Properties

Label 252.4.bm.a.173.7
Level $252$
Weight $4$
Character 252.173
Analytic conductor $14.868$
Analytic rank $0$
Dimension $48$
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] = [252,4,Mod(173,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.173");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.bm (of order \(6\), degree \(2\), 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: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 173.7
Character \(\chi\) \(=\) 252.173
Dual form 252.4.bm.a.185.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.15513 + 4.12858i) q^{3} +18.3795 q^{5} +(-11.4907 - 14.5246i) q^{7} +(-7.09031 - 26.0524i) q^{9} +O(q^{10})\) \(q+(-3.15513 + 4.12858i) q^{3} +18.3795 q^{5} +(-11.4907 - 14.5246i) q^{7} +(-7.09031 - 26.0524i) q^{9} -55.3725i q^{11} +(-68.0763 + 39.3039i) q^{13} +(-57.9898 + 75.8813i) q^{15} +(-9.99021 - 17.3036i) q^{17} +(-16.7611 - 9.67700i) q^{19} +(96.2206 - 1.61341i) q^{21} -124.143i q^{23} +212.807 q^{25} +(129.930 + 52.9258i) q^{27} +(-48.2851 - 27.8774i) q^{29} +(-275.387 - 158.995i) q^{31} +(228.610 + 174.707i) q^{33} +(-211.194 - 266.955i) q^{35} +(20.5303 - 35.5595i) q^{37} +(52.5205 - 405.067i) q^{39} +(53.1315 + 92.0264i) q^{41} +(279.238 - 483.655i) q^{43} +(-130.317 - 478.831i) q^{45} +(73.9580 + 128.099i) q^{47} +(-78.9267 + 333.796i) q^{49} +(102.959 + 13.3496i) q^{51} +(570.090 - 329.141i) q^{53} -1017.72i q^{55} +(92.8356 - 38.6672i) q^{57} +(-284.255 + 492.343i) q^{59} +(-122.068 + 70.4757i) q^{61} +(-296.927 + 402.345i) q^{63} +(-1251.21 + 722.387i) q^{65} +(147.142 - 254.857i) q^{67} +(512.535 + 391.688i) q^{69} +602.374i q^{71} +(541.701 - 312.751i) q^{73} +(-671.434 + 878.591i) q^{75} +(-804.262 + 636.270i) q^{77} +(267.022 + 462.495i) q^{79} +(-628.455 + 369.439i) q^{81} +(352.414 - 610.399i) q^{83} +(-183.615 - 318.031i) q^{85} +(267.440 - 111.392i) q^{87} +(38.6630 - 66.9663i) q^{89} +(1353.12 + 537.150i) q^{91} +(1525.30 - 635.307i) q^{93} +(-308.060 - 177.859i) q^{95} +(-382.120 - 220.617i) q^{97} +(-1442.59 + 392.608i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 6 q^{7} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 6 q^{7} - 30 q^{9} + 36 q^{13} + 66 q^{15} + 72 q^{17} + 126 q^{21} + 1200 q^{25} + 396 q^{27} + 42 q^{29} - 90 q^{31} + 108 q^{33} - 390 q^{35} + 84 q^{37} + 1014 q^{39} + 618 q^{41} - 42 q^{43} - 1014 q^{45} + 198 q^{47} - 276 q^{49} + 408 q^{51} + 1620 q^{53} + 492 q^{57} + 750 q^{59} - 1314 q^{61} + 1542 q^{63} + 564 q^{65} + 294 q^{67} + 924 q^{69} - 1410 q^{75} - 2448 q^{77} - 804 q^{79} - 666 q^{81} - 360 q^{85} + 1788 q^{87} - 1722 q^{89} + 540 q^{91} + 1128 q^{93} - 2946 q^{95} + 792 q^{97} - 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) −3.15513 + 4.12858i −0.607205 + 0.794545i
\(4\) 0 0
\(5\) 18.3795 1.64392 0.821958 0.569549i \(-0.192882\pi\)
0.821958 + 0.569549i \(0.192882\pi\)
\(6\) 0 0
\(7\) −11.4907 14.5246i −0.620441 0.784253i
\(8\) 0 0
\(9\) −7.09031 26.0524i −0.262604 0.964904i
\(10\) 0 0
\(11\) 55.3725i 1.51777i −0.651226 0.758883i \(-0.725745\pi\)
0.651226 0.758883i \(-0.274255\pi\)
\(12\) 0 0
\(13\) −68.0763 + 39.3039i −1.45238 + 0.838534i −0.998616 0.0525871i \(-0.983253\pi\)
−0.453766 + 0.891121i \(0.649920\pi\)
\(14\) 0 0
\(15\) −57.9898 + 75.8813i −0.998193 + 1.30616i
\(16\) 0 0
\(17\) −9.99021 17.3036i −0.142528 0.246866i 0.785920 0.618329i \(-0.212190\pi\)
−0.928448 + 0.371462i \(0.878857\pi\)
\(18\) 0 0
\(19\) −16.7611 9.67700i −0.202382 0.116845i 0.395384 0.918516i \(-0.370611\pi\)
−0.597766 + 0.801671i \(0.703945\pi\)
\(20\) 0 0
\(21\) 96.2206 1.61341i 0.999859 0.0167654i
\(22\) 0 0
\(23\) 124.143i 1.12546i −0.826640 0.562731i \(-0.809751\pi\)
0.826640 0.562731i \(-0.190249\pi\)
\(24\) 0 0
\(25\) 212.807 1.70246
\(26\) 0 0
\(27\) 129.930 + 52.9258i 0.926114 + 0.377243i
\(28\) 0 0
\(29\) −48.2851 27.8774i −0.309183 0.178507i 0.337378 0.941369i \(-0.390460\pi\)
−0.646561 + 0.762862i \(0.723793\pi\)
\(30\) 0 0
\(31\) −275.387 158.995i −1.59551 0.921170i −0.992337 0.123562i \(-0.960568\pi\)
−0.603176 0.797608i \(-0.706098\pi\)
\(32\) 0 0
\(33\) 228.610 + 174.707i 1.20593 + 0.921596i
\(34\) 0 0
\(35\) −211.194 266.955i −1.01995 1.28925i
\(36\) 0 0
\(37\) 20.5303 35.5595i 0.0912205 0.157999i −0.816804 0.576915i \(-0.804257\pi\)
0.908025 + 0.418916i \(0.137590\pi\)
\(38\) 0 0
\(39\) 52.5205 405.067i 0.215641 1.66315i
\(40\) 0 0
\(41\) 53.1315 + 92.0264i 0.202384 + 0.350539i 0.949296 0.314383i \(-0.101798\pi\)
−0.746912 + 0.664923i \(0.768464\pi\)
\(42\) 0 0
\(43\) 279.238 483.655i 0.990312 1.71527i 0.374898 0.927066i \(-0.377678\pi\)
0.615414 0.788204i \(-0.288989\pi\)
\(44\) 0 0
\(45\) −130.317 478.831i −0.431699 1.58622i
\(46\) 0 0
\(47\) 73.9580 + 128.099i 0.229529 + 0.397556i 0.957669 0.287873i \(-0.0929480\pi\)
−0.728139 + 0.685429i \(0.759615\pi\)
\(48\) 0 0
\(49\) −78.9267 + 333.796i −0.230107 + 0.973165i
\(50\) 0 0
\(51\) 102.959 + 13.3496i 0.282690 + 0.0366533i
\(52\) 0 0
\(53\) 570.090 329.141i 1.47751 0.853038i 0.477829 0.878453i \(-0.341424\pi\)
0.999677 + 0.0254144i \(0.00809053\pi\)
\(54\) 0 0
\(55\) 1017.72i 2.49508i
\(56\) 0 0
\(57\) 92.8356 38.6672i 0.215726 0.0898525i
\(58\) 0 0
\(59\) −284.255 + 492.343i −0.627234 + 1.08640i 0.360871 + 0.932616i \(0.382480\pi\)
−0.988104 + 0.153785i \(0.950854\pi\)
\(60\) 0 0
\(61\) −122.068 + 70.4757i −0.256215 + 0.147926i −0.622607 0.782535i \(-0.713926\pi\)
