Properties

Label 1050.3.c.c.449.16
Level $1050$
Weight $3$
Character 1050.449
Analytic conductor $28.610$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,3,Mod(449,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.c (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: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.16
Character \(\chi\) \(=\) 1050.449
Dual form 1050.3.c.c.449.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +(2.59182 + 1.51079i) q^{3} +2.00000 q^{4} +(-3.66538 - 2.13658i) q^{6} -2.64575i q^{7} -2.82843 q^{8} +(4.43502 + 7.83139i) q^{9} +O(q^{10})\) \(q-1.41421 q^{2} +(2.59182 + 1.51079i) q^{3} +2.00000 q^{4} +(-3.66538 - 2.13658i) q^{6} -2.64575i q^{7} -2.82843 q^{8} +(4.43502 + 7.83139i) q^{9} +14.6732i q^{11} +(5.18363 + 3.02158i) q^{12} +19.1112i q^{13} +3.74166i q^{14} +4.00000 q^{16} +29.7911 q^{17} +(-6.27207 - 11.0753i) q^{18} -14.7237 q^{19} +(3.99718 - 6.85730i) q^{21} -20.7510i q^{22} -22.7912 q^{23} +(-7.33076 - 4.27316i) q^{24} -27.0273i q^{26} +(-0.336829 + 26.9979i) q^{27} -5.29150i q^{28} -51.5923i q^{29} -38.6980 q^{31} -5.65685 q^{32} +(-22.1681 + 38.0302i) q^{33} -42.1310 q^{34} +(8.87004 + 15.6628i) q^{36} -29.2081i q^{37} +20.8224 q^{38} +(-28.8731 + 49.5328i) q^{39} +28.6850i q^{41} +(-5.65286 + 9.69769i) q^{42} +40.4802i q^{43} +29.3464i q^{44} +32.2317 q^{46} -10.5901 q^{47} +(10.3673 + 6.04316i) q^{48} -7.00000 q^{49} +(77.2131 + 45.0082i) q^{51} +38.2224i q^{52} +95.5975 q^{53} +(0.476348 - 38.1808i) q^{54} +7.48331i q^{56} +(-38.1611 - 22.2444i) q^{57} +72.9625i q^{58} +11.1354i q^{59} -104.131 q^{61} +54.7273 q^{62} +(20.7199 - 11.7340i) q^{63} +8.00000 q^{64} +(31.3504 - 53.7828i) q^{66} +45.2575i q^{67} +59.5823 q^{68} +(-59.0707 - 34.4328i) q^{69} +93.7964i q^{71} +(-12.5441 - 22.1505i) q^{72} +62.8141i q^{73} +41.3066i q^{74} -29.4474 q^{76} +38.8216 q^{77} +(40.8327 - 70.0499i) q^{78} -122.486 q^{79} +(-41.6612 + 69.4647i) q^{81} -40.5668i q^{82} +68.2772 q^{83} +(7.99435 - 13.7146i) q^{84} -57.2477i q^{86} +(77.9452 - 133.718i) q^{87} -41.5020i q^{88} +79.4213i q^{89} +50.5635 q^{91} -45.5825 q^{92} +(-100.298 - 58.4647i) q^{93} +14.9766 q^{94} +(-14.6615 - 8.54632i) q^{96} -28.2049i q^{97} +9.89949 q^{98} +(-114.911 + 65.0759i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 64 q^{4} + 32 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 64 q^{4} + 32 q^{6} + 8 q^{9} + 128 q^{16} - 96 q^{19} + 56 q^{21} + 64 q^{24} - 320 q^{34} + 16 q^{36} - 312 q^{39} + 64 q^{46} - 224 q^{49} + 168 q^{51} + 64 q^{54} + 224 q^{61} + 256 q^{64} - 16 q^{69} - 192 q^{76} - 16 q^{79} - 248 q^{81} + 112 q^{84} - 112 q^{91} - 64 q^{94} + 128 q^{96} + 104 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) 2.59182 + 1.51079i 0.863939 + 0.503597i
\(4\) 2.00000 0.500000
\(5\) 0 0
\(6\) −3.66538 2.13658i −0.610897 0.356097i
\(7\) 2.64575i 0.377964i
\(8\) −2.82843 −0.353553
\(9\) 4.43502 + 7.83139i 0.492780 + 0.870154i
\(10\) 0 0
\(11\) 14.6732i 1.33393i 0.745091 + 0.666963i \(0.232406\pi\)
−0.745091 + 0.666963i \(0.767594\pi\)
\(12\) 5.18363 + 3.02158i 0.431969 + 0.251799i
\(13\) 19.1112i 1.47009i 0.678016 + 0.735047i \(0.262840\pi\)
−0.678016 + 0.735047i \(0.737160\pi\)
\(14\) 3.74166i 0.267261i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 29.7911 1.75242 0.876210 0.481930i \(-0.160064\pi\)
0.876210 + 0.481930i \(0.160064\pi\)
\(18\) −6.27207 11.0753i −0.348448 0.615292i
\(19\) −14.7237 −0.774930 −0.387465 0.921884i \(-0.626649\pi\)
−0.387465 + 0.921884i \(0.626649\pi\)
\(20\) 0 0
\(21\) 3.99718 6.85730i 0.190342 0.326538i
\(22\) 20.7510i 0.943228i
\(23\) −22.7912 −0.990924 −0.495462 0.868630i \(-0.665001\pi\)
−0.495462 + 0.868630i \(0.665001\pi\)
\(24\) −7.33076 4.27316i −0.305448 0.178048i
\(25\) 0 0
\(26\) 27.0273i 1.03951i
\(27\) −0.336829 + 26.9979i −0.0124751 + 0.999922i
\(28\) 5.29150i 0.188982i
\(29\) 51.5923i 1.77904i −0.456891 0.889522i \(-0.651037\pi\)
0.456891 0.889522i \(-0.348963\pi\)
\(30\) 0 0
\(31\) −38.6980 −1.24832 −0.624162 0.781295i \(-0.714560\pi\)
−0.624162 + 0.781295i \(0.714560\pi\)
\(32\) −5.65685 −0.176777
\(33\) −22.1681 + 38.0302i −0.671761 + 1.15243i
\(34\) −42.1310 −1.23915
\(35\) 0 0
\(36\) 8.87004 + 15.6628i 0.246390 + 0.435077i
\(37\) 29.2081i 0.789409i −0.918808 0.394705i \(-0.870847\pi\)
0.918808 0.394705i \(-0.129153\pi\)
\(38\) 20.8224 0.547959
\(39\) −28.8731 + 49.5328i −0.740335 + 1.27007i
\(40\) 0 0
\(41\) 28.6850i 0.699635i 0.936818 + 0.349817i \(0.113756\pi\)
−0.936818 + 0.349817i \(0.886244\pi\)
\(42\) −5.65286 + 9.69769i −0.134592 + 0.230897i
\(43\) 40.4802i 0.941400i 0.882293 + 0.470700i \(0.155999\pi\)
−0.882293 + 0.470700i \(0.844001\pi\)
\(44\) 29.3464i 0.666963i
\(45\) 0 0
\(46\) 32.2317 0.700689
\(47\) −10.5901 −0.225321 −0.112660 0.993634i \(-0.535937\pi\)
−0.112660 + 0.993634i \(0.535937\pi\)
\(48\) 10.3673 + 6.04316i 0.215985 + 0.125899i
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 77.2131 + 45.0082i 1.51398 + 0.882513i
\(52\) 38.2224i 0.735047i
\(53\) 95.5975 1.80373 0.901863 0.432022i \(-0.142200\pi\)
0.901863 + 0.432022i \(0.142200\pi\)
\(54\) 0.476348 38.1808i 0.00882126 0.707052i
\(55\) 0 0
\(56\) 7.48331i 0.133631i
\(57\) −38.1611 22.2444i −0.669492 0.390253i
\(58\) 72.9625i 1.25797i
\(59\) 11.1354i 0.188735i 0.995537 + 0.0943675i \(0.0300829\pi\)
−0.995537 + 0.0943675i \(0.969917\pi\)
\(60\) 0 0
\(61\) −104.131 −1.70707 −0.853533 0.521038i \(-0.825545\pi\)
−0.853533 + 0.521038i \(0.825545\pi\)
\(62\) 54.7273 0.882698
\(63\) 20.7199 11.7340i 0.328887 0.186253i
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) 31.3504 53.7828i 0.475007 0.814891i
