Properties

Label 5225.2.a.y.1.12
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.a (trivial)

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: yes
Analytic conductor: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(1.89451\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.89451 q^{2} -3.43936 q^{3} +1.58918 q^{4} -6.51591 q^{6} +4.54180 q^{7} -0.778295 q^{8} +8.82918 q^{9} +O(q^{10})\) \(q+1.89451 q^{2} -3.43936 q^{3} +1.58918 q^{4} -6.51591 q^{6} +4.54180 q^{7} -0.778295 q^{8} +8.82918 q^{9} +1.00000 q^{11} -5.46577 q^{12} +3.12060 q^{13} +8.60451 q^{14} -4.65286 q^{16} +4.24799 q^{17} +16.7270 q^{18} +1.00000 q^{19} -15.6209 q^{21} +1.89451 q^{22} +7.93622 q^{23} +2.67684 q^{24} +5.91203 q^{26} -20.0486 q^{27} +7.21776 q^{28} +0.981983 q^{29} +5.23716 q^{31} -7.25832 q^{32} -3.43936 q^{33} +8.04788 q^{34} +14.0312 q^{36} +0.410380 q^{37} +1.89451 q^{38} -10.7329 q^{39} +6.31469 q^{41} -29.5940 q^{42} -3.20704 q^{43} +1.58918 q^{44} +15.0353 q^{46} -2.27417 q^{47} +16.0029 q^{48} +13.6280 q^{49} -14.6104 q^{51} +4.95921 q^{52} -9.47946 q^{53} -37.9824 q^{54} -3.53486 q^{56} -3.43936 q^{57} +1.86038 q^{58} -5.23047 q^{59} -12.9498 q^{61} +9.92187 q^{62} +40.1004 q^{63} -4.44527 q^{64} -6.51591 q^{66} +10.0523 q^{67} +6.75085 q^{68} -27.2955 q^{69} -13.0635 q^{71} -6.87171 q^{72} -2.99240 q^{73} +0.777472 q^{74} +1.58918 q^{76} +4.54180 q^{77} -20.3336 q^{78} -4.74136 q^{79} +42.4669 q^{81} +11.9633 q^{82} +12.4460 q^{83} -24.8245 q^{84} -6.07579 q^{86} -3.37739 q^{87} -0.778295 q^{88} -8.61817 q^{89} +14.1732 q^{91} +12.6121 q^{92} -18.0125 q^{93} -4.30844 q^{94} +24.9640 q^{96} -6.59008 q^{97} +25.8184 q^{98} +8.82918 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23} + q^{24} + 3 q^{26} + q^{27} + 46 q^{28} + 9 q^{29} + 14 q^{31} + 18 q^{32} + 4 q^{33} - 13 q^{34} + 12 q^{36} + 9 q^{37} + 5 q^{38} - 22 q^{39} + 4 q^{41} - 6 q^{42} + 28 q^{43} + 17 q^{44} + 27 q^{46} + 14 q^{47} - 4 q^{48} + 32 q^{49} - 40 q^{51} + 14 q^{52} + 3 q^{53} - 39 q^{54} + 34 q^{56} + 4 q^{57} + 26 q^{58} + q^{59} + 2 q^{61} - 3 q^{62} + 45 q^{63} + 5 q^{64} - q^{66} + 37 q^{67} + 26 q^{68} - 7 q^{69} - 7 q^{71} + 16 q^{72} + 42 q^{73} - 43 q^{74} + 17 q^{76} + 21 q^{77} - 64 q^{78} - 10 q^{79} + 31 q^{81} + 22 q^{82} + 14 q^{83} - 32 q^{84} + 37 q^{86} + 29 q^{87} + 9 q^{88} + 15 q^{89} - 22 q^{91} + 26 q^{92} - 18 q^{93} - 44 q^{94} + 71 q^{96} + 8 q^{97} - 10 q^{98} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.89451 1.33962 0.669812 0.742531i \(-0.266375\pi\)
0.669812 + 0.742531i \(0.266375\pi\)
\(3\) −3.43936 −1.98571 −0.992857 0.119311i \(-0.961932\pi\)
−0.992857 + 0.119311i \(0.961932\pi\)
\(4\) 1.58918 0.794592
\(5\) 0 0
\(6\) −6.51591 −2.66011
\(7\) 4.54180 1.71664 0.858320 0.513115i \(-0.171509\pi\)
0.858320 + 0.513115i \(0.171509\pi\)
\(8\) −0.778295 −0.275169
\(9\) 8.82918 2.94306
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −5.46577 −1.57783
\(13\) 3.12060 0.865499 0.432750 0.901514i \(-0.357543\pi\)
0.432750 + 0.901514i \(0.357543\pi\)
\(14\) 8.60451 2.29965
\(15\) 0 0
\(16\) −4.65286 −1.16322
\(17\) 4.24799 1.03029 0.515145 0.857103i \(-0.327738\pi\)
0.515145 + 0.857103i \(0.327738\pi\)
\(18\) 16.7270 3.94259
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −15.6209 −3.40875
\(22\) 1.89451 0.403912
\(23\) 7.93622 1.65482 0.827408 0.561602i \(-0.189815\pi\)
0.827408 + 0.561602i \(0.189815\pi\)
\(24\) 2.67684 0.546407
\(25\) 0 0
\(26\) 5.91203 1.15944
\(27\) −20.0486 −3.85836
\(28\) 7.21776 1.36403
\(29\) 0.981983 0.182350 0.0911748 0.995835i \(-0.470938\pi\)
0.0911748 + 0.995835i \(0.470938\pi\)
\(30\) 0 0
\(31\) 5.23716 0.940621 0.470311 0.882501i \(-0.344142\pi\)
0.470311 + 0.882501i \(0.344142\pi\)
\(32\) −7.25832 −1.28310
\(33\) −3.43936 −0.598715
\(34\) 8.04788 1.38020
\(35\) 0 0
\(36\) 14.0312 2.33853
\(37\) 0.410380 0.0674661 0.0337331 0.999431i \(-0.489260\pi\)
0.0337331 + 0.999431i \(0.489260\pi\)
\(38\) 1.89451 0.307331
\(39\) −10.7329 −1.71863
\(40\) 0 0
\(41\) 6.31469 0.986188 0.493094 0.869976i \(-0.335866\pi\)
0.493094 + 0.869976i \(0.335866\pi\)
\(42\) −29.5940 −4.56645
\(43\) −3.20704 −0.489070 −0.244535 0.969640i \(-0.578635\pi\)
−0.244535 + 0.969640i \(0.578635\pi\)
\(44\) 1.58918 0.239579
\(45\) 0 0
\(46\) 15.0353 2.21683
\(47\) −2.27417 −0.331721 −0.165861 0.986149i \(-0.553040\pi\)
−0.165861 + 0.986149i \(0.553040\pi\)
\(48\) 16.0029 2.30981
\(49\) 13.6280 1.94685
\(50\) 0 0
\(51\) −14.6104 −2.04586
\(52\) 4.95921 0.687719
\(53\) −9.47946 −1.30210 −0.651052 0.759033i \(-0.725672\pi\)
−0.651052 + 0.759033i \(0.725672\pi\)
\(54\) −37.9824 −5.16875
\(55\) 0 0
\(56\) −3.53486 −0.472366
\(57\) −3.43936 −0.455554
\(58\) 1.86038 0.244280
\(59\) −5.23047 −0.680949 −0.340474 0.940254i \(-0.610588\pi\)