0.366391 + 0.930461i \(0.380593\pi\)
\(62\) 0 0
\(63\) −296.927 + 402.345i −0.593799 + 0.804614i
\(64\) 0 0
\(65\) −1251.21 + 722.387i −2.38759 + 1.37848i
\(66\) 0 0
\(67\) 147.142 254.857i 0.268302 0.464712i −0.700122 0.714023i \(-0.746871\pi\)
0.968423 + 0.249312i \(0.0802043\pi\)
\(68\) 0 0
\(69\) 512.535 + 391.688i 0.894231 + 0.683386i
\(70\) 0 0
\(71\) 602.374i 1.00688i 0.864030 + 0.503441i \(0.167933\pi\)
−0.864030 + 0.503441i \(0.832067\pi\)
\(72\) 0 0
\(73\) 541.701 312.751i 0.868510 0.501435i 0.00165750 0.999999i \(-0.499472\pi\)
0.866853 + 0.498564i \(0.166139\pi\)
\(74\) 0 0
\(75\) −671.434 + 878.591i −1.03374 + 1.35268i
\(76\) 0 0
\(77\) −804.262 + 636.270i −1.19031 + 0.941684i
\(78\) 0 0
\(79\) 267.022 + 462.495i 0.380282 + 0.658668i 0.991102 0.133101i \(-0.0424935\pi\)
−0.610820 + 0.791769i \(0.709160\pi\)
\(80\) 0 0
\(81\) −628.455 + 369.439i −0.862078 + 0.506776i
\(82\) 0 0
\(83\) 352.414 610.399i 0.466054 0.807229i −0.533195 0.845993i \(-0.679009\pi\)
0.999248 + 0.0387640i \(0.0123420\pi\)
\(84\) 0 0
\(85\) −183.615 318.031i −0.234305 0.405827i
\(86\) 0 0
\(87\) 267.440 111.392i 0.329570 0.137270i
\(88\) 0 0
\(89\) 38.6630 66.9663i 0.0460480 0.0797574i −0.842083 0.539348i \(-0.818671\pi\)
0.888131 + 0.459591i \(0.152004\pi\)
\(90\) 0 0
\(91\) 1353.12 + 537.150i 1.55874 + 0.618776i
\(92\) 0 0
\(93\) 1525.30 635.307i 1.70071 0.708368i
\(94\) 0 0
\(95\) −308.060 177.859i −0.332698 0.192083i
\(96\) 0 0
\(97\) −382.120 220.617i −0.399984 0.230931i 0.286493 0.958082i \(-0.407510\pi\)
−0.686477 + 0.727151i \(0.740844\pi\)
\(98\) 0 0
\(99\) −1442.59 + 392.608i −1.46450 + 0.398572i
\(100\) 0 0
\(101\) −1118.35 −1.10179 −0.550893 0.834576i \(-0.685713\pi\)
−0.550893 + 0.834576i \(0.685713\pi\)
\(102\) 0 0
\(103\) 814.783i 0.779446i 0.920932 + 0.389723i \(0.127429\pi\)
−0.920932 + 0.389723i \(0.872571\pi\)
\(104\) 0 0
\(105\) 1768.49 29.6536i 1.64368 0.0275609i
\(106\) 0 0
\(107\) −1320.94 762.645i −1.19346 0.689044i −0.234370 0.972147i \(-0.575303\pi\)
−0.959089 + 0.283103i \(0.908636\pi\)
\(108\) 0 0
\(109\) 101.432 + 175.685i 0.0891319 + 0.154381i 0.907144 0.420819i \(-0.138257\pi\)
−0.818013 + 0.575200i \(0.804924\pi\)
\(110\) 0 0
\(111\) 82.0345 + 196.956i 0.0701475 + 0.168416i
\(112\) 0 0
\(113\) −75.2101 + 43.4226i −0.0626121 + 0.0361491i −0.530979 0.847385i \(-0.678176\pi\)
0.468367 + 0.883534i \(0.344842\pi\)
\(114\) 0 0
\(115\) 2281.69i 1.85016i
\(116\) 0 0
\(117\) 1506.64 + 1494.87i 1.19051 + 1.18121i
\(118\) 0 0
\(119\) −136.532 + 343.934i −0.105175 + 0.264944i
\(120\) 0 0
\(121\) −1735.11 −1.30362
\(122\) 0 0
\(123\) −547.575 70.9978i −0.401408 0.0520460i
\(124\) 0 0
\(125\) 1613.85 1.15478
\(126\) 0 0
\(127\) −486.709 −0.340066 −0.170033 0.985438i \(-0.554388\pi\)
−0.170033 + 0.985438i \(0.554388\pi\)
\(128\) 0 0
\(129\) 1115.77 + 2678.85i 0.761538 + 1.82837i
\(130\) 0 0
\(131\) 801.955 0.534864 0.267432 0.963577i \(-0.413825\pi\)
0.267432 + 0.963577i \(0.413825\pi\)
\(132\) 0 0
\(133\) 52.0423 + 354.643i 0.0339296 + 0.231214i
\(134\) 0 0
\(135\) 2388.06 + 972.751i 1.52245 + 0.620156i
\(136\) 0 0
\(137\) 918.829i 0.572999i 0.958080 + 0.286500i \(0.0924917\pi\)
−0.958080 + 0.286500i \(0.907508\pi\)
\(138\) 0 0
\(139\) 159.738 92.2248i 0.0974734 0.0562763i −0.450471 0.892791i \(-0.648744\pi\)
0.547944 + 0.836515i \(0.315411\pi\)
\(140\) 0 0
\(141\) −762.213 98.8276i −0.455248 0.0590268i
\(142\) 0 0
\(143\) 2176.35 + 3769.56i 1.27270 + 2.20438i
\(144\) 0 0
\(145\) −887.457 512.374i −0.508271 0.293450i
\(146\) 0 0
\(147\) −1129.08 1379.02i −0.633502 0.773741i
\(148\) 0 0
\(149\) 2240.73i 1.23200i −0.787748 0.615998i \(-0.788753\pi\)
0.787748 0.615998i \(-0.211247\pi\)
\(150\) 0 0
\(151\) −2129.10 −1.14744 −0.573721 0.819051i \(-0.694501\pi\)
−0.573721 + 0.819051i \(0.694501\pi\)
\(152\) 0 0
\(153\) −379.965 + 382.957i −0.200774 + 0.202354i
\(154\) 0 0
\(155\) −5061.48 2922.25i −2.62289 1.51433i
\(156\) 0 0
\(157\) 2403.02 + 1387.38i 1.22154 + 0.705256i 0.965246 0.261343i \(-0.0841653\pi\)
0.256294 + 0.966599i \(0.417499\pi\)
\(158\) 0 0
\(159\) −439.821 + 3392.14i −0.219371 + 1.69191i
\(160\) 0 0
\(161\) −1803.13 + 1426.49i −0.882648 + 0.698282i
\(162\) 0 0
\(163\) −536.860 + 929.870i −0.257976 + 0.446828i −0.965700 0.259661i \(-0.916389\pi\)
0.707723 + 0.706490i \(0.249722\pi\)
\(164\) 0 0
\(165\) 4201.74 + 3211.04i 1.98245 + 1.51503i
\(166\) 0 0
\(167\) 22.8325 + 39.5471i 0.0105798 + 0.0183248i 0.871267 0.490810i \(-0.163299\pi\)
−0.860687 + 0.509134i \(0.829966\pi\)
\(168\) 0 0
\(169\) 1991.09 3448.67i 0.906277 1.56972i
\(170\) 0 0
\(171\) −133.268 + 505.279i −0.0595980 + 0.225963i
\(172\) 0 0
\(173\) 454.541 + 787.289i 0.199758 + 0.345991i 0.948450 0.316927i \(-0.102651\pi\)
−0.748692 + 0.662918i \(0.769318\pi\)
\(174\) 0 0
\(175\) −2445.31 3090.93i −1.05627 1.33516i
\(176\) 0 0
\(177\) −1135.82 2726.97i −0.482335 1.15803i
\(178\) 0 0
\(179\) −3140.15 + 1812.96i −1.31120 + 0.757024i −0.982296 0.187338i \(-0.940014\pi\)
−0.328909 + 0.944362i \(0.606681\pi\)
\(180\) 0 0
\(181\) 1659.16i 0.681351i −0.940181 0.340676i \(-0.889344\pi\)
0.940181 0.340676i \(-0.110656\pi\)
\(182\) 0 0
\(183\) 94.1743 726.325i 0.0380414 0.293396i
\(184\) 0 0
\(185\) 377.337 653.567i 0.149959 0.259736i
\(186\) 0 0
\(187\) −958.141 + 553.183i −0.374686 + 0.216325i
\(188\) 0 0
\(189\) −724.267 2495.34i −0.278744 0.960365i
\(190\) 0 0
