Properties

Label 3025.2.a.bk.1.5
Level $3025$
Weight $2$
Character 3025.1
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $8$
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] = [3025,2,Mod(1,3025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3025, 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("3025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.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: \(24.1547466114\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.1480160000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.802699\) of defining polynomial
Character \(\chi\) \(=\) 3025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.802699 q^{2} -1.76074 q^{3} -1.35567 q^{4} -1.41335 q^{6} +0.592103 q^{7} -2.69360 q^{8} +0.100212 q^{9} +O(q^{10})\) \(q+0.802699 q^{2} -1.76074 q^{3} -1.35567 q^{4} -1.41335 q^{6} +0.592103 q^{7} -2.69360 q^{8} +0.100212 q^{9} +2.38699 q^{12} -1.79489 q^{13} +0.475281 q^{14} +0.549201 q^{16} +7.07712 q^{17} +0.0804405 q^{18} +2.28684 q^{19} -1.04254 q^{21} -1.49081 q^{23} +4.74273 q^{24} -1.44076 q^{26} +5.10578 q^{27} -0.802699 q^{28} -3.57549 q^{29} +6.16724 q^{31} +5.82804 q^{32} +5.68079 q^{34} -0.135855 q^{36} -7.33743 q^{37} +1.83565 q^{38} +3.16034 q^{39} -8.41020 q^{41} -0.836847 q^{42} +9.51936 q^{43} -1.19667 q^{46} -1.93165 q^{47} -0.967002 q^{48} -6.64941 q^{49} -12.4610 q^{51} +2.43329 q^{52} -2.38291 q^{53} +4.09840 q^{54} -1.59489 q^{56} -4.02654 q^{57} -2.87004 q^{58} -0.0382778 q^{59} +3.44158 q^{61} +4.95043 q^{62} +0.0593361 q^{63} +3.57976 q^{64} -6.79162 q^{67} -9.59426 q^{68} +2.62493 q^{69} -11.7935 q^{71} -0.269932 q^{72} +6.82275 q^{73} -5.88974 q^{74} -3.10021 q^{76} +2.53680 q^{78} -4.52605 q^{79} -9.29059 q^{81} -6.75086 q^{82} +5.94262 q^{83} +1.41335 q^{84} +7.64118 q^{86} +6.29552 q^{87} -6.21375 q^{89} -1.06276 q^{91} +2.02105 q^{92} -10.8589 q^{93} -1.55054 q^{94} -10.2617 q^{96} +5.37571 q^{97} -5.33748 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9} - 4 q^{14} - 22 q^{16} - 12 q^{19} - 4 q^{21} - 2 q^{24} + 10 q^{26} - 24 q^{29} + 14 q^{31} + 8 q^{34} + 20 q^{36} - 30 q^{39} - 34 q^{41} - 24 q^{46} - 30 q^{49} - 54 q^{51} - 20 q^{54} - 10 q^{56} - 6 q^{59} - 20 q^{61} + 14 q^{64} + 32 q^{69} - 42 q^{71} + 4 q^{74} - 28 q^{76} - 16 q^{79} - 36 q^{81} + 6 q^{84} + 46 q^{86} - 12 q^{89} + 20 q^{91} + 42 q^{94} + 8 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.802699 0.567594 0.283797 0.958884i \(-0.408406\pi\)
0.283797 + 0.958884i \(0.408406\pi\)
\(3\) −1.76074 −1.01656 −0.508282 0.861190i \(-0.669719\pi\)
−0.508282 + 0.861190i \(0.669719\pi\)
\(4\) −1.35567 −0.677837
\(5\) 0 0
\(6\) −1.41335 −0.576996
\(7\) 0.592103 0.223794 0.111897 0.993720i \(-0.464307\pi\)
0.111897 + 0.993720i \(0.464307\pi\)
\(8\) −2.69360 −0.952330
\(9\) 0.100212 0.0334042
\(10\) 0 0
\(11\) 0 0
\(12\) 2.38699 0.689065
\(13\) −1.79489 −0.497813 −0.248906 0.968528i \(-0.580071\pi\)
−0.248906 + 0.968528i \(0.580071\pi\)
\(14\) 0.475281 0.127024
\(15\) 0 0
\(16\) 0.549201 0.137300
\(17\) 7.07712 1.71645 0.858226 0.513271i \(-0.171567\pi\)
0.858226 + 0.513271i \(0.171567\pi\)
\(18\) 0.0804405 0.0189600
\(19\) 2.28684 0.524637 0.262319 0.964981i \(-0.415513\pi\)
0.262319 + 0.964981i \(0.415513\pi\)
\(20\) 0 0
\(21\) −1.04254 −0.227501
\(22\) 0 0
\(23\) −1.49081 −0.310855 −0.155428 0.987847i \(-0.549676\pi\)
−0.155428 + 0.987847i \(0.549676\pi\)
\(24\) 4.74273 0.968105
\(25\) 0 0
\(26\) −1.44076 −0.282556
\(27\) 5.10578 0.982607
\(28\) −0.802699 −0.151696
\(29\) −3.57549 −0.663952 −0.331976 0.943288i \(-0.607715\pi\)
−0.331976 + 0.943288i \(0.607715\pi\)
\(30\) 0 0
\(31\) 6.16724 1.10767 0.553834 0.832627i \(-0.313164\pi\)
0.553834 + 0.832627i \(0.313164\pi\)
\(32\) 5.82804 1.03026
\(33\) 0 0
\(34\) 5.68079 0.974248
\(35\) 0 0
\(36\) −0.135855 −0.0226426
\(37\) −7.33743 −1.20627 −0.603133 0.797641i \(-0.706081\pi\)
−0.603133 + 0.797641i \(0.706081\pi\)
\(38\) 1.83565 0.297781
\(39\) 3.16034 0.506059
\(40\) 0 0
\(41\) −8.41020 −1.31345 −0.656726 0.754129i \(-0.728059\pi\)
−0.656726 + 0.754129i \(0.728059\pi\)
\(42\) −0.836847 −0.129128
\(43\) 9.51936 1.45169 0.725844 0.687859i \(-0.241449\pi\)
0.725844 + 0.687859i \(0.241449\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.19667 −0.176440
\(47\) −1.93165 −0.281761 −0.140880 0.990027i \(-0.544993\pi\)
−0.140880 + 0.990027i \(0.544993\pi\)
\(48\) −0.967002 −0.139575
\(49\) −6.64941 −0.949916
\(50\) 0 0
\(51\) −12.4610 −1.74489
\(52\) 2.43329 0.337436
\(53\) −2.38291 −0.327318 −0.163659 0.986517i \(-0.552330\pi\)
−0.163659 + 0.986517i \(0.552330\pi\)
\(54\) 4.09840 0.557722
\(55\) 0 0
\(56\) −1.59489 −0.213126
\(57\) −4.02654 −0.533328
\(58\) −2.87004 −0.376855
\(59\) −0.0382778 −0.00498334 −0.00249167 0.999997i \(-0.500793\pi\)
−0.00249167 + 0.999997i \(0.500793\pi\)
\(60\) 0 0
\(61\) 3.44158 0.440649 0.220325 0.975427i \(-0.429288\pi\)
0.220325 + 0.975427i \(0.429288\pi\)
\(62\) 4.95043 0.628706
