Properties

Label 4527.2.a.k.1.2
Level $4527$
Weight $2$
Character 4527.1
Self dual yes
Analytic conductor $36.148$
Analytic rank $1$
Dimension $10$
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] = [4527,2,Mod(1,4527)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4527, 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("4527.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4527 = 3^{2} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4527.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: \(36.1482769950\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 503)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.40552\) of defining polynomial
Character \(\chi\) \(=\) 4527.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.36113 q^{2} -0.147314 q^{4} -0.590303 q^{5} +1.95900 q^{7} +2.92278 q^{8} +O(q^{10})\) \(q-1.36113 q^{2} -0.147314 q^{4} -0.590303 q^{5} +1.95900 q^{7} +2.92278 q^{8} +0.803481 q^{10} +1.52746 q^{11} -4.67738 q^{13} -2.66646 q^{14} -3.68367 q^{16} +3.04913 q^{17} +0.338159 q^{19} +0.0869597 q^{20} -2.07908 q^{22} +7.98874 q^{23} -4.65154 q^{25} +6.36654 q^{26} -0.288587 q^{28} -6.01794 q^{29} -4.17447 q^{31} -0.831593 q^{32} -4.15028 q^{34} -1.15640 q^{35} -11.0654 q^{37} -0.460279 q^{38} -1.72533 q^{40} +4.82359 q^{41} +12.4475 q^{43} -0.225016 q^{44} -10.8737 q^{46} -11.0094 q^{47} -3.16233 q^{49} +6.33137 q^{50} +0.689042 q^{52} +3.43684 q^{53} -0.901664 q^{55} +5.72572 q^{56} +8.19122 q^{58} -2.63490 q^{59} +11.4908 q^{61} +5.68202 q^{62} +8.49925 q^{64} +2.76107 q^{65} -10.2554 q^{67} -0.449179 q^{68} +1.57402 q^{70} -9.66931 q^{71} -6.33688 q^{73} +15.0615 q^{74} -0.0498154 q^{76} +2.99229 q^{77} -0.667323 q^{79} +2.17448 q^{80} -6.56556 q^{82} +2.26231 q^{83} -1.79991 q^{85} -16.9427 q^{86} +4.46443 q^{88} -2.21979 q^{89} -9.16297 q^{91} -1.17685 q^{92} +14.9853 q^{94} -0.199616 q^{95} +14.2878 q^{97} +4.30435 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8} - 4 q^{10} + 3 q^{11} - 18 q^{13} - q^{14} - 4 q^{16} + 11 q^{17} + 3 q^{20} - 18 q^{22} + 2 q^{23} - 27 q^{25} - 11 q^{26} - 22 q^{28} + 9 q^{29} - 22 q^{31} + 10 q^{32} - 10 q^{34} + 6 q^{35} - 35 q^{37} - 2 q^{38} - 19 q^{40} + 4 q^{41} - 20 q^{43} - 9 q^{44} - q^{46} - 7 q^{47} - 27 q^{49} - 16 q^{50} - 7 q^{52} + 24 q^{53} - 11 q^{55} - 12 q^{56} + 2 q^{58} - 17 q^{59} - 4 q^{61} - 8 q^{62} + 3 q^{64} + 16 q^{65} - 6 q^{67} - 28 q^{68} + 26 q^{70} + q^{71} - 31 q^{73} - 11 q^{74} + 20 q^{76} - 3 q^{77} - 10 q^{79} - 24 q^{80} - 9 q^{82} - 22 q^{83} - 6 q^{85} - 38 q^{86} - 3 q^{88} - q^{89} + 10 q^{91} - 27 q^{92} + 33 q^{94} - 39 q^{95} - 57 q^{97} - 40 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36113 −0.962467 −0.481234 0.876592i \(-0.659811\pi\)
