Properties

Label 7225.2.a.bg.1.4
Level $7225$
Weight $2$
Character 7225.1
Self dual yes
Analytic conductor $57.692$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7225,2,Mod(1,7225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7225 = 5^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7225.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: \(57.6919154604\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.93924352.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 17x^{4} + 73x^{2} - 67 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 425)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.12261\) of defining polynomial
Character \(\chi\) \(=\) 7225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.311108 q^{2} +1.12261 q^{3} -1.90321 q^{4} +0.349253 q^{6} +3.60843 q^{7} -1.21432 q^{8} -1.73975 q^{9} +O(q^{10})\) \(q+0.311108 q^{2} +1.12261 q^{3} -1.90321 q^{4} +0.349253 q^{6} +3.60843 q^{7} -1.21432 q^{8} -1.73975 q^{9} -5.74499 q^{11} -2.13656 q^{12} +2.59210 q^{13} +1.12261 q^{14} +3.42864 q^{16} -0.541249 q^{18} -4.28100 q^{19} +4.05086 q^{21} -1.78731 q^{22} +1.47186 q^{23} -1.36321 q^{24} +0.806424 q^{26} -5.32089 q^{27} -6.86760 q^{28} +5.50439 q^{29} +8.33946 q^{31} +3.49532 q^{32} -6.44938 q^{33} +3.31111 q^{36} -6.20290 q^{37} -1.33185 q^{38} +2.90992 q^{39} +3.95768 q^{41} +1.26025 q^{42} -9.76049 q^{43} +10.9339 q^{44} +0.457908 q^{46} +6.62222 q^{47} +3.84902 q^{48} +6.02074 q^{49} -4.93332 q^{52} -0.658781 q^{53} -1.65537 q^{54} -4.38178 q^{56} -4.80589 q^{57} +1.71246 q^{58} -5.95407 q^{59} +8.76357 q^{61} +2.59447 q^{62} -6.27775 q^{63} -5.76986 q^{64} -2.00645 q^{66} -0.428639 q^{67} +1.65233 q^{69} +1.36321 q^{71} +2.11261 q^{72} -3.25917 q^{73} -1.92977 q^{74} +8.14764 q^{76} -20.7304 q^{77} +0.905299 q^{78} +1.89597 q^{79} -0.754037 q^{81} +1.23127 q^{82} +16.6637 q^{83} -7.70964 q^{84} -3.03657 q^{86} +6.17929 q^{87} +6.97626 q^{88} -3.52543 q^{89} +9.35342 q^{91} -2.80127 q^{92} +9.36196 q^{93} +2.06022 q^{94} +3.92388 q^{96} +11.7073 q^{97} +1.87310 q^{98} +9.99483 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 16 q^{9} + 2 q^{13} - 6 q^{16} - 16 q^{18} - 12 q^{19} - 2 q^{21} - 22 q^{26} - 6 q^{32} + 28 q^{33} + 20 q^{36} + 32 q^{38} + 34 q^{42} + 8 q^{43} + 40 q^{47} - 4 q^{49} - 30 q^{52} + 10 q^{53} + 4 q^{59} - 22 q^{64} + 16 q^{66} + 24 q^{67} + 24 q^{69} + 52 q^{72} + 36 q^{76} - 44 q^{77} + 34 q^{81} + 20 q^{83} - 6 q^{84} - 4 q^{86} + 76 q^{87} - 8 q^{89} + 30 q^{93} + 40 q^{94} - 16 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.311108 0.219986 0.109993 0.993932i \(-0.464917\pi\)
0.109993 + 0.993932i \(0.464917\pi\)
\(3\) 1.12261 0.648139 0.324070 0.946033i \(-0.394949\pi\)
0.324070 + 0.946033i \(0.394949\pi\)
\(4\) −1.90321 −0.951606
\(5\) 0 0
\(6\) 0.349253 0.142582
\(7\) 3.60843 1.36386 0.681929 0.731419i \(-0.261141\pi\)
0.681929 + 0.731419i \(0.261141\pi\)
\(8\) −1.21432 −0.429327
\(9\) −1.73975 −0.579916
\(10\) 0 0
\(11\) −5.74499 −1.73218 −0.866090 0.499888i \(-0.833374\pi\)
−0.866090 + 0.499888i \(0.833374\pi\)
\(12\) −2.13656 −0.616773
\(13\) 2.59210 0.718920 0.359460 0.933160i \(-0.382961\pi\)
0.359460 + 0.933160i \(0.382961\pi\)
\(14\) 1.12261 0.300030
\(15\) 0 0
\(16\) 3.42864 0.857160
\(17\) 0 0
\(18\) −0.541249 −0.127574
\(19\) −4.28100 −0.982128 −0.491064 0.871124i \(-0.663392\pi\)
−0.491064 + 0.871124i \(0.663392\pi\)
\(20\) 0 0
\(21\) 4.05086 0.883969
\(22\) −1.78731 −0.381056
\(23\) 1.47186 0.306905 0.153452 0.988156i \(-0.450961\pi\)
0.153452 + 0.988156i \(0.450961\pi\)
\(24\) −1.36321 −0.278264
\(25\) 0 0
\(26\) 0.806424 0.158153
\(27\) −5.32089 −1.02401
\(28\) −6.86760 −1.29785
\(29\) 5.50439 1.02214 0.511070 0.859539i \(-0.329249\pi\)
0.511070 + 0.859539i \(0.329249\pi\)
\(30\) 0 0
\(31\) 8.33946 1.49781 0.748906 0.662676i \(-0.230579\pi\)
0.748906 + 0.662676i \(0.230579\pi\)
\(32\) 3.49532 0.617890
\(33\) −6.44938 −1.12269
\(34\) 0 0
\(35\) 0 0
\(36\) 3.31111 0.551851
\(37\) −6.20290 −1.01975 −0.509875 0.860248i \(-0.670308\pi\)
−0.509875 + 0.860248i \(0.670308\pi\)
\(38\) −1.33185 −0.216055
\(39\) 2.90992 0.465960
\(40\) 0 0
\(41\) 3.95768 0.618086 0.309043 0.951048i \(-0.399991\pi\)
0.309043 + 0.951048i \(0.399991\pi\)
\(42\) 1.26025 0.194461
\(43\) −9.76049 −1.48846 −0.744230 0.667923i \(-0.767184\pi\)
−0.744230 + 0.667923i \(0.767184\pi\)
\(44\) 10.9339 1.64835
\(45\) 0 0
\(46\) 0.457908 0.0675148
\(47\) 6.62222 0.965949 0.482975 0.875634i \(-0.339556\pi\)
0.482975 + 0.875634i \(0.339556\pi\)
\(48\) 3.84902 0.555559
\(49\) 6.02074 0.860106
\(50\) 0 0
\(51\) 0 0
\(52\) −4.93332 −0.684129
\(53\) −0.658781 −0.0904905 −0.0452452 0.998976i \(-0.514407\pi\)
−0.0452452 + 0.998976i \(0.514407\pi\)
\(54\) −1.65537 −0.225267
\(55\) 0 0
\(56\) −4.38178 −0.585540
\(57\) −4.80589 −0.636555