\(63\) 0.0593361 0.00747565
\(64\) 3.57976 0.447470
\(65\) 0 0
\(66\) 0 0
\(67\) −6.79162 −0.829728 −0.414864 0.909883i \(-0.636171\pi\)
−0.414864 + 0.909883i \(0.636171\pi\)
\(68\) −9.59426 −1.16348
\(69\) 2.62493 0.316005
\(70\) 0 0
\(71\) −11.7935 −1.39963 −0.699816 0.714324i \(-0.746735\pi\)
−0.699816 + 0.714324i \(0.746735\pi\)
\(72\) −0.269932 −0.0318118
\(73\) 6.82275 0.798543 0.399271 0.916833i \(-0.369263\pi\)
0.399271 + 0.916833i \(0.369263\pi\)
\(74\) −5.88974 −0.684669
\(75\) 0 0
\(76\) −3.10021 −0.355619
\(77\) 0 0
\(78\) 2.53680 0.287236
\(79\) −4.52605 −0.509221 −0.254610 0.967044i \(-0.581947\pi\)
−0.254610 + 0.967044i \(0.581947\pi\)
\(80\) 0 0
\(81\) −9.29059 −1.03229
\(82\) −6.75086 −0.745508
\(83\) 5.94262 0.652288 0.326144 0.945320i \(-0.394251\pi\)
0.326144 + 0.945320i \(0.394251\pi\)
\(84\) 1.41335 0.154209
\(85\) 0 0
\(86\) 7.64118 0.823970
\(87\) 6.29552 0.674951
\(88\) 0 0
\(89\) −6.21375 −0.658656 −0.329328 0.944216i \(-0.606822\pi\)
−0.329328 + 0.944216i \(0.606822\pi\)
\(90\) 0 0
\(91\) −1.06276 −0.111407
\(92\) 2.02105 0.210709
\(93\) −10.8589 −1.12602
\(94\) −1.55054 −0.159926
\(95\) 0 0
\(96\) −10.2617 −1.04733
\(97\) 5.37571 0.545821 0.272910 0.962039i \(-0.412014\pi\)
0.272910 + 0.962039i \(0.412014\pi\)
\(98\) −5.33748 −0.539167
\(99\) 0 0
\(100\) 0 0
\(101\) 9.93130 0.988201 0.494101 0.869405i \(-0.335497\pi\)
0.494101 + 0.869405i \(0.335497\pi\)
\(102\) −10.0024 −0.990386
\(103\) −13.5214 −1.33230 −0.666150 0.745818i \(-0.732059\pi\)
−0.666150 + 0.745818i \(0.732059\pi\)
\(104\) 4.83471 0.474082
\(105\) 0 0
\(106\) −1.91276 −0.185784
\(107\) −5.60440 −0.541798 −0.270899 0.962608i \(-0.587321\pi\)
−0.270899 + 0.962608i \(0.587321\pi\)
\(108\) −6.92177 −0.666048
\(109\) −18.6001 −1.78157 −0.890784 0.454428i \(-0.849844\pi\)
−0.890784 + 0.454428i \(0.849844\pi\)
\(110\) 0 0
\(111\) 12.9193 1.22625
\(112\) 0.325184 0.0307270
\(113\) 11.8014 1.11018 0.555091 0.831790i \(-0.312684\pi\)
0.555091 + 0.831790i \(0.312684\pi\)
\(114\) −3.23210 −0.302714
\(115\) 0 0
\(116\) 4.84720 0.450052
\(117\) −0.179870 −0.0166290
\(118\) −0.0307255 −0.00282851
\(119\) 4.19038 0.384132
\(120\) 0 0
\(121\) 0 0
\(122\) 2.76255 0.250110
\(123\) 14.8082 1.33521
\(124\) −8.36076 −0.750819
\(125\) 0 0
\(126\) 0.0476291 0.00424313
\(127\) −16.0566 −1.42479 −0.712397 0.701777i \(-0.752390\pi\)
−0.712397 + 0.701777i \(0.752390\pi\)
\(128\) −8.78261 −0.776280
\(129\) −16.7611 −1.47574
\(130\) 0 0
\(131\) −18.0296 −1.57525 −0.787625 0.616154i \(-0.788690\pi\)
−0.787625 + 0.616154i \(0.788690\pi\)
\(132\) 0 0
\(133\) 1.35405 0.117411
\(134\) −5.45162 −0.470949
\(135\) 0 0
\(136\) −19.0629 −1.63463
\(137\) 4.03208 0.344483 0.172242 0.985055i \(-0.444899\pi\)
0.172242 + 0.985055i \(0.444899\pi\)
\(138\) 2.10703 0.179362
\(139\) −7.93492 −0.673031 −0.336516 0.941678i \(-0.609248\pi\)
−0.336516 + 0.941678i \(0.609248\pi\)
\(140\) 0 0
\(141\) 3.40114 0.286428
\(142\) −9.46663 −0.794422
\(143\) 0 0
\(144\) 0.0550368 0.00458640
\(145\) 0 0
\(146\) 5.47662 0.453248
\(147\) 11.7079 0.965652
\(148\) 9.94716 0.817652
\(149\) −12.5009 −1.02411 −0.512056 0.858952i \(-0.671116\pi\)
−0.512056 + 0.858952i \(0.671116\pi\)
\(150\) 0 0
\(151\) −8.40248 −0.683784 −0.341892 0.939739i \(-0.611068\pi\)
−0.341892 + 0.939739i \(0.611068\pi\)
\(152\) −6.15983 −0.499628
\(153\) 0.709215 0.0573367
\(154\) 0 0
\(155\) 0 0
\(156\) −4.28439 −0.343026
\(157\) 13.9959 1.11699 0.558496 0.829507i \(-0.311379\pi\)
0.558496 + 0.829507i \(0.311379\pi\)
\(158\) −3.63306 −0.289031
\(159\) 4.19569 0.332740
\(160\) 0 0
\(161\) −0.882713 −0.0695676
\(162\) −7.45755 −0.585921
\(163\) 11.8415 0.927496 0.463748 0.885967i \(-0.346504\pi\)
0.463748 + 0.885967i \(0.346504\pi\)
\(164\) 11.4015 0.890307
\(165\) 0 0
\(166\) 4.77014 0.370235
\(167\) −8.72628 −0.675260 −0.337630 0.941279i \(-0.609625\pi\)
−0.337630 + 0.941279i \(0.609625\pi\)
\(168\) 2.80818 0.216656
\(169\) −9.77837 −0.752182
\(170\) 0 0
\(171\) 0.229170 0.0175251
\(172\) −12.9052 −0.984009
\(173\) 9.17861 0.697837 0.348918 0.937153i \(-0.386549\pi\)
0.348918 + 0.937153i \(0.386549\pi\)
\(174\) 5.05341 0.383098
\(175\) 0 0
\(176\) 0 0
\(177\) 0.0673973 0.00506589
\(178\) −4.98777 −0.373849
\(179\) 1.44816 0.108241 0.0541204 0.998534i \(-0.482765\pi\)
0.0541204 + 0.998534i \(0.482765\pi\)
\(180\) 0 0
\(181\) 0.793502 0.0589806 0.0294903 0.999565i \(-0.490612\pi\)
0.0294903 + 0.999565i \(0.490612\pi\)
\(182\) −0.853076 −0.0632342
\(183\) −6.05974 −0.447949
\(184\) 4.01564 0.296037
\(185\) 0 0
\(186\) −8.71644 −0.639120
\(187\) 0 0
\(188\) 2.61869 0.190988
\(189\) 3.02315 0.219902
\(190\) 0 0
\(191\) −8.15029 −0.589735 −0.294867 0.955538i \(-0.595275\pi\)
−0.294867 + 0.955538i \(0.595275\pi\)
\(192\) −6.30303 −0.454882
\(193\) 4.36836 0.314442 0.157221 0.987563i \(-0.449747\pi\)
0.157221 + 0.987563i \(0.449747\pi\)
\(194\) 4.31508 0.309804
\(195\) 0 0
\(196\) 9.01444 0.643889
\(197\) −15.6525 −1.11520 −0.557599 0.830111i \(-0.688277\pi\)