−0.481234 + 0.876592i \(0.659811\pi\)
\(3\) 0 0
\(4\) −0.147314 −0.0736568
\(5\) −0.590303 −0.263991 −0.131996 0.991250i \(-0.542139\pi\)
−0.131996 + 0.991250i \(0.542139\pi\)
\(6\) 0 0
\(7\) 1.95900 0.740431 0.370216 0.928946i \(-0.379284\pi\)
0.370216 + 0.928946i \(0.379284\pi\)
\(8\) 2.92278 1.03336
\(9\) 0 0
\(10\) 0.803481 0.254083
\(11\) 1.52746 0.460546 0.230273 0.973126i \(-0.426038\pi\)
0.230273 + 0.973126i \(0.426038\pi\)
\(12\) 0 0
\(13\) −4.67738 −1.29727 −0.648636 0.761099i \(-0.724660\pi\)
−0.648636 + 0.761099i \(0.724660\pi\)
\(14\) −2.66646 −0.712641
\(15\) 0 0
\(16\) −3.68367 −0.920918
\(17\) 3.04913 0.739523 0.369761 0.929127i \(-0.379439\pi\)
0.369761 + 0.929127i \(0.379439\pi\)
\(18\) 0 0
\(19\) 0.338159 0.0775789 0.0387895 0.999247i \(-0.487650\pi\)
0.0387895 + 0.999247i \(0.487650\pi\)
\(20\) 0.0869597 0.0194448
\(21\) 0 0
\(22\) −2.07908 −0.443261
\(23\) 7.98874 1.66577 0.832884 0.553448i \(-0.186688\pi\)
0.832884 + 0.553448i \(0.186688\pi\)
\(24\) 0 0
\(25\) −4.65154 −0.930309
\(26\) 6.36654 1.24858
\(27\) 0 0
\(28\) −0.288587 −0.0545378
\(29\) −6.01794 −1.11750 −0.558751 0.829335i \(-0.688719\pi\)
−0.558751 + 0.829335i \(0.688719\pi\)
\(30\) 0 0
\(31\) −4.17447 −0.749758 −0.374879 0.927074i \(-0.622316\pi\)
−0.374879 + 0.927074i \(0.622316\pi\)
\(32\) −0.831593 −0.147006
\(33\) 0 0
\(34\) −4.15028 −0.711767
\(35\) −1.15640 −0.195468
\(36\) 0 0
\(37\) −11.0654 −1.81914 −0.909569 0.415553i \(-0.863588\pi\)
−0.909569 + 0.415553i \(0.863588\pi\)
\(38\) −0.460279 −0.0746672
\(39\) 0 0
\(40\) −1.72533 −0.272798
\(41\) 4.82359 0.753319 0.376659 0.926352i \(-0.377073\pi\)
0.376659 + 0.926352i \(0.377073\pi\)
\(42\) 0 0
\(43\) 12.4475 1.89822 0.949111 0.314941i \(-0.101985\pi\)
0.949111 + 0.314941i \(0.101985\pi\)
\(44\) −0.225016 −0.0339224
\(45\) 0 0
\(46\) −10.8737 −1.60325
\(47\) −11.0094 −1.60589 −0.802945 0.596052i \(-0.796735\pi\)
−0.802945 + 0.596052i \(0.796735\pi\)
\(48\) 0 0
\(49\) −3.16233 −0.451761
\(50\) 6.33137 0.895391
\(51\) 0 0
\(52\) 0.689042 0.0955529
\(53\) 3.43684 0.472087 0.236043 0.971743i \(-0.424149\pi\)
0.236043 + 0.971743i \(0.424149\pi\)
\(54\) 0 0
\(55\) −0.901664 −0.121580
\(56\) 5.72572 0.765132
\(57\) 0 0
\(58\) 8.19122 1.07556
\(59\) −2.63490 −0.343034 −0.171517 0.985181i \(-0.554867\pi\)
−0.171517 + 0.985181i \(0.554867\pi\)
\(60\) 0 0
\(61\) 11.4908 1.47125 0.735625 0.677389i \(-0.236889\pi\)
