 + 0.971854i \(0.424299\pi\)
\(878\) 0 0
\(879\) 6.22940e8 + 6.96468e7i 0.917232 + 0.102550i
\(880\) 0 0
\(881\) 4.59366e8i 0.671786i −0.941900 0.335893i \(-0.890962\pi\)
0.941900 0.335893i \(-0.109038\pi\)
\(882\) 0 0
\(883\) −7.77967e8 −1.13000 −0.565001 0.825090i \(-0.691124\pi\)
−0.565001 + 0.825090i \(0.691124\pi\)
\(884\) 0 0
\(885\) −7.22390e7 + 6.46126e8i −0.104218 + 0.932153i
\(886\) 0 0
\(887\) 6.06049e8i 0.868435i −0.900808 0.434217i \(-0.857025\pi\)
0.900808 0.434217i \(-0.142975\pi\)
\(888\) 0 0
\(889\) 1.71662e8 0.244325
\(890\) 0 0
\(891\) −4.05441e8 + 8.49357e8i −0.573184 + 1.20076i
\(892\) 0 0
\(893\) 1.22962e8i 0.172669i
\(894\) 0 0
\(895\) −1.54667e9 −2.15739
\(896\) 0 0
\(897\) 6.55966e8 + 7.33392e7i 0.908875 + 0.101615i
\(898\) 0 0
\(899\) 2.53464e8i 0.348849i
\(900\) 0 0
\(901\) −1.21689e9 −1.66371
\(902\) 0 0
\(903\) −4.98167e7 + 4.45574e8i −0.0676568 + 0.605141i
\(904\) 0 0
\(905\) 6.20790e8i 0.837528i
\(906\) 0 0
\(907\) 7.52746e8 1.00885 0.504425 0.863455i \(-0.331704\pi\)
0.504425 + 0.863455i \(0.331704\pi\)
\(908\) 0 0
\(909\) 1.93026e8 8.52449e8i 0.256995 1.13495i
\(910\) 0 0
\(911\) 8.84130e8i 1.16939i −0.811252 0.584697i \(-0.801213\pi\)
0.811252 0.584697i \(-0.198787\pi\)
\(912\) 0 0
\(913\) −9.12669e8 −1.19923
\(914\) 0 0
\(915\) 1.07110e8 + 1.19753e7i 0.139819 + 0.0156323i
\(916\) 0 0
\(917\) 1.92430e8i 0.249554i
\(918\) 0 0
\(919\) −1.08334e9 −1.39578 −0.697892 0.716203i \(-0.745879\pi\)
−0.697892 + 0.716203i \(0.745879\pi\)
\(920\) 0 0
\(921\) −2.71544e7 + 2.42877e8i −0.0347586 + 0.310890i
\(922\) 0 0
\(923\) 8.48881e8i 1.07955i
\(924\) 0 0
\(925\) −4.81538e8 −0.608423
\(926\) 0 0
\(927\) 1.13547e9 + 2.57113e8i 1.42540 + 0.322764i
\(928\) 0 0
\(929\) 1.43691e9i 1.79218i 0.443875 + 0.896089i \(0.353603\pi\)
−0.443875 + 0.896089i \(0.646397\pi\)
\(930\) 0 0
\(931\) 3.41866e8 0.423649
\(932\) 0 0
\(933\) 5.82595e7 + 6.51361e6i 0.0717335 + 0.00802005i
\(934\) 0 0
\(935\) 2.01975e9i 2.47094i
\(936\) 0 0
\(937\) 1.12714e9 1.37013 0.685063 0.728484i \(-0.259775\pi\)
0.685063 + 0.728484i \(0.259775\pi\)
\(938\) 0 0
\(939\) 3.25537e7 2.91169e8i 0.0393190 0.351680i
\(940\) 0 0
\(941\) 1.23121e9i 1.47762i 0.673913 + 0.738810i \(0.264612\pi\)
−0.673913 + 0.738810i \(0.735388\pi\)
\(942\) 0 0
\(943\) 3.30739e7 0.0394412
\(944\) 0 0
\(945\) 7.24490e8 + 2.51417e8i 0.858493 + 0.297920i
\(946\) 0 0
\(947\) 6.93816e6i 0.00816948i −0.999992 0.00408474i \(-0.998700\pi\)
0.999992 0.00408474i \(-0.00130022\pi\)
\(948\) 0 0
\(949\) 2.97944e8 0.348607
\(950\) 0 0
\(951\) 8.56349e8 + 9.57428e7i 0.995656 + 0.111318i
\(952\) 0 0
\(953\) 4.99944e7i 0.0577620i 0.999583 + 0.0288810i \(0.00919439\pi\)
−0.999583 + 0.0288810i \(0.990806\pi\)
\(954\) 0 0
\(955\) −2.19387e8 −0.251884
\(956\) 0 0