\(67\) 45.2575i 0.675486i 0.941238 + 0.337743i \(0.109663\pi\)
−0.941238 + 0.337743i \(0.890337\pi\)
\(68\) 59.5823 0.876210
\(69\) −59.0707 34.4328i −0.856097 0.499026i
\(70\) 0 0
\(71\) 93.7964i 1.32108i 0.750792 + 0.660538i \(0.229672\pi\)
−0.750792 + 0.660538i \(0.770328\pi\)
\(72\) −12.5441 22.1505i −0.174224 0.307646i
\(73\) 62.8141i 0.860468i 0.902717 + 0.430234i \(0.141569\pi\)
−0.902717 + 0.430234i \(0.858431\pi\)
\(74\) 41.3066i 0.558197i
\(75\) 0 0
\(76\) −29.4474 −0.387465
\(77\) 38.8216 0.504176
\(78\) 40.8327 70.0499i 0.523496 0.898076i
\(79\) −122.486 −1.55046 −0.775231 0.631678i \(-0.782366\pi\)
−0.775231 + 0.631678i \(0.782366\pi\)
\(80\) 0 0
\(81\) −41.6612 + 69.4647i −0.514336 + 0.857589i
\(82\) 40.5668i 0.494717i
\(83\) 68.2772 0.822617 0.411309 0.911496i \(-0.365072\pi\)
0.411309 + 0.911496i \(0.365072\pi\)
\(84\) 7.99435 13.7146i 0.0951709 0.163269i
\(85\) 0 0
\(86\) 57.2477i 0.665671i
\(87\) 77.9452 133.718i 0.895922 1.53699i
\(88\) 41.5020i 0.471614i
\(89\) 79.4213i 0.892374i 0.894940 + 0.446187i \(0.147218\pi\)
−0.894940 + 0.446187i \(0.852782\pi\)
\(90\) 0 0
\(91\) 50.5635 0.555643
\(92\) −45.5825 −0.495462
\(93\) −100.298 58.4647i −1.07848 0.628652i
\(94\) 14.9766 0.159326
\(95\) 0 0
\(96\) −14.6615 8.54632i −0.152724 0.0890242i
\(97\) 28.2049i 0.290773i −0.989375 0.145386i \(-0.953558\pi\)
0.989375 0.145386i \(-0.0464425\pi\)
\(98\) 9.89949 0.101015
\(99\) −114.911 + 65.0759i −1.16072 + 0.657332i
\(100\) 0 0
\(101\) 10.6769i 0.105712i 0.998602 + 0.0528561i \(0.0168324\pi\)
−0.998602 + 0.0528561i \(0.983168\pi\)
\(102\) −109.196 63.6512i −1.07055 0.624031i
\(103\) 34.1042i 0.331109i −0.986201 0.165554i \(-0.947059\pi\)
0.986201 0.165554i \(-0.0529414\pi\)
\(104\) 54.0547i 0.519757i
\(105\) 0 0
\(106\) −135.195 −1.27543
\(107\) −15.4338 −0.144241 −0.0721205 0.997396i \(-0.522977\pi\)
−0.0721205 + 0.997396i \(0.522977\pi\)
\(108\) −0.673658 + 53.9958i −0.00623757 + 0.499961i
\(109\) 31.7494 0.291278 0.145639 0.989338i \(-0.453476\pi\)
0.145639 + 0.989338i \(0.453476\pi\)
\(110\) 0 0
\(111\) 44.1274 75.7021i 0.397544 0.682001i
\(112\) 10.5830i 0.0944911i
\(113\) 26.1742 0.231630 0.115815 0.993271i \(-0.463052\pi\)
0.115815 + 0.993271i \(0.463052\pi\)
\(114\) 53.9679 + 31.4583i 0.473403 + 0.275950i
\(115\) 0 0
\(116\) 103.185i 0.889522i
\(117\) −149.667 + 84.7587i −1.27921 + 0.724433i
\(118\) 15.7478i 0.133456i
\(119\) 78.8199i 0.662352i
\(120\) 0 0
\(121\) −94.3023 −0.779358
\(122\) 147.264 1.20708
\(123\) −43.3371 + 74.3463i −0.352334 + 0.604442i
\(124\) −77.3961 −0.624162
\(125\) 0 0
\(126\) −29.3024 + 16.5943i −0.232558 + 0.131701i
\(127\) 155.334i 1.22311i 0.791204 + 0.611553i \(0.209455\pi\)
−0.791204 + 0.611553i \(0.790545\pi\)
\(128\) −11.3137 −0.0883883
\(129\) −61.1572 + 104.917i −0.474086 + 0.813312i
\(130\) 0 0
\(131\) 100.859i 0.769912i −0.922935 0.384956i \(-0.874217\pi\)
0.922935 0.384956i \(-0.125783\pi\)
\(132\) −44.3362 + 76.0604i −0.335880 + 0.576215i
\(133\) 38.9552i 0.292896i
\(134\) 64.0038i 0.477640i
\(135\) 0 0
\(136\) −84.2620 −0.619574
\(137\) 117.672 0.858920 0.429460 0.903086i \(-0.358704\pi\)
0.429460 + 0.903086i \(0.358704\pi\)
\(138\) 83.5386 + 48.6953i 0.605352 + 0.352865i
\(139\) 74.8843 0.538736 0.269368 0.963037i \(-0.413185\pi\)
0.269368 + 0.963037i \(0.413185\pi\)
\(140\) 0 0
\(141\) −27.4475 15.9994i −0.194663 0.113471i
\(142\) 132.648i 0.934142i
\(143\) −280.422 −1.96100
\(144\) 17.7401 + 31.3255i 0.123195 + 0.217538i
\(145\) 0 0
\(146\) 88.8326i 0.608443i
\(147\) −18.1427 10.5755i −0.123420 0.0719424i
\(148\) 58.4163i 0.394705i
\(149\) 149.584i 1.00392i 0.864891 + 0.501960i \(0.167388\pi\)
−0.864891 + 0.501960i \(0.832612\pi\)
\(150\) 0 0
\(151\) 202.607 1.34177 0.670884 0.741562i \(-0.265915\pi\)
0.670884 + 0.741562i \(0.265915\pi\)
\(152\) 41.6449 0.273979
\(153\) 132.124 + 233.306i 0.863557 + 1.52487i
\(154\) −54.9020 −0.356507
\(155\) 0 0
\(156\) −57.7461 + 99.0655i −0.370167 + 0.635036i
\(157\) 79.2891i 0.505026i −0.967594 0.252513i \(-0.918743\pi\)
0.967594 0.252513i \(-0.0812570\pi\)
\(158\) 173.222 1.09634
\(159\) 247.771 + 144.428i 1.55831 + 0.908351i
\(160\) 0 0
\(161\) 60.3000i 0.374534i
\(162\) 58.9178 98.2379i 0.363690 0.606407i
\(163\) 72.9838i 0.447753i 0.974617 + 0.223877i \(0.0718713\pi\)
−0.974617 + 0.223877i \(0.928129\pi\)
\(164\) 57.3701i 0.349817i
\(165\) 0 0
\(166\) −96.5586 −0.581678
\(167\) −113.145 −0.677512 −0.338756 0.940874i \(-0.610006\pi\)
−0.338756 + 0.940874i \(0.610006\pi\)
\(168\) −11.3057 + 19.3954i −0.0672960 + 0.115449i
\(169\) −196.239 −1.16118
\(170\) 0 0
\(171\) −65.2998 115.307i −0.381870 0.674309i
\(172\) 80.9604i 0.470700i
\(173\) 115.843 0.669611 0.334806 0.942287i \(-0.391329\pi\)
0.334806 + 0.942287i \(0.391329\pi\)
\(174\) −110.231 + 189.105i −0.633512 + 1.08681i
\(175\) 0 0
\(176\) 58.6927i 0.333481i
\(177\) −16.8232 + 28.8608i −0.0950463 + 0.163055i
\(178\) 112.319i 0.631004i
\(179\) 76.5353i 0.427572i −0.976881 0.213786i \(-0.931421\pi\)
0.976881 0.213786i \(-0.0685794\pi\)
\(180\) 0 0
\(181\) 38.3458 0.211855 0.105928 0.994374i \(-0.466219\pi\)
0.105928 + 0.994374i \(0.466219\pi\)
\(182\) −71.5076 −0.392899
\(183\) −269.889 157.320i −1.47480 0.859674i
\(184\) 64.4634 0.350344
\(185\) 0 0
\(186\) 141.843 + 82.6815i 0.762597 + 0.444524i
\(187\) 437.131i 2.33760i
\(188\) −21.1802 −0.112660
\(189\) 71.4297 + 0.891165i 0.377935 + 0.00471516i
\(190\) 0 0
\(191\) 0.0103646i 5.42648e-5i −1.00000 2.71324e-5i \(-0.999991\pi\)
1.00000 2.71324e-5i \(-8.63652e-6\pi\)
\(192\) 20.7345 + 12.0863i 0.107992 + 0.0629496i
\(193\) 14.2760i 0.0739689i −0.999316 0.0369844i \(-0.988225\pi\)