\(191\) −1333.96 + 770.160i −0.505349 + 0.291763i −0.730920 0.682463i \(-0.760909\pi\)
0.225571 + 0.974227i \(0.427575\pi\)
\(192\) 0 0
\(193\) 1450.05 2511.55i 0.540811 0.936713i −0.458046 0.888928i \(-0.651451\pi\)
0.998858 0.0477845i \(-0.0152161\pi\)
\(194\) 0 0
\(195\) 965.302 7444.95i 0.354496 2.73407i
\(196\) 0 0
\(197\) 2096.85i 0.758347i −0.925326 0.379173i \(-0.876208\pi\)
0.925326 0.379173i \(-0.123792\pi\)
\(198\) 0 0
\(199\) 3937.78 2273.48i 1.40272 0.809862i 0.408051 0.912959i \(-0.366209\pi\)
0.994671 + 0.103097i \(0.0328752\pi\)
\(200\) 0 0
\(201\) 587.945 + 1411.59i 0.206321 + 0.495353i
\(202\) 0 0
\(203\) 149.923 + 1021.65i 0.0518351 + 0.353231i
\(204\) 0 0
\(205\) 976.532 + 1691.40i 0.332702 + 0.576257i
\(206\) 0 0
\(207\) −3234.23 + 880.214i −1.08596 + 0.295551i
\(208\) 0 0
\(209\) −535.840 + 928.102i −0.177344 + 0.307168i
\(210\) 0 0
\(211\) 1506.32 + 2609.03i 0.491467 + 0.851245i 0.999952 0.00982544i \(-0.00312758\pi\)
−0.508485 + 0.861071i \(0.669794\pi\)
\(212\) 0 0
\(213\) −2486.95 1900.57i −0.800013 0.611384i
\(214\) 0 0
\(215\) 5132.26 8889.34i 1.62799 2.81976i
\(216\) 0 0
\(217\) 855.062 + 5826.84i 0.267490 + 1.82282i
\(218\) 0 0
\(219\) −417.919 + 3223.22i −0.128951 + 0.994545i
\(220\) 0 0
\(221\) 1360.19 + 785.308i 0.414012 + 0.239030i
\(222\) 0 0
\(223\) −230.173 132.890i −0.0691188 0.0399058i 0.465042 0.885288i \(-0.346039\pi\)
−0.534161 + 0.845383i \(0.679372\pi\)
\(224\) 0 0
\(225\) −1508.87 5544.14i −0.447072 1.64271i
\(226\) 0 0
\(227\) −1158.84 −0.338834 −0.169417 0.985544i \(-0.554188\pi\)
−0.169417 + 0.985544i \(0.554188\pi\)
\(228\) 0 0
\(229\) 1142.85i 0.329790i 0.986311 + 0.164895i \(0.0527285\pi\)
−0.986311 + 0.164895i \(0.947272\pi\)
\(230\) 0 0
\(231\) −89.3383 5327.97i −0.0254460 1.51755i
\(232\) 0 0
\(233\) 4943.19 + 2853.95i 1.38987 + 0.802440i 0.993300 0.115565i \(-0.0368680\pi\)
0.396567 + 0.918006i \(0.370201\pi\)
\(234\) 0 0
\(235\) 1359.31 + 2354.40i 0.377327 + 0.653549i
\(236\) 0 0
\(237\) −2751.94 356.812i −0.754251 0.0977952i
\(238\) 0 0
\(239\) 310.843 179.466i 0.0841288 0.0485718i −0.457345 0.889289i \(-0.651200\pi\)
0.541474 + 0.840717i \(0.317866\pi\)
\(240\) 0 0
\(241\) 4439.82i 1.18670i −0.804946 0.593349i \(-0.797806\pi\)
0.804946 0.593349i \(-0.202194\pi\)
\(242\) 0 0
\(243\) 457.597 3760.25i 0.120802 0.992677i
\(244\) 0 0
\(245\) −1450.64 + 6135.01i −0.378277 + 1.59980i
\(246\) 0 0
\(247\) 1521.38 0.391914
\(248\) 0 0
\(249\) 1408.17 + 3380.86i 0.358390 + 0.860454i
\(250\) 0 0
\(251\) 3438.88 0.864782 0.432391 0.901686i \(-0.357670\pi\)
0.432391 + 0.901686i \(0.357670\pi\)
\(252\) 0 0
\(253\) −6874.12 −1.70819
\(254\) 0 0
\(255\) 1892.35 + 245.359i 0.464719 + 0.0602549i
\(256\) 0 0
\(257\) −1821.67 −0.442150 −0.221075 0.975257i \(-0.570957\pi\)
−0.221075 + 0.975257i \(0.570957\pi\)
\(258\) 0 0
\(259\) −752.395 + 110.411i −0.180508 + 0.0264887i
\(260\) 0 0
\(261\) −383.917 + 1455.60i −0.0910493 + 0.345209i
\(262\) 0 0
\(263\) 4243.16i 0.994845i −0.867508 0.497422i \(-0.834280\pi\)
0.867508 0.497422i \(-0.165720\pi\)
\(264\) 0 0
\(265\) 10478.0 6049.46i 2.42889 1.40232i
\(266\) 0 0
\(267\) 154.489 + 370.910i 0.0354103 + 0.0850163i
\(268\) 0 0
\(269\) 1538.72 + 2665.14i 0.348763 + 0.604075i 0.986030 0.166568i \(-0.0532685\pi\)
−0.637267 + 0.770643i \(0.719935\pi\)
\(270\) 0 0
\(271\) −623.012 359.696i −0.139651 0.0806273i 0.428547 0.903520i \(-0.359026\pi\)
−0.568197 + 0.822892i \(0.692359\pi\)
\(272\) 0 0
\(273\) −6486.93 + 3891.68i −1.43812 + 0.862766i
\(274\) 0 0
\(275\) 11783.7i 2.58393i
\(276\) 0 0
\(277\) −911.179 −0.197644 −0.0988221 0.995105i \(-0.531507\pi\)
−0.0988221 + 0.995105i \(0.531507\pi\)
\(278\) 0 0
\(279\) −2189.61 + 8301.81i −0.469852 + 1.78142i
\(280\) 0 0
\(281\) 4142.21 + 2391.50i 0.879372 + 0.507705i 0.870451 0.492255i \(-0.163827\pi\)
0.00892043 + 0.999960i \(0.497161\pi\)
\(282\) 0 0
\(283\) −3471.59 2004.33i −0.729204 0.421006i 0.0889266 0.996038i \(-0.471656\pi\)
−0.818131 + 0.575032i \(0.804990\pi\)
\(284\) 0 0
\(285\) 1706.27 710.684i 0.354635 0.147710i
\(286\) 0 0
\(287\) 726.126 1829.16i 0.149344 0.376209i
\(288\) 0 0
\(289\) 2256.89 3909.05i 0.459371 0.795654i
\(290\) 0 0
\(291\) 2116.48 881.538i 0.426357 0.177583i
\(292\) 0 0
\(293\) −3758.85 6510.51i −0.749468 1.29812i −0.948078 0.318038i \(-0.896976\pi\)
0.198610 0.980079i \(-0.436357\pi\)
\(294\) 0 0
\(295\) −5224.47 + 9049.04i −1.03112 + 1.78595i
\(296\) 0 0
\(297\) 2930.63 7194.56i 0.572567 1.40563i
\(298\) 0 0
\(299\) 4879.31 + 8451.21i 0.943738 + 1.63460i
\(300\) 0 0
\(301\) −10233.5 + 1501.72i −1.95964 + 0.287568i
\(302\) 0 0
\(303\) 3528.55 4617.21i 0.669009 0.875418i
\(304\) 0 0
\(305\) −2243.54 + 1295.31i −0.421196 + 0.243178i
\(306\) 0 0
\(307\) 6149.80i 1.14328i 0.820504 + 0.571641i \(0.193693\pi\)
−0.820504 + 0.571641i \(0.806307\pi\)
\(308\) 0 0
\(309\) −3363.90 2570.75i −0.619305 0.473284i
\(310\) 0 0
\(311\) 1059.67 1835.41i 0.193211 0.334650i −0.753102 0.657904i \(-0.771443\pi\)
0.946312 + 0.323253i \(0.104777\pi\)
\(312\) 0 0
\(313\) −126.231 + 72.8795i −0.0227955 + 0.0131610i −0.511354 0.859370i \(-0.670856\pi\)
0.488559 + 0.872531i \(0.337523\pi\)
\(314\) 0 0
\(315\) −5457.38 + 7394.91i −0.976155 + 1.32272i
\(316\) 0 0
\(317\) 2257.86 1303.58i 0.400044 0.230966i −0.286459 0.958093i \(-0.592478\pi\)
0.686503 + 0.727127i \(0.259145\pi\)
\(318\) 0 0
\(319\) −1543.64 + 2673.67i −0.270932 + 0.469268i
\(320\) 0 0