\(58\) 1.71246 0.224857
\(59\) −5.95407 −0.775154 −0.387577 0.921837i \(-0.626688\pi\)
−0.387577 + 0.921837i \(0.626688\pi\)
\(60\) 0 0
\(61\) 8.76357 1.12206 0.561030 0.827796i \(-0.310405\pi\)
0.561030 + 0.827796i \(0.310405\pi\)
\(62\) 2.59447 0.329498
\(63\) −6.27775 −0.790922
\(64\) −5.76986 −0.721232
\(65\) 0 0
\(66\) −2.00645 −0.246977
\(67\) −0.428639 −0.0523666 −0.0261833 0.999657i \(-0.508335\pi\)
−0.0261833 + 0.999657i \(0.508335\pi\)
\(68\) 0 0
\(69\) 1.65233 0.198917
\(70\) 0 0
\(71\) 1.36321 0.161783 0.0808915 0.996723i \(-0.474223\pi\)
0.0808915 + 0.996723i \(0.474223\pi\)
\(72\) 2.11261 0.248973
\(73\) −3.25917 −0.381457 −0.190729 0.981643i \(-0.561085\pi\)
−0.190729 + 0.981643i \(0.561085\pi\)
\(74\) −1.92977 −0.224331
\(75\) 0 0
\(76\) 8.14764 0.934599
\(77\) −20.7304 −2.36245
\(78\) 0.905299 0.102505
\(79\) 1.89597 0.213313 0.106656 0.994296i \(-0.465985\pi\)
0.106656 + 0.994296i \(0.465985\pi\)
\(80\) 0 0
\(81\) −0.754037 −0.0837819
\(82\) 1.23127 0.135970
\(83\) 16.6637 1.82908 0.914540 0.404497i \(-0.132553\pi\)
0.914540 + 0.404497i \(0.132553\pi\)
\(84\) −7.70964 −0.841190
\(85\) 0 0
\(86\) −3.03657 −0.327441
\(87\) 6.17929 0.662489
\(88\) 6.97626 0.743671
\(89\) −3.52543 −0.373695 −0.186847 0.982389i \(-0.559827\pi\)
−0.186847 + 0.982389i \(0.559827\pi\)
\(90\) 0 0
\(91\) 9.35342 0.980505
\(92\) −2.80127 −0.292052
\(93\) 9.36196 0.970790
\(94\) 2.06022 0.212496
\(95\) 0 0
\(96\) 3.92388 0.400479
\(97\) 11.7073 1.18870 0.594348 0.804208i \(-0.297410\pi\)
0.594348 + 0.804208i \(0.297410\pi\)
\(98\) 1.87310 0.189212
\(99\) 9.99483 1.00452
\(100\) 0 0
\(101\) 3.96989 0.395019 0.197509 0.980301i \(-0.436715\pi\)
0.197509 + 0.980301i \(0.436715\pi\)
\(102\) 0 0
\(103\) −4.23506 −0.417293 −0.208647 0.977991i \(-0.566906\pi\)
−0.208647 + 0.977991i \(0.566906\pi\)
\(104\) −3.14764 −0.308652
\(105\) 0 0
\(106\) −0.204952 −0.0199067
\(107\) 11.8392 1.14454 0.572271 0.820065i \(-0.306063\pi\)
0.572271 + 0.820065i \(0.306063\pi\)
\(108\) 10.1268 0.974449
\(109\) 12.0227 1.15157 0.575785 0.817601i \(-0.304697\pi\)
0.575785 + 0.817601i \(0.304697\pi\)
\(110\) 0 0
\(111\) −6.96343 −0.660940
\(112\) 12.3720 1.16904
\(113\) 10.7915 1.01518 0.507588 0.861600i \(-0.330537\pi\)
0.507588 + 0.861600i \(0.330537\pi\)
\(114\) −1.49515 −0.140034
\(115\) 0 0
\(116\) −10.4760 −0.972675
\(117\) −4.50961 −0.416913
\(118\) −1.85236 −0.170523
\(119\) 0 0
\(120\) 0 0
\(121\) 22.0049 2.00045
\(122\) 2.72641 0.246838
\(123\) 4.44293 0.400605
\(124\) −15.8718 −1.42533
\(125\) 0 0
\(126\) −1.95306 −0.173992
\(127\) 17.1383 1.52078 0.760388 0.649469i \(-0.225009\pi\)
0.760388 + 0.649469i \(0.225009\pi\)
\(128\) −8.78568 −0.776552
\(129\) −10.9572 −0.964730
\(130\) 0 0
\(131\) 8.89551 0.777204 0.388602 0.921406i \(-0.372958\pi\)
0.388602 + 0.921406i \(0.372958\pi\)
\(132\) 12.2745 1.06836
\(133\) −15.4477 −1.33948
\(134\) −0.133353 −0.0115200
\(135\) 0 0
\(136\) 0 0
\(137\) −14.0716 −1.20222 −0.601109 0.799167i \(-0.705274\pi\)
−0.601109 + 0.799167i \(0.705274\pi\)
\(138\) 0.514052 0.0437590
\(139\) −16.7876 −1.42390 −0.711952 0.702228i \(-0.752189\pi\)
−0.711952 + 0.702228i \(0.752189\pi\)
\(140\) 0 0
\(141\) 7.43416 0.626070
\(142\) 0.424104 0.0355901
\(143\) −14.8916 −1.24530
\(144\) −5.96497 −0.497081
\(145\) 0 0
\(146\) −1.01395 −0.0839155
\(147\) 6.75895 0.557468
\(148\) 11.8054 0.970400
\(149\) −4.76494 −0.390359 −0.195179 0.980768i \(-0.562529\pi\)
−0.195179 + 0.980768i \(0.562529\pi\)
\(150\) 0 0
\(151\) −2.99063 −0.243374 −0.121687 0.992569i \(-0.538830\pi\)
−0.121687 + 0.992569i \(0.538830\pi\)
\(152\) 5.19850 0.421654
\(153\) 0 0
\(154\) −6.44938 −0.519706
\(155\) 0 0
\(156\) −5.53820 −0.443411
\(157\) 13.9447 1.11291 0.556454 0.830878i \(-0.312162\pi\)
0.556454 + 0.830878i \(0.312162\pi\)
\(158\) 0.589850 0.0469260
\(159\) −0.739554 −0.0586504
\(160\) 0 0
\(161\) 5.31111 0.418574
\(162\) −0.234587 −0.0184309
\(163\) −8.65491 −0.677905 −0.338953 0.940803i \(-0.610073\pi\)
−0.338953 + 0.940803i \(0.610073\pi\)
\(164\) −7.53230 −0.588174
\(165\) 0 0
\(166\) 5.18421 0.402373
\(167\) 0.773357 0.0598442 0.0299221 0.999552i \(-0.490474\pi\)
0.0299221 + 0.999552i \(0.490474\pi\)
\(168\) −4.91903 −0.379512
\(169\) −6.28100 −0.483154
\(170\) 0 0
\(171\) 7.44785 0.569551
\(172\) 18.5763 1.41643
\(173\) −2.41097 −0.183302 −0.0916511 0.995791i \(-0.529214\pi\)
−0.0916511 + 0.995791i \(0.529214\pi\)
\(174\) 1.92242 0.145739
\(175\) 0 0
\(176\) −19.6975 −1.48476
\(177\) −6.68409 −0.502407
\(178\) −1.09679 −0.0822077
\(179\) 18.4558 1.37945 0.689727 0.724070i \(-0.257731\pi\)
0.689727 + 0.724070i \(0.257731\pi\)
\(180\) 0 0
\(181\) 20.6366 1.53391 0.766953 0.641703i \(-0.221772\pi\)
0.766953 + 0.641703i \(0.221772\pi\)
\(182\) 2.90992 0.215698
\(183\) 9.83807 0.727251
\(184\) −1.78731 −0.131762