−0.557599 + 0.830111i \(0.688277\pi\)
\(198\) 0 0
\(199\) 1.43830 0.101959 0.0509793 0.998700i \(-0.483766\pi\)
0.0509793 + 0.998700i \(0.483766\pi\)
\(200\) 0 0
\(201\) 11.9583 0.843472
\(202\) 7.97185 0.560897
\(203\) −2.11706 −0.148589
\(204\) 16.8930 1.18275
\(205\) 0 0
\(206\) −10.8536 −0.756206
\(207\) −0.149398 −0.0103839
\(208\) −0.985756 −0.0683499
\(209\) 0 0
\(210\) 0 0
\(211\) −8.68130 −0.597646 −0.298823 0.954309i \(-0.596594\pi\)
−0.298823 + 0.954309i \(0.596594\pi\)
\(212\) 3.23045 0.221868
\(213\) 20.7653 1.42282
\(214\) −4.49864 −0.307521
\(215\) 0 0
\(216\) −13.7529 −0.935767
\(217\) 3.65164 0.247889
\(218\) −14.9303 −1.01121
\(219\) −12.0131 −0.811770
\(220\) 0 0
\(221\) −12.7026 −0.854472
\(222\) 10.3703 0.696010
\(223\) −6.30604 −0.422284 −0.211142 0.977455i \(-0.567718\pi\)
−0.211142 + 0.977455i \(0.567718\pi\)
\(224\) 3.45080 0.230566
\(225\) 0 0
\(226\) 9.47296 0.630132
\(227\) 1.11681 0.0741252 0.0370626 0.999313i \(-0.488200\pi\)
0.0370626 + 0.999313i \(0.488200\pi\)
\(228\) 5.45867 0.361510
\(229\) −4.19616 −0.277290 −0.138645 0.990342i \(-0.544275\pi\)
−0.138645 + 0.990342i \(0.544275\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 9.63094 0.632302
\(233\) 6.77947 0.444138 0.222069 0.975031i \(-0.428719\pi\)
0.222069 + 0.975031i \(0.428719\pi\)
\(234\) −0.144382 −0.00943853
\(235\) 0 0
\(236\) 0.0518922 0.00337789
\(237\) 7.96921 0.517656
\(238\) 3.36362 0.218031
\(239\) −4.39808 −0.284488 −0.142244 0.989832i \(-0.545432\pi\)
−0.142244 + 0.989832i \(0.545432\pi\)
\(240\) 0 0
\(241\) 9.61218 0.619175 0.309587 0.950871i \(-0.399809\pi\)
0.309587 + 0.950871i \(0.399809\pi\)
\(242\) 0 0
\(243\) 1.04101 0.0667807
\(244\) −4.66566 −0.298688
\(245\) 0 0
\(246\) 11.8865 0.757857
\(247\) −4.10463 −0.261171
\(248\) −16.6120 −1.05487
\(249\) −10.4634 −0.663093
\(250\) 0 0
\(251\) 13.4206 0.847100 0.423550 0.905873i \(-0.360784\pi\)
0.423550 + 0.905873i \(0.360784\pi\)
\(252\) −0.0804405 −0.00506727
\(253\) 0 0
\(254\) −12.8886 −0.808704
\(255\) 0 0
\(256\) −14.2093 −0.888081
\(257\) −10.8174 −0.674769 −0.337384 0.941367i \(-0.609542\pi\)
−0.337384 + 0.941367i \(0.609542\pi\)
\(258\) −13.4541 −0.837619
\(259\) −4.34451 −0.269955
\(260\) 0 0
\(261\) −0.358309 −0.0221788
\(262\) −14.4723 −0.894103
\(263\) −24.6351 −1.51906 −0.759531 0.650471i \(-0.774572\pi\)
−0.759531 + 0.650471i \(0.774572\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.08689 0.0666416
\(267\) 10.9408 0.669566
\(268\) 9.20722 0.562420
\(269\) −4.80101 −0.292723 −0.146361 0.989231i \(-0.546756\pi\)
−0.146361 + 0.989231i \(0.546756\pi\)
\(270\) 0 0
\(271\) −23.3303 −1.41722 −0.708609 0.705602i \(-0.750677\pi\)
−0.708609 + 0.705602i \(0.750677\pi\)
\(272\) 3.88676 0.235670
\(273\) 1.87125 0.113253
\(274\) 3.23654 0.195527
\(275\) 0 0
\(276\) −3.55855 −0.214200
\(277\) 21.8973 1.31568 0.657840 0.753158i \(-0.271470\pi\)
0.657840 + 0.753158i \(0.271470\pi\)
\(278\) −6.36935 −0.382008
\(279\) 0.618034 0.0370007
\(280\) 0 0
\(281\) −15.7754 −0.941084 −0.470542 0.882378i \(-0.655942\pi\)
−0.470542 + 0.882378i \(0.655942\pi\)
\(282\) 2.73009 0.162575
\(283\) −22.6091 −1.34397 −0.671987 0.740563i \(-0.734559\pi\)
−0.671987 + 0.740563i \(0.734559\pi\)
\(284\) 15.9881 0.948722
\(285\) 0 0
\(286\) 0 0
\(287\) −4.97971 −0.293943
\(288\) 0.584042 0.0344150
\(289\) 33.0856 1.94621
\(290\) 0 0
\(291\) −9.46524 −0.554862
\(292\) −9.24943 −0.541282
\(293\) 13.2596 0.774635 0.387317 0.921946i \(-0.373402\pi\)
0.387317 + 0.921946i \(0.373402\pi\)
\(294\) 9.39792 0.548098
\(295\) 0 0
\(296\) 19.7641 1.14876
\(297\) 0 0
\(298\) −10.0344 −0.581280
\(299\) 2.67584 0.154748
\(300\) 0 0
\(301\) 5.63644 0.324879
\(302\) −6.74466 −0.388112
\(303\) −17.4865 −1.00457
\(304\) 1.25594 0.0720329
\(305\) 0 0
\(306\) 0.569286 0.0325439
\(307\) −10.0161 −0.571650 −0.285825 0.958282i \(-0.592268\pi\)
−0.285825 + 0.958282i \(0.592268\pi\)
\(308\) 0 0
\(309\) 23.8076 1.35437
\(310\) 0 0
\(311\) −3.40826 −0.193265 −0.0966323 0.995320i \(-0.530807\pi\)
−0.0966323 + 0.995320i \(0.530807\pi\)
\(312\) −8.51267 −0.481935
\(313\) 24.5008 1.38487 0.692434 0.721481i \(-0.256538\pi\)
0.692434 + 0.721481i \(0.256538\pi\)
\(314\) 11.2345 0.633998
\(315\) 0 0
\(316\) 6.13586 0.345169
\(317\) 6.98851 0.392514 0.196257 0.980553i \(-0.437121\pi\)
0.196257 + 0.980553i \(0.437121\pi\)
\(318\) 3.36787 0.188861
\(319\) 0 0
\(320\) 0 0
\(321\) 9.86790 0.550772
\(322\) −0.708553 −0.0394861
\(323\) 16.1842 0.900515
\(324\) 12.5950 0.699723
\(325\) 0 0
\(326\) 9.50514 0.526441
\(327\) 32.7500 1.81108
\(328\) 22.6537 1.25084
\(329\) −1.14374 −0.0630563
\(330\) 0 0
\(331\) 5.64321 0.310179 0.155089 0.987900i \(-0.450433\pi\)
0.155089 + 0.987900i \(0.450433\pi\)
\(332\) −8.05626 −0.442145
\(333\) −0.735302 −0.0402943
\(334\) −7.00458 −0.383273
\(335\) 0 0
\(336\) −0.572565 −0.0312360
\(337\) −22.6164 −1.23199 −0.615996 0.787750i \(-0.711246\pi\)
−0.615996 + 0.787750i \(0.711246\pi\)
\(338\) −7.84909 −0.426934