0.999316 0.0369844i \(-0.0117752\pi\)
\(194\) 39.8878i 0.205607i
\(195\) 0 0
\(196\) −14.0000 −0.0714286
\(197\) −123.543 −0.627120 −0.313560 0.949568i \(-0.601522\pi\)
−0.313560 + 0.949568i \(0.601522\pi\)
\(198\) 162.509 92.0312i 0.820753 0.464804i
\(199\) 187.525 0.942335 0.471167 0.882044i \(-0.343833\pi\)
0.471167 + 0.882044i \(0.343833\pi\)
\(200\) 0 0
\(201\) −68.3747 + 117.299i −0.340173 + 0.583578i
\(202\) 15.0995i 0.0747498i
\(203\) −136.500 −0.672416
\(204\) 154.426 + 90.0164i 0.756992 + 0.441257i
\(205\) 0 0
\(206\) 48.2307i 0.234129i
\(207\) −101.080 178.487i −0.488307 0.862256i
\(208\) 76.4449i 0.367523i
\(209\) 216.043i 1.03370i
\(210\) 0 0
\(211\) −280.253 −1.32821 −0.664106 0.747638i \(-0.731188\pi\)
−0.664106 + 0.747638i \(0.731188\pi\)
\(212\) 191.195 0.901863
\(213\) −141.707 + 243.103i −0.665290 + 1.14133i
\(214\) 21.8267 0.101994
\(215\) 0 0
\(216\) 0.952696 76.3616i 0.00441063 0.353526i
\(217\) 102.385i 0.471822i
\(218\) −44.9004 −0.205965
\(219\) −94.8990 + 162.803i −0.433329 + 0.743391i
\(220\) 0 0
\(221\) 569.345i 2.57622i
\(222\) −62.4056 + 107.059i −0.281106 + 0.482248i
\(223\) 100.764i 0.451858i 0.974144 + 0.225929i \(0.0725418\pi\)
−0.974144 + 0.225929i \(0.927458\pi\)
\(224\) 14.9666i 0.0668153i
\(225\) 0 0
\(226\) −37.0159 −0.163787
\(227\) −154.057 −0.678664 −0.339332 0.940667i \(-0.610201\pi\)
−0.339332 + 0.940667i \(0.610201\pi\)
\(228\) −76.3221 44.4888i −0.334746 0.195126i
\(229\) −395.080 −1.72524 −0.862621 0.505852i \(-0.831178\pi\)
−0.862621 + 0.505852i \(0.831178\pi\)
\(230\) 0 0
\(231\) 100.618 + 58.6513i 0.435578 + 0.253902i
\(232\) 145.925i 0.628987i
\(233\) 192.917 0.827972 0.413986 0.910283i \(-0.364136\pi\)
0.413986 + 0.910283i \(0.364136\pi\)
\(234\) 211.662 119.867i 0.904537 0.512251i
\(235\) 0 0
\(236\) 22.2707i 0.0943675i
\(237\) −317.462 185.051i −1.33950 0.780808i
\(238\) 111.468i 0.468354i
\(239\) 312.840i 1.30896i −0.756081 0.654478i \(-0.772888\pi\)
0.756081 0.654478i \(-0.227112\pi\)
\(240\) 0 0
\(241\) 328.130 1.36153 0.680767 0.732500i \(-0.261647\pi\)
0.680767 + 0.732500i \(0.261647\pi\)
\(242\) 133.364 0.551089
\(243\) −212.925 + 117.098i −0.876234 + 0.481886i
\(244\) −208.262 −0.853533
\(245\) 0 0
\(246\) 61.2879 105.142i 0.249138 0.427405i
\(247\) 281.387i 1.13922i
\(248\) 109.455 0.441349
\(249\) 176.962 + 103.153i 0.710691 + 0.414268i
\(250\) 0 0
\(251\) 146.160i 0.582310i −0.956676 0.291155i \(-0.905960\pi\)
0.956676 0.291155i \(-0.0940396\pi\)
\(252\) 41.4398 23.4679i 0.164444 0.0931267i
\(253\) 334.420i 1.32182i
\(254\) 219.676i 0.864866i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 306.185 1.19138 0.595691 0.803214i \(-0.296878\pi\)
0.595691 + 0.803214i \(0.296878\pi\)
\(258\) 86.4893 148.375i 0.335230 0.575099i
\(259\) −77.2775 −0.298369
\(260\) 0 0
\(261\) 404.039 228.813i 1.54804 0.876678i
\(262\) 142.635i 0.544410i
\(263\) 158.182 0.601452 0.300726 0.953711i \(-0.402771\pi\)
0.300726 + 0.953711i \(0.402771\pi\)
\(264\) 62.7009 107.566i 0.237503 0.407446i
\(265\) 0 0
\(266\) 55.0910i 0.207109i
\(267\) −119.989 + 205.845i −0.449397 + 0.770956i
\(268\) 90.5151i 0.337743i
\(269\) 158.158i 0.587948i −0.955813 0.293974i \(-0.905022\pi\)
0.955813 0.293974i \(-0.0949779\pi\)
\(270\) 0 0
\(271\) 361.490 1.33391 0.666956 0.745097i \(-0.267597\pi\)
0.666956 + 0.745097i \(0.267597\pi\)
\(272\) 119.165 0.438105
\(273\) 131.051 + 76.3909i 0.480042 + 0.279820i
\(274\) −166.413 −0.607348
\(275\) 0 0
\(276\) −118.141 68.8656i −0.428049 0.249513i
\(277\) 77.1087i 0.278371i −0.990266 0.139185i \(-0.955552\pi\)
0.990266 0.139185i \(-0.0444484\pi\)
\(278\) −105.902 −0.380944
\(279\) −171.627 303.059i −0.615149 1.08623i
\(280\) 0 0
\(281\) 96.2437i 0.342504i −0.985227 0.171252i \(-0.945219\pi\)
0.985227 0.171252i \(-0.0547813\pi\)
\(282\) 38.8167 + 22.6266i 0.137648 + 0.0802361i
\(283\) 210.868i 0.745118i −0.928008 0.372559i \(-0.878480\pi\)
0.928008 0.372559i \(-0.121520\pi\)
\(284\) 187.593i 0.660538i
\(285\) 0 0
\(286\) 396.577 1.38663
\(287\) 75.8935 0.264437
\(288\) −25.0883 44.3010i −0.0871120 0.153823i
\(289\) 598.512 2.07097
\(290\) 0 0
\(291\) 42.6118 73.1020i 0.146432 0.251210i
\(292\) 125.628i 0.430234i
\(293\) −441.017 −1.50518 −0.752588 0.658491i \(-0.771195\pi\)
−0.752588 + 0.658491i \(0.771195\pi\)
\(294\) 25.6577 + 14.9561i 0.0872710 + 0.0508710i
\(295\) 0 0
\(296\) 82.6131i 0.279098i
\(297\) −396.145 4.94235i −1.33382 0.0166409i
\(298\) 211.544i 0.709879i
\(299\) 435.569i 1.45675i
\(300\) 0 0
\(301\) 107.101 0.355816
\(302\) −286.529 −0.948773
\(303\) −16.1306 + 27.6726i −0.0532363 + 0.0913288i
\(304\) −58.8947 −0.193733
\(305\) 0 0
\(306\) −186.852 329.944i −0.610627 1.07825i
\(307\) 253.872i 0.826944i 0.910517 + 0.413472i \(0.135684\pi\)
−0.910517 + 0.413472i \(0.864316\pi\)
\(308\) 77.6432 0.252088
\(309\) 51.5244 88.3919i 0.166745 0.286058i
\(310\) 0 0
\(311\) 402.715i 1.29490i 0.762106 + 0.647452i \(0.224165\pi\)
−0.762106 + 0.647452i \(0.775835\pi\)
\(312\) 81.6654 140.100i 0.261748 0.449038i
\(313\) 58.4446i 0.186724i −0.995632 0.0933620i \(-0.970239\pi\)
0.995632 0.0933620i \(-0.0297614\pi\)
\(314\) 112.132i 0.357107i
\(315\) 0 0
\(316\) −244.973 −0.775231
\(317\) 181.889 0.573783 0.286891 0.957963i \(-0.407378\pi\)
0.286891 + 0.957963i \(0.407378\pi\)
\(318\) −350.401 204.252i −1.10189 0.642301i
\(319\) 757.023 2.37311
\(320\) 0 0
\(321\) −40.0015 23.3172i −0.124615 0.0726393i
\(322\) 85.2770i 0.264835i
\(323\) −438.635 −1.35800
\(324\) −83.3224 + 138.929i −0.257168 + 0.428795i
\(325\) 0 0
\(326\) 103.215i 0.316609i
\(327\) 82.2885 + 47.9666i 0.251647 + 0.146687i