\(321\) 7316.38 3047.36i 1.27215 0.529866i
\(322\) 0 0
\(323\) 386.701i 0.0666150i
\(324\) 0 0
\(325\) −14487.1 + 8364.15i −2.47262 + 1.42757i
\(326\) 0 0
\(327\) −1045.36 135.540i −0.176784 0.0229216i
\(328\) 0 0
\(329\) 1010.75 2546.16i 0.169376 0.426669i
\(330\) 0 0
\(331\) 5445.02 + 9431.04i 0.904185 + 1.56609i 0.822008 + 0.569476i \(0.192854\pi\)
0.0821769 + 0.996618i \(0.473813\pi\)
\(332\) 0 0
\(333\) −1071.98 282.735i −0.176408 0.0465279i
\(334\) 0 0
\(335\) 2704.39 4684.15i 0.441065 0.763947i
\(336\) 0 0
\(337\) 1171.50 + 2029.09i 0.189364 + 0.327987i 0.945038 0.326960i \(-0.106024\pi\)
−0.755675 + 0.654947i \(0.772691\pi\)
\(338\) 0 0
\(339\) 58.0241 447.514i 0.00929628 0.0716981i
\(340\) 0 0
\(341\) −8803.93 + 15248.8i −1.39812 + 2.42162i
\(342\) 0 0
\(343\) 5755.17 2689.18i 0.905976 0.423329i
\(344\) 0 0
\(345\) 9420.15 + 7199.04i 1.47004 + 1.12343i
\(346\) 0 0
\(347\) −3380.66 1951.83i −0.523007 0.301958i 0.215157 0.976579i \(-0.430974\pi\)
−0.738164 + 0.674621i \(0.764307\pi\)
\(348\) 0 0
\(349\) −1985.23 1146.17i −0.304490 0.175797i 0.339968 0.940437i \(-0.389584\pi\)
−0.644458 + 0.764640i \(0.722917\pi\)
\(350\) 0 0
\(351\) −10925.4 + 1503.77i −1.66140 + 0.228676i
\(352\) 0 0
\(353\) 7745.90 1.16791 0.583956 0.811785i \(-0.301504\pi\)
0.583956 + 0.811785i \(0.301504\pi\)
\(354\) 0 0
\(355\) 11071.3i 1.65523i
\(356\) 0 0
\(357\) −989.182 1648.84i −0.146647 0.244442i
\(358\) 0 0
\(359\) 5235.74 + 3022.86i 0.769727 + 0.444402i 0.832777 0.553608i \(-0.186749\pi\)
−0.0630502 + 0.998010i \(0.520083\pi\)
\(360\) 0 0
\(361\) −3242.21 5615.67i −0.472694 0.818731i
\(362\) 0 0
\(363\) 5474.51 7163.55i 0.791562 1.03578i
\(364\) 0 0
\(365\) 9956.20 5748.22i 1.42776 0.824316i
\(366\) 0 0
\(367\) 10933.5i 1.55510i 0.628820 + 0.777551i \(0.283538\pi\)
−0.628820 + 0.777551i \(0.716462\pi\)
\(368\) 0 0
\(369\) 2020.79 2036.70i 0.285090 0.287334i
\(370\) 0 0
\(371\) −11331.4 4498.24i −1.58570 0.629480i
\(372\) 0 0
\(373\) 2865.12 0.397722 0.198861 0.980028i \(-0.436276\pi\)
0.198861 + 0.980028i \(0.436276\pi\)
\(374\) 0 0
\(375\) −5091.92 + 6662.92i −0.701188 + 0.917525i
\(376\) 0 0
\(377\) 4382.76 0.598737
\(378\) 0 0
\(379\) −2071.78 −0.280792 −0.140396 0.990095i \(-0.544838\pi\)
−0.140396 + 0.990095i \(0.544838\pi\)
\(380\) 0 0
\(381\) 1535.63 2009.42i 0.206490 0.270198i
\(382\) 0 0
\(383\) 7428.00 0.991001 0.495500 0.868608i \(-0.334985\pi\)
0.495500 + 0.868608i \(0.334985\pi\)
\(384\) 0 0
\(385\) −14782.0 + 11694.3i −1.95678 + 1.54805i
\(386\) 0 0
\(387\) −14580.2 3845.56i −1.91513 0.505118i
\(388\) 0 0
\(389\) 7584.01i 0.988496i 0.869321 + 0.494248i \(0.164556\pi\)
−0.869321 + 0.494248i \(0.835444\pi\)
\(390\) 0 0
\(391\) −2148.12 + 1240.22i −0.277839 + 0.160410i
\(392\) 0 0
\(393\) −2530.27 + 3310.94i −0.324772 + 0.424973i
\(394\) 0 0
\(395\) 4907.74 + 8500.45i 0.625152 + 1.08279i
\(396\) 0 0
\(397\) 1031.73 + 595.671i 0.130431 + 0.0753044i 0.563796 0.825914i \(-0.309340\pi\)
−0.433365 + 0.901219i \(0.642674\pi\)
\(398\) 0 0
\(399\) −1628.37 904.084i −0.204312 0.113436i
\(400\) 0 0
\(401\) 6457.05i 0.804114i 0.915615 + 0.402057i \(0.131705\pi\)
−0.915615 + 0.402057i \(0.868295\pi\)
\(402\) 0 0
\(403\) 24996.4 3.08973
\(404\) 0 0
\(405\) −11550.7 + 6790.12i −1.41718 + 0.833096i
\(406\) 0 0
\(407\) −1969.02 1136.81i −0.239805 0.138451i
\(408\) 0 0
\(409\) −1642.41 948.247i −0.198563 0.114640i 0.397422 0.917636i \(-0.369905\pi\)
−0.595985 + 0.802996i \(0.703238\pi\)
\(410\) 0 0
\(411\) −3793.46 2899.03i −0.455274 0.347928i
\(412\) 0 0
\(413\) 10417.4 1528.70i 1.24117 0.182137i
\(414\) 0 0
\(415\) 6477.20 11218.8i 0.766153 1.32702i
\(416\) 0 0
\(417\) −123.237 + 950.473i −0.0144723 + 0.111618i
\(418\) 0 0
\(419\) −7002.37 12128.5i −0.816440 1.41412i −0.908289 0.418342i \(-0.862611\pi\)
0.0918495 0.995773i \(-0.470722\pi\)
\(420\) 0 0
\(421\) 4485.43 7769.00i 0.519256 0.899377i −0.480494 0.876998i \(-0.659543\pi\)
0.999750 0.0223789i \(-0.00712403\pi\)
\(422\) 0 0
\(423\) 2812.90 2835.04i 0.323328 0.325874i
\(424\) 0 0
\(425\) −2125.99 3682.32i −0.242648 0.420279i
\(426\) 0 0
\(427\) 2426.27 + 963.162i 0.274978 + 0.109159i
\(428\) 0 0
\(429\) −22429.6 2908.19i −2.52427 0.327293i
\(430\) 0 0
\(431\) 8722.70 5036.05i 0.974844 0.562826i 0.0741343 0.997248i \(-0.476381\pi\)
0.900709 + 0.434422i \(0.143047\pi\)
\(432\) 0 0
\(433\) 10574.2i 1.17358i −0.809738 0.586791i \(-0.800391\pi\)
0.809738 0.586791i \(-0.199609\pi\)
\(434\) 0 0
\(435\) 4915.42 2047.33i 0.541784 0.225660i
\(436\) 0 0
\(437\) −1201.33 + 2080.77i −0.131505 + 0.227773i
\(438\) 0 0
\(439\) 7842.45 4527.84i 0.852619 0.492260i −0.00891464 0.999960i \(-0.502838\pi\)
0.861534 + 0.507700i \(0.169504\pi\)
\(440\) 0 0
\(441\) 9255.79 310.486i 0.999438 0.0335262i
\(442\) 0 0
\(443\) −3071.48 + 1773.32i −0.329414 + 0.190187i −0.655581 0.755125i \(-0.727576\pi\)
0.326167 + 0.945312i \(0.394243\pi\)
\(444\) 0 0
\(445\) 710.608 1230.81i 0.0756989 0.131114i
\(446\) 0 0
\(447\) 9251.01 + 7069.78i 0.978876 + 0.748074i
\(448\) 0 0
\(449\) 200.727i 0.0210977i −0.999944 0.0105489i \(-0.996642\pi\)
0.999944 0.0105489i \(-0.00335787\pi\)
\(450\) 0 0
\(451\) 5095.73 2942.02i 0.532037 0.307172i
\(452\) 0 0
\(453\) 6717.59 8790.16i 0.696732 0.911694i
\(454\) 0 0
\(455\) 24869.7 + 9872.56i 2.56244 + 1.01722i
\(456\) 0 0
\(457\) 4292.02 + 7434.00i 0.439327 + 0.760936i 0.997638 0.0686955i \(-0.0218837\pi\)
−0.558311 + 0.829632i \(0.688550\pi\)