\(339\) −20.7792 −1.12857
\(340\) 0 0
\(341\) 0 0
\(342\) 0.183955 0.00994713
\(343\) −8.08186 −0.436379
\(344\) −25.6413 −1.38249
\(345\) 0 0
\(346\) 7.36766 0.396088
\(347\) 0.182395 0.00979145 0.00489573 0.999988i \(-0.498442\pi\)
0.00489573 + 0.999988i \(0.498442\pi\)
\(348\) −8.53468 −0.457507
\(349\) 15.4273 0.825803 0.412902 0.910776i \(-0.364515\pi\)
0.412902 + 0.910776i \(0.364515\pi\)
\(350\) 0 0
\(351\) −9.16431 −0.489155
\(352\) 0 0
\(353\) −23.9103 −1.27262 −0.636308 0.771435i \(-0.719539\pi\)
−0.636308 + 0.771435i \(0.719539\pi\)
\(354\) 0.0540997 0.00287537
\(355\) 0 0
\(356\) 8.42382 0.446461
\(357\) −7.37818 −0.390495
\(358\) 1.16244 0.0614368
\(359\) 8.76734 0.462723 0.231361 0.972868i \(-0.425682\pi\)
0.231361 + 0.972868i \(0.425682\pi\)
\(360\) 0 0
\(361\) −13.7704 −0.724756
\(362\) 0.636943 0.0334770
\(363\) 0 0
\(364\) 1.44076 0.0755161
\(365\) 0 0
\(366\) −4.86414 −0.254253
\(367\) 5.38232 0.280955 0.140477 0.990084i \(-0.455136\pi\)
0.140477 + 0.990084i \(0.455136\pi\)
\(368\) −0.818755 −0.0426806
\(369\) −0.842807 −0.0438748
\(370\) 0 0
\(371\) −1.41093 −0.0732517
\(372\) 14.7211 0.763256
\(373\) 3.22450 0.166958 0.0834792 0.996510i \(-0.473397\pi\)
0.0834792 + 0.996510i \(0.473397\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 5.20309 0.268329
\(377\) 6.41762 0.330524
\(378\) 2.42668 0.124815
\(379\) −19.6634 −1.01004 −0.505020 0.863108i \(-0.668515\pi\)
−0.505020 + 0.863108i \(0.668515\pi\)
\(380\) 0 0
\(381\) 28.2716 1.44840
\(382\) −6.54223 −0.334730
\(383\) 27.9751 1.42946 0.714731 0.699400i \(-0.246549\pi\)
0.714731 + 0.699400i \(0.246549\pi\)
\(384\) 15.4639 0.789139
\(385\) 0 0
\(386\) 3.50648 0.178475
\(387\) 0.953959 0.0484924
\(388\) −7.28771 −0.369977
\(389\) 13.0400 0.661154 0.330577 0.943779i \(-0.392757\pi\)
0.330577 + 0.943779i \(0.392757\pi\)
\(390\) 0 0
\(391\) −10.5506 −0.533569
\(392\) 17.9108 0.904634
\(393\) 31.7454 1.60134
\(394\) −12.5643 −0.632980
\(395\) 0 0
\(396\) 0 0
\(397\) −1.82243 −0.0914651 −0.0457325 0.998954i \(-0.514562\pi\)
−0.0457325 + 0.998954i \(0.514562\pi\)
\(398\) 1.15452 0.0578711
\(399\) −2.38413 −0.119356
\(400\) 0 0
\(401\) 5.38085 0.268707 0.134353 0.990933i \(-0.457104\pi\)
0.134353 + 0.990933i \(0.457104\pi\)
\(402\) 9.59890 0.478750
\(403\) −11.0695 −0.551411
\(404\) −13.4636 −0.669840
\(405\) 0 0
\(406\) −1.69936 −0.0843379
\(407\) 0 0
\(408\) 33.5648 1.66171
\(409\) 36.6821 1.81381 0.906906 0.421333i \(-0.138438\pi\)
0.906906 + 0.421333i \(0.138438\pi\)
\(410\) 0 0
\(411\) −7.09944 −0.350190
\(412\) 18.3306 0.903083
\(413\) −0.0226644 −0.00111524
\(414\) −0.119921 −0.00589382
\(415\) 0 0
\(416\) −10.4607 −0.512877
\(417\) 13.9713 0.684180
\(418\) 0 0
\(419\) 2.86630 0.140028 0.0700141 0.997546i \(-0.477696\pi\)
0.0700141 + 0.997546i \(0.477696\pi\)
\(420\) 0 0
\(421\) 4.65975 0.227102 0.113551 0.993532i \(-0.463777\pi\)
0.113551 + 0.993532i \(0.463777\pi\)
\(422\) −6.96847 −0.339220
\(423\) −0.193576 −0.00941198
\(424\) 6.41859 0.311714
\(425\) 0 0
\(426\) 16.6683 0.807582
\(427\) 2.03777 0.0986147
\(428\) 7.59774 0.367250
\(429\) 0 0
\(430\) 0 0
\(431\) −20.7691 −1.00041 −0.500207 0.865906i \(-0.666743\pi\)
−0.500207 + 0.865906i \(0.666743\pi\)
\(432\) 2.80410 0.134912
\(433\) −12.7972 −0.614993 −0.307496 0.951549i \(-0.599491\pi\)
−0.307496 + 0.951549i \(0.599491\pi\)
\(434\) 2.93117 0.140701
\(435\) 0 0
\(436\) 25.2157 1.20761
\(437\) −3.40925 −0.163086
\(438\) −9.64291 −0.460756
\(439\) 10.6208 0.506905 0.253452 0.967348i \(-0.418434\pi\)
0.253452 + 0.967348i \(0.418434\pi\)
\(440\) 0 0
\(441\) −0.666354 −0.0317312
\(442\) −10.1964 −0.484993
\(443\) 6.59894 0.313525 0.156763 0.987636i \(-0.449894\pi\)
0.156763 + 0.987636i \(0.449894\pi\)
\(444\) −17.5144 −0.831196
\(445\) 0 0
\(446\) −5.06185 −0.239686
\(447\) 22.0108 1.04108
\(448\) 2.11958 0.100141
\(449\) 13.6281 0.643147 0.321574 0.946885i \(-0.395788\pi\)
0.321574 + 0.946885i \(0.395788\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −15.9988 −0.752522
\(453\) 14.7946 0.695111
\(454\) 0.896462 0.0420730
\(455\) 0 0
\(456\) 10.8459 0.507904
\(457\) 13.4999 0.631498 0.315749 0.948843i \(-0.397744\pi\)
0.315749 + 0.948843i \(0.397744\pi\)
\(458\) −3.36825 −0.157388
\(459\) 36.1342 1.68660
\(460\) 0 0
\(461\) −11.3217 −0.527303 −0.263652 0.964618i \(-0.584927\pi\)
−0.263652 + 0.964618i \(0.584927\pi\)
\(462\) 0 0
\(463\) −4.82990 −0.224464 −0.112232 0.993682i \(-0.535800\pi\)
−0.112232 + 0.993682i \(0.535800\pi\)
\(464\) −1.96367 −0.0911609
\(465\) 0 0
\(466\) 5.44187 0.252090
\(467\) 24.0173 1.11139 0.555694 0.831387i \(-0.312453\pi\)
0.555694 + 0.831387i \(0.312453\pi\)
\(468\) 0.243846 0.0112718
\(469\) −4.02134 −0.185688
\(470\) 0 0
\(471\) −24.6431 −1.13550
\(472\) 0.103105 0.00474579
\(473\) 0 0
\(474\) 6.39688 0.293818
\(475\) 0 0
\(476\) −5.68079 −0.260379
\(477\) −0.238797 −0.0109338
\(478\) −3.53034 −0.161474
\(479\) −43.2250 −1.97500 −0.987501 0.157611i \(-0.949621\pi\)