\(328\) 81.1335i 0.247358i
\(329\) 28.0187i 0.0851633i
\(330\) 0 0
\(331\) 359.994 1.08760 0.543798 0.839216i \(-0.316986\pi\)
0.543798 + 0.839216i \(0.316986\pi\)
\(332\) 136.554 0.411309
\(333\) 228.740 129.539i 0.686908 0.389005i
\(334\) 160.010 0.479073
\(335\) 0 0
\(336\) 15.9887 27.4292i 0.0475854 0.0816345i
\(337\) 637.901i 1.89288i −0.322878 0.946441i \(-0.604650\pi\)
0.322878 0.946441i \(-0.395350\pi\)
\(338\) 277.524 0.821076
\(339\) 67.8388 + 39.5438i 0.200114 + 0.116648i
\(340\) 0 0
\(341\) 567.823i 1.66517i
\(342\) 92.3479 + 163.068i 0.270023 + 0.476808i
\(343\) 18.5203i 0.0539949i
\(344\) 114.495i 0.332835i
\(345\) 0 0
\(346\) −163.826 −0.473487
\(347\) 131.343 0.378509 0.189255 0.981928i \(-0.439393\pi\)
0.189255 + 0.981928i \(0.439393\pi\)
\(348\) 155.890 267.436i 0.447961 0.768493i
\(349\) 462.012 1.32382 0.661908 0.749585i \(-0.269747\pi\)
0.661908 + 0.749585i \(0.269747\pi\)
\(350\) 0 0
\(351\) −515.963 6.43721i −1.46998 0.0183396i
\(352\) 83.0041i 0.235807i
\(353\) −43.9668 −0.124552 −0.0622759 0.998059i \(-0.519836\pi\)
−0.0622759 + 0.998059i \(0.519836\pi\)
\(354\) 23.7916 40.8153i 0.0672079 0.115298i
\(355\) 0 0
\(356\) 158.843i 0.446187i
\(357\) 119.080 204.287i 0.333559 0.572232i
\(358\) 108.237i 0.302339i
\(359\) 106.165i 0.295724i 0.989008 + 0.147862i \(0.0472392\pi\)
−0.989008 + 0.147862i \(0.952761\pi\)
\(360\) 0 0
\(361\) −144.213 −0.399483
\(362\) −54.2291 −0.149804
\(363\) −244.414 142.471i −0.673317 0.392482i
\(364\) 101.127 0.277822
\(365\) 0 0
\(366\) 381.680 + 222.484i 1.04284 + 0.607881i
\(367\) 181.497i 0.494543i −0.968946 0.247272i \(-0.920466\pi\)
0.968946 0.247272i \(-0.0795340\pi\)
\(368\) −91.1650 −0.247731
\(369\) −224.644 + 127.219i −0.608790 + 0.344766i
\(370\) 0 0
\(371\) 252.927i 0.681744i
\(372\) −200.596 116.929i −0.539238 0.314326i
\(373\) 12.3681i 0.0331585i 0.999863 + 0.0165792i \(0.00527757\pi\)
−0.999863 + 0.0165792i \(0.994722\pi\)
\(374\) 618.196i 1.65293i
\(375\) 0 0
\(376\) 29.9533 0.0796630
\(377\) 985.992 2.61536
\(378\) −101.017 1.26030i −0.267240 0.00333412i
\(379\) 423.772 1.11813 0.559067 0.829123i \(-0.311160\pi\)
0.559067 + 0.829123i \(0.311160\pi\)
\(380\) 0 0
\(381\) −234.678 + 402.598i −0.615952 + 1.05669i
\(382\) 0.0146577i 3.83710e-5i
\(383\) 485.717 1.26819 0.634096 0.773255i \(-0.281373\pi\)
0.634096 + 0.773255i \(0.281373\pi\)
\(384\) −29.3231 17.0926i −0.0763621 0.0445121i
\(385\) 0 0
\(386\) 20.1893i 0.0523039i
\(387\) −317.016 + 179.531i −0.819163 + 0.463903i
\(388\) 56.4099i 0.145386i
\(389\) 205.829i 0.529123i −0.964369 0.264562i \(-0.914773\pi\)
0.964369 0.264562i \(-0.0852273\pi\)
\(390\) 0 0
\(391\) −678.977 −1.73651
\(392\) 19.7990 0.0505076
\(393\) 152.376 261.407i 0.387726 0.665157i
\(394\) 174.716 0.443441
\(395\) 0 0
\(396\) −229.823 + 130.152i −0.580360 + 0.328666i
\(397\) 266.721i 0.671842i −0.941890 0.335921i \(-0.890952\pi\)
0.941890 0.335921i \(-0.109048\pi\)
\(398\) −265.200 −0.666331
\(399\) −58.8532 + 100.965i −0.147502 + 0.253044i
\(400\) 0 0
\(401\) 251.023i 0.625991i −0.949755 0.312996i \(-0.898667\pi\)
0.949755 0.312996i \(-0.101333\pi\)
\(402\) 96.6964 165.886i 0.240538 0.412652i
\(403\) 739.567i 1.83515i
\(404\) 21.3538i 0.0528561i
\(405\) 0 0
\(406\) 193.041 0.475470
\(407\) 428.576 1.05301
\(408\) −218.392 127.302i −0.535274 0.312016i
\(409\) −588.064 −1.43781 −0.718905 0.695109i \(-0.755356\pi\)
−0.718905 + 0.695109i \(0.755356\pi\)
\(410\) 0 0
\(411\) 304.984 + 177.778i 0.742054 + 0.432550i
\(412\) 68.2084i 0.165554i
\(413\) 29.4614 0.0713351
\(414\) 142.948 + 252.419i 0.345285 + 0.609707i
\(415\) 0 0
\(416\) 108.109i 0.259878i
\(417\) 194.086 + 113.135i 0.465435 + 0.271306i
\(418\) 305.531i 0.730936i
\(419\) 21.3662i 0.0509934i −0.999675 0.0254967i \(-0.991883\pi\)
0.999675 0.0254967i \(-0.00811673\pi\)
\(420\) 0 0
\(421\) 29.7516 0.0706688 0.0353344 0.999376i \(-0.488750\pi\)
0.0353344 + 0.999376i \(0.488750\pi\)
\(422\) 396.337 0.939188
\(423\) −46.9672 82.9350i −0.111034 0.196064i
\(424\) −270.390 −0.637713
\(425\) 0 0
\(426\) 200.404 343.800i 0.470431 0.807041i
\(427\) 275.505i 0.645210i
\(428\) −30.8676 −0.0721205
\(429\) −726.803 423.660i −1.69418 0.987552i
\(430\) 0 0
\(431\) 44.2977i 0.102779i −0.998679 0.0513894i \(-0.983635\pi\)
0.998679 0.0513894i \(-0.0163650\pi\)
\(432\) −1.34732 + 107.992i −0.00311879 + 0.249981i
\(433\) 742.259i 1.71422i 0.515131 + 0.857112i \(0.327743\pi\)
−0.515131 + 0.857112i \(0.672257\pi\)
\(434\) 144.795i 0.333629i
\(435\) 0 0
\(436\) 63.4987 0.145639
\(437\) 335.571 0.767897
\(438\) 134.208 230.238i 0.306410 0.525657i
\(439\) 339.668 0.773732 0.386866 0.922136i \(-0.373558\pi\)
0.386866 + 0.922136i \(0.373558\pi\)
\(440\) 0 0
\(441\) −31.0451 54.8197i −0.0703972 0.124308i
\(442\) 805.175i 1.82166i
\(443\) −7.56065 −0.0170669 −0.00853347 0.999964i \(-0.502716\pi\)
−0.00853347 + 0.999964i \(0.502716\pi\)
\(444\) 88.2548 151.404i 0.198772 0.341001i
\(445\) 0 0
\(446\) 142.502i 0.319512i
\(447\) −225.990 + 387.695i −0.505571 + 0.867326i
\(448\) 21.1660i 0.0472456i
\(449\) 480.937i 1.07113i −0.844494 0.535564i \(-0.820099\pi\)
0.844494 0.535564i \(-0.179901\pi\)
\(450\) 0 0
\(451\) −420.901 −0.933261
\(452\) 52.3484 0.115815
\(453\) 525.120 + 306.097i 1.15921 + 0.675710i
\(454\) 217.869 0.479888
\(455\) 0 0
\(456\) 107.936 + 62.9167i 0.236701 + 0.137975i
\(457\) 593.531i 1.29876i 0.760466 + 0.649378i \(0.224970\pi\)
−0.760466 + 0.649378i \(0.775030\pi\)
\(458\) 558.728 1.21993
\(459\) −10.0345 + 804.298i −0.0218617 + 1.75228i
\(460\) 0 0
\(461\) 211.732i 0.459289i 0.973275 + 0.229645i \(0.0737564\pi\)
−0.973275 + 0.229645i \(0.926244\pi\)