−0.340474 + 0.940254i \(0.610588\pi\)
\(60\) 0 0
\(61\) −12.9498 −1.65806 −0.829028 0.559206i \(-0.811106\pi\)
−0.829028 + 0.559206i \(0.811106\pi\)
\(62\) 9.92187 1.26008
\(63\) 40.1004 5.05217
\(64\) −4.44527 −0.555659
\(65\) 0 0
\(66\) −6.51591 −0.802053
\(67\) 10.0523 1.22809 0.614043 0.789273i \(-0.289542\pi\)
0.614043 + 0.789273i \(0.289542\pi\)
\(68\) 6.75085 0.818660
\(69\) −27.2955 −3.28599
\(70\) 0 0
\(71\) −13.0635 −1.55035 −0.775174 0.631748i \(-0.782338\pi\)
−0.775174 + 0.631748i \(0.782338\pi\)
\(72\) −6.87171 −0.809839
\(73\) −2.99240 −0.350234 −0.175117 0.984548i \(-0.556030\pi\)
−0.175117 + 0.984548i \(0.556030\pi\)
\(74\) 0.777472 0.0903792
\(75\) 0 0
\(76\) 1.58918 0.182292
\(77\) 4.54180 0.517586
\(78\) −20.3336 −2.30232
\(79\) −4.74136 −0.533445 −0.266723 0.963773i \(-0.585941\pi\)
−0.266723 + 0.963773i \(0.585941\pi\)
\(80\) 0 0
\(81\) 42.4669 4.71854
\(82\) 11.9633 1.32112
\(83\) 12.4460 1.36613 0.683063 0.730359i \(-0.260647\pi\)
0.683063 + 0.730359i \(0.260647\pi\)
\(84\) −24.8245 −2.70857
\(85\) 0 0
\(86\) −6.07579 −0.655170
\(87\) −3.37739 −0.362094
\(88\) −0.778295 −0.0829666
\(89\) −8.61817 −0.913524 −0.456762 0.889589i \(-0.650991\pi\)
−0.456762 + 0.889589i \(0.650991\pi\)
\(90\) 0 0
\(91\) 14.1732 1.48575
\(92\) 12.6121 1.31490
\(93\) −18.0125 −1.86780
\(94\) −4.30844 −0.444382
\(95\) 0 0
\(96\) 24.9640 2.54787
\(97\) −6.59008 −0.669122 −0.334561 0.942374i \(-0.608588\pi\)
−0.334561 + 0.942374i \(0.608588\pi\)
\(98\) 25.8184 2.60805
\(99\) 8.82918 0.887366
\(100\) 0 0
\(101\) −15.8124 −1.57340 −0.786698 0.617338i \(-0.788211\pi\)
−0.786698 + 0.617338i \(0.788211\pi\)
\(102\) −27.6795 −2.74068
\(103\) 19.7762 1.94861 0.974306 0.225229i \(-0.0723132\pi\)
0.974306 + 0.225229i \(0.0723132\pi\)
\(104\) −2.42875 −0.238159
\(105\) 0 0
\(106\) −17.9590 −1.74433
\(107\) 14.9861 1.44876 0.724379 0.689402i \(-0.242127\pi\)
0.724379 + 0.689402i \(0.242127\pi\)
\(108\) −31.8610 −3.06582
\(109\) 6.17987 0.591924 0.295962 0.955200i \(-0.404360\pi\)
0.295962 + 0.955200i \(0.404360\pi\)
\(110\) 0 0
\(111\) −1.41144 −0.133968
\(112\) −21.1324 −1.99682
\(113\) 2.90486 0.273267 0.136633 0.990622i \(-0.456372\pi\)
0.136633 + 0.990622i \(0.456372\pi\)
\(114\) −6.51591 −0.610271
\(115\) 0 0
\(116\) 1.56055 0.144894
\(117\) 27.5524 2.54722
\(118\) −9.90920 −0.912216
\(119\) 19.2935 1.76864
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −24.5337 −2.22117
\(123\) −21.7185 −1.95829
\(124\) 8.32281 0.747411
\(125\) 0 0
\(126\) 75.9707 6.76801
\(127\) 8.76524 0.777790 0.388895 0.921282i \(-0.372857\pi\)
0.388895 + 0.921282i \(0.372857\pi\)
\(128\) 6.09501 0.538728
\(129\) 11.0302 0.971153
\(130\) 0 0
\(131\) −21.9654 −1.91913 −0.959565 0.281488i \(-0.909172\pi\)
−0.959565 + 0.281488i \(0.909172\pi\)
\(132\) −5.46577 −0.475735
\(133\) 4.54180 0.393824
\(134\) 19.0443 1.64517
\(135\) 0 0
\(136\) −3.30619 −0.283504
\(137\) 0.169026 0.0144409 0.00722044 0.999974i \(-0.497702\pi\)
0.00722044 + 0.999974i \(0.497702\pi\)
\(138\) −51.7117 −4.40199
\(139\) 16.2125 1.37512 0.687562 0.726125i \(-0.258681\pi\)
0.687562 + 0.726125i \(0.258681\pi\)
\(140\) 0 0
\(141\) 7.82168 0.658704
\(142\) −24.7489 −2.07688
\(143\) 3.12060 0.260958
\(144\) −41.0809 −3.42341
\(145\) 0 0
\(146\) −5.66914 −0.469181
\(147\) −46.8714 −3.86589
\(148\) 0.652170 0.0536081
\(149\) 7.82405 0.640971 0.320486 0.947253i \(-0.396154\pi\)
0.320486 + 0.947253i \(0.396154\pi\)
\(150\) 0 0
\(151\) −6.25375 −0.508923 −0.254461 0.967083i \(-0.581898\pi\)
−0.254461 + 0.967083i \(0.581898\pi\)
\(152\) −0.778295 −0.0631281
\(153\) 37.5063 3.03220
\(154\) 8.60451 0.693371
\(155\) 0 0
\(156\) −17.0565 −1.36561
\(157\) 9.87376 0.788012 0.394006 0.919108i \(-0.371089\pi\)
0.394006 + 0.919108i \(0.371089\pi\)
\(158\) −8.98258 −0.714616
\(159\) 32.6032 2.58561
\(160\) 0 0
\(161\) 36.0447 2.84072
\(162\) 80.4541 6.32107
\(163\) 5.20698 0.407842 0.203921 0.978987i \(-0.434631\pi\)
0.203921 + 0.978987i \(0.434631\pi\)
\(164\) 10.0352 0.783618
\(165\) 0 0
\(166\) 23.5791 1.83010
\(167\) 10.5716 0.818053 0.409026 0.912523i \(-0.365868\pi\)
0.409026 + 0.912523i \(0.365868\pi\)
\(168\) 12.1577 0.937983
\(169\) −3.26184 −0.250911
\(170\) 0 0
\(171\) 8.82918 0.675184
\(172\) −5.09659 −0.388611
\(173\) −18.3844 −1.39774 −0.698872 0.715247i \(-0.746314\pi\)
−0.698872 + 0.715247i \(0.746314\pi\)
\(174\) −6.39851 −0.485070
\(175\) 0 0
\(176\) −4.65286 −0.350723
\(177\) 17.9894 1.35217
\(178\) −16.3272 −1.22378
\(179\) −9.66434 −0.722347 −0.361173 0.932499i \(-0.617624\pi\)
−0.361173 + 0.932499i \(0.617624\pi\)
\(180\) 0 0
\(181\) −4.55344 −0.338455 −0.169227 0.985577i \(-0.554127\pi\)
−0.169227 + 0.985577i \(0.554127\pi\)
\(182\) 26.8512 1.99035
\(183\) 44.5391 3.29243