\(458\) 0 0
\(459\) −382.226 2776.99i −0.0388688 0.282394i
\(460\) 0 0
\(461\) −5807.63 + 10059.1i −0.586742 + 1.01627i 0.407913 + 0.913021i \(0.366257\pi\)
−0.994656 + 0.103247i \(0.967077\pi\)
\(462\) 0 0
\(463\) 5426.15 + 9398.36i 0.544653 + 0.943367i 0.998629 + 0.0523527i \(0.0166720\pi\)
−0.453976 + 0.891014i \(0.649995\pi\)
\(464\) 0 0
\(465\) 28034.3 11676.6i 2.79583 1.16450i
\(466\) 0 0
\(467\) −3071.32 + 5319.68i −0.304333 + 0.527120i −0.977113 0.212723i \(-0.931767\pi\)
0.672780 + 0.739843i \(0.265100\pi\)
\(468\) 0 0
\(469\) −5392.45 + 791.317i −0.530917 + 0.0779097i
\(470\) 0 0
\(471\) −13309.8 + 5543.67i −1.30208 + 0.542333i
\(472\) 0 0
\(473\) −26781.2 15462.1i −2.60338 1.50306i
\(474\) 0 0
\(475\) −3566.87 2059.34i −0.344546 0.198924i
\(476\) 0 0
\(477\) −12617.0 12518.5i −1.21110 1.20164i
\(478\) 0 0
\(479\) −15213.6 −1.45121 −0.725603 0.688113i \(-0.758439\pi\)
−0.725603 + 0.688113i \(0.758439\pi\)
\(480\) 0 0
\(481\) 3227.68i 0.305966i
\(482\) 0 0
\(483\) −200.293 11945.1i −0.0188689 1.12530i
\(484\) 0 0
\(485\) −7023.19 4054.84i −0.657540 0.379631i
\(486\) 0 0
\(487\) −780.525 1351.91i −0.0726262 0.125792i 0.827425 0.561576i \(-0.189805\pi\)
−0.900052 + 0.435783i \(0.856471\pi\)
\(488\) 0 0
\(489\) −2145.17 5150.33i −0.198381 0.476290i
\(490\) 0 0
\(491\) 5107.22 2948.66i 0.469421 0.271020i −0.246576 0.969123i \(-0.579306\pi\)
0.715997 + 0.698103i \(0.245972\pi\)
\(492\) 0 0
\(493\) 1114.00i 0.101769i
\(494\) 0 0
\(495\) −26514.1 + 7215.96i −2.40751 + 0.655219i
\(496\) 0 0
\(497\) 8749.22 6921.71i 0.789651 0.624710i
\(498\) 0 0
\(499\) −11691.6 −1.04888 −0.524438 0.851449i \(-0.675724\pi\)
−0.524438 + 0.851449i \(0.675724\pi\)
\(500\) 0 0
\(501\) −235.313 30.5103i −0.0209840 0.00272076i
\(502\) 0 0
\(503\) −2180.14 −0.193255 −0.0966277 0.995321i \(-0.530806\pi\)
−0.0966277 + 0.995321i \(0.530806\pi\)
\(504\) 0 0
\(505\) −20554.8 −1.81124
\(506\) 0 0
\(507\) 7955.96 + 19101.4i 0.696916 + 1.67322i
\(508\) 0 0
\(509\) −1595.05 −0.138899 −0.0694494 0.997585i \(-0.522124\pi\)
−0.0694494 + 0.997585i \(0.522124\pi\)
\(510\) 0 0
\(511\) −10767.1 4274.24i −0.932111 0.370022i
\(512\) 0 0
\(513\) −1665.61 2144.43i −0.143349 0.184559i
\(514\) 0 0
\(515\) 14975.3i 1.28134i
\(516\) 0 0
\(517\) 7093.16 4095.24i 0.603398 0.348372i
\(518\) 0 0
\(519\) −4684.52 607.389i −0.396200 0.0513707i
\(520\) 0 0
\(521\) −2974.63 5152.20i −0.250136 0.433248i 0.713427 0.700729i \(-0.247142\pi\)
−0.963563 + 0.267481i \(0.913809\pi\)
\(522\) 0 0
\(523\) 12106.0 + 6989.39i 1.01216 + 0.584369i 0.911822 0.410585i \(-0.134675\pi\)
0.100334 + 0.994954i \(0.468009\pi\)
\(524\) 0 0
\(525\) 20476.4 343.344i 1.70222 0.0285424i
\(526\) 0 0
\(527\) 6353.56i 0.525171i
\(528\) 0 0
\(529\) −3244.52 −0.266665
\(530\) 0 0
\(531\) 14842.2 + 3914.64i 1.21299 + 0.319927i
\(532\) 0 0
\(533\) −7233.99 4176.55i −0.587878 0.339412i
\(534\) 0 0
\(535\) −24278.3 14017.1i −1.96195 1.13273i
\(536\) 0 0
\(537\) 2422.60 18684.5i 0.194680 1.50148i
\(538\) 0 0
\(539\) 18483.1 + 4370.37i 1.47704 + 0.349249i
\(540\) 0 0
\(541\) 6174.78 10695.0i 0.490710 0.849935i −0.509232 0.860629i \(-0.670071\pi\)
0.999943 + 0.0106936i \(0.00340395\pi\)
\(542\) 0 0
\(543\) 6849.98 + 5234.87i 0.541364 + 0.413720i
\(544\) 0 0
\(545\) 1864.26 + 3229.00i 0.146525 + 0.253789i
\(546\) 0 0
\(547\) 8771.44 15192.6i 0.685630 1.18755i −0.287609 0.957748i \(-0.592860\pi\)
0.973238 0.229798i \(-0.0738064\pi\)
\(548\) 0 0
\(549\) 2701.56 + 2680.46i 0.210018 + 0.208377i
\(550\) 0 0
\(551\) 539.540 + 934.510i 0.0417154 + 0.0722531i
\(552\) 0 0
\(553\) 3649.28 9192.78i 0.280620 0.706902i
\(554\) 0 0
\(555\) 1507.76 + 3619.95i 0.115316 + 0.276862i
\(556\) 0 0
\(557\) −19166.8 + 11065.9i −1.45803 + 0.841794i −0.998914 0.0465824i \(-0.985167\pi\)
−0.459116 + 0.888376i \(0.651834\pi\)
\(558\) 0 0
\(559\) 43900.6i 3.32164i
\(560\) 0 0
\(561\) 739.200 5701.12i 0.0556311 0.429058i
\(562\) 0 0
\(563\) 4808.10 8327.88i 0.359924 0.623407i −0.628024 0.778194i \(-0.716136\pi\)
0.987948 + 0.154787i \(0.0494692\pi\)
\(564\) 0 0
\(565\) −1382.33 + 798.086i −0.102929 + 0.0594261i
\(566\) 0 0
\(567\) 12587.3 + 4882.92i 0.932309 + 0.361664i
\(568\) 0 0
\(569\) 9884.02 5706.54i 0.728225 0.420441i −0.0895477 0.995983i \(-0.528542\pi\)
0.817772 + 0.575542i \(0.195209\pi\)
\(570\) 0 0
\(571\) 9591.91 16613.7i 0.702993 1.21762i −0.264418 0.964408i \(-0.585180\pi\)
0.967411 0.253211i \(-0.0814868\pi\)
\(572\) 0 0
\(573\) 1029.14 7937.30i 0.0750312 0.578683i
\(574\) 0 0
\(575\) 26418.5i 1.91605i
\(576\) 0 0
\(577\) 9697.32 5598.75i 0.699661 0.403950i −0.107560 0.994199i \(-0.534304\pi\)
0.807221 + 0.590249i \(0.200970\pi\)
\(578\) 0 0
\(579\) 5794.06 + 13910.9i 0.415877 + 0.998476i
\(580\) 0 0
\(581\) −12915.3 + 1895.26i −0.922230 + 0.135333i
\(582\) 0 0
\(583\) −18225.4 31567.3i −1.29471 2.24251i
\(584\) 0 0
\(585\) 27691.4 + 27475.1i 1.95709 + 1.94180i
\(586\) 0 0
\(587\) 5285.80 9155.27i 0.371666 0.643745i −0.618156 0.786056i \(-0.712120\pi\)
0.989822 + 0.142311i \(0.0454532\pi\)
\(588\) 0 0
\(589\) 3077.18 + 5329.84i 0.215268 + 0.372856i
\(590\) 0 0
\(591\) 8657.01 + 6615.83i 0.602541 + 0.460472i
\(592\) 0 0
\(593\) 7667.36 13280.3i 0.530963 0.919655i −0.468384 0.883525i \(-0.655164\pi\)
0.999347 0.0361297i \(-0.0115029\pi\)
\(594\) 0 0
\(595\) −2509.40 + 6321.34i −0.172900 + 0.435546i
\(596\) 0 0
\(597\) −3037.97 + 23430.5i −0.208268 + 1.60628i