\(462\) −142.296 82.9455i −0.308000 0.179536i
\(463\) 618.360i 1.33555i −0.744363 0.667776i \(-0.767247\pi\)
0.744363 0.667776i \(-0.232753\pi\)
\(464\) 206.369i 0.444761i
\(465\) 0 0
\(466\) −272.827 −0.585465
\(467\) −365.286 −0.782197 −0.391099 0.920349i \(-0.627905\pi\)
−0.391099 + 0.920349i \(0.627905\pi\)
\(468\) −299.335 + 169.517i −0.639604 + 0.362217i
\(469\) 119.740 0.255310
\(470\) 0 0
\(471\) 119.789 205.503i 0.254330 0.436311i
\(472\) 31.4956i 0.0667279i
\(473\) −593.974 −1.25576
\(474\) 448.959 + 261.702i 0.947172 + 0.552114i
\(475\) 0 0
\(476\) 157.640i 0.331176i
\(477\) 423.977 + 748.661i 0.888840 + 1.56952i
\(478\) 442.423i 0.925571i
\(479\) 167.464i 0.349611i 0.984603 + 0.174805i \(0.0559296\pi\)
−0.984603 + 0.174805i \(0.944070\pi\)
\(480\) 0 0
\(481\) 558.203 1.16051
\(482\) −464.046 −0.962750
\(483\) −91.1006 + 156.286i −0.188614 + 0.323574i
\(484\) −188.605 −0.389679
\(485\) 0 0
\(486\) 301.121 165.602i 0.619591 0.340745i
\(487\) 239.295i 0.491366i −0.969350 0.245683i \(-0.920988\pi\)
0.969350 0.245683i \(-0.0790122\pi\)
\(488\) 294.527 0.603539
\(489\) −110.263 + 189.161i −0.225487 + 0.386831i
\(490\) 0 0
\(491\) 255.880i 0.521140i −0.965455 0.260570i \(-0.916090\pi\)
0.965455 0.260570i \(-0.0839105\pi\)
\(492\) −86.6742 + 148.693i −0.176167 + 0.302221i
\(493\) 1536.99i 3.11763i
\(494\) 397.942i 0.805551i
\(495\) 0 0
\(496\) −154.792 −0.312081
\(497\) 248.162 0.499320
\(498\) −250.262 145.880i −0.502534 0.292931i
\(499\) 78.1567 0.156627 0.0783133 0.996929i \(-0.475047\pi\)
0.0783133 + 0.996929i \(0.475047\pi\)
\(500\) 0 0
\(501\) −293.250 170.938i −0.585329 0.341193i
\(502\) 206.701i 0.411755i
\(503\) 756.325 1.50363 0.751814 0.659375i \(-0.229179\pi\)
0.751814 + 0.659375i \(0.229179\pi\)
\(504\) −58.6047 + 33.1887i −0.116279 + 0.0658505i
\(505\) 0 0
\(506\) 472.941i 0.934667i
\(507\) −508.615 296.476i −1.00319 0.584765i
\(508\) 310.669i 0.611553i
\(509\) 797.969i 1.56772i 0.620938 + 0.783859i \(0.286752\pi\)
−0.620938 + 0.783859i \(0.713248\pi\)
\(510\) 0 0
\(511\) 166.191 0.325226
\(512\) −22.6274 −0.0441942
\(513\) 4.95936 397.508i 0.00966737 0.774870i
\(514\) −433.011 −0.842434
\(515\) 0 0
\(516\) −122.314 + 209.835i −0.237043 + 0.406656i
\(517\) 155.390i 0.300561i
\(518\) 109.287 0.210979
\(519\) 300.243 + 175.014i 0.578503 + 0.337214i
\(520\) 0 0
\(521\) 22.5301i 0.0432440i −0.999766 0.0216220i \(-0.993117\pi\)
0.999766 0.0216220i \(-0.00688303\pi\)
\(522\) −571.398 + 323.590i −1.09463 + 0.619905i
\(523\) 250.580i 0.479121i −0.970881 0.239560i \(-0.922997\pi\)
0.970881 0.239560i \(-0.0770033\pi\)
\(524\) 201.717i 0.384956i
\(525\) 0 0
\(526\) −223.703 −0.425291
\(527\) −1152.86 −2.18759
\(528\) −88.6724 + 152.121i −0.167940 + 0.288107i
\(529\) −9.55921 −0.0180703
\(530\) 0 0
\(531\) −87.2053 + 49.3856i −0.164228 + 0.0930048i
\(532\) 77.9104i 0.146448i
\(533\) −548.206 −1.02853
\(534\) 169.690 291.109i 0.317772 0.545149i
\(535\) 0 0
\(536\) 128.008i 0.238820i
\(537\) 115.629 198.365i 0.215324 0.369396i
\(538\) 223.669i 0.415742i
\(539\) 102.712i 0.190561i
\(540\) 0 0
\(541\) −438.380 −0.810314 −0.405157 0.914247i \(-0.632783\pi\)
−0.405157 + 0.914247i \(0.632783\pi\)
\(542\) −511.224 −0.943218
\(543\) 99.3851 + 57.9324i 0.183030 + 0.106690i
\(544\) −168.524 −0.309787
\(545\) 0 0
\(546\) −185.335 108.033i −0.339441 0.197863i
\(547\) 212.261i 0.388045i −0.980997 0.194023i \(-0.937846\pi\)
0.980997 0.194023i \(-0.0621535\pi\)
\(548\) 235.344 0.429460
\(549\) −461.823 815.490i −0.841208 1.48541i
\(550\) 0 0
\(551\) 759.628i 1.37864i
\(552\) 167.077 + 97.3907i 0.302676 + 0.176432i
\(553\) 324.069i 0.586019i
\(554\) 109.048i 0.196838i
\(555\) 0 0
\(556\) 149.769 0.269368
\(557\) −303.846 −0.545504 −0.272752 0.962084i \(-0.587934\pi\)
−0.272752 + 0.962084i \(0.587934\pi\)
\(558\) 242.717 + 428.591i 0.434976 + 0.768083i
\(559\) −773.626 −1.38395
\(560\) 0 0
\(561\) −660.413 + 1132.96i −1.17721 + 2.01954i
\(562\) 136.109i 0.242187i
\(563\) 59.7131 0.106062 0.0530312 0.998593i \(-0.483112\pi\)
0.0530312 + 0.998593i \(0.483112\pi\)
\(564\) −54.8951 31.9988i −0.0973317 0.0567355i
\(565\) 0 0
\(566\) 298.213i 0.526878i
\(567\) 183.786 + 110.225i 0.324138 + 0.194401i
\(568\) 265.296i 0.467071i
\(569\) 125.358i 0.220313i −0.993914 0.110157i \(-0.964865\pi\)
0.993914 0.110157i \(-0.0351352\pi\)
\(570\) 0 0
\(571\) 865.616 1.51597 0.757983 0.652275i \(-0.226185\pi\)
0.757983 + 0.652275i \(0.226185\pi\)
\(572\) −560.845 −0.980498
\(573\) 0.0156587 0.0268631i 2.73276e−5 4.68815e-5i
\(574\) −107.330 −0.186985
\(575\) 0 0
\(576\) 35.4802 + 62.6511i 0.0615975 + 0.108769i
\(577\) 176.822i 0.306451i 0.988191 + 0.153225i \(0.0489660\pi\)
−0.988191 + 0.153225i \(0.951034\pi\)
\(578\) −846.423 −1.46440
\(579\) 21.5680 37.0007i 0.0372505 0.0639046i
\(580\) 0 0
\(581\) 180.645i 0.310920i
\(582\) −60.2621 + 103.382i −0.103543 + 0.177632i
\(583\) 1402.72i 2.40604i
\(584\) 177.665i 0.304221i
\(585\) 0 0
\(586\) 623.692 1.06432
\(587\) 330.200 0.562521 0.281260 0.959631i \(-0.409248\pi\)
0.281260 + 0.959631i \(0.409248\pi\)
\(588\) −36.2854 21.1511i −0.0617099 0.0359712i
\(589\) 569.778 0.967364
\(590\) 0 0
\(591\) −320.200 186.647i −0.541793 0.315816i
\(592\) 116.833i 0.197352i
\(593\) 996.327 1.68015 0.840073 0.542473i \(-0.182512\pi\)
0.840073 + 0.542473i \(0.182512\pi\)
\(594\) 560.234 + 6.98954i 0.943154 + 0.0117669i
\(595\) 0 0
\(596\) 299.168i 0.501960i
\(597\) 486.029 + 283.311i 0.814119 + 0.474557i
\(598\) 615.987i 1.03008i
\(599\) 416.236i 0.694885i 0.937701 + 0.347443i \(0.112950\pi\)
−0.937701 + 0.347443i \(0.887050\pi\)
\(600\) 0 0
\(601\) 1141.63 1.89955 0.949777 0.312929i \(-0.101310\pi\)