\(185\) 0 0
\(186\) 2.91258 0.213561
\(187\) 0 0
\(188\) −12.6035 −0.919203
\(189\) −19.2000 −1.39660
\(190\) 0 0
\(191\) 22.8113 1.65057 0.825286 0.564716i \(-0.191014\pi\)
0.825286 + 0.564716i \(0.191014\pi\)
\(192\) −6.47730 −0.467459
\(193\) −19.7208 −1.41953 −0.709767 0.704437i \(-0.751200\pi\)
−0.709767 + 0.704437i \(0.751200\pi\)
\(194\) 3.64223 0.261497
\(195\) 0 0
\(196\) −11.4588 −0.818482
\(197\) 12.8870 0.918160 0.459080 0.888395i \(-0.348179\pi\)
0.459080 + 0.888395i \(0.348179\pi\)
\(198\) 3.10947 0.220980
\(199\) 6.44350 0.456767 0.228384 0.973571i \(-0.426656\pi\)
0.228384 + 0.973571i \(0.426656\pi\)
\(200\) 0 0
\(201\) −0.481195 −0.0339409
\(202\) 1.23506 0.0868988
\(203\) 19.8622 1.39405
\(204\) 0 0
\(205\) 0 0
\(206\) −1.31756 −0.0917988
\(207\) −2.56067 −0.177979
\(208\) 8.88739 0.616230
\(209\) 24.5943 1.70122
\(210\) 0 0
\(211\) −11.3580 −0.781920 −0.390960 0.920408i \(-0.627857\pi\)
−0.390960 + 0.920408i \(0.627857\pi\)
\(212\) 1.25380 0.0861113
\(213\) 1.53035 0.104858
\(214\) 3.68328 0.251784
\(215\) 0 0
\(216\) 6.46126 0.439633
\(217\) 30.0923 2.04280
\(218\) 3.74037 0.253330
\(219\) −3.65878 −0.247237
\(220\) 0 0
\(221\) 0 0
\(222\) −2.16638 −0.145398
\(223\) 0.888922 0.0595266 0.0297633 0.999557i \(-0.490525\pi\)
0.0297633 + 0.999557i \(0.490525\pi\)
\(224\) 12.6126 0.842714
\(225\) 0 0
\(226\) 3.35731 0.223325
\(227\) 12.9280 0.858064 0.429032 0.903289i \(-0.358855\pi\)
0.429032 + 0.903289i \(0.358855\pi\)
\(228\) 9.14662 0.605750
\(229\) −2.83161 −0.187118 −0.0935591 0.995614i \(-0.529824\pi\)
−0.0935591 + 0.995614i \(0.529824\pi\)
\(230\) 0 0
\(231\) −23.2721 −1.53119
\(232\) −6.68409 −0.438832
\(233\) −19.9381 −1.30619 −0.653094 0.757277i \(-0.726529\pi\)
−0.653094 + 0.757277i \(0.726529\pi\)
\(234\) −1.40297 −0.0917153
\(235\) 0 0
\(236\) 11.3319 0.737641
\(237\) 2.12843 0.138256
\(238\) 0 0
\(239\) 27.2859 1.76498 0.882490 0.470332i \(-0.155866\pi\)
0.882490 + 0.470332i \(0.155866\pi\)
\(240\) 0 0
\(241\) −3.57462 −0.230262 −0.115131 0.993350i \(-0.536729\pi\)
−0.115131 + 0.993350i \(0.536729\pi\)
\(242\) 6.84590 0.440071
\(243\) 15.1162 0.969703
\(244\) −16.6789 −1.06776
\(245\) 0 0
\(246\) 1.38223 0.0881278
\(247\) −11.0968 −0.706072
\(248\) −10.1268 −0.643051
\(249\) 18.7068 1.18550
\(250\) 0 0
\(251\) 17.6128 1.11171 0.555857 0.831278i \(-0.312390\pi\)
0.555857 + 0.831278i \(0.312390\pi\)
\(252\) 11.9479 0.752646
\(253\) −8.45584 −0.531614
\(254\) 5.33185 0.334550
\(255\) 0 0
\(256\) 8.80642 0.550401
\(257\) 20.5462 1.28163 0.640817 0.767693i \(-0.278596\pi\)
0.640817 + 0.767693i \(0.278596\pi\)
\(258\) −3.40888 −0.212227
\(259\) −22.3827 −1.39079
\(260\) 0 0
\(261\) −9.57625 −0.592755
\(262\) 2.76746 0.170974
\(263\) 8.38715 0.517174 0.258587 0.965988i \(-0.416743\pi\)
0.258587 + 0.965988i \(0.416743\pi\)
\(264\) 7.83161 0.482002
\(265\) 0 0
\(266\) −4.80589 −0.294668
\(267\) −3.95768 −0.242206
\(268\) 0.815792 0.0498324
\(269\) −12.4058 −0.756395 −0.378197 0.925725i \(-0.623456\pi\)
−0.378197 + 0.925725i \(0.623456\pi\)
\(270\) 0 0
\(271\) −7.13828 −0.433619 −0.216810 0.976214i \(-0.569565\pi\)
−0.216810 + 0.976214i \(0.569565\pi\)
\(272\) 0 0
\(273\) 10.5002 0.635503
\(274\) −4.37778 −0.264472
\(275\) 0 0
\(276\) −3.14473 −0.189290
\(277\) −20.4193 −1.22688 −0.613438 0.789743i \(-0.710214\pi\)
−0.613438 + 0.789743i \(0.710214\pi\)
\(278\) −5.22275 −0.313240
\(279\) −14.5086 −0.868605
\(280\) 0 0
\(281\) −29.1432 −1.73854 −0.869269 0.494340i \(-0.835410\pi\)
−0.869269 + 0.494340i \(0.835410\pi\)
\(282\) 2.31283 0.137727
\(283\) −9.16991 −0.545095 −0.272547 0.962142i \(-0.587866\pi\)
−0.272547 + 0.962142i \(0.587866\pi\)
\(284\) −2.59447 −0.153954
\(285\) 0 0
\(286\) −4.63290 −0.273949
\(287\) 14.2810 0.842981
\(288\) −6.08097 −0.358324
\(289\) 0 0
\(290\) 0 0
\(291\) 13.1427 0.770440
\(292\) 6.20290 0.362997
\(293\) 16.2351 0.948463 0.474231 0.880400i \(-0.342726\pi\)
0.474231 + 0.880400i \(0.342726\pi\)
\(294\) 2.10276 0.122636
\(295\) 0 0
\(296\) 7.53230 0.437806
\(297\) 30.5684 1.77376
\(298\) −1.48241 −0.0858737
\(299\) 3.81522 0.220640
\(300\) 0 0
\(301\) −35.2200 −2.03005
\(302\) −0.930409 −0.0535390
\(303\) 4.45664 0.256027
\(304\) −14.6780 −0.841841
\(305\) 0 0
\(306\) 0 0
\(307\) −3.37778 −0.192780 −0.0963902 0.995344i \(-0.530730\pi\)
−0.0963902 + 0.995344i \(0.530730\pi\)
\(308\) 39.4543 2.24812
\(309\) −4.75432 −0.270464
\(310\) 0 0
\(311\) 7.95641 0.451166 0.225583 0.974224i \(-0.427571\pi\)
0.225583 + 0.974224i \(0.427571\pi\)
\(312\) −3.53358 −0.200049
\(313\) 6.83380 0.386269 0.193135 0.981172i \(-0.438135\pi\)
0.193135 + 0.981172i \(0.438135\pi\)
\(314\) 4.33830 0.244825
\(315\) 0 0
\(316\) −3.60843 −0.202990
\(317\) −27.1550 −1.52517 −0.762587 0.646886i \(-0.776071\pi\)
−0.762587 + 0.646886i \(0.776071\pi\)
\(318\) −0.230081 −0.0129023