\(957\) −6.58627e7 + 5.89094e8i −0.0751457 + 0.672123i
\(958\) 0 0
\(959\) 3.55249e8i 0.402788i
\(960\) 0 0
\(961\) −4.69465e8 −0.528972
\(962\) 0 0
\(963\) 3.44995e7 1.52358e8i 0.0386308 0.170603i
\(964\) 0 0
\(965\) 7.29571e8i 0.811868i
\(966\) 0 0
\(967\) −1.77551e9 −1.96356 −0.981780 0.190022i \(-0.939144\pi\)
−0.981780 + 0.190022i \(0.939144\pi\)
\(968\) 0 0
\(969\) −1.09980e9 1.22962e8i −1.20877 0.135145i
\(970\) 0 0
\(971\) 4.37887e8i 0.478304i −0.970982 0.239152i \(-0.923131\pi\)
0.970982 0.239152i \(-0.0768694\pi\)
\(972\) 0 0
\(973\) 1.22793e9 1.33302
\(974\) 0 0
\(975\) 8.08569e7 7.23206e8i 0.0872375 0.780276i
\(976\) 0 0
\(977\) 1.00202e9i 1.07447i 0.843433 + 0.537235i \(0.180531\pi\)
−0.843433 + 0.537235i \(0.819469\pi\)
\(978\) 0 0
\(979\) −2.22451e9 −2.37075
\(980\) 0 0
\(981\) 1.57707e9 + 3.57108e8i 1.67049 + 0.378262i
\(982\) 0 0
\(983\) 1.13260e9i 1.19238i −0.802844 0.596189i \(-0.796681\pi\)
0.802844 0.596189i \(-0.203319\pi\)
\(984\) 0 0
\(985\) 9.79957e8 1.02541
\(986\) 0 0
\(987\) −1.37998e8 1.54287e7i −0.143523 0.0160464i
\(988\) 0 0
\(989\) 6.40743e8i 0.662361i
\(990\) 0 0
\(991\) 1.34604e9 1.38305 0.691525 0.722352i \(-0.256939\pi\)
0.691525 + 0.722352i \(0.256939\pi\)
\(992\) 0 0
\(993\) 1.70580e8 1.52571e9i 0.174213 1.55821i
\(994\) 0 0
\(995\) 4.69753e8i 0.476870i
\(996\) 0 0
\(997\) −2.62968e8 −0.265349 −0.132675 0.991160i \(-0.542357\pi\)
−0.132675 + 0.991160i \(0.542357\pi\)
\(998\) 0 0
\(999\) −3.01833e8 + 8.69769e8i −0.302740 + 0.872384i
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 192.7.e.e.65.1 2
3.2 odd 2 inner 192.7.e.e.65.2 2
4.3 odd 2 192.7.e.d.65.2 2
8.3 odd 2 48.7.e.c.17.1 2
8.5 even 2 12.7.c.a.5.2 yes 2
12.11 even 2 192.7.e.d.65.1 2
24.5 odd 2 12.7.c.a.5.1 2
24.11 even 2 48.7.e.c.17.2 2
40.13 odd 4 300.7.b.c.149.1 4
40.29 even 2 300.7.g.e.101.1 2
40.37 odd 4 300.7.b.c.149.4 4
56.13 odd 2 588.7.c.e.197.1 2
72.5 odd 6 324.7.g.c.269.1 4
72.13 even 6 324.7.g.c.269.2 4
72.29 odd 6 324.7.g.c.53.2 4
72.61 even 6 324.7.g.c.53.1 4
120.29 odd 2 300.7.g.e.101.2 2
120.53 even 4 300.7.b.c.149.3 4
120.77 even 4 300.7.b.c.149.2 4
168.125 even 2 588.7.c.e.197.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.7.c.a.5.1 2 24.5 odd 2
12.7.c.a.5.2 yes 2 8.5 even 2
48.7.e.c.17.1 2 8.3 odd 2
48.7.e.c.17.2 2 24.11 even 2
192.7.e.d.65.1 2 12.11 even 2
192.7.e.d.65.2 2 4.3 odd 2
192.7.e.e.65.1 2 1.1 even 1 trivial
192.7.e.e.65.2 2 3.2 odd 2 inner
300.7.b.c.149.1 4 40.13 odd 4
300.7.b.c.149.2 4 120.77 even 4
300.7.b.c.149.3 4 120.53 even 4
300.7.b.c.149.4 4 40.37 odd 4
300.7.g.e.101.1 2 40.29 even 2
300.7.g.e.101.2 2 120.29 odd 2
324.7.g.c.53.1 4 72.61 even 6
324.7.g.c.53.2 4 72.29 odd 6
324.7.g.c.269.1 4 72.5 odd 6
324.7.g.c.269.2 4 72.13 even 6
588.7.c.e.197.1 2 56.13 odd 2
588.7.c.e.197.2 2 168.125 even 2