\(184\) −6.17672 −0.455354
\(185\) 0 0
\(186\) −34.1249 −2.50216
\(187\) 4.24799 0.310644
\(188\) −3.61407 −0.263583
\(189\) −91.0569 −6.62341
\(190\) 0 0
\(191\) 14.3707 1.03983 0.519915 0.854218i \(-0.325963\pi\)
0.519915 + 0.854218i \(0.325963\pi\)
\(192\) 15.2889 1.10338
\(193\) −0.688168 −0.0495354 −0.0247677 0.999693i \(-0.507885\pi\)
−0.0247677 + 0.999693i \(0.507885\pi\)
\(194\) −12.4850 −0.896371
\(195\) 0 0
\(196\) 21.6573 1.54695
\(197\) 12.3873 0.882555 0.441278 0.897371i \(-0.354525\pi\)
0.441278 + 0.897371i \(0.354525\pi\)
\(198\) 16.7270 1.18874
\(199\) −10.7275 −0.760451 −0.380226 0.924894i \(-0.624154\pi\)
−0.380226 + 0.924894i \(0.624154\pi\)
\(200\) 0 0
\(201\) −34.5735 −2.43863
\(202\) −29.9569 −2.10776
\(203\) 4.45997 0.313029
\(204\) −23.2186 −1.62563
\(205\) 0 0
\(206\) 37.4664 2.61041
\(207\) 70.0703 4.87022
\(208\) −14.5197 −1.00676
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 4.20060 0.289181 0.144591 0.989492i \(-0.453813\pi\)
0.144591 + 0.989492i \(0.453813\pi\)
\(212\) −15.0646 −1.03464
\(213\) 44.9299 3.07855
\(214\) 28.3913 1.94079
\(215\) 0 0
\(216\) 15.6038 1.06170
\(217\) 23.7861 1.61471
\(218\) 11.7078 0.792955
\(219\) 10.2919 0.695464
\(220\) 0 0
\(221\) 13.2563 0.891715
\(222\) −2.67400 −0.179467
\(223\) 5.25583 0.351956 0.175978 0.984394i \(-0.443691\pi\)
0.175978 + 0.984394i \(0.443691\pi\)
\(224\) −32.9659 −2.20262
\(225\) 0 0
\(226\) 5.50331 0.366075
\(227\) −2.44091 −0.162009 −0.0810043 0.996714i \(-0.525813\pi\)
−0.0810043 + 0.996714i \(0.525813\pi\)
\(228\) −5.46577 −0.361980
\(229\) 2.68082 0.177154 0.0885769 0.996069i \(-0.471768\pi\)
0.0885769 + 0.996069i \(0.471768\pi\)
\(230\) 0 0
\(231\) −15.6209 −1.02778
\(232\) −0.764273 −0.0501770
\(233\) 22.6306 1.48258 0.741290 0.671185i \(-0.234214\pi\)
0.741290 + 0.671185i \(0.234214\pi\)
\(234\) 52.1983 3.41231
\(235\) 0 0
\(236\) −8.31218 −0.541077
\(237\) 16.3072 1.05927
\(238\) 36.5519 2.36931
\(239\) 21.2527 1.37472 0.687360 0.726317i \(-0.258770\pi\)
0.687360 + 0.726317i \(0.258770\pi\)
\(240\) 0 0
\(241\) −9.14068 −0.588803 −0.294401 0.955682i \(-0.595120\pi\)
−0.294401 + 0.955682i \(0.595120\pi\)
\(242\) 1.89451 0.121784
\(243\) −85.9128 −5.51131
\(244\) −20.5797 −1.31748
\(245\) 0 0
\(246\) −41.1459 −2.62337
\(247\) 3.12060 0.198559
\(248\) −4.07606 −0.258830
\(249\) −42.8062 −2.71274
\(250\) 0 0
\(251\) 13.3030 0.839678 0.419839 0.907599i \(-0.362087\pi\)
0.419839 + 0.907599i \(0.362087\pi\)
\(252\) 63.7269 4.01442
\(253\) 7.93622 0.498946
\(254\) 16.6059 1.04195
\(255\) 0 0
\(256\) 20.4376 1.27735
\(257\) −3.01036 −0.187781 −0.0938906 0.995583i \(-0.529930\pi\)
−0.0938906 + 0.995583i \(0.529930\pi\)
\(258\) 20.8968 1.30098
\(259\) 1.86387 0.115815
\(260\) 0 0
\(261\) 8.67010 0.536666
\(262\) −41.6138 −2.57091
\(263\) −3.18414 −0.196342 −0.0981712 0.995170i \(-0.531299\pi\)
−0.0981712 + 0.995170i \(0.531299\pi\)
\(264\) 2.67684 0.164748
\(265\) 0 0
\(266\) 8.60451 0.527576
\(267\) 29.6410 1.81400
\(268\) 15.9750 0.975828
\(269\) 2.49977 0.152413 0.0762067 0.997092i \(-0.475719\pi\)
0.0762067 + 0.997092i \(0.475719\pi\)
\(270\) 0 0
\(271\) −12.5192 −0.760486 −0.380243 0.924887i \(-0.624160\pi\)
−0.380243 + 0.924887i \(0.624160\pi\)
\(272\) −19.7653 −1.19845
\(273\) −48.7466 −2.95028
\(274\) 0.320222 0.0193453
\(275\) 0 0
\(276\) −43.3776 −2.61102
\(277\) 17.4012 1.04554 0.522768 0.852475i \(-0.324899\pi\)
0.522768 + 0.852475i \(0.324899\pi\)
\(278\) 30.7148 1.84215
\(279\) 46.2398 2.76830
\(280\) 0 0
\(281\) 4.60634 0.274791 0.137396 0.990516i \(-0.456127\pi\)
0.137396 + 0.990516i \(0.456127\pi\)
\(282\) 14.8183 0.882416
\(283\) −15.6538 −0.930523 −0.465262 0.885173i \(-0.654040\pi\)
−0.465262 + 0.885173i \(0.654040\pi\)
\(284\) −20.7602 −1.23189
\(285\) 0 0
\(286\) 5.91203 0.349585
\(287\) 28.6801 1.69293
\(288\) −64.0850 −3.77625
\(289\) 1.04544 0.0614966
\(290\) 0 0
\(291\) 22.6656 1.32868
\(292\) −4.75547 −0.278293
\(293\) −15.2255 −0.889482 −0.444741 0.895659i \(-0.646704\pi\)
−0.444741 + 0.895659i \(0.646704\pi\)
\(294\) −88.7986 −5.17884
\(295\) 0 0
\(296\) −0.319397 −0.0185646
\(297\) −20.0486 −1.16334
\(298\) 14.8228 0.858661
\(299\) 24.7658 1.43224
\(300\) 0 0
\(301\) −14.5658 −0.839556
\(302\) −11.8478 −0.681765
\(303\) 54.3846 3.12431
\(304\) −4.65286 −0.266860
\(305\) 0 0
\(306\) 71.0562 4.06201
\(307\) −17.7630 −1.01379 −0.506894 0.862009i \(-0.669206\pi\)
−0.506894 + 0.862009i \(0.669206\pi\)
\(308\) 7.21776 0.411270
\(309\) −68.0176 −3.86938
\(310\) 0 0
\(311\) −1.64083 −0.0930427 −0.0465214 0.998917i \(-0.514814\pi\)
−0.0465214 + 0.998917i \(0.514814\pi\)
\(312\) 8.35334 0.472915
\(313\) 0.0889283 0.00502653 0.00251326 0.999997i \(-0.499200\pi\)
0.00251326 + 0.999997i \(0.499200\pi\)