\(319\) −31.6227 −1.77053
\(320\) 0 0
\(321\) 13.2908 0.741822
\(322\) 1.65233 0.0920806
\(323\) 0 0
\(324\) 1.43509 0.0797274
\(325\) 0 0
\(326\) −2.69261 −0.149130
\(327\) 13.4968 0.746377
\(328\) −4.80589 −0.265361
\(329\) 23.8958 1.31742
\(330\) 0 0
\(331\) 20.3783 1.12009 0.560045 0.828462i \(-0.310784\pi\)
0.560045 + 0.828462i \(0.310784\pi\)
\(332\) −31.7146 −1.74056
\(333\) 10.7915 0.591369
\(334\) 0.240597 0.0131649
\(335\) 0 0
\(336\) 13.8889 0.757703
\(337\) −4.70775 −0.256447 −0.128224 0.991745i \(-0.540928\pi\)
−0.128224 + 0.991745i \(0.540928\pi\)
\(338\) −1.95407 −0.106287
\(339\) 12.1146 0.657976
\(340\) 0 0
\(341\) −47.9101 −2.59448
\(342\) 2.31708 0.125294
\(343\) −3.53358 −0.190795
\(344\) 11.8524 0.639036
\(345\) 0 0
\(346\) −0.750070 −0.0403240
\(347\) 35.3520 1.89779 0.948896 0.315588i \(-0.102202\pi\)
0.948896 + 0.315588i \(0.102202\pi\)
\(348\) −11.7605 −0.630428
\(349\) 25.0558 1.34121 0.670603 0.741817i \(-0.266036\pi\)
0.670603 + 0.741817i \(0.266036\pi\)
\(350\) 0 0
\(351\) −13.7923 −0.736178
\(352\) −20.0806 −1.07030
\(353\) −13.6953 −0.728930 −0.364465 0.931217i \(-0.618748\pi\)
−0.364465 + 0.931217i \(0.618748\pi\)
\(354\) −2.07947 −0.110523
\(355\) 0 0
\(356\) 6.70964 0.355610
\(357\) 0 0
\(358\) 5.74176 0.303461
\(359\) −2.53480 −0.133781 −0.0668907 0.997760i \(-0.521308\pi\)
−0.0668907 + 0.997760i \(0.521308\pi\)
\(360\) 0 0
\(361\) −0.673071 −0.0354248
\(362\) 6.42021 0.337439
\(363\) 24.7029 1.29657
\(364\) −17.8015 −0.933054
\(365\) 0 0
\(366\) 3.06070 0.159985
\(367\) −36.3426 −1.89707 −0.948535 0.316673i \(-0.897434\pi\)
−0.948535 + 0.316673i \(0.897434\pi\)
\(368\) 5.04649 0.263066
\(369\) −6.88536 −0.358438
\(370\) 0 0
\(371\) −2.37716 −0.123416
\(372\) −17.8178 −0.923810
\(373\) 12.7714 0.661278 0.330639 0.943757i \(-0.392736\pi\)
0.330639 + 0.943757i \(0.392736\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.04149 −0.414708
\(377\) 14.2680 0.734837
\(378\) −5.97328 −0.307232
\(379\) −22.1495 −1.13774 −0.568872 0.822426i \(-0.692620\pi\)
−0.568872 + 0.822426i \(0.692620\pi\)
\(380\) 0 0
\(381\) 19.2396 0.985674
\(382\) 7.09679 0.363103
\(383\) −1.95407 −0.0998482 −0.0499241 0.998753i \(-0.515898\pi\)
−0.0499241 + 0.998753i \(0.515898\pi\)
\(384\) −9.86289 −0.503314
\(385\) 0 0
\(386\) −6.13529 −0.312278
\(387\) 16.9808 0.863182
\(388\) −22.2815 −1.13117
\(389\) 7.91258 0.401184 0.200592 0.979675i \(-0.435713\pi\)
0.200592 + 0.979675i \(0.435713\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −7.31111 −0.369267
\(393\) 9.98619 0.503736
\(394\) 4.00924 0.201983
\(395\) 0 0
\(396\) −19.0223 −0.955906
\(397\) −9.24476 −0.463981 −0.231991 0.972718i \(-0.574524\pi\)
−0.231991 + 0.972718i \(0.574524\pi\)
\(398\) 2.00462 0.100483
\(399\) −17.3417 −0.868171
\(400\) 0 0
\(401\) 21.4848 1.07290 0.536450 0.843932i \(-0.319765\pi\)
0.536450 + 0.843932i \(0.319765\pi\)
\(402\) −0.149703 −0.00746653
\(403\) 21.6168 1.07681
\(404\) −7.55554 −0.375902
\(405\) 0 0
\(406\) 6.17929 0.306673
\(407\) 35.6356 1.76639
\(408\) 0 0
\(409\) 18.1669 0.898293 0.449147 0.893458i \(-0.351728\pi\)
0.449147 + 0.893458i \(0.351728\pi\)
\(410\) 0 0
\(411\) −15.7969 −0.779204
\(412\) 8.06022 0.397099
\(413\) −21.4848 −1.05720
\(414\) −0.796644 −0.0391529
\(415\) 0 0
\(416\) 9.06022 0.444214
\(417\) −18.8459 −0.922888
\(418\) 7.65147 0.374246
\(419\) −14.3589 −0.701476 −0.350738 0.936474i \(-0.614069\pi\)
−0.350738 + 0.936474i \(0.614069\pi\)
\(420\) 0 0
\(421\) 3.96989 0.193481 0.0967403 0.995310i \(-0.469158\pi\)
0.0967403 + 0.995310i \(0.469158\pi\)
\(422\) −3.53358 −0.172012
\(423\) −11.5210 −0.560169
\(424\) 0.799970 0.0388500
\(425\) 0 0
\(426\) 0.476104 0.0230673
\(427\) 31.6227 1.53033
\(428\) −22.5326 −1.08915
\(429\) −16.7175 −0.807127
\(430\) 0 0
\(431\) 32.9337 1.58636 0.793181 0.608985i \(-0.208423\pi\)
0.793181 + 0.608985i \(0.208423\pi\)
\(432\) −18.2434 −0.877736
\(433\) 38.2306 1.83725 0.918623 0.395135i \(-0.129302\pi\)
0.918623 + 0.395135i \(0.129302\pi\)
\(434\) 9.36196 0.449389
\(435\) 0 0
\(436\) −22.8818 −1.09584
\(437\) −6.30104 −0.301420
\(438\) −1.13828 −0.0543889
\(439\) −0.664702 −0.0317245 −0.0158622 0.999874i \(-0.505049\pi\)
−0.0158622 + 0.999874i \(0.505049\pi\)
\(440\) 0 0
\(441\) −10.4746 −0.498789
\(442\) 0 0
\(443\) 27.5669 1.30974 0.654872 0.755740i \(-0.272723\pi\)
0.654872 + 0.755740i \(0.272723\pi\)
\(444\) 13.2529 0.628954
\(445\) 0 0
\(446\) 0.276551 0.0130950
\(447\) −5.34916 −0.253007
\(448\) −20.8201 −0.983658
\(449\) 17.4756 0.824723 0.412362 0.911020i \(-0.364704\pi\)
0.412362 + 0.911020i \(0.364704\pi\)
\(450\) 0 0
\(451\) −22.7368 −1.07064
\(452\) −20.5385 −0.966048
\(453\) −3.35731 −0.157740
\(454\) 4.02201 0.188762
\(455\) 0 0
\(456\) 5.83589 0.273290
\(457\) −13.0350 −0.609753 −0.304877 0.952392i \(-0.598615\pi\)