−0.987501 + 0.157611i \(0.949621\pi\)
\(480\) 0 0
\(481\) 13.1699 0.600494
\(482\) 7.71569 0.351440
\(483\) 1.55423 0.0707199
\(484\) 0 0
\(485\) 0 0
\(486\) 0.835616 0.0379043
\(487\) −41.9609 −1.90143 −0.950715 0.310067i \(-0.899648\pi\)
−0.950715 + 0.310067i \(0.899648\pi\)
\(488\) −9.27023 −0.419644
\(489\) −20.8498 −0.942860
\(490\) 0 0
\(491\) 9.05983 0.408864 0.204432 0.978881i \(-0.434465\pi\)
0.204432 + 0.978881i \(0.434465\pi\)
\(492\) −20.0751 −0.905055
\(493\) −25.3042 −1.13964
\(494\) −3.29478 −0.148239
\(495\) 0 0
\(496\) 3.38705 0.152083
\(497\) −6.98297 −0.313229
\(498\) −8.39898 −0.376367
\(499\) 30.2793 1.35549 0.677744 0.735298i \(-0.262958\pi\)
0.677744 + 0.735298i \(0.262958\pi\)
\(500\) 0 0
\(501\) 15.3647 0.686446
\(502\) 10.7727 0.480809
\(503\) −18.0638 −0.805426 −0.402713 0.915326i \(-0.631933\pi\)
−0.402713 + 0.915326i \(0.631933\pi\)
\(504\) −0.159828 −0.00711929
\(505\) 0 0
\(506\) 0 0
\(507\) 17.2172 0.764642
\(508\) 21.7675 0.965778
\(509\) −24.9157 −1.10437 −0.552184 0.833722i \(-0.686206\pi\)
−0.552184 + 0.833722i \(0.686206\pi\)
\(510\) 0 0
\(511\) 4.03977 0.178709
\(512\) 6.15942 0.272210
\(513\) 11.6761 0.515513
\(514\) −8.68309 −0.382995
\(515\) 0 0
\(516\) 22.7226 1.00031
\(517\) 0 0
\(518\) −3.48734 −0.153225
\(519\) −16.1612 −0.709396
\(520\) 0 0
\(521\) 36.2831 1.58959 0.794797 0.606876i \(-0.207578\pi\)
0.794797 + 0.606876i \(0.207578\pi\)
\(522\) −0.287614 −0.0125885
\(523\) 24.7070 1.08036 0.540181 0.841549i \(-0.318356\pi\)
0.540181 + 0.841549i \(0.318356\pi\)
\(524\) 24.4422 1.06776
\(525\) 0 0
\(526\) −19.7745 −0.862211
\(527\) 43.6462 1.90126
\(528\) 0 0
\(529\) −20.7775 −0.903369
\(530\) 0 0
\(531\) −0.00383591 −0.000166464 0
\(532\) −1.83565 −0.0795853
\(533\) 15.0954 0.653854
\(534\) 8.78217 0.380042
\(535\) 0 0
\(536\) 18.2939 0.790175
\(537\) −2.54984 −0.110034
\(538\) −3.85376 −0.166148
\(539\) 0 0
\(540\) 0 0
\(541\) 12.5420 0.539221 0.269610 0.962970i \(-0.413105\pi\)
0.269610 + 0.962970i \(0.413105\pi\)
\(542\) −18.7272 −0.804404
\(543\) −1.39715 −0.0599576
\(544\) 41.2457 1.76839
\(545\) 0 0
\(546\) 1.50205 0.0642817
\(547\) −30.7407 −1.31438 −0.657189 0.753726i \(-0.728255\pi\)
−0.657189 + 0.753726i \(0.728255\pi\)
\(548\) −5.46618 −0.233504
\(549\) 0.344889 0.0147195
\(550\) 0 0
\(551\) −8.17659 −0.348334
\(552\) −7.07051 −0.300941
\(553\) −2.67989 −0.113961
\(554\) 17.5769 0.746772
\(555\) 0 0
\(556\) 10.7572 0.456206
\(557\) −13.4294 −0.569021 −0.284510 0.958673i \(-0.591831\pi\)
−0.284510 + 0.958673i \(0.591831\pi\)
\(558\) 0.496095 0.0210014
\(559\) −17.0862 −0.722669
\(560\) 0 0
\(561\) 0 0
\(562\) −12.6629 −0.534154
\(563\) −30.5401 −1.28711 −0.643556 0.765399i \(-0.722542\pi\)
−0.643556 + 0.765399i \(0.722542\pi\)
\(564\) −4.61084 −0.194152
\(565\) 0 0
\(566\) −18.1483 −0.762831
\(567\) −5.50099 −0.231020
\(568\) 31.7669 1.33291
\(569\) −25.7204 −1.07826 −0.539128 0.842224i \(-0.681246\pi\)
−0.539128 + 0.842224i \(0.681246\pi\)
\(570\) 0 0
\(571\) 27.1115 1.13458 0.567291 0.823518i \(-0.307992\pi\)
0.567291 + 0.823518i \(0.307992\pi\)
\(572\) 0 0
\(573\) 14.3506 0.599504
\(574\) −3.99721 −0.166840
\(575\) 0 0
\(576\) 0.358736 0.0149473
\(577\) 2.87015 0.119486 0.0597430 0.998214i \(-0.480972\pi\)
0.0597430 + 0.998214i \(0.480972\pi\)
\(578\) 26.5578 1.10466
\(579\) −7.69156 −0.319650
\(580\) 0 0
\(581\) 3.51865 0.145978
\(582\) −7.59774 −0.314936
\(583\) 0 0
\(584\) −18.3777 −0.760476
\(585\) 0 0
\(586\) 10.6435 0.439678
\(587\) 44.8360 1.85058 0.925289 0.379262i \(-0.123822\pi\)
0.925289 + 0.379262i \(0.123822\pi\)
\(588\) −15.8721 −0.654554
\(589\) 14.1035 0.581124
\(590\) 0 0
\(591\) 27.5601 1.13367
\(592\) −4.02972 −0.165621
\(593\) 6.09322 0.250219 0.125109 0.992143i \(-0.460072\pi\)
0.125109 + 0.992143i \(0.460072\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.9471 0.694181
\(597\) −2.53248 −0.103648
\(598\) 2.14789 0.0878339
\(599\) −12.9337 −0.528457 −0.264229 0.964460i \(-0.585117\pi\)
−0.264229 + 0.964460i \(0.585117\pi\)
\(600\) 0 0
\(601\) 11.0471 0.450621 0.225310 0.974287i \(-0.427660\pi\)
0.225310 + 0.974287i \(0.427660\pi\)
\(602\) 4.52437 0.184399
\(603\) −0.680605 −0.0277164
\(604\) 11.3910 0.463494
\(605\) 0 0
\(606\) −14.0364 −0.570188
\(607\) −0.115912 −0.00470474 −0.00235237 0.999997i \(-0.500749\pi\)
−0.00235237 + 0.999997i \(0.500749\pi\)
\(608\) 13.3278 0.540514
\(609\) 3.72760 0.151050
\(610\) 0 0
\(611\) 3.46710 0.140264
\(612\) −0.961465 −0.0388649
\(613\) 19.0445 0.769202 0.384601 0.923083i \(-0.374339\pi\)
0.384601 + 0.923083i \(0.374339\pi\)
\(614\) −8.03993 −0.324465
\(615\) 0 0
\(616\) 0 0
\(617\) 30.8894 1.24356 0.621780 0.783192i \(-0.286410\pi\)
0.621780 + 0.783192i \(0.286410\pi\)
\(618\) 19.1104 0.768732
\(619\) 33.5697 1.34928 0.674639 0.738148i \(-0.264299\pi\)
0.674639 + 0.738148i \(0.264299\pi\)
\(620\) 0 0
\(621\) −7.61174 −0.305449
\(622\) −2.73581 −0.109696
\(623\) −3.67918 −0.147403
\(624\) 1.73566 0.0694821
\(625\) 0 0