\(314\) 18.7060 1.05564
\(315\) 0 0
\(316\) −7.53490 −0.423871
\(317\) −6.14634 −0.345213 −0.172606 0.984991i \(-0.555219\pi\)
−0.172606 + 0.984991i \(0.555219\pi\)
\(318\) 61.7673 3.46374
\(319\) 0.981983 0.0549805
\(320\) 0 0
\(321\) −51.5424 −2.87682
\(322\) 68.2872 3.80550
\(323\) 4.24799 0.236365
\(324\) 67.4877 3.74932
\(325\) 0 0
\(326\) 9.86469 0.546355
\(327\) −21.2548 −1.17539
\(328\) −4.91469 −0.271368
\(329\) −10.3288 −0.569446
\(330\) 0 0
\(331\) −7.51963 −0.413316 −0.206658 0.978413i \(-0.566259\pi\)
−0.206658 + 0.978413i \(0.566259\pi\)
\(332\) 19.7790 1.08551
\(333\) 3.62332 0.198557
\(334\) 20.0280 1.09588
\(335\) 0 0
\(336\) 72.6818 3.96512
\(337\) −8.13119 −0.442934 −0.221467 0.975168i \(-0.571085\pi\)
−0.221467 + 0.975168i \(0.571085\pi\)
\(338\) −6.17960 −0.336126
\(339\) −9.99087 −0.542629
\(340\) 0 0
\(341\) 5.23716 0.283608
\(342\) 16.7270 0.904493
\(343\) 30.1029 1.62540
\(344\) 2.49603 0.134577
\(345\) 0 0
\(346\) −34.8296 −1.87245
\(347\) 3.18776 0.171128 0.0855640 0.996333i \(-0.472731\pi\)
0.0855640 + 0.996333i \(0.472731\pi\)
\(348\) −5.36730 −0.287717
\(349\) 11.4958 0.615355 0.307678 0.951491i \(-0.400448\pi\)
0.307678 + 0.951491i \(0.400448\pi\)
\(350\) 0 0
\(351\) −62.5638 −3.33941
\(352\) −7.25832 −0.386870
\(353\) −17.4372 −0.928086 −0.464043 0.885813i \(-0.653602\pi\)
−0.464043 + 0.885813i \(0.653602\pi\)
\(354\) 34.0813 1.81140
\(355\) 0 0
\(356\) −13.6959 −0.725879
\(357\) −66.3574 −3.51200
\(358\) −18.3092 −0.967673
\(359\) 16.1036 0.849913 0.424956 0.905214i \(-0.360289\pi\)
0.424956 + 0.905214i \(0.360289\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −8.62657 −0.453402
\(363\) −3.43936 −0.180519
\(364\) 22.5238 1.18057
\(365\) 0 0
\(366\) 84.3800 4.41061
\(367\) 3.44894 0.180033 0.0900167 0.995940i \(-0.471308\pi\)
0.0900167 + 0.995940i \(0.471308\pi\)
\(368\) −36.9261 −1.92491
\(369\) 55.7535 2.90241
\(370\) 0 0
\(371\) −43.0538 −2.23524
\(372\) −28.6251 −1.48414
\(373\) −1.39855 −0.0724142 −0.0362071 0.999344i \(-0.511528\pi\)
−0.0362071 + 0.999344i \(0.511528\pi\)
\(374\) 8.04788 0.416146
\(375\) 0 0
\(376\) 1.76997 0.0912794
\(377\) 3.06438 0.157824
\(378\) −172.509 −8.87288
\(379\) 6.72162 0.345267 0.172633 0.984986i \(-0.444772\pi\)
0.172633 + 0.984986i \(0.444772\pi\)
\(380\) 0 0
\(381\) −30.1468 −1.54447
\(382\) 27.2256 1.39298
\(383\) −15.1956 −0.776461 −0.388230 0.921562i \(-0.626914\pi\)
−0.388230 + 0.921562i \(0.626914\pi\)
\(384\) −20.9629 −1.06976
\(385\) 0 0
\(386\) −1.30374 −0.0663589
\(387\) −28.3156 −1.43936
\(388\) −10.4729 −0.531679
\(389\) 0.847266 0.0429581 0.0214790 0.999769i \(-0.493162\pi\)
0.0214790 + 0.999769i \(0.493162\pi\)
\(390\) 0 0
\(391\) 33.7130 1.70494
\(392\) −10.6066 −0.535713
\(393\) 75.5470 3.81084
\(394\) 23.4678 1.18229
\(395\) 0 0
\(396\) 14.0312 0.705094
\(397\) 19.3480 0.971047 0.485524 0.874224i \(-0.338629\pi\)
0.485524 + 0.874224i \(0.338629\pi\)
\(398\) −20.3234 −1.01872
\(399\) −15.6209 −0.782022
\(400\) 0 0
\(401\) −20.6662 −1.03202 −0.516009 0.856583i \(-0.672583\pi\)
−0.516009 + 0.856583i \(0.672583\pi\)
\(402\) −65.5000 −3.26684
\(403\) 16.3431 0.814107
\(404\) −25.1289 −1.25021
\(405\) 0 0
\(406\) 8.44948 0.419341
\(407\) 0.410380 0.0203418
\(408\) 11.3712 0.562957
\(409\) −11.6120 −0.574178 −0.287089 0.957904i \(-0.592688\pi\)
−0.287089 + 0.957904i \(0.592688\pi\)
\(410\) 0 0
\(411\) −0.581341 −0.0286754
\(412\) 31.4281 1.54835
\(413\) −23.7557 −1.16894
\(414\) 132.749 6.52426
\(415\) 0 0
\(416\) −22.6503 −1.11052
\(417\) −55.7605 −2.73060
\(418\) 1.89451 0.0926637
\(419\) 30.8500 1.50712 0.753561 0.657378i \(-0.228334\pi\)
0.753561 + 0.657378i \(0.228334\pi\)
\(420\) 0 0
\(421\) −12.2925 −0.599101 −0.299550 0.954080i \(-0.596837\pi\)
−0.299550 + 0.954080i \(0.596837\pi\)
\(422\) 7.95810 0.387394
\(423\) −20.0790 −0.976276
\(424\) 7.37782 0.358299
\(425\) 0 0
\(426\) 85.1203 4.12409
\(427\) −58.8156 −2.84629
\(428\) 23.8156 1.15117
\(429\) −10.7329 −0.518188
\(430\) 0 0
\(431\) −3.08294 −0.148500 −0.0742500 0.997240i \(-0.523656\pi\)
−0.0742500 + 0.997240i \(0.523656\pi\)
\(432\) 93.2835 4.48810
\(433\) 27.6734 1.32990 0.664950 0.746888i \(-0.268453\pi\)
0.664950 + 0.746888i \(0.268453\pi\)
\(434\) 45.0632 2.16310
\(435\) 0 0
\(436\) 9.82095 0.470338
\(437\) 7.93622 0.379641
\(438\) 19.4982 0.931660
\(439\) −17.0897 −0.815649 −0.407824 0.913060i \(-0.633712\pi\)
−0.407824 + 0.913060i \(0.633712\pi\)
\(440\) 0 0
\(441\) 120.324 5.72970
\(442\) 25.1142 1.19456
\(443\) 11.4820 0.545527 0.272764 0.962081i \(-0.412062\pi\)
0.272764 + 0.962081i \(0.412062\pi\)
\(444\) −2.24305 −0.106450
\(445\) 0 0
\(446\) 9.95724 0.471489
\(447\) −26.9097 −1.27279
\(448\) −20.1895 −0.953866
\(449\) 10.0089 0.472347 0.236173 0.971711i \(-0.424107\pi\)