−0.304877 + 0.952392i \(0.598615\pi\)
\(458\) −0.880937 −0.0411635
\(459\) 0 0
\(460\) 0 0
\(461\) 10.9491 0.509953 0.254976 0.966947i \(-0.417932\pi\)
0.254976 + 0.966947i \(0.417932\pi\)
\(462\) −7.24014 −0.336842
\(463\) −17.2400 −0.801210 −0.400605 0.916251i \(-0.631200\pi\)
−0.400605 + 0.916251i \(0.631200\pi\)
\(464\) 18.8726 0.876138
\(465\) 0 0
\(466\) −6.20290 −0.287344
\(467\) −0.0414872 −0.00191980 −0.000959899 1.00000i \(-0.500306\pi\)
−0.000959899 1.00000i \(0.500306\pi\)
\(468\) 8.58274 0.396737
\(469\) −1.54671 −0.0714206
\(470\) 0 0
\(471\) 15.6545 0.721319
\(472\) 7.23014 0.332794
\(473\) 56.0739 2.57828
\(474\) 0.662171 0.0304145
\(475\) 0 0
\(476\) 0 0
\(477\) 1.14611 0.0524769
\(478\) 8.48886 0.388272
\(479\) 35.4606 1.62024 0.810118 0.586266i \(-0.199403\pi\)
0.810118 + 0.586266i \(0.199403\pi\)
\(480\) 0 0
\(481\) −16.0786 −0.733119
\(482\) −1.11209 −0.0506545
\(483\) 5.96230 0.271294
\(484\) −41.8800 −1.90364
\(485\) 0 0
\(486\) 4.70276 0.213321
\(487\) 24.6691 1.11787 0.558933 0.829213i \(-0.311211\pi\)
0.558933 + 0.829213i \(0.311211\pi\)
\(488\) −10.6418 −0.481730
\(489\) −9.71609 −0.439377
\(490\) 0 0
\(491\) −8.58073 −0.387243 −0.193621 0.981076i \(-0.562023\pi\)
−0.193621 + 0.981076i \(0.562023\pi\)
\(492\) −8.45584 −0.381219
\(493\) 0 0
\(494\) −3.45230 −0.155326
\(495\) 0 0
\(496\) 28.5930 1.28386
\(497\) 4.91903 0.220649
\(498\) 5.81984 0.260793
\(499\) 8.48917 0.380027 0.190014 0.981781i \(-0.439147\pi\)
0.190014 + 0.981781i \(0.439147\pi\)
\(500\) 0 0
\(501\) 0.868178 0.0387873
\(502\) 5.47949 0.244562
\(503\) 24.1879 1.07849 0.539244 0.842150i \(-0.318710\pi\)
0.539244 + 0.842150i \(0.318710\pi\)
\(504\) 7.62320 0.339564
\(505\) 0 0
\(506\) −2.63068 −0.116948
\(507\) −7.05111 −0.313151
\(508\) −32.6178 −1.44718
\(509\) −25.8671 −1.14654 −0.573270 0.819366i \(-0.694325\pi\)
−0.573270 + 0.819366i \(0.694325\pi\)
\(510\) 0 0
\(511\) −11.7605 −0.520253
\(512\) 20.3111 0.897633
\(513\) 22.7787 1.00570
\(514\) 6.39207 0.281942
\(515\) 0 0
\(516\) 20.8539 0.918042
\(517\) −38.0446 −1.67320
\(518\) −6.96343 −0.305956
\(519\) −2.70657 −0.118805
\(520\) 0 0
\(521\) 11.4900 0.503385 0.251693 0.967807i \(-0.419013\pi\)
0.251693 + 0.967807i \(0.419013\pi\)
\(522\) −2.97925 −0.130398
\(523\) 33.0509 1.44521 0.722606 0.691260i \(-0.242944\pi\)
0.722606 + 0.691260i \(0.242944\pi\)
\(524\) −16.9300 −0.739592
\(525\) 0 0
\(526\) 2.60931 0.113771
\(527\) 0 0
\(528\) −22.1126 −0.962328
\(529\) −20.8336 −0.905810
\(530\) 0 0
\(531\) 10.3586 0.449524
\(532\) 29.4002 1.27466
\(533\) 10.2587 0.444354
\(534\) −1.23127 −0.0532820
\(535\) 0 0
\(536\) 0.520505 0.0224824
\(537\) 20.7187 0.894078
\(538\) −3.85954 −0.166397
\(539\) −34.5891 −1.48986
\(540\) 0 0
\(541\) 19.7884 0.850770 0.425385 0.905013i \(-0.360139\pi\)
0.425385 + 0.905013i \(0.360139\pi\)
\(542\) −2.22077 −0.0953904
\(543\) 23.1669 0.994185
\(544\) 0 0
\(545\) 0 0
\(546\) 3.26671 0.139802
\(547\) −15.4815 −0.661940 −0.330970 0.943641i \(-0.607376\pi\)
−0.330970 + 0.943641i \(0.607376\pi\)
\(548\) 26.7812 1.14404
\(549\) −15.2464 −0.650700
\(550\) 0 0
\(551\) −23.5643 −1.00387
\(552\) −2.00645 −0.0854003
\(553\) 6.84146 0.290928
\(554\) −6.35260 −0.269896
\(555\) 0 0
\(556\) 31.9503 1.35500
\(557\) −15.3477 −0.650302 −0.325151 0.945662i \(-0.605415\pi\)
−0.325151 + 0.945662i \(0.605415\pi\)
\(558\) −4.51373 −0.191081
\(559\) −25.3002 −1.07008
\(560\) 0 0
\(561\) 0 0
\(562\) −9.06668 −0.382455
\(563\) −14.3827 −0.606159 −0.303079 0.952965i \(-0.598015\pi\)
−0.303079 + 0.952965i \(0.598015\pi\)
\(564\) −14.1488 −0.595772
\(565\) 0 0
\(566\) −2.85283 −0.119913
\(567\) −2.72089 −0.114267
\(568\) −1.65537 −0.0694578
\(569\) 12.1432 0.509069 0.254535 0.967064i \(-0.418078\pi\)
0.254535 + 0.967064i \(0.418078\pi\)
\(570\) 0 0
\(571\) 2.88663 0.120802 0.0604009 0.998174i \(-0.480762\pi\)
0.0604009 + 0.998174i \(0.480762\pi\)
\(572\) 28.3419 1.18503
\(573\) 25.6082 1.06980
\(574\) 4.44293 0.185444
\(575\) 0 0
\(576\) 10.0381 0.418254
\(577\) −3.10970 −0.129458 −0.0647291 0.997903i \(-0.520618\pi\)
−0.0647291 + 0.997903i \(0.520618\pi\)
\(578\) 0 0
\(579\) −22.1388 −0.920055
\(580\) 0 0
\(581\) 60.1298 2.49460
\(582\) 4.08880 0.169486
\(583\) 3.78469 0.156746
\(584\) 3.95768 0.163770
\(585\) 0 0
\(586\) 5.05086 0.208649
\(587\) −10.2034 −0.421140 −0.210570 0.977579i \(-0.567532\pi\)
−0.210570 + 0.977579i \(0.567532\pi\)
\(588\) −12.8637 −0.530490
\(589\) −35.7012 −1.47104
\(590\) 0 0
\(591\) 14.4671 0.595095
\(592\) −21.2675 −0.874089
\(593\) 16.2301 0.666492 0.333246 0.942840i \(-0.391856\pi\)
0.333246 + 0.942840i \(0.391856\pi\)
\(594\) 9.51008 0.390203
\(595\) 0 0
\(596\) 9.06868 0.371468
\(597\) 7.23353 0.296049
\(598\) 1.18694 0.0485378
\(599\) −8.43309 −0.344567 −0.172283 0.985047i \(-0.555114\pi\)