\(626\) 19.6668 0.786043
\(627\) 0 0
\(628\) −18.9738 −0.757139
\(629\) −51.9278 −2.07050
\(630\) 0 0
\(631\) 24.6573 0.981590 0.490795 0.871275i \(-0.336706\pi\)
0.490795 + 0.871275i \(0.336706\pi\)
\(632\) 12.1914 0.484946
\(633\) 15.2855 0.607546
\(634\) 5.60967 0.222788
\(635\) 0 0
\(636\) −5.68799 −0.225543
\(637\) 11.9350 0.472880
\(638\) 0 0
\(639\) −1.18186 −0.0467535
\(640\) 0 0
\(641\) 7.01647 0.277134 0.138567 0.990353i \(-0.455750\pi\)
0.138567 + 0.990353i \(0.455750\pi\)
\(642\) 7.92095 0.312615
\(643\) 12.2525 0.483192 0.241596 0.970377i \(-0.422329\pi\)
0.241596 + 0.970377i \(0.422329\pi\)
\(644\) 1.19667 0.0471555
\(645\) 0 0
\(646\) 12.9911 0.511127
\(647\) 6.12014 0.240608 0.120304 0.992737i \(-0.461613\pi\)
0.120304 + 0.992737i \(0.461613\pi\)
\(648\) 25.0251 0.983079
\(649\) 0 0
\(650\) 0 0
\(651\) −6.42960 −0.251996
\(652\) −16.0532 −0.628691
\(653\) −38.0316 −1.48829 −0.744145 0.668018i \(-0.767143\pi\)
−0.744145 + 0.668018i \(0.767143\pi\)
\(654\) 26.2884 1.02796
\(655\) 0 0
\(656\) −4.61889 −0.180338
\(657\) 0.683725 0.0266746
\(658\) −0.918077 −0.0357904
\(659\) −15.7879 −0.615011 −0.307505 0.951546i \(-0.599494\pi\)
−0.307505 + 0.951546i \(0.599494\pi\)
\(660\) 0 0
\(661\) −24.5794 −0.956027 −0.478014 0.878352i \(-0.658643\pi\)
−0.478014 + 0.878352i \(0.658643\pi\)
\(662\) 4.52980 0.176056
\(663\) 22.3661 0.868626
\(664\) −16.0070 −0.621193
\(665\) 0 0
\(666\) −0.590226 −0.0228708
\(667\) 5.33038 0.206393
\(668\) 11.8300 0.457716
\(669\) 11.1033 0.429279
\(670\) 0 0
\(671\) 0 0
\(672\) −6.07597 −0.234385
\(673\) −29.9733 −1.15539 −0.577693 0.816254i \(-0.696047\pi\)
−0.577693 + 0.816254i \(0.696047\pi\)
\(674\) −18.1541 −0.699271
\(675\) 0 0
\(676\) 13.2563 0.509857
\(677\) −3.75709 −0.144397 −0.0721984 0.997390i \(-0.523001\pi\)
−0.0721984 + 0.997390i \(0.523001\pi\)
\(678\) −16.6794 −0.640570
\(679\) 3.18297 0.122151
\(680\) 0 0
\(681\) −1.96641 −0.0753531
\(682\) 0 0
\(683\) 21.0157 0.804144 0.402072 0.915608i \(-0.368290\pi\)
0.402072 + 0.915608i \(0.368290\pi\)
\(684\) −0.310680 −0.0118791
\(685\) 0 0
\(686\) −6.48730 −0.247686
\(687\) 7.38836 0.281883
\(688\) 5.22805 0.199317
\(689\) 4.27706 0.162943
\(690\) 0 0
\(691\) −38.2825 −1.45633 −0.728167 0.685399i \(-0.759628\pi\)
−0.728167 + 0.685399i \(0.759628\pi\)
\(692\) −12.4432 −0.473020
\(693\) 0 0
\(694\) 0.146408 0.00555757
\(695\) 0 0
\(696\) −16.9576 −0.642776
\(697\) −59.5200 −2.25448
\(698\) 12.3835 0.468721
\(699\) −11.9369 −0.451495
\(700\) 0 0
\(701\) 34.2344 1.29302 0.646509 0.762907i \(-0.276228\pi\)
0.646509 + 0.762907i \(0.276228\pi\)
\(702\) −7.35618 −0.277641
\(703\) −16.7795 −0.632852
\(704\) 0 0
\(705\) 0 0
\(706\) −19.1928 −0.722329
\(707\) 5.88035 0.221153
\(708\) −0.0913687 −0.00343385
\(709\) −4.12477 −0.154909 −0.0774545 0.996996i \(-0.524679\pi\)
−0.0774545 + 0.996996i \(0.524679\pi\)
\(710\) 0 0
\(711\) −0.453567 −0.0170101
\(712\) 16.7373 0.627258
\(713\) −9.19418 −0.344325
\(714\) −5.92246 −0.221642
\(715\) 0 0
\(716\) −1.96324 −0.0733697
\(717\) 7.74389 0.289201
\(718\) 7.03754 0.262639
\(719\) 29.3596 1.09493 0.547463 0.836830i \(-0.315594\pi\)
0.547463 + 0.836830i \(0.315594\pi\)
\(720\) 0 0
\(721\) −8.00605 −0.298161
\(722\) −11.0535 −0.411367
\(723\) −16.9246 −0.629431
\(724\) −1.07573 −0.0399792
\(725\) 0 0
\(726\) 0 0
\(727\) −44.0893 −1.63518 −0.817591 0.575799i \(-0.804691\pi\)
−0.817591 + 0.575799i \(0.804691\pi\)
\(728\) 2.86265 0.106097
\(729\) 26.0388 0.964401
\(730\) 0 0
\(731\) 67.3696 2.49176
\(732\) 8.21503 0.303636
\(733\) −48.9490 −1.80797 −0.903987 0.427561i \(-0.859373\pi\)
−0.903987 + 0.427561i \(0.859373\pi\)
\(734\) 4.32038 0.159468
\(735\) 0 0
\(736\) −8.68849 −0.320262
\(737\) 0 0
\(738\) −0.676520 −0.0249031
\(739\) 20.6622 0.760072 0.380036 0.924972i \(-0.375912\pi\)
0.380036 + 0.924972i \(0.375912\pi\)
\(740\) 0 0
\(741\) 7.22719 0.265498
\(742\) −1.13255 −0.0415772
\(743\) −30.1631 −1.10658 −0.553288 0.832990i \(-0.686627\pi\)
−0.553288 + 0.832990i \(0.686627\pi\)
\(744\) 29.2495 1.07234
\(745\) 0 0
\(746\) 2.58830 0.0947645
\(747\) 0.595525 0.0217891
\(748\) 0 0
\(749\) −3.31838 −0.121251
\(750\) 0 0
\(751\) 12.0966 0.441412 0.220706 0.975340i \(-0.429164\pi\)
0.220706 + 0.975340i \(0.429164\pi\)
\(752\) −1.06087 −0.0386858
\(753\) −23.6302 −0.861133
\(754\) 5.15141 0.187603
\(755\) 0 0
\(756\) −4.09840 −0.149057
\(757\) 20.6101 0.749086 0.374543 0.927210i \(-0.377800\pi\)
0.374543 + 0.927210i \(0.377800\pi\)
\(758\) −15.7838 −0.573293
\(759\) 0 0
\(760\) 0 0
\(761\) −20.8450 −0.755631 −0.377816 0.925881i \(-0.623325\pi\)
−0.377816 + 0.925881i \(0.623325\pi\)
\(762\) 22.6935 0.822100
\(763\) −11.0132 −0.398704
\(764\) 11.0491 0.399744
\(765\) 0 0
\(766\) 22.4556 0.811354
\(767\) 0.0687044 0.00248077
\(768\) 25.0189 0.902792
\(769\) −10.7167 −0.386455 −0.193228 0.981154i \(-0.561896\pi\)
−0.193228 + 0.981154i \(0.561896\pi\)
\(770\) 0 0
\(771\) 19.0466 0.685946
\(772\) −5.92207 −0.213140