0.236173 + 0.971711i \(0.424107\pi\)
\(450\) 0 0
\(451\) 6.31469 0.297347
\(452\) 4.61637 0.217136
\(453\) 21.5089 1.01057
\(454\) −4.62433 −0.217031
\(455\) 0 0
\(456\) 2.67684 0.125354
\(457\) −8.49503 −0.397381 −0.198690 0.980062i \(-0.563669\pi\)
−0.198690 + 0.980062i \(0.563669\pi\)
\(458\) 5.07886 0.237319
\(459\) −85.1664 −3.97523
\(460\) 0 0
\(461\) 18.8075 0.875953 0.437977 0.898986i \(-0.355695\pi\)
0.437977 + 0.898986i \(0.355695\pi\)
\(462\) −29.5940 −1.37684
\(463\) −35.2732 −1.63929 −0.819643 0.572874i \(-0.805828\pi\)
−0.819643 + 0.572874i \(0.805828\pi\)
\(464\) −4.56903 −0.212112
\(465\) 0 0
\(466\) 42.8740 1.98610
\(467\) 19.3530 0.895549 0.447774 0.894147i \(-0.352217\pi\)
0.447774 + 0.894147i \(0.352217\pi\)
\(468\) 43.7858 2.02400
\(469\) 45.6556 2.10818
\(470\) 0 0
\(471\) −33.9594 −1.56477
\(472\) 4.07085 0.187376
\(473\) −3.20704 −0.147460
\(474\) 30.8943 1.41902
\(475\) 0 0
\(476\) 30.6610 1.40534
\(477\) −83.6958 −3.83217
\(478\) 40.2635 1.84161
\(479\) −28.7710 −1.31458 −0.657289 0.753638i \(-0.728297\pi\)
−0.657289 + 0.753638i \(0.728297\pi\)
\(480\) 0 0
\(481\) 1.28063 0.0583919
\(482\) −17.3171 −0.788775
\(483\) −123.971 −5.64086
\(484\) 1.58918 0.0722357
\(485\) 0 0
\(486\) −162.763 −7.38309
\(487\) 6.67404 0.302429 0.151215 0.988501i \(-0.451682\pi\)
0.151215 + 0.988501i \(0.451682\pi\)
\(488\) 10.0788 0.456246
\(489\) −17.9087 −0.809857
\(490\) 0 0
\(491\) 7.32389 0.330522 0.165261 0.986250i \(-0.447153\pi\)
0.165261 + 0.986250i \(0.447153\pi\)
\(492\) −34.5147 −1.55604
\(493\) 4.17146 0.187873
\(494\) 5.91203 0.265995
\(495\) 0 0
\(496\) −24.3678 −1.09415
\(497\) −59.3316 −2.66139
\(498\) −81.0970 −3.63405
\(499\) −17.4305 −0.780298 −0.390149 0.920752i \(-0.627577\pi\)
−0.390149 + 0.920752i \(0.627577\pi\)
\(500\) 0 0
\(501\) −36.3594 −1.62442
\(502\) 25.2027 1.12485
\(503\) −37.1696 −1.65731 −0.828656 0.559758i \(-0.810894\pi\)
−0.828656 + 0.559758i \(0.810894\pi\)
\(504\) −31.2099 −1.39020
\(505\) 0 0
\(506\) 15.0353 0.668400
\(507\) 11.2186 0.498237
\(508\) 13.9296 0.618026
\(509\) −16.8034 −0.744797 −0.372398 0.928073i \(-0.621465\pi\)
−0.372398 + 0.928073i \(0.621465\pi\)
\(510\) 0 0
\(511\) −13.5909 −0.601225
\(512\) 26.5294 1.17244
\(513\) −20.0486 −0.885169
\(514\) −5.70318 −0.251556
\(515\) 0 0
\(516\) 17.5290 0.771670
\(517\) −2.27417 −0.100018
\(518\) 3.53112 0.155149
\(519\) 63.2307 2.77552
\(520\) 0 0
\(521\) −37.2735 −1.63298 −0.816492 0.577357i \(-0.804084\pi\)
−0.816492 + 0.577357i \(0.804084\pi\)
\(522\) 16.4256 0.718930
\(523\) 31.2215 1.36522 0.682610 0.730783i \(-0.260845\pi\)
0.682610 + 0.730783i \(0.260845\pi\)
\(524\) −34.9071 −1.52493
\(525\) 0 0
\(526\) −6.03240 −0.263025
\(527\) 22.2474 0.969112
\(528\) 16.0029 0.696435
\(529\) 39.9835 1.73841
\(530\) 0 0
\(531\) −46.1807 −2.00407
\(532\) 7.21776 0.312930
\(533\) 19.7056 0.853546
\(534\) 56.1552 2.43007
\(535\) 0 0
\(536\) −7.82367 −0.337931
\(537\) 33.2391 1.43437
\(538\) 4.73584 0.204177
\(539\) 13.6280 0.586998
\(540\) 0 0
\(541\) 35.7719 1.53796 0.768978 0.639275i \(-0.220766\pi\)
0.768978 + 0.639275i \(0.220766\pi\)
\(542\) −23.7178 −1.01876
\(543\) 15.6609 0.672075
\(544\) −30.8333 −1.32197
\(545\) 0 0
\(546\) −92.3510 −3.95226
\(547\) 23.1497 0.989809 0.494905 0.868947i \(-0.335203\pi\)
0.494905 + 0.868947i \(0.335203\pi\)
\(548\) 0.268614 0.0114746
\(549\) −114.336 −4.87976
\(550\) 0 0
\(551\) 0.981983 0.0418339
\(552\) 21.2439 0.904203
\(553\) −21.5343 −0.915733
\(554\) 32.9668 1.40063
\(555\) 0 0
\(556\) 25.7646 1.09266
\(557\) 4.94276 0.209432 0.104716 0.994502i \(-0.466607\pi\)
0.104716 + 0.994502i \(0.466607\pi\)
\(558\) 87.6020 3.70849
\(559\) −10.0079 −0.423290
\(560\) 0 0
\(561\) −14.6104 −0.616850
\(562\) 8.72678 0.368117
\(563\) −42.3236 −1.78373 −0.891863 0.452306i \(-0.850601\pi\)
−0.891863 + 0.452306i \(0.850601\pi\)
\(564\) 12.4301 0.523401
\(565\) 0 0
\(566\) −29.6564 −1.24655
\(567\) 192.876 8.10003
\(568\) 10.1672 0.426607
\(569\) −6.21601 −0.260589 −0.130294 0.991475i \(-0.541592\pi\)
−0.130294 + 0.991475i \(0.541592\pi\)
\(570\) 0 0
\(571\) 7.31329 0.306052 0.153026 0.988222i \(-0.451098\pi\)
0.153026 + 0.988222i \(0.451098\pi\)
\(572\) 4.95921 0.207355
\(573\) −49.4261 −2.06481
\(574\) 54.3348 2.26789
\(575\) 0 0
\(576\) −39.2481 −1.63534
\(577\) −40.8779 −1.70177 −0.850885 0.525352i \(-0.823934\pi\)
−0.850885 + 0.525352i \(0.823934\pi\)
\(578\) 1.98060 0.0823823
\(579\) 2.36686 0.0983632
\(580\) 0 0
\(581\) 56.5273 2.34515
\(582\) 42.9404 1.77994
\(583\) −9.47946 −0.392599
\(584\) 2.32897 0.0963734
\(585\) 0 0
\(586\) −28.8449 −1.19157
\(587\) −38.4431 −1.58672 −0.793358 0.608756i \(-0.791669\pi\)
−0.793358 + 0.608756i \(0.791669\pi\)
\(588\) −74.4873 −3.07181
\(589\) 5.23716 0.215793