−0.172283 + 0.985047i \(0.555114\pi\)
\(600\) 0 0
\(601\) −20.2535 −0.826160 −0.413080 0.910695i \(-0.635547\pi\)
−0.413080 + 0.910695i \(0.635547\pi\)
\(602\) −10.9572 −0.446583
\(603\) 0.745724 0.0303682
\(604\) 5.69181 0.231596
\(605\) 0 0
\(606\) 1.38649 0.0563225
\(607\) 22.7732 0.924334 0.462167 0.886793i \(-0.347072\pi\)
0.462167 + 0.886793i \(0.347072\pi\)
\(608\) −14.9634 −0.606847
\(609\) 22.2975 0.903540
\(610\) 0 0
\(611\) 17.1655 0.694441
\(612\) 0 0
\(613\) −8.40636 −0.339530 −0.169765 0.985485i \(-0.554301\pi\)
−0.169765 + 0.985485i \(0.554301\pi\)
\(614\) −1.05086 −0.0424091
\(615\) 0 0
\(616\) 25.1733 1.01426
\(617\) 14.8168 0.596500 0.298250 0.954488i \(-0.403597\pi\)
0.298250 + 0.954488i \(0.403597\pi\)
\(618\) −1.47911 −0.0594984
\(619\) −31.6221 −1.27100 −0.635500 0.772101i \(-0.719206\pi\)
−0.635500 + 0.772101i \(0.719206\pi\)
\(620\) 0 0
\(621\) −7.83161 −0.314272
\(622\) 2.47530 0.0992505
\(623\) −12.7212 −0.509666
\(624\) 9.97707 0.399403
\(625\) 0 0
\(626\) 2.12605 0.0849740
\(627\) 27.6098 1.10263
\(628\) −26.5397 −1.05905
\(629\) 0 0
\(630\) 0 0
\(631\) 17.3733 0.691622 0.345811 0.938304i \(-0.387604\pi\)
0.345811 + 0.938304i \(0.387604\pi\)
\(632\) −2.30231 −0.0915810
\(633\) −12.7506 −0.506793
\(634\) −8.44812 −0.335518
\(635\) 0 0
\(636\) 1.40753 0.0558121
\(637\) 15.6064 0.618348
\(638\) −9.83807 −0.389493
\(639\) −2.37164 −0.0938205
\(640\) 0 0
\(641\) −7.26842 −0.287085 −0.143543 0.989644i \(-0.545849\pi\)
−0.143543 + 0.989644i \(0.545849\pi\)
\(642\) 4.13488 0.163191
\(643\) 4.40507 0.173719 0.0868595 0.996221i \(-0.472317\pi\)
0.0868595 + 0.996221i \(0.472317\pi\)
\(644\) −10.1082 −0.398317
\(645\) 0 0
\(646\) 0 0
\(647\) 0.198022 0.00778504 0.00389252 0.999992i \(-0.498761\pi\)
0.00389252 + 0.999992i \(0.498761\pi\)
\(648\) 0.915643 0.0359698
\(649\) 34.2061 1.34271
\(650\) 0 0
\(651\) 33.7820 1.32402
\(652\) 16.4721 0.645098
\(653\) −9.92723 −0.388482 −0.194241 0.980954i \(-0.562224\pi\)
−0.194241 + 0.980954i \(0.562224\pi\)
\(654\) 4.19897 0.164193
\(655\) 0 0
\(656\) 13.5695 0.529798
\(657\) 5.67014 0.221213
\(658\) 7.43416 0.289814
\(659\) 3.00492 0.117055 0.0585276 0.998286i \(-0.481359\pi\)
0.0585276 + 0.998286i \(0.481359\pi\)
\(660\) 0 0
\(661\) −23.6923 −0.921523 −0.460762 0.887524i \(-0.652424\pi\)
−0.460762 + 0.887524i \(0.652424\pi\)
\(662\) 6.33984 0.246405
\(663\) 0 0
\(664\) −20.2351 −0.785273
\(665\) 0 0
\(666\) 3.35731 0.130093
\(667\) 8.10171 0.313699
\(668\) −1.47186 −0.0569481
\(669\) 0.997912 0.0385815
\(670\) 0 0
\(671\) −50.3466 −1.94361
\(672\) 14.1590 0.546196
\(673\) 30.1968 1.16400 0.582001 0.813188i \(-0.302270\pi\)
0.582001 + 0.813188i \(0.302270\pi\)
\(674\) −1.46462 −0.0564150
\(675\) 0 0
\(676\) 11.9541 0.459772
\(677\) −47.9413 −1.84253 −0.921266 0.388933i \(-0.872844\pi\)
−0.921266 + 0.388933i \(0.872844\pi\)
\(678\) 3.76895 0.144746
\(679\) 42.2449 1.62121
\(680\) 0 0
\(681\) 14.5131 0.556145
\(682\) −14.9052 −0.570750
\(683\) 1.41477 0.0541347 0.0270674 0.999634i \(-0.491383\pi\)
0.0270674 + 0.999634i \(0.491383\pi\)
\(684\) −14.1748 −0.541989
\(685\) 0 0
\(686\) −1.09932 −0.0419723
\(687\) −3.17880 −0.121279
\(688\) −33.4652 −1.27585
\(689\) −1.70763 −0.0650554
\(690\) 0 0
\(691\) 1.36321 0.0518588 0.0259294 0.999664i \(-0.491745\pi\)
0.0259294 + 0.999664i \(0.491745\pi\)
\(692\) 4.58858 0.174432
\(693\) 36.0656 1.37002
\(694\) 10.9983 0.417489
\(695\) 0 0
\(696\) −7.50363 −0.284424
\(697\) 0 0
\(698\) 7.79505 0.295047
\(699\) −22.3827 −0.846592
\(700\) 0 0
\(701\) −40.1990 −1.51829 −0.759147 0.650919i \(-0.774384\pi\)
−0.759147 + 0.650919i \(0.774384\pi\)
\(702\) −4.29089 −0.161949
\(703\) 26.5546 1.00153
\(704\) 33.1478 1.24930
\(705\) 0 0
\(706\) −4.26073 −0.160355
\(707\) 14.3251 0.538749
\(708\) 12.7212 0.478094
\(709\) −19.4729 −0.731322 −0.365661 0.930748i \(-0.619157\pi\)
−0.365661 + 0.930748i \(0.619157\pi\)
\(710\) 0 0
\(711\) −3.29850 −0.123704
\(712\) 4.28100 0.160437
\(713\) 12.2745 0.459685
\(714\) 0 0
\(715\) 0 0
\(716\) −35.1254 −1.31270
\(717\) 30.6314 1.14395
\(718\) −0.788595 −0.0294301
\(719\) −21.9765 −0.819586 −0.409793 0.912179i \(-0.634399\pi\)
−0.409793 + 0.912179i \(0.634399\pi\)
\(720\) 0 0
\(721\) −15.2819 −0.569128
\(722\) −0.209398 −0.00779297
\(723\) −4.01291 −0.149242
\(724\) −39.2758 −1.45967
\(725\) 0 0
\(726\) 7.68528 0.285227
\(727\) 36.7007 1.36116 0.680578 0.732676i \(-0.261729\pi\)
0.680578 + 0.732676i \(0.261729\pi\)
\(728\) −11.3580 −0.420957
\(729\) 19.2317 0.712284
\(730\) 0 0
\(731\) 0 0
\(732\) −18.7239 −0.692056
\(733\) −24.8666 −0.918471 −0.459235 0.888315i \(-0.651877\pi\)
−0.459235 + 0.888315i \(0.651877\pi\)
\(734\) −11.3065 −0.417330
\(735\) 0 0
\(736\) 5.14462 0.189633
\(737\) 2.46253 0.0907085
\(738\) −2.14209 −0.0788514
\(739\) 32.7368 1.20424 0.602122 0.798404i \(-0.294322\pi\)