\(773\) −20.3563 −0.732164 −0.366082 0.930583i \(-0.619301\pi\)
−0.366082 + 0.930583i \(0.619301\pi\)
\(774\) 0.765742 0.0275240
\(775\) 0 0
\(776\) −14.4800 −0.519801
\(777\) 7.64957 0.274427
\(778\) 10.4672 0.375267
\(779\) −19.2328 −0.689087
\(780\) 0 0
\(781\) 0 0
\(782\) −8.46898 −0.302850
\(783\) −18.2557 −0.652405
\(784\) −3.65187 −0.130424
\(785\) 0 0
\(786\) 25.4820 0.908914
\(787\) −14.2222 −0.506967 −0.253484 0.967340i \(-0.581576\pi\)
−0.253484 + 0.967340i \(0.581576\pi\)
\(788\) 21.2198 0.755923
\(789\) 43.3760 1.54423
\(790\) 0 0
\(791\) 6.98764 0.248452
\(792\) 0 0
\(793\) −6.17726 −0.219361
\(794\) −1.46286 −0.0519150
\(795\) 0 0
\(796\) −1.94987 −0.0691113
\(797\) −45.0384 −1.59534 −0.797671 0.603093i \(-0.793935\pi\)
−0.797671 + 0.603093i \(0.793935\pi\)
\(798\) −1.91374 −0.0677455
\(799\) −13.6705 −0.483629
\(800\) 0 0
\(801\) −0.622695 −0.0220018
\(802\) 4.31920 0.152516
\(803\) 0 0
\(804\) −16.2115 −0.571737
\(805\) 0 0
\(806\) −8.88548 −0.312978
\(807\) 8.45334 0.297572
\(808\) −26.7509 −0.941094
\(809\) −23.7753 −0.835896 −0.417948 0.908471i \(-0.637251\pi\)
−0.417948 + 0.908471i \(0.637251\pi\)
\(810\) 0 0
\(811\) 8.19869 0.287895 0.143948 0.989585i \(-0.454020\pi\)
0.143948 + 0.989585i \(0.454020\pi\)
\(812\) 2.87004 0.100719
\(813\) 41.0787 1.44069
\(814\) 0 0
\(815\) 0 0
\(816\) −6.84358 −0.239573
\(817\) 21.7693 0.761610
\(818\) 29.4447 1.02951
\(819\) −0.106502 −0.00372147
\(820\) 0 0
\(821\) 43.4373 1.51597 0.757986 0.652271i \(-0.226184\pi\)
0.757986 + 0.652271i \(0.226184\pi\)
\(822\) −5.69872 −0.198766
\(823\) −51.6801 −1.80145 −0.900727 0.434386i \(-0.856966\pi\)
−0.900727 + 0.434386i \(0.856966\pi\)
\(824\) 36.4211 1.26879
\(825\) 0 0
\(826\) −0.0181927 −0.000633004 0
\(827\) −25.3079 −0.880042 −0.440021 0.897987i \(-0.645029\pi\)
−0.440021 + 0.897987i \(0.645029\pi\)
\(828\) 0.202535 0.00703857
\(829\) −41.8760 −1.45441 −0.727207 0.686418i \(-0.759182\pi\)
−0.727207 + 0.686418i \(0.759182\pi\)
\(830\) 0 0
\(831\) −38.5555 −1.33747
\(832\) −6.42527 −0.222756
\(833\) −47.0587 −1.63049
\(834\) 11.2148 0.388336
\(835\) 0 0
\(836\) 0 0
\(837\) 31.4885 1.08840
\(838\) 2.30078 0.0794791
\(839\) −14.3848 −0.496619 −0.248309 0.968681i \(-0.579875\pi\)
−0.248309 + 0.968681i \(0.579875\pi\)
\(840\) 0 0
\(841\) −16.2158 −0.559167
\(842\) 3.74038 0.128902
\(843\) 27.7765 0.956673
\(844\) 11.7690 0.405106
\(845\) 0 0
\(846\) −0.155383 −0.00534218
\(847\) 0 0
\(848\) −1.30870 −0.0449408
\(849\) 39.8088 1.36624
\(850\) 0 0
\(851\) 10.9387 0.374974
\(852\) −28.1510 −0.964437
\(853\) −43.0014 −1.47234 −0.736170 0.676797i \(-0.763368\pi\)
−0.736170 + 0.676797i \(0.763368\pi\)
\(854\) 1.63572 0.0559731
\(855\) 0 0
\(856\) 15.0960 0.515970
\(857\) −54.3052 −1.85503 −0.927516 0.373784i \(-0.878060\pi\)
−0.927516 + 0.373784i \(0.878060\pi\)
\(858\) 0 0
\(859\) 24.3361 0.830336 0.415168 0.909745i \(-0.363723\pi\)
0.415168 + 0.909745i \(0.363723\pi\)
\(860\) 0 0
\(861\) 8.76798 0.298812
\(862\) −16.6714 −0.567828
\(863\) −39.1501 −1.33268 −0.666342 0.745646i \(-0.732141\pi\)
−0.666342 + 0.745646i \(0.732141\pi\)
\(864\) 29.7567 1.01234
\(865\) 0 0
\(866\) −10.2723 −0.349066
\(867\) −58.2551 −1.97845
\(868\) −4.95043 −0.168029
\(869\) 0 0
\(870\) 0 0
\(871\) 12.1902 0.413049
\(872\) 50.1012 1.69664
\(873\) 0.538713 0.0182327
\(874\) −2.73660 −0.0925668
\(875\) 0 0
\(876\) 16.2859 0.550248
\(877\) 43.0882 1.45499 0.727493 0.686116i \(-0.240686\pi\)
0.727493 + 0.686116i \(0.240686\pi\)
\(878\) 8.52534 0.287716
\(879\) −23.3468 −0.787466
\(880\) 0 0
\(881\) −38.1083 −1.28390 −0.641950 0.766746i \(-0.721874\pi\)
−0.641950 + 0.766746i \(0.721874\pi\)
\(882\) −0.534882 −0.0180104
\(883\) −29.0261 −0.976805 −0.488403 0.872618i \(-0.662420\pi\)
−0.488403 + 0.872618i \(0.662420\pi\)
\(884\) 17.2206 0.579193
\(885\) 0 0
\(886\) 5.29696 0.177955
\(887\) 2.21174 0.0742631 0.0371315 0.999310i \(-0.488178\pi\)
0.0371315 + 0.999310i \(0.488178\pi\)
\(888\) −34.7994 −1.16779
\(889\) −9.50717 −0.318860
\(890\) 0 0
\(891\) 0 0
\(892\) 8.54894 0.286240
\(893\) −4.41739 −0.147822
\(894\) 17.6681 0.590909
\(895\) 0 0
\(896\) −5.20021 −0.173727
\(897\) −4.71146 −0.157311
\(898\) 10.9392 0.365047
\(899\) −22.0509 −0.735439
\(900\) 0 0
\(901\) −16.8641 −0.561825
\(902\) 0 0
\(903\) −9.92432 −0.330261
\(904\) −31.7882 −1.05726
\(905\) 0 0
\(906\) 11.8756 0.394541
\(907\) −26.6863 −0.886106 −0.443053 0.896496i \(-0.646105\pi\)
−0.443053 + 0.896496i \(0.646105\pi\)
\(908\) −1.51403 −0.0502448
\(909\) 0.995240 0.0330100
\(910\) 0 0
\(911\) −36.2736 −1.20180 −0.600899 0.799325i \(-0.705191\pi\)
−0.600899 + 0.799325i \(0.705191\pi\)
\(912\) −2.21138 −0.0732261
\(913\) 0 0
\(914\) 10.8363 0.358434
\(915\) 0 0
\(916\) 5.68863 0.187958
\(917\) −10.6754 −0.352532
\(918\) 29.0049 0.957303
\(919\) 29.6718 0.978781 0.489391 0.872065i \(-0.337219\pi\)
0.489391 + 0.872065i \(0.337219\pi\)
\(920\) 0 0
\(921\) 17.6358 0.581119
\(922\) −9.08790 −0.299294
\(923\) 21.1680 0.696754
\(924\) 0 0