0.735625 + 0.677389i \(0.236889\pi\)
\(62\) 5.68202 0.721617
\(63\) 0 0
\(64\) 8.49925 1.06241
\(65\) 2.76107 0.342469
\(66\) 0 0
\(67\) −10.2554 −1.25289 −0.626445 0.779465i \(-0.715491\pi\)
−0.626445 + 0.779465i \(0.715491\pi\)
\(68\) −0.449179 −0.0544709
\(69\) 0 0
\(70\) 1.57402 0.188131
\(71\) −9.66931 −1.14754 −0.573768 0.819018i \(-0.694519\pi\)
−0.573768 + 0.819018i \(0.694519\pi\)
\(72\) 0 0
\(73\) −6.33688 −0.741676 −0.370838 0.928698i \(-0.620929\pi\)
−0.370838 + 0.928698i \(0.620929\pi\)
\(74\) 15.0615 1.75086
\(75\) 0 0
\(76\) −0.0498154 −0.00571422
\(77\) 2.99229 0.341003
\(78\) 0 0
\(79\) −0.667323 −0.0750798 −0.0375399 0.999295i \(-0.511952\pi\)
−0.0375399 + 0.999295i \(0.511952\pi\)
\(80\) 2.17448 0.243114
\(81\) 0 0
\(82\) −6.56556 −0.725045
\(83\) 2.26231 0.248321 0.124161 0.992262i \(-0.460376\pi\)
0.124161 + 0.992262i \(0.460376\pi\)
\(84\) 0 0
\(85\) −1.79991 −0.195228
\(86\) −16.9427 −1.82698
\(87\) 0 0
\(88\) 4.46443 0.475910
\(89\) −2.21979 −0.235297 −0.117649 0.993055i \(-0.537536\pi\)
−0.117649 + 0.993055i \(0.537536\pi\)
\(90\) 0 0
\(91\) −9.16297 −0.960541
\(92\) −1.17685 −0.122695
\(93\) 0 0
\(94\) 14.9853 1.54562
\(95\) −0.199616 −0.0204802
\(96\) 0 0
\(97\) 14.2878 1.45070 0.725352 0.688378i \(-0.241677\pi\)
0.725352 + 0.688378i \(0.241677\pi\)
\(98\) 4.30435 0.434806
\(99\) 0 0
\(100\) 0.685236 0.0685236
\(101\) 0.629834 0.0626708 0.0313354 0.999509i \(-0.490024\pi\)
0.0313354 + 0.999509i \(0.490024\pi\)
\(102\) 0 0
\(103\) −4.51410 −0.444788 −0.222394 0.974957i \(-0.571387\pi\)
−0.222394 + 0.974957i \(0.571387\pi\)
\(104\) −13.6710 −1.34055
\(105\) 0 0
\(106\) −4.67801 −0.454368
\(107\) −3.43562 −0.332134 −0.166067 0.986115i \(-0.553107\pi\)
−0.166067 + 0.986115i \(0.553107\pi\)
\(108\) 0 0
\(109\) 9.90039 0.948285 0.474142 0.880448i \(-0.342758\pi\)
0.474142 + 0.880448i \(0.342758\pi\)
\(110\) 1.22729 0.117017
\(111\) 0 0
\(112\) −7.21630 −0.681876
\(113\) 9.47684 0.891506 0.445753 0.895156i \(-0.352936\pi\)
0.445753 + 0.895156i \(0.352936\pi\)
\(114\) 0 0
\(115\) −4.71578 −0.439748
\(116\) 0.886524 0.0823117
\(117\) 0 0
\(118\) 3.58645 0.330159
\(119\) 5.97324 0.547566
\(120\) 0 0
\(121\) −8.66687 −0.787897
\(122\) −15.6406 −1.41603
\(123\) 0 0
\(124\) 0.614957 0.0552248
\(125\) 5.69733 0.509585
\(126\) 0 0
\(127\) −11.3574 −1.00781 −0.503904 0.863760i \(-0.668104\pi\)
−0.503904 + 0.863760i \(0.668104\pi\)
\(128\) −9.90544 −0.875525
\(129\) 0 0
\(130\) −3.75819 −0.329615
\(131\) −15.9119 −1.39023 −0.695113 0.718900i \(-0.744646\pi\)
−0.695113 + 0.718900i \(0.744646\pi\)
\(132\) 0 0