0.602122 + 0.798404i \(0.294322\pi\)
\(740\) 0 0
\(741\) −12.4574 −0.457633
\(742\) −0.739554 −0.0271499
\(743\) 21.3762 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(744\) −11.3684 −0.416786
\(745\) 0 0
\(746\) 3.97328 0.145472
\(747\) −28.9906 −1.06071
\(748\) 0 0
\(749\) 42.7210 1.56099
\(750\) 0 0
\(751\) −44.5557 −1.62586 −0.812930 0.582362i \(-0.802129\pi\)
−0.812930 + 0.582362i \(0.802129\pi\)
\(752\) 22.7052 0.827973
\(753\) 19.7724 0.720545
\(754\) 4.43887 0.161654
\(755\) 0 0
\(756\) 36.5417 1.32901
\(757\) −12.1793 −0.442664 −0.221332 0.975199i \(-0.571040\pi\)
−0.221332 + 0.975199i \(0.571040\pi\)
\(758\) −6.89089 −0.250288
\(759\) −9.49260 −0.344560
\(760\) 0 0
\(761\) 7.18712 0.260533 0.130266 0.991479i \(-0.458417\pi\)
0.130266 + 0.991479i \(0.458417\pi\)
\(762\) 5.98559 0.216835
\(763\) 43.3832 1.57058
\(764\) −43.4148 −1.57069
\(765\) 0 0
\(766\) −0.607926 −0.0219652
\(767\) −15.4336 −0.557274
\(768\) 9.88618 0.356737
\(769\) 14.9748 0.540005 0.270003 0.962860i \(-0.412975\pi\)
0.270003 + 0.962860i \(0.412975\pi\)
\(770\) 0 0
\(771\) 23.0653 0.830678
\(772\) 37.5328 1.35084
\(773\) 26.2938 0.945721 0.472860 0.881137i \(-0.343222\pi\)
0.472860 + 0.881137i \(0.343222\pi\)
\(774\) 5.28286 0.189888
\(775\) 0 0
\(776\) −14.2164 −0.510339
\(777\) −25.1270 −0.901428
\(778\) 2.46167 0.0882550
\(779\) −16.9428 −0.607039
\(780\) 0 0
\(781\) −7.83161 −0.280237
\(782\) 0 0
\(783\) −29.2883 −1.04668
\(784\) 20.6430 0.737249
\(785\) 0 0
\(786\) 3.10678 0.110815
\(787\) −9.35342 −0.333413 −0.166707 0.986007i \(-0.553313\pi\)
−0.166707 + 0.986007i \(0.553313\pi\)
\(788\) −24.5267 −0.873727
\(789\) 9.41550 0.335201
\(790\) 0 0
\(791\) 38.9403 1.38456
\(792\) −12.1369 −0.431267
\(793\) 22.7161 0.806672
\(794\) −2.87612 −0.102070
\(795\) 0 0
\(796\) −12.2633 −0.434663
\(797\) 29.4800 1.04423 0.522117 0.852874i \(-0.325142\pi\)
0.522117 + 0.852874i \(0.325142\pi\)
\(798\) −5.39514 −0.190986
\(799\) 0 0
\(800\) 0 0
\(801\) 6.13335 0.216711
\(802\) 6.68409 0.236024
\(803\) 18.7239 0.660753
\(804\) 0.915816 0.0322983
\(805\) 0 0
\(806\) 6.72514 0.236883
\(807\) −13.9269 −0.490249
\(808\) −4.82071 −0.169592
\(809\) 35.1219 1.23482 0.617410 0.786642i \(-0.288182\pi\)
0.617410 + 0.786642i \(0.288182\pi\)
\(810\) 0 0
\(811\) −17.5682 −0.616902 −0.308451 0.951240i \(-0.599811\pi\)
−0.308451 + 0.951240i \(0.599811\pi\)
\(812\) −37.8020 −1.32659
\(813\) −8.01350 −0.281046
\(814\) 11.0865 0.388582
\(815\) 0 0
\(816\) 0 0
\(817\) 41.7846 1.46186
\(818\) 5.65185 0.197612
\(819\) −16.2726 −0.568610
\(820\) 0 0
\(821\) −13.1043 −0.457343 −0.228672 0.973504i \(-0.573438\pi\)
−0.228672 + 0.973504i \(0.573438\pi\)
\(822\) −4.91454 −0.171414
\(823\) 23.7638 0.828355 0.414178 0.910196i \(-0.364069\pi\)
0.414178 + 0.910196i \(0.364069\pi\)
\(824\) 5.14272 0.179155
\(825\) 0 0
\(826\) −6.68409 −0.232569
\(827\) −28.9423 −1.00642 −0.503211 0.864164i \(-0.667848\pi\)
−0.503211 + 0.864164i \(0.667848\pi\)
\(828\) 4.87350 0.169366
\(829\) 48.3131 1.67798 0.838992 0.544144i \(-0.183145\pi\)
0.838992 + 0.544144i \(0.183145\pi\)
\(830\) 0 0
\(831\) −22.9229 −0.795187
\(832\) −14.9561 −0.518509
\(833\) 0 0
\(834\) −5.86311 −0.203023
\(835\) 0 0
\(836\) −46.8081 −1.61889
\(837\) −44.3733 −1.53377
\(838\) −4.46715 −0.154315
\(839\) −36.8238 −1.27130 −0.635650 0.771978i \(-0.719268\pi\)
−0.635650 + 0.771978i \(0.719268\pi\)
\(840\) 0 0
\(841\) 1.29835 0.0447707
\(842\) 1.23506 0.0425631
\(843\) −32.7164 −1.12681
\(844\) 21.6168 0.744079
\(845\) 0 0
\(846\) −3.58427 −0.123230
\(847\) 79.4031 2.72832
\(848\) −2.25872 −0.0775648
\(849\) −10.2942 −0.353297
\(850\) 0 0
\(851\) −9.12981 −0.312966
\(852\) −2.91258 −0.0997833
\(853\) −29.8138 −1.02080 −0.510402 0.859936i \(-0.670503\pi\)
−0.510402 + 0.859936i \(0.670503\pi\)
\(854\) 9.83807 0.336652
\(855\) 0 0
\(856\) −14.3766 −0.491383
\(857\) −1.71246 −0.0584965 −0.0292483 0.999572i \(-0.509311\pi\)
−0.0292483 + 0.999572i \(0.509311\pi\)
\(858\) −5.20094 −0.177557
\(859\) −22.5749 −0.770246 −0.385123 0.922865i \(-0.625841\pi\)
−0.385123 + 0.922865i \(0.625841\pi\)
\(860\) 0 0
\(861\) 16.0320 0.546369
\(862\) 10.2459 0.348978
\(863\) −54.2578 −1.84696 −0.923479 0.383650i \(-0.874667\pi\)
−0.923479 + 0.383650i \(0.874667\pi\)
\(864\) −18.5982 −0.632723
\(865\) 0 0
\(866\) 11.8938 0.404169
\(867\) 0 0
\(868\) −57.2721 −1.94394
\(869\) −10.8923 −0.369496
\(870\) 0 0
\(871\) −1.11108 −0.0376474
\(872\) −14.5995 −0.494400
\(873\) −20.3677 −0.689343
\(874\) −1.96030 −0.0663082
\(875\) 0 0
\(876\) 6.96343 0.235273
\(877\) −14.8989 −0.503099 −0.251549 0.967844i \(-0.580940\pi\)
−0.251549 + 0.967844i \(0.580940\pi\)
\(878\) −0.206794 −0.00697896
\(879\) 18.2256 0.614736
\(880\) 0 0
\(881\) −22.3330 −0.752419 −0.376209 0.926535i \(-0.622773\pi\)