0.949777 + 0.312929i \(0.101310\pi\)
\(602\) −151.463 −0.251600
\(603\) −354.429 + 200.718i −0.587776 + 0.332866i
\(604\) 405.214 0.670884
\(605\) 0 0
\(606\) 22.8121 39.1350i 0.0376438 0.0645792i
\(607\) 664.913i 1.09541i −0.836672 0.547704i \(-0.815502\pi\)
0.836672 0.547704i \(-0.184498\pi\)
\(608\) 83.2897 0.136990
\(609\) −353.784 206.224i −0.580926 0.338627i
\(610\) 0 0
\(611\) 202.389i 0.331243i
\(612\) 264.249 + 466.612i 0.431779 + 0.762437i
\(613\) 588.435i 0.959927i 0.877289 + 0.479963i \(0.159350\pi\)
−0.877289 + 0.479963i \(0.840650\pi\)
\(614\) 359.029i 0.584738i
\(615\) 0 0
\(616\) −109.804 −0.178253
\(617\) 631.416 1.02336 0.511682 0.859175i \(-0.329022\pi\)
0.511682 + 0.859175i \(0.329022\pi\)
\(618\) −72.8664 + 125.005i −0.117907 + 0.202273i
\(619\) 335.858 0.542582 0.271291 0.962497i \(-0.412549\pi\)
0.271291 + 0.962497i \(0.412549\pi\)
\(620\) 0 0
\(621\) 7.67675 615.316i 0.0123619 0.990847i
\(622\) 569.525i 0.915635i
\(623\) 210.129 0.337286
\(624\) −115.492 + 198.131i −0.185084 + 0.317518i
\(625\) 0 0
\(626\) 82.6532i 0.132034i
\(627\) 326.396 559.944i 0.520568 0.893053i
\(628\) 158.578i 0.252513i
\(629\) 870.144i 1.38338i
\(630\) 0 0
\(631\) −641.185 −1.01614 −0.508070 0.861316i \(-0.669641\pi\)
−0.508070 + 0.861316i \(0.669641\pi\)
\(632\) 346.444 0.548171
\(633\) −726.364 423.403i −1.14749 0.668884i
\(634\) −257.230 −0.405726
\(635\) 0 0
\(636\) 495.542 + 288.856i 0.779154 + 0.454175i
\(637\) 133.779i 0.210013i
\(638\) −1070.59 −1.67804
\(639\) −734.556 + 415.989i −1.14954 + 0.651000i
\(640\) 0 0
\(641\) 358.234i 0.558867i 0.960165 + 0.279433i \(0.0901466\pi\)
−0.960165 + 0.279433i \(0.909853\pi\)
\(642\) 56.5707 + 32.9755i 0.0881164 + 0.0513638i
\(643\) 1218.12i 1.89443i 0.320606 + 0.947213i \(0.396114\pi\)
−0.320606 + 0.947213i \(0.603886\pi\)
\(644\) 120.600i 0.187267i
\(645\) 0 0
\(646\) 620.324 0.960253
\(647\) −706.621 −1.09215 −0.546075 0.837737i \(-0.683879\pi\)
−0.546075 + 0.837737i \(0.683879\pi\)
\(648\) 117.836 196.476i 0.181845 0.303204i
\(649\) −163.391 −0.251758
\(650\) 0 0
\(651\) −154.683 + 265.364i −0.237608 + 0.407625i
\(652\) 145.968i 0.223877i
\(653\) 647.469 0.991531 0.495765 0.868457i \(-0.334888\pi\)
0.495765 + 0.868457i \(0.334888\pi\)
\(654\) −116.373 67.8351i −0.177941 0.103723i
\(655\) 0 0
\(656\) 114.740i 0.174909i
\(657\) −491.922 + 278.582i −0.748739 + 0.424021i
\(658\) 39.6245i 0.0602196i
\(659\) 925.263i 1.40404i 0.712157 + 0.702020i \(0.247718\pi\)
−0.712157 + 0.702020i \(0.752282\pi\)
\(660\) 0 0
\(661\) 516.011 0.780652 0.390326 0.920677i \(-0.372362\pi\)
0.390326 + 0.920677i \(0.372362\pi\)
\(662\) −509.109 −0.769047
\(663\) −860.161 + 1475.64i −1.29738 + 2.22570i
\(664\) −193.117 −0.290839
\(665\) 0 0
\(666\) −323.488 + 183.195i −0.485717 + 0.275068i
\(667\) 1175.85i 1.76290i
\(668\) −226.289 −0.338756
\(669\) −152.234 + 261.163i −0.227555 + 0.390378i
\(670\) 0 0
\(671\) 1527.93i 2.27710i
\(672\) −22.6115 + 38.7908i −0.0336480 + 0.0577243i
\(673\) 1260.47i 1.87291i −0.350782 0.936457i \(-0.614084\pi\)
0.350782 0.936457i \(-0.385916\pi\)
\(674\) 902.128i 1.33847i
\(675\) 0 0
\(676\) −392.478 −0.580588
\(677\) 364.477 0.538370 0.269185 0.963088i \(-0.413246\pi\)
0.269185 + 0.963088i \(0.413246\pi\)
\(678\) −95.9385 55.9233i −0.141502 0.0824828i
\(679\) −74.6232 −0.109902
\(680\) 0 0
\(681\) −399.286 232.747i −0.586324 0.341773i
\(682\) 803.024i 1.17745i
\(683\) 335.908 0.491813 0.245906 0.969294i \(-0.420914\pi\)
0.245906 + 0.969294i \(0.420914\pi\)
\(684\) −130.600 230.614i −0.190935 0.337154i
\(685\) 0 0
\(686\) 26.1916i 0.0381802i
\(687\) −1023.98 596.884i −1.49050 0.868826i
\(688\) 161.921i 0.235350i
\(689\) 1826.98i 2.65165i
\(690\) 0 0
\(691\) 56.3269 0.0815150 0.0407575 0.999169i \(-0.487023\pi\)
0.0407575 + 0.999169i \(0.487023\pi\)
\(692\) 231.686 0.334806
\(693\) 172.175 + 304.027i 0.248448 + 0.438711i
\(694\) −185.747 −0.267647
\(695\) 0 0
\(696\) −220.462 + 378.211i −0.316756 + 0.543407i
\(697\) 854.560i 1.22605i
\(698\) −653.383 −0.936079
\(699\) 500.007 + 291.458i 0.715317 + 0.416964i
\(700\) 0 0
\(701\) 795.844i 1.13530i −0.823270 0.567649i \(-0.807853\pi\)
0.823270 0.567649i \(-0.192147\pi\)
\(702\) 729.682 + 9.10359i 1.03943 + 0.0129681i
\(703\) 430.051i 0.611737i
\(704\) 117.385i 0.166741i
\(705\) 0 0
\(706\) 62.1784 0.0880714
\(707\) 28.2485 0.0399554
\(708\) −33.6464 + 57.7216i −0.0475232 + 0.0815277i
\(709\) −127.721 −0.180142 −0.0900712 0.995935i \(-0.528709\pi\)
−0.0900712 + 0.995935i \(0.528709\pi\)
\(710\) 0 0
\(711\) −543.230 959.238i −0.764036 1.34914i
\(712\) 224.637i 0.315502i
\(713\) 881.977 1.23699
\(714\) −168.405 + 288.905i −0.235862 + 0.404629i
\(715\) 0 0
\(716\) 153.071i 0.213786i
\(717\) 472.637 810.825i 0.659186 1.13086i
\(718\) 150.140i 0.209108i
\(719\) 652.748i 0.907855i 0.891039 + 0.453927i \(0.149977\pi\)
−0.891039 + 0.453927i \(0.850023\pi\)
\(720\) 0 0
\(721\) −90.2313 −0.125147
\(722\) 203.948 0.282477
\(723\) 850.452 + 495.736i 1.17628 + 0.685665i
\(724\) 76.6915 0.105928
\(725\) 0 0
\(726\) 345.654 + 201.484i 0.476107 + 0.277527i
\(727\) 181.338i 0.249433i 0.992192 + 0.124717i \(0.0398022\pi\)
−0.992192 + 0.124717i \(0.960198\pi\)
\(728\) −143.015 −0.196450
\(729\) −728.773 18.1873i −0.999689 0.0249483i
\(730\) 0 0
\(731\) 1205.95i 1.64973i
\(732\) −539.777 314.641i −0.737400 0.429837i
\(733\) 980.665i 1.33788i 0.743317 + 0.668939i \(0.233251\pi\)
−0.743317 + 0.668939i \(0.766749\pi\)
\(734\) 256.676i 0.349695i
\(735\) 0 0
\(736\) 128.927 0.175172
\(737\) −664.072 −0.901047
\(738\) 317.694 179.914i 0.430480 0.243786i