\(133\) 0.662452 0.0574419
\(134\) 13.9589 1.20587
\(135\) 0 0
\(136\) 8.91195 0.764193
\(137\) 15.0390 1.28486 0.642432 0.766343i \(-0.277925\pi\)
0.642432 + 0.766343i \(0.277925\pi\)
\(138\) 0 0
\(139\) −16.6486 −1.41212 −0.706059 0.708153i \(-0.749529\pi\)
−0.706059 + 0.708153i \(0.749529\pi\)
\(140\) 0.170354 0.0143975
\(141\) 0 0
\(142\) 13.1612 1.10447
\(143\) −7.14451 −0.597454
\(144\) 0 0
\(145\) 3.55241 0.295011
\(146\) 8.62534 0.713838
\(147\) 0 0
\(148\) 1.63008 0.133992
\(149\) 12.0657 0.988463 0.494232 0.869330i \(-0.335450\pi\)
0.494232 + 0.869330i \(0.335450\pi\)
\(150\) 0 0
\(151\) −16.4897 −1.34192 −0.670958 0.741495i \(-0.734117\pi\)
−0.670958 + 0.741495i \(0.734117\pi\)
\(152\) 0.988364 0.0801669
\(153\) 0 0
\(154\) −4.07291 −0.328204
\(155\) 2.46420 0.197930
\(156\) 0 0
\(157\) 4.10607 0.327700 0.163850 0.986485i \(-0.447609\pi\)
0.163850 + 0.986485i \(0.447609\pi\)
\(158\) 0.908317 0.0722618
\(159\) 0 0
\(160\) 0.490892 0.0388084
\(161\) 15.6499 1.23339
\(162\) 0 0
\(163\) 21.2459 1.66411 0.832053 0.554696i \(-0.187165\pi\)
0.832053 + 0.554696i \(0.187165\pi\)
\(164\) −0.710581 −0.0554871
\(165\) 0 0
\(166\) −3.07931 −0.239001
\(167\) 5.96545 0.461620 0.230810 0.972999i \(-0.425862\pi\)
0.230810 + 0.972999i \(0.425862\pi\)
\(168\) 0 0
\(169\) 8.87787 0.682913
\(170\) 2.44992 0.187900
\(171\) 0 0
\(172\) −1.83368 −0.139817
\(173\) 16.2621 1.23638 0.618192 0.786027i \(-0.287865\pi\)
0.618192 + 0.786027i \(0.287865\pi\)
\(174\) 0 0
\(175\) −9.11236 −0.688830
\(176\) −5.62666 −0.424125
\(177\) 0 0
\(178\) 3.02143 0.226466
\(179\) −9.36724 −0.700140 −0.350070 0.936723i \(-0.613842\pi\)
−0.350070 + 0.936723i \(0.613842\pi\)
\(180\) 0 0
\(181\) 1.62885 0.121072 0.0605358 0.998166i \(-0.480719\pi\)
0.0605358 + 0.998166i \(0.480719\pi\)
\(182\) 12.4720 0.924489
\(183\) 0 0
\(184\) 23.3493 1.72134
\(185\) 6.53193 0.480237
\(186\) 0 0
\(187\) 4.65742 0.340585
\(188\) 1.62184 0.118285
\(189\) 0 0
\(190\) 0.271704 0.0197115
\(191\) 8.41952 0.609215 0.304607 0.952478i \(-0.401475\pi\)
0.304607 + 0.952478i \(0.401475\pi\)
\(192\) 0 0
\(193\) −24.9514 −1.79604 −0.898022 0.439950i \(-0.854996\pi\)
−0.898022 + 0.439950i \(0.854996\pi\)
\(194\) −19.4476 −1.39625
\(195\) 0 0
\(196\) 0.465854 0.0332753
\(197\) −2.36559 −0.168541 −0.0842705 0.996443i \(-0.526856\pi\)
−0.0842705 + 0.996443i \(0.526856\pi\)
\(198\) 0 0
\(199\) 11.7817 0.835180 0.417590 0.908636i \(-0.362875\pi\)
0.417590 + 0.908636i \(0.362875\pi\)
\(200\) −13.5954 −0.961343
\(201\) 0 0
\(202\) −0.857288 −0.0603186
\(203\) −11.7891 −0.827434
\(204\) 0 0