\(598\) 0 0
\(599\) −7727.40 4461.42i −0.527100 0.304321i 0.212735 0.977110i \(-0.431763\pi\)
−0.739835 + 0.672789i \(0.765096\pi\)
\(600\) 0 0
\(601\) −6221.25 3591.84i −0.422246 0.243784i 0.273792 0.961789i \(-0.411722\pi\)
−0.696038 + 0.718005i \(0.745055\pi\)
\(602\) 0 0
\(603\) −7682.91 2026.38i −0.518859 0.136850i
\(604\) 0 0
\(605\) −31890.6 −2.14303
\(606\) 0 0
\(607\) 5512.13i 0.368584i −0.982872 0.184292i \(-0.941001\pi\)
0.982872 0.184292i \(-0.0589992\pi\)
\(608\) 0 0
\(609\) −4691.00 2604.48i −0.312133 0.173298i
\(610\) 0 0
\(611\) −10069.6 5813.67i −0.666729 0.384936i
\(612\) 0 0
\(613\) −12611.1 21843.0i −0.830925 1.43920i −0.897306 0.441410i \(-0.854478\pi\)
0.0663806 0.997794i \(-0.478855\pi\)
\(614\) 0 0
\(615\) −10064.2 1304.91i −0.659881 0.0855592i
\(616\) 0 0
\(617\) 22013.2 12709.3i 1.43633 0.829268i 0.438741 0.898613i \(-0.355424\pi\)
0.997593 + 0.0693457i \(0.0220912\pi\)
\(618\) 0 0
\(619\) 10829.0i 0.703155i −0.936159 0.351577i \(-0.885645\pi\)
0.936159 0.351577i \(-0.114355\pi\)
\(620\) 0 0
\(621\) 6570.37 16129.9i 0.424573 1.04231i
\(622\) 0 0
\(623\) −1416.92 + 207.927i −0.0911200 + 0.0133715i
\(624\) 0 0
\(625\) 3060.98 0.195903
\(626\) 0 0
\(627\) −2141.10 5140.54i −0.136375 0.327422i
\(628\) 0 0
\(629\) −820.408 −0.0520061
\(630\) 0 0
\(631\) 25750.8 1.62460 0.812301 0.583239i \(-0.198215\pi\)
0.812301 + 0.583239i \(0.198215\pi\)
\(632\) 0 0
\(633\) −15524.2 2012.85i −0.974774 0.126388i
\(634\) 0 0
\(635\) −8945.48 −0.559040
\(636\) 0 0
\(637\) −7746.43 25825.7i −0.481828 1.60636i
\(638\) 0 0
\(639\) 15693.3 4271.02i 0.971544 0.264411i
\(640\) 0 0
\(641\) 7576.45i 0.466851i −0.972375 0.233426i \(-0.925006\pi\)
0.972375 0.233426i \(-0.0749936\pi\)
\(642\) 0 0
\(643\) 15581.8 8996.16i 0.955655 0.551748i 0.0608220 0.998149i \(-0.480628\pi\)
0.894833 + 0.446401i \(0.147294\pi\)
\(644\) 0 0
\(645\) 20507.4 + 49236.0i 1.25190 + 3.00568i
\(646\) 0 0
\(647\) 8457.22 + 14648.3i 0.513891 + 0.890086i 0.999870 + 0.0161153i \(0.00512987\pi\)
−0.485979 + 0.873971i \(0.661537\pi\)
\(648\) 0 0
\(649\) 27262.3 + 15739.9i 1.64890 + 0.951995i
\(650\) 0 0
\(651\) −26754.4 14854.2i −1.61073 0.894291i
\(652\) 0 0
\(653\) 17449.4i 1.04571i 0.852422 + 0.522854i \(0.175133\pi\)
−0.852422 + 0.522854i \(0.824867\pi\)
\(654\) 0 0
\(655\) 14739.6 0.879271
\(656\) 0 0
\(657\) −11988.7 11895.1i −0.711911 0.706350i
\(658\) 0 0
\(659\) −1730.83 999.298i −0.102312 0.0590700i 0.447971 0.894048i \(-0.352147\pi\)
−0.550283 + 0.834978i \(0.685480\pi\)
\(660\) 0 0
\(661\) −29006.3 16746.8i −1.70683 0.985440i −0.938428 0.345476i \(-0.887718\pi\)
−0.768404 0.639964i \(-0.778949\pi\)
\(662\) 0 0
\(663\) −7533.80 + 3137.92i −0.441310 + 0.183811i
\(664\) 0 0
\(665\) 956.513 + 6518.17i 0.0557774 + 0.380096i
\(666\) 0 0
\(667\) −3460.79 + 5994.26i −0.200903 + 0.347974i
\(668\) 0 0
\(669\) 1274.87 531.000i 0.0736762 0.0306870i
\(670\) 0 0
\(671\) 3902.42 + 6759.18i 0.224517 + 0.388875i
\(672\) 0 0
\(673\) −16342.6 + 28306.3i −0.936051 + 1.62129i −0.163303 + 0.986576i \(0.552215\pi\)
−0.772749 + 0.634712i \(0.781119\pi\)
\(674\) 0 0
\(675\) 27650.1 + 11263.0i 1.57667 + 0.642241i
\(676\) 0 0
\(677\) −3663.87 6346.00i −0.207997 0.360261i 0.743087 0.669195i \(-0.233361\pi\)
−0.951083 + 0.308934i \(0.900028\pi\)
\(678\) 0 0
\(679\) 1186.46 + 8085.19i 0.0670579 + 0.456968i
\(680\) 0 0
\(681\) 3656.31 4784.38i 0.205741 0.269219i
\(682\) 0 0
\(683\) 12311.2 7107.88i 0.689715 0.398207i −0.113790 0.993505i \(-0.536299\pi\)
0.803505 + 0.595298i \(0.202966\pi\)
\(684\) 0 0
\(685\) 16887.7i 0.941962i
\(686\) 0 0
\(687\) −4718.36 3605.85i −0.262033 0.200250i
\(688\) 0 0
\(689\) −25873.1 + 44813.5i −1.43060 + 2.47788i
\(690\) 0 0
\(691\) −12403.5 + 7161.17i −0.682854 + 0.394246i −0.800929 0.598759i \(-0.795661\pi\)
0.118076 + 0.993005i \(0.462328\pi\)
\(692\) 0 0
\(693\) 22278.8 + 16441.6i 1.22122 + 0.901248i
\(694\) 0 0
\(695\) 2935.91 1695.05i 0.160238 0.0925135i
\(696\) 0 0
\(697\) 1061.59 1838.73i 0.0576909 0.0999236i
\(698\) 0 0
\(699\) −27379.2 + 11403.8i −1.48151 + 0.617067i
\(700\) 0 0
\(701\) 3371.89i 0.181675i 0.995866 + 0.0908377i \(0.0289544\pi\)
−0.995866 + 0.0908377i \(0.971046\pi\)
\(702\) 0 0
\(703\) −688.219 + 397.344i −0.0369227 + 0.0213173i
\(704\) 0 0
\(705\) −14009.1 1816.40i −0.748389 0.0970351i
\(706\) 0 0
\(707\) 12850.7 + 16243.6i 0.683592 + 0.864079i
\(708\) 0 0
\(709\) 15358.4 + 26601.6i 0.813537 + 1.40909i 0.910374 + 0.413787i \(0.135794\pi\)
−0.0968372 + 0.995300i \(0.530873\pi\)
\(710\) 0 0
\(711\) 10155.8 10235.8i 0.535688 0.539905i
\(712\) 0 0
\(713\) −19738.1 + 34187.4i −1.03674 + 1.79569i
\(714\) 0 0
\(715\) 40000.4 + 69282.7i 2.09221 + 3.62381i
\(716\) 0 0
\(717\) −239.814 + 1849.58i −0.0124910 + 0.0963372i
\(718\) 0 0
\(719\) −5218.17 + 9038.14i −0.270661 + 0.468798i −0.969031 0.246939i \(-0.920575\pi\)
0.698371 + 0.715736i \(0.253909\pi\)
\(720\) 0 0
\(721\) 11834.4 9362.45i 0.611284 0.483600i
\(722\) 0 0
\(723\) 18330.1 + 14008.2i 0.942884 + 0.720568i
\(724\) 0 0
\(725\) −10275.4 5932.51i −0.526371 0.303901i
\(726\) 0 0
\(727\) −22053.0 12732.3i −1.12503 0.649538i −0.182352 0.983233i \(-0.558371\pi\)
−0.942681 + 0.333696i \(0.891704\pi\)
\(728\) 0 0
\(729\) 14080.7 + 13753.3i 0.715375 + 0.698741i
\(730\) 0 0
\(731\) −11158.6 −0.564590
\(732\) 0 0
\(733\) 10275.7i 0.517793i −0.965905 0.258896i \(-0.916641\pi\)
0.965905 0.258896i \(-0.0833588\pi\)