\(739\) −115.426 −0.156193 −0.0780963 0.996946i \(-0.524884\pi\)
−0.0780963 + 0.996946i \(0.524884\pi\)
\(740\) 0 0
\(741\) 425.118 729.305i 0.573708 0.984217i
\(742\) 357.693i 0.482066i
\(743\) −131.260 −0.176662 −0.0883311 0.996091i \(-0.528153\pi\)
−0.0883311 + 0.996091i \(0.528153\pi\)
\(744\) 283.686 + 165.363i 0.381299 + 0.222262i
\(745\) 0 0
\(746\) 17.4911i 0.0234466i
\(747\) 302.811 + 534.705i 0.405369 + 0.715804i
\(748\) 874.261i 1.16880i
\(749\) 40.8340i 0.0545180i
\(750\) 0 0
\(751\) −1088.21 −1.44902 −0.724508 0.689266i \(-0.757933\pi\)
−0.724508 + 0.689266i \(0.757933\pi\)
\(752\) −42.3603 −0.0563302
\(753\) 220.817 378.819i 0.293250 0.503080i
\(754\) −1394.40 −1.84934
\(755\) 0 0
\(756\) 142.859 + 1.78233i 0.188968 + 0.00235758i
\(757\) 1115.02i 1.47295i −0.676467 0.736473i \(-0.736490\pi\)
0.676467 0.736473i \(-0.263510\pi\)
\(758\) −599.305 −0.790639
\(759\) 505.239 866.755i 0.665664 1.14197i
\(760\) 0 0
\(761\) 632.008i 0.830497i 0.909708 + 0.415249i \(0.136305\pi\)
−0.909708 + 0.415249i \(0.863695\pi\)
\(762\) 331.885 569.360i 0.435544 0.747191i
\(763\) 84.0009i 0.110093i
\(764\) 0.0207292i 2.71324e-5i
\(765\) 0 0
\(766\) −686.908 −0.896746
\(767\) −212.810 −0.277458
\(768\) 41.4691 + 24.1727i 0.0539962 + 0.0314748i
\(769\) −183.616 −0.238773 −0.119386 0.992848i \(-0.538093\pi\)
−0.119386 + 0.992848i \(0.538093\pi\)
\(770\) 0 0
\(771\) 793.575 + 462.582i 1.02928 + 0.599976i
\(772\) 28.5520i 0.0369844i
\(773\) 239.771 0.310182 0.155091 0.987900i \(-0.450433\pi\)
0.155091 + 0.987900i \(0.450433\pi\)
\(774\) 448.329 253.895i 0.579236 0.328029i
\(775\) 0 0
\(776\) 79.7756i 0.102804i
\(777\) −200.289 116.750i −0.257772 0.150258i
\(778\) 291.086i 0.374147i
\(779\) 422.349i 0.542168i
\(780\) 0 0
\(781\) −1376.29 −1.76222
\(782\) 960.218 1.22790
\(783\) 1392.88 + 17.3778i 1.77891 + 0.0221938i
\(784\) −28.0000 −0.0357143
\(785\) 0 0
\(786\) −215.492 + 369.685i −0.274163 + 0.470337i
\(787\) 430.055i 0.546449i −0.961950 0.273224i \(-0.911910\pi\)
0.961950 0.273224i \(-0.0880902\pi\)
\(788\) −247.085 −0.313560
\(789\) 409.978 + 238.980i 0.519618 + 0.302889i
\(790\) 0 0
\(791\) 69.2505i 0.0875480i
\(792\) 325.018 184.062i 0.410377 0.232402i
\(793\) 1990.07i 2.50955i
\(794\) 377.201i 0.475064i
\(795\) 0 0
\(796\) 375.049 0.471167
\(797\) −454.406 −0.570146 −0.285073 0.958506i \(-0.592018\pi\)
−0.285073 + 0.958506i \(0.592018\pi\)
\(798\) 83.2309 142.786i 0.104299 0.178929i
\(799\) −315.491 −0.394857
\(800\) 0 0
\(801\) −621.979 + 352.235i −0.776503 + 0.439744i
\(802\) 354.999i 0.442643i
\(803\) −921.683 −1.14780
\(804\) −136.749 + 234.598i −0.170086 + 0.291789i
\(805\) 0 0
\(806\) 1045.91i 1.29765i
\(807\) 238.944 409.916i 0.296089 0.507951i
\(808\) 30.1989i 0.0373749i
\(809\) 1016.61i 1.25662i −0.777963 0.628310i \(-0.783747\pi\)
0.777963 0.628310i \(-0.216253\pi\)
\(810\) 0 0
\(811\) 756.253 0.932494 0.466247 0.884654i \(-0.345606\pi\)
0.466247 + 0.884654i \(0.345606\pi\)
\(812\) −273.001 −0.336208
\(813\) 936.916 + 546.136i 1.15242 + 0.671754i
\(814\) −606.099 −0.744593
\(815\) 0 0
\(816\) 308.853 + 180.033i 0.378496 + 0.220628i
\(817\) 596.018i 0.729520i
\(818\) 831.648 1.01668
\(819\) 224.250 + 395.983i 0.273810 + 0.483495i
\(820\) 0 0
\(821\) 1207.04i 1.47021i −0.677955 0.735104i \(-0.737134\pi\)
0.677955 0.735104i \(-0.262866\pi\)
\(822\) −431.313 251.416i −0.524712 0.305859i
\(823\) 169.449i 0.205891i −0.994687 0.102946i \(-0.967173\pi\)
0.994687 0.102946i \(-0.0328268\pi\)
\(824\) 96.4613i 0.117065i
\(825\) 0 0
\(826\) −41.6647 −0.0504415
\(827\) −953.176 −1.15257 −0.576285 0.817248i \(-0.695498\pi\)
−0.576285 + 0.817248i \(0.695498\pi\)
\(828\) −202.159 356.974i −0.244154 0.431128i
\(829\) 6.32036 0.00762407 0.00381204 0.999993i \(-0.498787\pi\)
0.00381204 + 0.999993i \(0.498787\pi\)
\(830\) 0 0
\(831\) 116.495 199.852i 0.140187 0.240495i
\(832\) 152.890i 0.183762i
\(833\) −208.538 −0.250346
\(834\) −274.480 159.996i −0.329112 0.191842i
\(835\) 0 0
\(836\) 432.086i 0.516850i
\(837\) 13.0346 1044.77i 0.0155730 1.24823i
\(838\) 30.2164i 0.0360578i
\(839\) 1230.70i 1.46686i 0.679764 + 0.733430i \(0.262082\pi\)
−0.679764 + 0.733430i \(0.737918\pi\)
\(840\) 0 0
\(841\) −1820.77 −2.16500
\(842\) −42.0751 −0.0499704
\(843\) 145.404 249.446i 0.172484 0.295903i
\(844\) −560.506 −0.664106
\(845\) 0 0
\(846\) 66.4217 + 117.288i 0.0785127 + 0.138638i
\(847\) 249.500i 0.294569i
\(848\) 382.390 0.450931
\(849\) 318.578 546.532i 0.375239 0.643737i
\(850\) 0 0
\(851\) 665.690i 0.782244i
\(852\) −283.414 + 486.206i −0.332645 + 0.570664i
\(853\) 759.822i 0.890764i −0.895341 0.445382i \(-0.853068\pi\)
0.895341 0.445382i \(-0.146932\pi\)
\(854\) 389.623i 0.456233i
\(855\) 0 0
\(856\) 43.6533 0.0509969
\(857\) 531.002 0.619606 0.309803 0.950801i \(-0.399737\pi\)
0.309803 + 0.950801i \(0.399737\pi\)
\(858\) 1027.86 + 599.145i 1.19797 + 0.698305i
\(859\) −765.291 −0.890910 −0.445455 0.895304i \(-0.646958\pi\)
−0.445455 + 0.895304i \(0.646958\pi\)
\(860\) 0 0
\(861\) 196.702 + 114.659i 0.228457 + 0.133170i
\(862\) 62.6464i 0.0726756i
\(863\) 1329.43 1.54048 0.770240 0.637754i \(-0.220136\pi\)
0.770240 + 0.637754i \(0.220136\pi\)
\(864\) 1.90539 152.723i 0.00220531 0.176763i
\(865\) 0 0
\(866\) 1049.71i 1.21214i
\(867\) 1551.23 + 904.226i 1.78919 + 1.04294i
\(868\) 204.771i 0.235911i
\(869\) 1797.27i 2.06820i
\(870\) 0 0
\(871\) −864.927 −0.993027
\(872\) −89.8007 −0.102982
\(873\) 220.884 125.089i 0.253017 0.143287i
\(874\) −474.569 −0.542985
\(875\) 0 0
\(876\) −189.798 + 325.605i −0.216664 + 0.371696i
\(877\) 919.329i 1.04827i −0.851637 0.524133i \(-0.824390\pi\)
0.851637 0.524133i \(-0.175610\pi\)