\(205\) −2.84738 −0.198870
\(206\) 6.14430 0.428094
\(207\) 0 0
\(208\) 17.2299 1.19468
\(209\) 0.516524 0.0357287
\(210\) 0 0
\(211\) −16.8124 −1.15742 −0.578708 0.815535i \(-0.696443\pi\)
−0.578708 + 0.815535i \(0.696443\pi\)
\(212\) −0.506294 −0.0347724
\(213\) 0 0
\(214\) 4.67633 0.319668
\(215\) −7.34778 −0.501115
\(216\) 0 0
\(217\) −8.17778 −0.555144
\(218\) −13.4758 −0.912693
\(219\) 0 0
\(220\) 0.132827 0.00895522
\(221\) −14.2619 −0.959362
\(222\) 0 0
\(223\) 8.32490 0.557476 0.278738 0.960367i \(-0.410084\pi\)
0.278738 + 0.960367i \(0.410084\pi\)
\(224\) −1.62909 −0.108848
\(225\) 0 0
\(226\) −12.8992 −0.858045
\(227\) −6.35919 −0.422074 −0.211037 0.977478i \(-0.567684\pi\)
−0.211037 + 0.977478i \(0.567684\pi\)
\(228\) 0 0
\(229\) 11.4619 0.757426 0.378713 0.925514i \(-0.376367\pi\)
0.378713 + 0.925514i \(0.376367\pi\)
\(230\) 6.41880 0.423243
\(231\) 0 0
\(232\) −17.5891 −1.15478
\(233\) −21.5891 −1.41435 −0.707175 0.707039i \(-0.750031\pi\)
−0.707175 + 0.707039i \(0.750031\pi\)
\(234\) 0 0
\(235\) 6.49890 0.423941
\(236\) 0.388156 0.0252668
\(237\) 0 0
\(238\) −8.13038 −0.527014
\(239\) 4.52568 0.292742 0.146371 0.989230i \(-0.453241\pi\)
0.146371 + 0.989230i \(0.453241\pi\)
\(240\) 0 0
\(241\) −9.84982 −0.634483 −0.317241 0.948345i \(-0.602757\pi\)
−0.317241 + 0.948345i \(0.602757\pi\)
\(242\) 11.7968 0.758325
\(243\) 0 0
\(244\) −1.69276 −0.108368
\(245\) 1.86673 0.119261
\(246\) 0 0
\(247\) −1.58170 −0.100641
\(248\) −12.2011 −0.774769
\(249\) 0 0
\(250\) −7.75483 −0.490459
\(251\) −22.0558 −1.39215 −0.696075 0.717969i \(-0.745072\pi\)
−0.696075 + 0.717969i \(0.745072\pi\)
\(252\) 0 0
\(253\) 12.2025 0.767163
\(254\) 15.4590 0.969982
\(255\) 0 0
\(256\) −3.51587 −0.219742
\(257\) −25.3211 −1.57948 −0.789742 0.613439i \(-0.789786\pi\)
−0.789742 + 0.613439i \(0.789786\pi\)
\(258\) 0 0
\(259\) −21.6771 −1.34695
\(260\) −0.406743 −0.0252251
\(261\) 0 0
\(262\) 21.6582 1.33805
\(263\) −10.7243 −0.661289 −0.330644 0.943755i \(-0.607266\pi\)
−0.330644 + 0.943755i \(0.607266\pi\)
\(264\) 0 0
\(265\) −2.02878 −0.124627
\(266\) −0.901686 −0.0552859
\(267\) 0 0
\(268\) 1.51075 0.0922839
\(269\) 11.9609 0.729266 0.364633 0.931151i \(-0.381194\pi\)
0.364633 + 0.931151i \(0.381194\pi\)
\(270\) 0 0
\(271\) −11.7258 −0.712295 −0.356147 0.934430i \(-0.615910\pi\)
−0.356147 + 0.934430i \(0.615910\pi\)
\(272\) −11.2320 −0.681040
\(273\) 0 0
\(274\) −20.4700 −1.23664
\(275\) −7.10504 −0.428450
\(276\) 0 0
\(277\) −0.916745 −0.0550819 −0.0275409 0.999621i \(-0.508768\pi\)
−0.0275409 + 0.999621i \(0.508768\pi\)
\(278\) 22.6610 1.35912