\(590\) 0 0
\(591\) −42.6042 −1.75250
\(592\) −1.90944 −0.0784776
\(593\) −33.2828 −1.36676 −0.683381 0.730062i \(-0.739491\pi\)
−0.683381 + 0.730062i \(0.739491\pi\)
\(594\) −37.9824 −1.55844
\(595\) 0 0
\(596\) 12.4339 0.509311
\(597\) 36.8957 1.51004
\(598\) 46.9191 1.91867
\(599\) 3.75100 0.153262 0.0766308 0.997060i \(-0.475584\pi\)
0.0766308 + 0.997060i \(0.475584\pi\)
\(600\) 0 0
\(601\) 7.66714 0.312749 0.156375 0.987698i \(-0.450019\pi\)
0.156375 + 0.987698i \(0.450019\pi\)
\(602\) −27.5950 −1.12469
\(603\) 88.7537 3.61433
\(604\) −9.93836 −0.404386
\(605\) 0 0
\(606\) 103.032 4.18541
\(607\) 22.0775 0.896100 0.448050 0.894009i \(-0.352119\pi\)
0.448050 + 0.894009i \(0.352119\pi\)
\(608\) −7.25832 −0.294364
\(609\) −15.3394 −0.621585
\(610\) 0 0
\(611\) −7.09677 −0.287105
\(612\) 59.6044 2.40937
\(613\) 8.39128 0.338921 0.169460 0.985537i \(-0.445798\pi\)
0.169460 + 0.985537i \(0.445798\pi\)
\(614\) −33.6522 −1.35809
\(615\) 0 0
\(616\) −3.53486 −0.142424
\(617\) −27.3231 −1.09999 −0.549994 0.835169i \(-0.685370\pi\)
−0.549994 + 0.835169i \(0.685370\pi\)
\(618\) −128.860 −5.18352
\(619\) −30.4668 −1.22457 −0.612283 0.790639i \(-0.709749\pi\)
−0.612283 + 0.790639i \(0.709749\pi\)
\(620\) 0 0
\(621\) −159.110 −6.38488
\(622\) −3.10857 −0.124642
\(623\) −39.1420 −1.56819
\(624\) 49.9385 1.99914
\(625\) 0 0
\(626\) 0.168476 0.00673366
\(627\) −3.43936 −0.137355
\(628\) 15.6912 0.626148
\(629\) 1.74329 0.0695096
\(630\) 0 0
\(631\) −18.9798 −0.755573 −0.377787 0.925893i \(-0.623315\pi\)
−0.377787 + 0.925893i \(0.623315\pi\)
\(632\) 3.69018 0.146788
\(633\) −14.4474 −0.574232
\(634\) −11.6443 −0.462455
\(635\) 0 0
\(636\) 51.8126 2.05450
\(637\) 42.5274 1.68500
\(638\) 1.86038 0.0736532
\(639\) −115.340 −4.56276
\(640\) 0 0
\(641\) −41.3689 −1.63397 −0.816987 0.576657i \(-0.804357\pi\)
−0.816987 + 0.576657i \(0.804357\pi\)
\(642\) −97.6479 −3.85386
\(643\) 2.46165 0.0970782 0.0485391 0.998821i \(-0.484543\pi\)
0.0485391 + 0.998821i \(0.484543\pi\)
\(644\) 57.2817 2.25722
\(645\) 0 0
\(646\) 8.04788 0.316640
\(647\) 42.3960 1.66676 0.833379 0.552702i \(-0.186403\pi\)
0.833379 + 0.552702i \(0.186403\pi\)
\(648\) −33.0518 −1.29840
\(649\) −5.23047 −0.205314
\(650\) 0 0
\(651\) −81.8090 −3.20635
\(652\) 8.27485 0.324068
\(653\) −15.5160 −0.607188 −0.303594 0.952801i \(-0.598187\pi\)
−0.303594 + 0.952801i \(0.598187\pi\)
\(654\) −40.2675 −1.57458
\(655\) 0 0
\(656\) −29.3814 −1.14715
\(657\) −26.4204 −1.03076
\(658\) −19.5681 −0.762844
\(659\) 5.37760 0.209481 0.104741 0.994500i \(-0.466599\pi\)
0.104741 + 0.994500i \(0.466599\pi\)
\(660\) 0 0
\(661\) −6.89055 −0.268011 −0.134006 0.990981i \(-0.542784\pi\)
−0.134006 + 0.990981i \(0.542784\pi\)
\(662\) −14.2460 −0.553688
\(663\) −45.5931 −1.77069
\(664\) −9.68666 −0.375915
\(665\) 0 0
\(666\) 6.86444 0.265991
\(667\) 7.79323 0.301755
\(668\) 16.8002 0.650019
\(669\) −18.0767 −0.698884
\(670\) 0 0
\(671\) −12.9498 −0.499923
\(672\) 113.381 4.37378
\(673\) 18.5921 0.716675 0.358337 0.933592i \(-0.383344\pi\)
0.358337 + 0.933592i \(0.383344\pi\)
\(674\) −15.4047 −0.593365
\(675\) 0 0
\(676\) −5.18366 −0.199372
\(677\) −29.0743 −1.11742 −0.558708 0.829364i \(-0.688703\pi\)
−0.558708 + 0.829364i \(0.688703\pi\)
\(678\) −18.9278 −0.726919
\(679\) −29.9308 −1.14864
\(680\) 0 0
\(681\) 8.39515 0.321703
\(682\) 9.92187 0.379928
\(683\) 12.9142 0.494148 0.247074 0.968997i \(-0.420531\pi\)
0.247074 + 0.968997i \(0.420531\pi\)
\(684\) 14.0312 0.536496
\(685\) 0 0
\(686\) 57.0303 2.17743
\(687\) −9.22031 −0.351777
\(688\) 14.9219 0.568893
\(689\) −29.5816 −1.12697
\(690\) 0 0
\(691\) −2.84035 −0.108052 −0.0540261 0.998540i \(-0.517205\pi\)
−0.0540261 + 0.998540i \(0.517205\pi\)
\(692\) −29.2163 −1.11064
\(693\) 40.1004 1.52329
\(694\) 6.03926 0.229247
\(695\) 0 0
\(696\) 2.62861 0.0996371
\(697\) 26.8247 1.01606
\(698\) 21.7789 0.824344
\(699\) −77.8347 −2.94398
\(700\) 0 0
\(701\) 34.6946 1.31040 0.655198 0.755458i \(-0.272585\pi\)
0.655198 + 0.755458i \(0.272585\pi\)
\(702\) −118.528 −4.47355
\(703\) 0.410380 0.0154778
\(704\) −4.44527 −0.167538
\(705\) 0 0
\(706\) −33.0349 −1.24329
\(707\) −71.8169 −2.70095
\(708\) 28.5886 1.07442
\(709\) −22.3861 −0.840727 −0.420363 0.907356i \(-0.638097\pi\)
−0.420363 + 0.907356i \(0.638097\pi\)
\(710\) 0 0
\(711\) −41.8624 −1.56996
\(712\) 6.70748 0.251374
\(713\) 41.5632 1.55655
\(714\) −125.715 −4.70477
\(715\) 0 0
\(716\) −15.3584 −0.573971
\(717\) −73.0955 −2.72980
\(718\) 30.5084 1.13856
\(719\) 3.90517 0.145638 0.0728191 0.997345i \(-0.476800\pi\)
0.0728191 + 0.997345i \(0.476800\pi\)
\(720\) 0 0
\(721\) 89.8198 3.34506
\(722\) 1.89451 0.0705065
\(723\) 31.4381 1.16919
\(724\) −7.23626 −0.268934
\(725\) 0 0
\(726\) −6.51591 −0.241828