\(878\) −480.364 −0.547111
\(879\) −1143.03 666.284i −1.30038 0.758002i
\(880\) 0 0
\(881\) 832.315i 0.944739i −0.881401 0.472369i \(-0.843399\pi\)
0.881401 0.472369i \(-0.156601\pi\)
\(882\) 43.9045 + 77.5268i 0.0497783 + 0.0878988i
\(883\) 1425.51i 1.61439i −0.590284 0.807196i \(-0.700984\pi\)
0.590284 0.807196i \(-0.299016\pi\)
\(884\) 1138.69i 1.28811i
\(885\) 0 0
\(886\) 10.6924 0.0120681
\(887\) −641.162 −0.722843 −0.361422 0.932402i \(-0.617709\pi\)
−0.361422 + 0.932402i \(0.617709\pi\)
\(888\) −124.811 + 214.118i −0.140553 + 0.241124i
\(889\) 410.976 0.462290
\(890\) 0 0
\(891\) −1019.27 611.302i −1.14396 0.686085i
\(892\) 201.529i 0.225929i
\(893\) 155.925 0.174608
\(894\) 319.599 548.283i 0.357493 0.613292i
\(895\) 0 0
\(896\) 29.9333i 0.0334077i
\(897\) 658.053 1128.91i 0.733615 1.25854i
\(898\) 680.147i 0.757402i
\(899\) 1996.52i 2.22082i
\(900\) 0 0
\(901\) 2847.96 3.16088
\(902\) 595.243 0.659915
\(903\) 277.585 + 161.807i 0.307403 + 0.179188i
\(904\) −74.0319 −0.0818937
\(905\) 0 0
\(906\) −742.632 432.886i −0.819682 0.477799i
\(907\) 1569.63i 1.73057i 0.501278 + 0.865287i \(0.332864\pi\)
−0.501278 + 0.865287i \(0.667136\pi\)
\(908\) −308.113 −0.339332
\(909\) −83.6151 + 47.3524i −0.0919858 + 0.0520928i
\(910\) 0 0
\(911\) 1742.53i 1.91277i 0.292108 + 0.956385i \(0.405643\pi\)
−0.292108 + 0.956385i \(0.594357\pi\)
\(912\) −152.644 88.9776i −0.167373 0.0975632i
\(913\) 1001.84i 1.09731i
\(914\) 839.380i 0.918359i
\(915\) 0 0
\(916\) −790.160 −0.862621
\(917\) −266.847 −0.291000
\(918\) 14.1909 1137.45i 0.0154585 1.23905i
\(919\) 944.373 1.02761 0.513805 0.857907i \(-0.328236\pi\)
0.513805 + 0.857907i \(0.328236\pi\)
\(920\) 0 0
\(921\) −383.547 + 657.989i −0.416447 + 0.714429i
\(922\) 299.435i 0.324767i
\(923\) −1792.56 −1.94211
\(924\) 201.237 + 117.303i 0.217789 + 0.126951i
\(925\) 0 0
\(926\) 874.494i 0.944377i
\(927\) 267.083 151.253i 0.288116 0.163164i
\(928\) 291.850i 0.314494i
\(929\) 600.958i 0.646887i −0.946248 0.323443i \(-0.895159\pi\)
0.946248 0.323443i \(-0.104841\pi\)
\(930\) 0 0
\(931\) 103.066 0.110704
\(932\) 385.835 0.413986
\(933\) −608.418 + 1043.76i −0.652110 + 1.11872i
\(934\) 516.593 0.553097
\(935\) 0 0
\(936\) 423.323 239.734i 0.452268 0.256126i
\(937\) 19.4399i 0.0207470i −0.999946 0.0103735i \(-0.996698\pi\)
0.999946 0.0103735i \(-0.00330204\pi\)
\(938\) −169.338 −0.180531
\(939\) 88.2976 151.478i 0.0940336 0.161318i
\(940\) 0 0
\(941\) 709.230i 0.753698i −0.926275 0.376849i \(-0.877008\pi\)
0.926275 0.376849i \(-0.122992\pi\)
\(942\) −169.408 + 290.625i −0.179838 + 0.308519i
\(943\) 653.768i 0.693285i
\(944\) 44.5414i 0.0471837i
\(945\) 0 0
\(946\) 840.006 0.887955
\(947\) 228.266 0.241041 0.120521 0.992711i \(-0.461544\pi\)
0.120521 + 0.992711i \(0.461544\pi\)
\(948\) −634.925 370.103i −0.669752 0.390404i
\(949\) −1200.45 −1.26497
\(950\) 0 0
\(951\) 471.423 + 274.796i 0.495713 + 0.288955i
\(952\) 222.936i 0.234177i
\(953\) −830.703 −0.871672 −0.435836 0.900026i \(-0.643547\pi\)
−0.435836 + 0.900026i \(0.643547\pi\)
\(954\) −599.594 1058.77i −0.628505 1.10982i
\(955\) 0 0
\(956\) 625.681i 0.654478i
\(957\) 1962.06 + 1143.70i 2.05022 + 1.19509i
\(958\) 236.829i 0.247212i
\(959\) 311.331i 0.324641i
\(960\) 0 0
\(961\) 536.539 0.558313
\(962\) −789.419 −0.820602
\(963\) −68.4491 120.868i −0.0710791 0.125512i
\(964\) 656.260 0.680767
\(965\) 0 0
\(966\) 128.836 221.022i 0.133370 0.228802i
\(967\) 1120.26i 1.15849i 0.815153 + 0.579246i \(0.196653\pi\)
−0.815153 + 0.579246i \(0.803347\pi\)
\(968\) 266.727 0.275544
\(969\) −1136.86 662.686i −1.17323 0.683886i
\(970\) 0 0
\(971\) 533.802i 0.549745i 0.961481 + 0.274872i \(0.0886356\pi\)
−0.961481 + 0.274872i \(0.911364\pi\)
\(972\) −425.850 + 234.197i −0.438117 + 0.240943i
\(973\) 198.125i 0.203623i
\(974\) 338.414i 0.347448i
\(975\) 0 0
\(976\) −416.524 −0.426767
\(977\) 1232.14 1.26115 0.630575 0.776129i \(-0.282819\pi\)
0.630575 + 0.776129i \(0.282819\pi\)
\(978\) 155.936 267.513i 0.159444 0.273531i
\(979\) −1165.36 −1.19036
\(980\) 0 0
\(981\) 140.809 + 248.641i 0.143536 + 0.253457i
\(982\) 361.868i 0.368502i
\(983\) 1098.64 1.11764 0.558820 0.829289i \(-0.311254\pi\)
0.558820 + 0.829289i \(0.311254\pi\)
\(984\) 122.576 210.283i 0.124569 0.213702i
\(985\) 0 0
\(986\) 2173.64i 2.20450i
\(987\) −42.3304 + 72.6194i −0.0428880 + 0.0735759i
\(988\) 562.775i 0.569610i
\(989\) 922.595i 0.932856i
\(990\) 0 0
\(991\) −1388.10 −1.40070 −0.700352 0.713798i \(-0.746973\pi\)
−0.700352 + 0.713798i \(0.746973\pi\)
\(992\) 218.909 0.220675
\(993\) 933.039 + 543.876i 0.939617 + 0.547710i
\(994\) −350.954 −0.353073
\(995\) 0 0
\(996\) 353.924 + 206.305i 0.355345 + 0.207134i
\(997\) 818.089i 0.820551i −0.911962 0.410275i \(-0.865433\pi\)
0.911962 0.410275i \(-0.134567\pi\)
\(998\) −110.530 −0.110752
\(999\) 788.558 + 9.83814i 0.789348 + 0.00984799i
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 1050.3.c.c.449.16 32
3.2 odd 2 inner 1050.3.c.c.449.18 32
5.2 odd 4 1050.3.e.d.701.5 16
5.3 odd 4 210.3.e.a.71.12 yes 16
5.4 even 2 inner 1050.3.c.c.449.17 32
15.2 even 4 1050.3.e.d.701.13 16
15.8 even 4 210.3.e.a.71.4 16
15.14 odd 2 inner 1050.3.c.c.449.15 32
20.3 even 4 1680.3.l.c.1121.9 16
60.23 odd 4 1680.3.l.c.1121.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.3.e.a.71.4 16 15.8 even 4
210.3.e.a.71.12 yes 16 5.3 odd 4
1050.3.c.c.449.15 32 15.14 odd 2 inner
1050.3.c.c.449.16 32 1.1 even 1 trivial
1050.3.c.c.449.17 32 5.4 even 2 inner
1050.3.c.c.449.18 32 3.2 odd 2 inner
1050.3.e.d.701.5 16 5.2 odd 4
1050.3.e.d.701.13 16 15.2 even 4
1680.3.l.c.1121.9 16 20.3 even 4
1680.3.l.c.1121.10 16 60.23 odd 4