\(734\) 0 0
\(735\) −20751.9 25345.8i −1.04142 1.27197i
\(736\) 0 0
\(737\) −14112.1 8147.60i −0.705324 0.407219i
\(738\) 0 0
\(739\) −8232.96 14259.9i −0.409816 0.709823i 0.585053 0.810995i \(-0.301074\pi\)
−0.994869 + 0.101173i \(0.967741\pi\)
\(740\) 0 0
\(741\) −4800.14 + 6281.12i −0.237972 + 0.311394i
\(742\) 0 0
\(743\) −34426.3 + 19876.0i −1.69984 + 0.981400i −0.753932 + 0.656953i \(0.771845\pi\)
−0.945904 + 0.324448i \(0.894822\pi\)
\(744\) 0 0
\(745\) 41183.5i 2.02530i
\(746\) 0 0
\(747\) −18401.1 4853.31i −0.901285 0.237715i
\(748\) 0 0
\(749\) 4101.46 + 27949.5i 0.200085 + 1.36349i
\(750\) 0 0
\(751\) 18477.7 0.897818 0.448909 0.893578i \(-0.351813\pi\)
0.448909 + 0.893578i \(0.351813\pi\)
\(752\) 0 0
\(753\) −10850.1 + 14197.7i −0.525100 + 0.687109i
\(754\) 0 0
\(755\) −39131.9 −1.88630
\(756\) 0 0
\(757\) 14830.5 0.712053 0.356026 0.934476i \(-0.384131\pi\)
0.356026 + 0.934476i \(0.384131\pi\)
\(758\) 0 0
\(759\) 21688.7 28380.3i 1.03722 1.35723i
\(760\) 0 0
\(761\) −5974.38 −0.284587 −0.142294 0.989824i \(-0.545448\pi\)
−0.142294 + 0.989824i \(0.545448\pi\)
\(762\) 0 0
\(763\) 1386.22 3491.99i 0.0657728 0.165686i
\(764\) 0 0
\(765\) −6983.59 + 7038.56i −0.330055 + 0.332653i
\(766\) 0 0
\(767\) 44689.2i 2.10383i
\(768\) 0 0
\(769\) 16782.8 9689.54i 0.786999 0.454374i −0.0519057 0.998652i \(-0.516530\pi\)
0.838905 + 0.544278i \(0.183196\pi\)
\(770\) 0 0
\(771\) 5747.60 7520.91i 0.268476 0.351308i
\(772\) 0 0
\(773\) 8932.67 + 15471.8i 0.415635 + 0.719900i 0.995495 0.0948156i \(-0.0302261\pi\)
−0.579860 + 0.814716i \(0.696893\pi\)
\(774\) 0 0
\(775\) −58604.3 33835.2i −2.71629 1.56825i
\(776\) 0 0
\(777\) 1918.06 3454.68i 0.0885588 0.159506i
\(778\) 0 0
\(779\) 2056.61i 0.0945903i
\(780\) 0 0
\(781\) 33354.9 1.52821
\(782\) 0 0
\(783\) −4798.26 6177.64i −0.218998 0.281955i
\(784\) 0 0
\(785\) 44166.3 + 25499.5i 2.00811 + 1.15938i
\(786\) 0 0
\(787\) −10416.8 6014.14i −0.471816 0.272403i 0.245184 0.969477i \(-0.421152\pi\)
−0.716999 + 0.697074i \(0.754485\pi\)
\(788\) 0 0
\(789\) 17518.2 + 13387.7i 0.790449 + 0.604075i
\(790\) 0 0
\(791\) 1494.91 + 593.438i 0.0671972 + 0.0266754i
\(792\) 0 0
\(793\) 5539.94 9595.45i 0.248082 0.429691i
\(794\) 0 0
\(795\) −8083.70 + 62346.0i −0.360628 + 2.78136i
\(796\) 0 0
\(797\) 13117.1 + 22719.5i 0.582976 + 1.00974i 0.995124 + 0.0986270i \(0.0314451\pi\)
−0.412149 + 0.911117i \(0.635222\pi\)
\(798\) 0 0
\(799\) 1477.71 2559.47i 0.0654289 0.113326i
\(800\) 0 0
\(801\) −2018.76 532.452i −0.0890506 0.0234872i
\(802\) 0 0
\(803\) −17317.8 29995.3i −0.761061 1.31820i
\(804\) 0 0
\(805\) −33140.6 + 26218.3i −1.45100 + 1.14792i
\(806\) 0 0
\(807\) −15858.1 2056.14i −0.691736 0.0896895i
\(808\) 0 0
\(809\) −24743.3 + 14285.5i −1.07531 + 0.620832i −0.929628 0.368499i \(-0.879872\pi\)
−0.145684 + 0.989331i \(0.546538\pi\)
\(810\) 0 0
\(811\) 36790.7i 1.59297i −0.604661 0.796483i \(-0.706692\pi\)
0.604661 0.796483i \(-0.293308\pi\)
\(812\) 0 0
\(813\) 3450.72 1437.27i 0.148859 0.0620014i
\(814\) 0 0
\(815\) −9867.24 + 17090.6i −0.424091 + 0.734548i
\(816\) 0 0
\(817\) −9360.65 + 5404.38i −0.400842 + 0.231426i
\(818\) 0 0
\(819\) 4400.01 39060.5i 0.187728 1.66653i
\(820\) 0 0
\(821\) 23764.4 13720.4i 1.01021 0.583246i 0.0989586 0.995092i \(-0.468449\pi\)
0.911254 + 0.411845i \(0.135116\pi\)
\(822\) 0 0
\(823\) 14717.5 25491.5i 0.623354 1.07968i −0.365503 0.930810i \(-0.619103\pi\)
0.988857 0.148871i \(-0.0475638\pi\)
\(824\) 0 0
\(825\) 48649.8 + 37179.0i 2.05305 + 1.56898i
\(826\) 0 0
\(827\) 9108.48i 0.382990i 0.981494 + 0.191495i \(0.0613336\pi\)
−0.981494 + 0.191495i \(0.938666\pi\)
\(828\) 0 0
\(829\) 745.006 430.129i 0.0312124 0.0180205i −0.484313 0.874895i \(-0.660930\pi\)
0.515525 + 0.856875i \(0.327597\pi\)
\(830\) 0 0
\(831\) 2874.89 3761.88i 0.120011 0.157037i
\(832\) 0 0
\(833\) 6564.35 1968.98i 0.273039 0.0818980i
\(834\) 0 0
\(835\) 419.651 + 726.856i 0.0173923 + 0.0301244i
\(836\) 0 0
\(837\) −27366.1 35233.3i −1.13012 1.45501i
\(838\) 0 0
\(839\) −13869.3 + 24022.3i −0.570704 + 0.988488i 0.425790 + 0.904822i \(0.359996\pi\)
−0.996494 + 0.0836658i \(0.973337\pi\)
\(840\) 0 0
\(841\) −10640.2 18429.4i −0.436270 0.755643i
\(842\) 0 0
\(843\) −22942.7 + 9555.92i −0.937354 + 0.390419i
\(844\) 0 0
\(845\) 36595.3 63384.9i 1.48984 2.58048i
\(846\) 0 0
\(847\) 19937.7 + 25201.8i 0.808816 + 1.02237i
\(848\) 0 0
\(849\) 19228.3 8008.84i 0.777285 0.323749i
\(850\) 0 0
\(851\) −4414.47 2548.70i −0.177821 0.102665i
\(852\) 0 0
\(853\) 13605.1 + 7854.94i 0.546109 + 0.315296i 0.747551 0.664204i \(-0.231229\pi\)
−0.201442 + 0.979500i \(0.564563\pi\)
\(854\) 0 0
\(855\) −2449.40 + 9286.79i −0.0979740 + 0.371464i
\(856\) 0 0
\(857\) 41065.2 1.63683 0.818414 0.574629i \(-0.194854\pi\)
0.818414 + 0.574629i \(0.194854\pi\)
\(858\) 0 0
\(859\) 3839.19i 0.152493i −0.997089 0.0762466i \(-0.975706\pi\)
0.997089 0.0762466i \(-0.0242936\pi\)
\(860\) 0 0
\(861\) 5260.82 + 8769.11i 0.208233 + 0.347097i
\(862\) 0 0
\(863\) −24681.4 14249.8i −0.973540 0.562074i −0.0732263 0.997315i \(-0.523330\pi\)
−0.900314 + 0.435242i \(0.856663\pi\)
\(864\) 0 0
\(865\) 8354.26 + 14470.0i 0.328385 + 0.568780i
\(866\) 0 0
\(867\) 9018.04 + 21651.3i 0.353251 + 0.848117i
\(868\) 0 0
\(869\) 25609.5 14785.7i 0.999705 0.577180i
\(870\) 0 0
\(871\) 23132.9i 0.899919i
\(872\) 0 0
\(873\) −3038.26 + 11519.4i −0.117789 + 0.446589i
\(874\) 0 0