\(279\) 0 0
\(280\) −3.37991 −0.201988
\(281\) −21.7464 −1.29728 −0.648640 0.761096i \(-0.724662\pi\)
−0.648640 + 0.761096i \(0.724662\pi\)
\(282\) 0 0
\(283\) −8.06684 −0.479524 −0.239762 0.970832i \(-0.577069\pi\)
−0.239762 + 0.970832i \(0.577069\pi\)
\(284\) 1.42442 0.0845238
\(285\) 0 0
\(286\) 9.72463 0.575030
\(287\) 9.44941 0.557781
\(288\) 0 0
\(289\) −7.70280 −0.453106
\(290\) −4.83530 −0.283939
\(291\) 0 0
\(292\) 0.933509 0.0546295
\(293\) 2.16609 0.126544 0.0632722 0.997996i \(-0.479846\pi\)
0.0632722 + 0.997996i \(0.479846\pi\)
\(294\) 0 0
\(295\) 1.55539 0.0905581
\(296\) −32.3417 −1.87982
\(297\) 0 0
\(298\) −16.4231 −0.951363
\(299\) −37.3664 −2.16095
\(300\) 0 0
\(301\) 24.3846 1.40550
\(302\) 22.4448 1.29155
\(303\) 0 0
\(304\) −1.24567 −0.0714438
\(305\) −6.78307 −0.388397
\(306\) 0 0
\(307\) 6.60523 0.376980 0.188490 0.982075i \(-0.439641\pi\)
0.188490 + 0.982075i \(0.439641\pi\)
\(308\) −0.440805 −0.0251172
\(309\) 0 0
\(310\) −3.35411 −0.190501
\(311\) −3.84184 −0.217851 −0.108925 0.994050i \(-0.534741\pi\)
−0.108925 + 0.994050i \(0.534741\pi\)
\(312\) 0 0
\(313\) −25.4617 −1.43918 −0.719591 0.694398i \(-0.755671\pi\)
−0.719591 + 0.694398i \(0.755671\pi\)
\(314\) −5.58891 −0.315401
\(315\) 0 0
\(316\) 0.0983058 0.00553014
\(317\) 8.23817 0.462702 0.231351 0.972870i \(-0.425686\pi\)
0.231351 + 0.972870i \(0.425686\pi\)
\(318\) 0 0
\(319\) −9.19216 −0.514662
\(320\) −5.01713 −0.280466
\(321\) 0 0
\(322\) −21.3016 −1.18709
\(323\) 1.03109 0.0573714
\(324\) 0 0
\(325\) 21.7570 1.20686
\(326\) −28.9185 −1.60165
\(327\) 0 0
\(328\) 14.0983 0.778449
\(329\) −21.5675 −1.18905
\(330\) 0 0
\(331\) −6.68597 −0.367494 −0.183747 0.982974i \(-0.558823\pi\)
−0.183747 + 0.982974i \(0.558823\pi\)
\(332\) −0.333270 −0.0182906
\(333\) 0 0
\(334\) −8.11977 −0.444294
\(335\) 6.05376 0.330752
\(336\) 0 0
\(337\) −34.9491 −1.90380 −0.951899 0.306411i \(-0.900872\pi\)
−0.951899 + 0.306411i \(0.900872\pi\)
\(338\) −12.0840 −0.657282
\(339\) 0 0
\(340\) 0.265151 0.0143799
\(341\) −6.37634 −0.345298
\(342\) 0 0
\(343\) −19.9080 −1.07493
\(344\) 36.3813 1.96155
\(345\) 0 0
\(346\) −22.1349 −1.18998
\(347\) 16.1317 0.865998 0.432999 0.901394i \(-0.357455\pi\)
0.432999 + 0.901394i \(0.357455\pi\)
\(348\) 0 0
\(349\) −16.2651 −0.870653 −0.435326 0.900273i \(-0.643367\pi\)
−0.435326 + 0.900273i \(0.643367\pi\)
\(350\) 12.4031 0.662976
\(351\) 0 0
\(352\) −1.27022 −0.0677032
\(353\) −15.5427 −0.827254 −0.413627 0.910446i \(-0.635738\pi\)
−0.413627 + 0.910446i \(0.635738\pi\)
\(354\) 0 0
\(355\) 5.70782 0.302940