\(727\) 20.4701 0.759194 0.379597 0.925152i \(-0.376063\pi\)
0.379597 + 0.925152i \(0.376063\pi\)
\(728\) −11.0309 −0.408832
\(729\) 168.084 6.22535
\(730\) 0 0
\(731\) −13.6235 −0.503883
\(732\) 70.7809 2.61614
\(733\) 19.9022 0.735105 0.367553 0.930003i \(-0.380196\pi\)
0.367553 + 0.930003i \(0.380196\pi\)
\(734\) 6.53407 0.241177
\(735\) 0 0
\(736\) −57.6036 −2.12330
\(737\) 10.0523 0.370282
\(738\) 105.626 3.88814
\(739\) 7.13637 0.262516 0.131258 0.991348i \(-0.458098\pi\)
0.131258 + 0.991348i \(0.458098\pi\)
\(740\) 0 0
\(741\) −10.7329 −0.394282
\(742\) −81.5661 −2.99438
\(743\) −14.9005 −0.546647 −0.273323 0.961922i \(-0.588123\pi\)
−0.273323 + 0.961922i \(0.588123\pi\)
\(744\) 14.0190 0.513962
\(745\) 0 0
\(746\) −2.64957 −0.0970077
\(747\) 109.888 4.02059
\(748\) 6.75085 0.246835
\(749\) 68.0637 2.48700
\(750\) 0 0
\(751\) 44.6388 1.62889 0.814447 0.580238i \(-0.197040\pi\)
0.814447 + 0.580238i \(0.197040\pi\)
\(752\) 10.5814 0.385863
\(753\) −45.7538 −1.66736
\(754\) 5.80551 0.211424
\(755\) 0 0
\(756\) −144.706 −5.26291
\(757\) 41.9585 1.52501 0.762503 0.646984i \(-0.223970\pi\)
0.762503 + 0.646984i \(0.223970\pi\)
\(758\) 12.7342 0.462527
\(759\) −27.2955 −0.990763
\(760\) 0 0
\(761\) 53.9530 1.95580 0.977898 0.209084i \(-0.0670481\pi\)
0.977898 + 0.209084i \(0.0670481\pi\)
\(762\) −57.1136 −2.06901
\(763\) 28.0677 1.01612
\(764\) 22.8378 0.826242
\(765\) 0 0
\(766\) −28.7884 −1.04017
\(767\) −16.3222 −0.589361
\(768\) −70.2923 −2.53646
\(769\) 19.9891 0.720826 0.360413 0.932793i \(-0.382636\pi\)
0.360413 + 0.932793i \(0.382636\pi\)
\(770\) 0 0
\(771\) 10.3537 0.372880
\(772\) −1.09363 −0.0393605
\(773\) −39.2968 −1.41341 −0.706704 0.707510i \(-0.749819\pi\)
−0.706704 + 0.707510i \(0.749819\pi\)
\(774\) −53.6443 −1.92820
\(775\) 0 0
\(776\) 5.12903 0.184121
\(777\) −6.41050 −0.229975
\(778\) 1.60516 0.0575477
\(779\) 6.31469 0.226247
\(780\) 0 0
\(781\) −13.0635 −0.467447
\(782\) 63.8697 2.28398
\(783\) −19.6874 −0.703571
\(784\) −63.4090 −2.26461
\(785\) 0 0
\(786\) 143.125 5.10510
\(787\) 18.3768 0.655062 0.327531 0.944840i \(-0.393783\pi\)
0.327531 + 0.944840i \(0.393783\pi\)
\(788\) 19.6856 0.701272
\(789\) 10.9514 0.389880
\(790\) 0 0
\(791\) 13.1933 0.469100
\(792\) −6.87171 −0.244176
\(793\) −40.4113 −1.43505
\(794\) 36.6550 1.30084
\(795\) 0 0
\(796\) −17.0480 −0.604249
\(797\) −49.4820 −1.75274 −0.876371 0.481637i \(-0.840042\pi\)
−0.876371 + 0.481637i \(0.840042\pi\)
\(798\) −29.5940 −1.04762
\(799\) −9.66065 −0.341769
\(800\) 0 0
\(801\) −76.0914 −2.68856
\(802\) −39.1523 −1.38252
\(803\) −2.99240 −0.105599
\(804\) −54.9437 −1.93771
\(805\) 0 0
\(806\) 30.9622 1.09060
\(807\) −8.59759 −0.302650
\(808\) 12.3067 0.432950
\(809\) 15.1639 0.533133 0.266567 0.963817i \(-0.414111\pi\)
0.266567 + 0.963817i \(0.414111\pi\)
\(810\) 0 0
\(811\) 21.6823 0.761369 0.380685 0.924705i \(-0.375688\pi\)
0.380685 + 0.924705i \(0.375688\pi\)
\(812\) 7.08772 0.248730
\(813\) 43.0579 1.51011
\(814\) 0.777472 0.0272504
\(815\) 0 0
\(816\) 67.9800 2.37978
\(817\) −3.20704 −0.112200
\(818\) −21.9992 −0.769182
\(819\) 125.137 4.37265
\(820\) 0 0
\(821\) −1.81334 −0.0632860 −0.0316430 0.999499i \(-0.510074\pi\)
−0.0316430 + 0.999499i \(0.510074\pi\)
\(822\) −1.10136 −0.0384143
\(823\) 44.0014 1.53379 0.766897 0.641770i \(-0.221800\pi\)
0.766897 + 0.641770i \(0.221800\pi\)
\(824\) −15.3918 −0.536197
\(825\) 0 0
\(826\) −45.0056 −1.56595
\(827\) 18.5928 0.646536 0.323268 0.946307i \(-0.395218\pi\)
0.323268 + 0.946307i \(0.395218\pi\)
\(828\) 111.355 3.86984
\(829\) −35.3044 −1.22617 −0.613087 0.790015i \(-0.710072\pi\)
−0.613087 + 0.790015i \(0.710072\pi\)
\(830\) 0 0
\(831\) −59.8489 −2.07614
\(832\) −13.8719 −0.480923
\(833\) 57.8915 2.00582
\(834\) −105.639 −3.65798
\(835\) 0 0
\(836\) 1.58918 0.0549631
\(837\) −104.998 −3.62926
\(838\) 58.4458 2.01898
\(839\) 49.5868 1.71193 0.855964 0.517036i \(-0.172965\pi\)
0.855964 + 0.517036i \(0.172965\pi\)
\(840\) 0 0
\(841\) −28.0357 −0.966749
\(842\) −23.2884 −0.802570
\(843\) −15.8429 −0.545657
\(844\) 6.67554 0.229781
\(845\) 0 0
\(846\) −38.0400 −1.30784
\(847\) 4.54180 0.156058
\(848\) 44.1066 1.51463
\(849\) 53.8391 1.84775
\(850\) 0 0
\(851\) 3.25687 0.111644
\(852\) 71.4019 2.44619
\(853\) −2.10368 −0.0720288 −0.0360144 0.999351i \(-0.511466\pi\)
−0.0360144 + 0.999351i \(0.511466\pi\)
\(854\) −111.427 −3.81295
\(855\) 0 0
\(856\) −11.6636 −0.398653
\(857\) 33.3610 1.13959 0.569795 0.821787i \(-0.307023\pi\)
0.569795 + 0.821787i \(0.307023\pi\)
\(858\) −20.3336 −0.694177
\(859\) 42.1500 1.43814 0.719070 0.694937i \(-0.244568\pi\)
0.719070 + 0.694937i \(0.244568\pi\)
\(860\) 0 0
\(861\) −98.6410 −3.36167
\(862\) −5.84067 −0.198934
\(863\) −24.2017 −0.823837 −0.411918 0.911221i \(-0.635141\pi\)