−0.376209 + 0.926535i \(0.622773\pi\)
\(882\) −3.25872 −0.109727
\(883\) −25.3319 −0.852485 −0.426242 0.904609i \(-0.640163\pi\)
−0.426242 + 0.904609i \(0.640163\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 8.57628 0.288126
\(887\) 6.35813 0.213485 0.106743 0.994287i \(-0.465958\pi\)
0.106743 + 0.994287i \(0.465958\pi\)
\(888\) 8.45584 0.283759
\(889\) 61.8422 2.07412
\(890\) 0 0
\(891\) 4.33194 0.145125
\(892\) −1.69181 −0.0566459
\(893\) −28.3497 −0.948686
\(894\) −1.66417 −0.0556581
\(895\) 0 0
\(896\) −31.7025 −1.05911
\(897\) 4.28300 0.143005
\(898\) 5.43679 0.181428
\(899\) 45.9037 1.53097
\(900\) 0 0
\(901\) 0 0
\(902\) −7.07361 −0.235525
\(903\) −39.5383 −1.31575
\(904\) −13.1043 −0.435843
\(905\) 0 0
\(906\) −1.04449 −0.0347007
\(907\) −13.0295 −0.432636 −0.216318 0.976323i \(-0.569405\pi\)
−0.216318 + 0.976323i \(0.569405\pi\)
\(908\) −24.6048 −0.816539
\(909\) −6.90660 −0.229078
\(910\) 0 0
\(911\) −20.7685 −0.688093 −0.344046 0.938953i \(-0.611798\pi\)
−0.344046 + 0.938953i \(0.611798\pi\)
\(912\) −16.4777 −0.545630
\(913\) −95.7328 −3.16829
\(914\) −4.05530 −0.134137
\(915\) 0 0
\(916\) 5.38916 0.178063
\(917\) 32.0988 1.06000
\(918\) 0 0
\(919\) −5.10970 −0.168553 −0.0842766 0.996442i \(-0.526858\pi\)
−0.0842766 + 0.996442i \(0.526858\pi\)
\(920\) 0 0
\(921\) −3.79193 −0.124948
\(922\) 3.40636 0.112183
\(923\) 3.53358 0.116309
\(924\) 44.2918 1.45709
\(925\) 0 0
\(926\) −5.36349 −0.176255
\(927\) 7.36794 0.241995
\(928\) 19.2396 0.631571
\(929\) 39.6073 1.29947 0.649737 0.760159i \(-0.274879\pi\)
0.649737 + 0.760159i \(0.274879\pi\)
\(930\) 0 0
\(931\) −25.7748 −0.844734
\(932\) 37.9464 1.24298
\(933\) 8.93194 0.292419
\(934\) −0.0129070 −0.000422330 0
\(935\) 0 0
\(936\) 5.47610 0.178992
\(937\) −39.5575 −1.29229 −0.646144 0.763215i \(-0.723620\pi\)
−0.646144 + 0.763215i \(0.723620\pi\)
\(938\) −0.481195 −0.0157116
\(939\) 7.67169 0.250356
\(940\) 0 0
\(941\) −53.2283 −1.73519 −0.867597 0.497268i \(-0.834337\pi\)
−0.867597 + 0.497268i \(0.834337\pi\)
\(942\) 4.87022 0.158680
\(943\) 5.82516 0.189693
\(944\) −20.4143 −0.664430
\(945\) 0 0
\(946\) 17.4450 0.567187
\(947\) 0.773357 0.0251307 0.0125654 0.999921i \(-0.496000\pi\)
0.0125654 + 0.999921i \(0.496000\pi\)
\(948\) −4.05086 −0.131566
\(949\) −8.44812 −0.274238
\(950\) 0 0
\(951\) −30.4844 −0.988525
\(952\) 0 0
\(953\) 36.8731 1.19444 0.597218 0.802079i \(-0.296273\pi\)
0.597218 + 0.802079i \(0.296273\pi\)
\(954\) 0.356564 0.0115442
\(955\) 0 0
\(956\) −51.9309 −1.67956
\(957\) −35.4999 −1.14755
\(958\) 11.0321 0.356430
\(959\) −50.7763 −1.63965
\(960\) 0 0
\(961\) 38.5466 1.24344
\(962\) −5.00217 −0.161276
\(963\) −20.5973 −0.663738
\(964\) 6.80327 0.219118
\(965\) 0 0
\(966\) 1.85492 0.0596810
\(967\) −17.8306 −0.573392 −0.286696 0.958022i \(-0.592557\pi\)
−0.286696 + 0.958022i \(0.592557\pi\)
\(968\) −26.7210 −0.858846
\(969\) 0 0
\(970\) 0 0
\(971\) 20.3783 0.653970 0.326985 0.945030i \(-0.393967\pi\)
0.326985 + 0.945030i \(0.393967\pi\)
\(972\) −28.7693 −0.922775
\(973\) −60.5768 −1.94200
\(974\) 7.67476 0.245915
\(975\) 0 0
\(976\) 30.0471 0.961785
\(977\) 8.64587 0.276606 0.138303 0.990390i \(-0.455835\pi\)
0.138303 + 0.990390i \(0.455835\pi\)
\(978\) −3.02275 −0.0966569
\(979\) 20.2535 0.647306
\(980\) 0 0
\(981\) −20.9165 −0.667813
\(982\) −2.66953 −0.0851882
\(983\) −61.1615 −1.95075 −0.975374 0.220558i \(-0.929212\pi\)
−0.975374 + 0.220558i \(0.929212\pi\)
\(984\) −5.39514 −0.171991
\(985\) 0 0
\(986\) 0 0
\(987\) 26.8256 0.853869
\(988\) 21.1195 0.671902
\(989\) −14.3661 −0.456815
\(990\) 0 0
\(991\) −27.0979 −0.860792 −0.430396 0.902640i \(-0.641626\pi\)
−0.430396 + 0.902640i \(0.641626\pi\)
\(992\) 29.1491 0.925484
\(993\) 22.8768 0.725974
\(994\) 1.53035 0.0485397
\(995\) 0 0
\(996\) −35.6031 −1.12813
\(997\) −48.3759 −1.53208 −0.766040 0.642793i \(-0.777775\pi\)
−0.766040 + 0.642793i \(0.777775\pi\)
\(998\) 2.64105 0.0836009
\(999\) 33.0049 1.04423
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 7225.2.a.bg.1.4 6
5.4 even 2 7225.2.a.ba.1.3 6
17.4 even 4 425.2.d.a.101.3 6
17.13 even 4 425.2.d.a.101.4 yes 6
17.16 even 2 inner 7225.2.a.bg.1.3 6
85.4 even 4 425.2.d.b.101.4 yes 6
85.13 odd 4 425.2.c.c.424.7 12
85.38 odd 4 425.2.c.c.424.8 12
85.47 odd 4 425.2.c.c.424.6 12
85.64 even 4 425.2.d.b.101.3 yes 6
85.72 odd 4 425.2.c.c.424.5 12
85.84 even 2 7225.2.a.ba.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.c.c.424.5 12 85.72 odd 4
425.2.c.c.424.6 12 85.47 odd 4
425.2.c.c.424.7 12 85.13 odd 4
425.2.c.c.424.8 12 85.38 odd 4
425.2.d.a.101.3 6 17.4 even 4
425.2.d.a.101.4 yes 6 17.13 even 4
425.2.d.b.101.3 yes 6 85.64 even 4
425.2.d.b.101.4 yes 6 85.4 even 4
7225.2.a.ba.1.3 6 5.4 even 2
7225.2.a.ba.1.4 6 85.84 even 2
7225.2.a.bg.1.3 6 17.16 even 2 inner
7225.2.a.bg.1.4 6 1.1 even 1 trivial