\(875\) −18544.3 23440.5i −0.716472 0.905640i
\(876\) 0 0
\(877\) −4008.65 −0.154347 −0.0771737 0.997018i \(-0.524590\pi\)
−0.0771737 + 0.997018i \(0.524590\pi\)
\(878\) 0 0
\(879\) 38738.8 + 5022.82i 1.48649 + 0.192737i
\(880\) 0 0
\(881\) 20416.0 0.780742 0.390371 0.920658i \(-0.372347\pi\)
0.390371 + 0.920658i \(0.372347\pi\)
\(882\) 0 0
\(883\) 16822.4 0.641133 0.320566 0.947226i \(-0.396127\pi\)
0.320566 + 0.947226i \(0.396127\pi\)
\(884\) 0 0
\(885\) −20875.8 50120.5i −0.792918 1.90371i
\(886\) 0 0
\(887\) 31455.3 1.19072 0.595358 0.803461i \(-0.297010\pi\)
0.595358 + 0.803461i \(0.297010\pi\)
\(888\) 0 0
\(889\) 5592.64 + 7069.24i 0.210991 + 0.266698i
\(890\) 0 0
\(891\) 20456.8 + 34799.1i 0.769167 + 1.30843i
\(892\) 0 0
\(893\) 2862.77i 0.107278i
\(894\) 0 0
\(895\) −57714.4 + 33321.4i −2.15551 + 1.24448i
\(896\) 0 0
\(897\) −50286.3 6520.06i −1.87181 0.242696i
\(898\) 0 0
\(899\) 8864.71 + 15354.1i 0.328871 + 0.569621i
\(900\) 0 0
\(901\) −11390.6 6576.38i −0.421173 0.243164i
\(902\) 0 0
\(903\) 26088.1 46988.0i 0.961415 1.73163i
\(904\) 0 0
\(905\) 30494.6i 1.12008i
\(906\) 0 0
\(907\) −216.082 −0.00791058 −0.00395529 0.999992i \(-0.501259\pi\)
−0.00395529 + 0.999992i \(0.501259\pi\)
\(908\) 0 0
\(909\) 7929.48 + 29135.8i 0.289333 + 1.06312i
\(910\) 0 0
\(911\) 9169.28 + 5293.88i 0.333471 + 0.192529i 0.657381 0.753558i \(-0.271664\pi\)
−0.323910 + 0.946088i \(0.604998\pi\)
\(912\) 0 0
\(913\) −33799.3 19514.0i −1.22518 0.707361i
\(914\) 0 0
\(915\) 1730.88 13349.5i 0.0625368 0.482318i
\(916\) 0 0
\(917\) −9215.04 11648.1i −0.331851 0.419469i
\(918\) 0 0
\(919\) −14001.8 + 24251.8i −0.502586 + 0.870504i 0.497410 + 0.867516i \(0.334285\pi\)
−0.999996 + 0.00298831i \(0.999049\pi\)
\(920\) 0 0
\(921\) −25389.9 19403.4i −0.908390 0.694207i
\(922\) 0 0
\(923\) −23675.6 41007.4i −0.844304 1.46238i
\(924\) 0 0
\(925\) 4368.99 7567.32i 0.155299 0.268986i
\(926\) 0 0
\(927\) 21227.1 5777.07i 0.752091 0.204686i
\(928\) 0 0
\(929\) 13648.2 + 23639.4i 0.482007 + 0.834860i 0.999787 0.0206539i \(-0.00657480\pi\)
−0.517780 + 0.855514i \(0.673241\pi\)
\(930\) 0 0
\(931\) 4553.04 4831.00i 0.160279 0.170064i
\(932\) 0 0
\(933\) 4234.21 + 10165.9i 0.148577 + 0.356716i
\(934\) 0 0
\(935\) −17610.2 + 10167.2i −0.615951 + 0.355620i
\(936\) 0 0
\(937\) 6443.03i 0.224637i 0.993672 + 0.112318i \(0.0358277\pi\)
−0.993672 + 0.112318i \(0.964172\pi\)
\(938\) 0 0
\(939\) 97.3864 751.098i 0.00338454 0.0261035i
\(940\) 0 0
\(941\) −17839.1 + 30898.2i −0.617999 + 1.07041i 0.371851 + 0.928292i \(0.378723\pi\)
−0.989850 + 0.142114i \(0.954610\pi\)
\(942\) 0 0
\(943\) 11424.5 6595.91i 0.394519 0.227776i
\(944\) 0 0
\(945\) −13311.7 45863.1i −0.458232 1.57876i
\(946\) 0 0
\(947\) −4451.12 + 2569.86i −0.152737 + 0.0881828i −0.574421 0.818560i \(-0.694773\pi\)
0.421684 + 0.906743i \(0.361439\pi\)
\(948\) 0 0
\(949\) −24584.7 + 42581.9i −0.840940 + 1.45655i
\(950\) 0 0
\(951\) −1741.93 + 13434.7i −0.0593962 + 0.458097i
\(952\) 0 0
\(953\) 12018.0i 0.408502i −0.978919 0.204251i \(-0.934524\pi\)
0.978919 0.204251i \(-0.0654758\pi\)
\(954\) 0 0
\(955\) −24517.5 + 14155.2i −0.830751 + 0.479634i
\(956\) 0 0
\(957\) −6168.05 14808.8i −0.208343 0.500210i
\(958\) 0 0
\(959\) 13345.6 10558.0i 0.449377 0.355512i
\(960\) 0 0
\(961\) 35663.1 + 61770.2i 1.19711 + 2.07345i
\(962\) 0 0
\(963\) −10502.9 + 39821.1i −0.351454 + 1.33252i
\(964\) 0 0
\(965\) 26651.2 46161.2i 0.889048 1.53988i
\(966\) 0 0
\(967\) 7474.18 + 12945.7i 0.248556 + 0.430511i 0.963125 0.269053i \(-0.0867108\pi\)
−0.714570 + 0.699564i \(0.753377\pi\)
\(968\) 0 0
\(969\) −1596.53 1220.09i −0.0529286 0.0404489i
\(970\) 0 0
\(971\) 21188.3 36699.3i 0.700274 1.21291i −0.268096 0.963392i \(-0.586395\pi\)
0.968370 0.249518i \(-0.0802721\pi\)
\(972\) 0 0
\(973\) −3175.03 1260.40i −0.104611 0.0415278i
\(974\) 0 0
\(975\) 11176.7 86201.2i 0.367120 2.83143i
\(976\) 0 0
\(977\) 13715.1 + 7918.44i 0.449116 + 0.259297i 0.707457 0.706757i \(-0.249842\pi\)
−0.258341 + 0.966054i \(0.583176\pi\)
\(978\) 0 0
\(979\) −3708.09 2140.87i −0.121053 0.0698901i
\(980\) 0 0
\(981\) 3857.82 3888.19i 0.125556 0.126545i
\(982\) 0 0
\(983\) 14254.8 0.462519 0.231260 0.972892i \(-0.425715\pi\)
0.231260 + 0.972892i \(0.425715\pi\)
\(984\) 0 0
\(985\) 38539.1i 1.24666i
\(986\) 0 0
\(987\) 7322.95 + 12206.4i 0.236162 + 0.393652i
\(988\) 0 0
\(989\) −60042.4 34665.5i −1.93047 1.11456i
\(990\) 0 0
\(991\) 1262.03 + 2185.91i 0.0404539 + 0.0700682i 0.885543 0.464557i \(-0.153786\pi\)
−0.845090 + 0.534625i \(0.820453\pi\)
\(992\) 0 0
\(993\) −56116.5 7275.99i −1.79336 0.232524i
\(994\) 0 0
\(995\) 72374.5 41785.5i 2.30596 1.33134i
\(996\) 0 0
\(997\) 50024.6i 1.58906i −0.607224 0.794531i \(-0.707717\pi\)
0.607224 0.794531i \(-0.292283\pi\)
\(998\) 0 0
\(999\) 4549.52 3533.67i 0.144085 0.111912i
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 252.4.bm.a.173.7 yes 48
3.2 odd 2 756.4.bm.a.89.2 48
7.3 odd 6 252.4.w.a.101.2 yes 48
9.4 even 3 756.4.w.a.341.2 48
9.5 odd 6 252.4.w.a.5.2 48
21.17 even 6 756.4.w.a.521.2 48
63.31 odd 6 756.4.bm.a.17.2 48
63.59 even 6 inner 252.4.bm.a.185.7 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.w.a.5.2 48 9.5 odd 6
252.4.w.a.101.2 yes 48 7.3 odd 6
252.4.bm.a.173.7 yes 48 1.1 even 1 trivial
252.4.bm.a.185.7 yes 48 63.59 even 6 inner
756.4.w.a.341.2 48 9.4 even 3
756.4.w.a.521.2 48 21.17 even 6
756.4.bm.a.17.2 48 63.31 odd 6
756.4.bm.a.89.2 48 3.2 odd 2