\(925\) 0 0
\(926\) −3.87696 −0.127405
\(927\) −1.35501 −0.0445044
\(928\) −20.8381 −0.684044
\(929\) 27.7726 0.911189 0.455595 0.890187i \(-0.349427\pi\)
0.455595 + 0.890187i \(0.349427\pi\)
\(930\) 0 0
\(931\) −15.2062 −0.498362
\(932\) −9.19075 −0.301053
\(933\) 6.00106 0.196466
\(934\) 19.2786 0.630817
\(935\) 0 0
\(936\) 0.484498 0.0158363
\(937\) 30.2421 0.987967 0.493983 0.869471i \(-0.335540\pi\)
0.493983 + 0.869471i \(0.335540\pi\)
\(938\) −3.22792 −0.105395
\(939\) −43.1396 −1.40781
\(940\) 0 0
\(941\) 6.00672 0.195813 0.0979067 0.995196i \(-0.468785\pi\)
0.0979067 + 0.995196i \(0.468785\pi\)
\(942\) −19.7810 −0.644500
\(943\) 12.5380 0.408294
\(944\) −0.0210222 −0.000684214 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.0218 0.325665 0.162833 0.986654i \(-0.447937\pi\)
0.162833 + 0.986654i \(0.447937\pi\)
\(948\) −10.8037 −0.350887
\(949\) −12.2461 −0.397525
\(950\) 0 0
\(951\) −12.3050 −0.399016
\(952\) −11.2872 −0.365820
\(953\) 9.78136 0.316849 0.158425 0.987371i \(-0.449359\pi\)
0.158425 + 0.987371i \(0.449359\pi\)
\(954\) −0.191682 −0.00620594
\(955\) 0 0
\(956\) 5.96237 0.192837
\(957\) 0 0
\(958\) −34.6967 −1.12100
\(959\) 2.38740 0.0770933
\(960\) 0 0
\(961\) 7.03479 0.226929
\(962\) 10.5714 0.340837
\(963\) −0.561631 −0.0180983
\(964\) −13.0310 −0.419700
\(965\) 0 0
\(966\) 1.24758 0.0401402
\(967\) 1.22635 0.0394367 0.0197184 0.999806i \(-0.493723\pi\)
0.0197184 + 0.999806i \(0.493723\pi\)
\(968\) 0 0
\(969\) −28.4963 −0.915432
\(970\) 0 0
\(971\) −44.6467 −1.43278 −0.716390 0.697700i \(-0.754207\pi\)
−0.716390 + 0.697700i \(0.754207\pi\)
\(972\) −1.41127 −0.0452664
\(973\) −4.69829 −0.150620
\(974\) −33.6820 −1.07924
\(975\) 0 0
\(976\) 1.89012 0.0605013
\(977\) −47.2451 −1.51151 −0.755753 0.654857i \(-0.772729\pi\)
−0.755753 + 0.654857i \(0.772729\pi\)
\(978\) −16.7361 −0.535162
\(979\) 0 0
\(980\) 0 0
\(981\) −1.86396 −0.0595118
\(982\) 7.27232 0.232069
\(983\) 13.9351 0.444459 0.222230 0.974994i \(-0.428667\pi\)
0.222230 + 0.974994i \(0.428667\pi\)
\(984\) −39.8873 −1.27156
\(985\) 0 0
\(986\) −20.3116 −0.646854
\(987\) 2.01383 0.0641008
\(988\) 5.56454 0.177032
\(989\) −14.1916 −0.451265
\(990\) 0 0
\(991\) 36.5755 1.16186 0.580930 0.813953i \(-0.302689\pi\)
0.580930 + 0.813953i \(0.302689\pi\)
\(992\) 35.9429 1.14119
\(993\) −9.93623 −0.315317
\(994\) −5.60522 −0.177787
\(995\) 0 0
\(996\) 14.1850 0.449469
\(997\) −19.3415 −0.612552 −0.306276 0.951943i \(-0.599083\pi\)
−0.306276 + 0.951943i \(0.599083\pi\)
\(998\) 24.3052 0.769367
\(999\) −37.4633 −1.18529
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 3025.2.a.bk.1.5 8
5.2 odd 4 605.2.b.f.364.5 8
5.3 odd 4 605.2.b.f.364.4 8
5.4 even 2 inner 3025.2.a.bk.1.4 8
11.7 odd 10 275.2.h.d.126.2 16
11.8 odd 10 275.2.h.d.251.2 16
11.10 odd 2 3025.2.a.bl.1.4 8
55.2 even 20 605.2.j.h.444.2 16
55.3 odd 20 605.2.j.d.9.2 16
55.7 even 20 55.2.j.a.49.3 yes 16
55.8 even 20 55.2.j.a.9.3 yes 16
55.13 even 20 605.2.j.h.444.3 16
55.17 even 20 605.2.j.h.124.3 16
55.18 even 20 55.2.j.a.49.2 yes 16
55.19 odd 10 275.2.h.d.251.3 16
55.27 odd 20 605.2.j.g.124.2 16
55.28 even 20 605.2.j.h.124.2 16
55.29 odd 10 275.2.h.d.126.3 16
55.32 even 4 605.2.b.g.364.4 8
55.37 odd 20 605.2.j.d.269.2 16
55.38 odd 20 605.2.j.g.124.3 16
55.42 odd 20 605.2.j.g.444.3 16
55.43 even 4 605.2.b.g.364.5 8
55.47 odd 20 605.2.j.d.9.3 16
55.48 odd 20 605.2.j.d.269.3 16
55.52 even 20 55.2.j.a.9.2 16
55.53 odd 20 605.2.j.g.444.2 16
55.54 odd 2 3025.2.a.bl.1.5 8
165.8 odd 20 495.2.ba.a.64.2 16
165.62 odd 20 495.2.ba.a.379.2 16
165.107 odd 20 495.2.ba.a.64.3 16
165.128 odd 20 495.2.ba.a.379.3 16
220.7 odd 20 880.2.cd.c.49.2 16
220.63 odd 20 880.2.cd.c.449.2 16
220.107 odd 20 880.2.cd.c.449.3 16
220.183 odd 20 880.2.cd.c.49.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.j.a.9.2 16 55.52 even 20
55.2.j.a.9.3 yes 16 55.8 even 20
55.2.j.a.49.2 yes 16 55.18 even 20
55.2.j.a.49.3 yes 16 55.7 even 20
275.2.h.d.126.2 16 11.7 odd 10
275.2.h.d.126.3 16 55.29 odd 10
275.2.h.d.251.2 16 11.8 odd 10
275.2.h.d.251.3 16 55.19 odd 10
495.2.ba.a.64.2 16 165.8 odd 20
495.2.ba.a.64.3 16 165.107 odd 20
495.2.ba.a.379.2 16 165.62 odd 20
495.2.ba.a.379.3 16 165.128 odd 20
605.2.b.f.364.4 8 5.3 odd 4
605.2.b.f.364.5 8 5.2 odd 4
605.2.b.g.364.4 8 55.32 even 4
605.2.b.g.364.5 8 55.43 even 4
605.2.j.d.9.2 16 55.3 odd 20
605.2.j.d.9.3 16 55.47 odd 20
605.2.j.d.269.2 16 55.37 odd 20
605.2.j.d.269.3 16 55.48 odd 20
605.2.j.g.124.2 16 55.27 odd 20
605.2.j.g.124.3 16 55.38 odd 20
605.2.j.g.444.2 16 55.53 odd 20
605.2.j.g.444.3 16 55.42 odd 20
605.2.j.h.124.2 16 55.28 even 20
605.2.j.h.124.3 16 55.17 even 20
605.2.j.h.444.2 16 55.2 even 20
605.2.j.h.444.3 16 55.13 even 20
880.2.cd.c.49.2 16 220.7 odd 20
880.2.cd.c.49.3 16 220.183 odd 20
880.2.cd.c.449.2 16 220.63 odd 20
880.2.cd.c.449.3 16 220.107 odd 20
3025.2.a.bk.1.4 8 5.4 even 2 inner
3025.2.a.bk.1.5 8 1.1 even 1 trivial
3025.2.a.bl.1.4 8 11.10 odd 2
3025.2.a.bl.1.5 8 55.54 odd 2