−0.411918 + 0.911221i \(0.635141\pi\)
\(864\) 145.519 4.95067
\(865\) 0 0
\(866\) 52.4277 1.78157
\(867\) −3.59565 −0.122115
\(868\) 37.8006 1.28303
\(869\) −4.74136 −0.160840
\(870\) 0 0
\(871\) 31.3693 1.06291
\(872\) −4.80976 −0.162879
\(873\) −58.1850 −1.96926
\(874\) 15.0353 0.508576
\(875\) 0 0
\(876\) 16.3558 0.552610
\(877\) −56.2252 −1.89859 −0.949296 0.314384i \(-0.898202\pi\)
−0.949296 + 0.314384i \(0.898202\pi\)
\(878\) −32.3768 −1.09266
\(879\) 52.3659 1.76626
\(880\) 0 0
\(881\) 25.8936 0.872379 0.436189 0.899855i \(-0.356328\pi\)
0.436189 + 0.899855i \(0.356328\pi\)
\(882\) 227.955 7.67564
\(883\) 21.2965 0.716686 0.358343 0.933590i \(-0.383342\pi\)
0.358343 + 0.933590i \(0.383342\pi\)
\(884\) 21.0667 0.708550
\(885\) 0 0
\(886\) 21.7529 0.730801
\(887\) 28.5513 0.958657 0.479329 0.877635i \(-0.340880\pi\)
0.479329 + 0.877635i \(0.340880\pi\)
\(888\) 1.09852 0.0368639
\(889\) 39.8100 1.33518
\(890\) 0 0
\(891\) 42.4669 1.42269
\(892\) 8.35248 0.279662
\(893\) −2.27417 −0.0761021
\(894\) −50.9808 −1.70505
\(895\) 0 0
\(896\) 27.6823 0.924802
\(897\) −85.1784 −2.84402
\(898\) 18.9619 0.632767
\(899\) 5.14280 0.171522
\(900\) 0 0
\(901\) −40.2687 −1.34154
\(902\) 11.9633 0.398333
\(903\) 50.0968 1.66712
\(904\) −2.26084 −0.0751945
\(905\) 0 0
\(906\) 40.7489 1.35379
\(907\) 42.5078 1.41145 0.705724 0.708487i \(-0.250622\pi\)
0.705724 + 0.708487i \(0.250622\pi\)
\(908\) −3.87905 −0.128731
\(909\) −139.611 −4.63060
\(910\) 0 0
\(911\) −22.2364 −0.736725 −0.368362 0.929682i \(-0.620081\pi\)
−0.368362 + 0.929682i \(0.620081\pi\)
\(912\) 16.0029 0.529907
\(913\) 12.4460 0.411902
\(914\) −16.0940 −0.532341
\(915\) 0 0
\(916\) 4.26032 0.140765
\(917\) −99.7626 −3.29445
\(918\) −161.349 −5.32531
\(919\) 20.4010 0.672968 0.336484 0.941689i \(-0.390762\pi\)
0.336484 + 0.941689i \(0.390762\pi\)
\(920\) 0 0
\(921\) 61.0933 2.01309
\(922\) 35.6311 1.17345
\(923\) −40.7659 −1.34182
\(924\) −24.8245 −0.816665
\(925\) 0 0
\(926\) −66.8257 −2.19603
\(927\) 174.608 5.73488
\(928\) −7.12755 −0.233973
\(929\) 2.87019 0.0941678 0.0470839 0.998891i \(-0.485007\pi\)
0.0470839 + 0.998891i \(0.485007\pi\)
\(930\) 0 0
\(931\) 13.6280 0.446638
\(932\) 35.9642 1.17805
\(933\) 5.64339 0.184756
\(934\) 36.6645 1.19970
\(935\) 0 0
\(936\) −21.4439 −0.700915
\(937\) 25.1123 0.820382 0.410191 0.912000i \(-0.365462\pi\)
0.410191 + 0.912000i \(0.365462\pi\)
\(938\) 86.4952 2.82417
\(939\) −0.305856 −0.00998124
\(940\) 0 0
\(941\) −3.17217 −0.103410 −0.0517049 0.998662i \(-0.516466\pi\)
−0.0517049 + 0.998662i \(0.516466\pi\)
\(942\) −64.3366 −2.09620
\(943\) 50.1147 1.63196
\(944\) 24.3366 0.792090
\(945\) 0 0
\(946\) −6.07579 −0.197541
\(947\) −42.9183 −1.39466 −0.697328 0.716752i \(-0.745628\pi\)
−0.697328 + 0.716752i \(0.745628\pi\)
\(948\) 25.9152 0.841687
\(949\) −9.33808 −0.303127
\(950\) 0 0
\(951\) 21.1395 0.685494
\(952\) −15.0161 −0.486674
\(953\) −13.5339 −0.438407 −0.219203 0.975679i \(-0.570346\pi\)
−0.219203 + 0.975679i \(0.570346\pi\)
\(954\) −158.563 −5.13367
\(955\) 0 0
\(956\) 33.7744 1.09234
\(957\) −3.37739 −0.109176
\(958\) −54.5070 −1.76104
\(959\) 0.767683 0.0247898
\(960\) 0 0
\(961\) −3.57218 −0.115232
\(962\) 2.42618 0.0782232
\(963\) 132.315 4.26378
\(964\) −14.5262 −0.467858
\(965\) 0 0
\(966\) −234.864 −7.55663
\(967\) −37.1903 −1.19596 −0.597980 0.801511i \(-0.704030\pi\)
−0.597980 + 0.801511i \(0.704030\pi\)
\(968\) −0.778295 −0.0250154
\(969\) −14.6104 −0.469353
\(970\) 0 0
\(971\) −7.23925 −0.232319 −0.116159 0.993231i \(-0.537058\pi\)
−0.116159 + 0.993231i \(0.537058\pi\)
\(972\) −136.531 −4.37925
\(973\) 73.6339 2.36059
\(974\) 12.6441 0.405142
\(975\) 0 0
\(976\) 60.2538 1.92868
\(977\) 5.16023 0.165090 0.0825451 0.996587i \(-0.473695\pi\)
0.0825451 + 0.996587i \(0.473695\pi\)
\(978\) −33.9282 −1.08490
\(979\) −8.61817 −0.275438
\(980\) 0 0
\(981\) 54.5631 1.74207
\(982\) 13.8752 0.442776
\(983\) 5.28429 0.168543 0.0842713 0.996443i \(-0.473144\pi\)
0.0842713 + 0.996443i \(0.473144\pi\)
\(984\) 16.9034 0.538860
\(985\) 0 0
\(986\) 7.90288 0.251679
\(987\) 35.5245 1.13076
\(988\) 4.95921 0.157774
\(989\) −25.4518 −0.809320
\(990\) 0 0
\(991\) 54.2190 1.72232 0.861161 0.508332i \(-0.169738\pi\)
0.861161 + 0.508332i \(0.169738\pi\)
\(992\) −38.0130 −1.20691
\(993\) 25.8627 0.820728
\(994\) −112.405 −3.56526
\(995\) 0 0
\(996\) −68.0270 −2.15552
\(997\) −46.5273 −1.47353 −0.736767 0.676147i \(-0.763648\pi\)
−0.736767 + 0.676147i \(0.763648\pi\)
\(998\) −33.0224 −1.04531
\(999\) −8.22756 −0.260309
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 5225.2.a.y.1.12 yes 15
5.4 even 2 5225.2.a.r.1.4 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.r.1.4 15 5.4 even 2
5225.2.a.y.1.12 yes 15 1.1 even 1 trivial