\(356\) 0.327005 0.0173312
\(357\) 0 0
\(358\) 12.7501 0.673862
\(359\) 16.5339 0.872625 0.436312 0.899795i \(-0.356284\pi\)
0.436312 + 0.899795i \(0.356284\pi\)
\(360\) 0 0
\(361\) −18.8856 −0.993982
\(362\) −2.21708 −0.116527
\(363\) 0 0
\(364\) 1.34983 0.0707504
\(365\) 3.74068 0.195796
\(366\) 0 0
\(367\) 5.62500 0.293623 0.146811 0.989165i \(-0.453099\pi\)
0.146811 + 0.989165i \(0.453099\pi\)
\(368\) −29.4279 −1.53403
\(369\) 0 0
\(370\) −8.89083 −0.462212
\(371\) 6.73277 0.349548
\(372\) 0 0
\(373\) −18.0602 −0.935122 −0.467561 0.883961i \(-0.654867\pi\)
−0.467561 + 0.883961i \(0.654867\pi\)
\(374\) −6.33938 −0.327802
\(375\) 0 0
\(376\) −32.1782 −1.65946
\(377\) 28.1482 1.44970
\(378\) 0 0
\(379\) −2.42773 −0.124704 −0.0623521 0.998054i \(-0.519860\pi\)
−0.0623521 + 0.998054i \(0.519860\pi\)
\(380\) 0.0294062 0.00150850
\(381\) 0 0
\(382\) −11.4601 −0.586349
\(383\) 16.2778 0.831757 0.415878 0.909420i \(-0.363474\pi\)
0.415878 + 0.909420i \(0.363474\pi\)
\(384\) 0 0
\(385\) −1.76636 −0.0900219
\(386\) 33.9623 1.72863
\(387\) 0 0
\(388\) −2.10478 −0.106854
\(389\) 15.0862 0.764899 0.382450 0.923976i \(-0.375081\pi\)
0.382450 + 0.923976i \(0.375081\pi\)
\(390\) 0 0
\(391\) 24.3587 1.23187
\(392\) −9.24280 −0.466832
\(393\) 0 0
\(394\) 3.21988 0.162215
\(395\) 0.393923 0.0198204
\(396\) 0 0
\(397\) −26.7523 −1.34266 −0.671329 0.741160i \(-0.734276\pi\)
−0.671329 + 0.741160i \(0.734276\pi\)
\(398\) −16.0364 −0.803834
\(399\) 0 0
\(400\) 17.1348 0.856738
\(401\) 21.1930 1.05833 0.529165 0.848519i \(-0.322505\pi\)
0.529165 + 0.848519i \(0.322505\pi\)
\(402\) 0 0
\(403\) 19.5256 0.972639
\(404\) −0.0927831 −0.00461613
\(405\) 0 0
\(406\) 16.0466 0.796378
\(407\) −16.9019 −0.837797
\(408\) 0 0
\(409\) 14.2178 0.703023 0.351511 0.936184i \(-0.385668\pi\)
0.351511 + 0.936184i \(0.385668\pi\)
\(410\) 3.87567 0.191406
\(411\) 0 0
\(412\) 0.664989 0.0327617
\(413\) −5.16176 −0.253993
\(414\) 0 0
\(415\) −1.33545 −0.0655547
\(416\) 3.88968 0.190707
\(417\) 0 0
\(418\) −0.703058 −0.0343877
\(419\) −40.7220 −1.98940 −0.994700 0.102818i \(-0.967214\pi\)
−0.994700 + 0.102818i \(0.967214\pi\)
\(420\) 0 0
\(421\) 30.2685 1.47519 0.737597 0.675241i \(-0.235960\pi\)
0.737597 + 0.675241i \(0.235960\pi\)
\(422\) 22.8840 1.11397
\(423\) 0 0
\(424\) 10.0451 0.487835
\(425\) −14.1832 −0.687984
\(426\) 0 0
\(427\) 22.5105 1.08936
\(428\) 0.506113 0.0244639
\(429\) 0 0
\(430\) 10.0013 0.482306
\(431\) −14.3367 −0.690572 −0.345286 0.938497i \(-0.612218\pi\)
−0.345286 + 0.938497i \(0.612218\pi\)
\(432\) 0 0
\(433\) −26.4985 −1.27344 −0.636719 0.771096i \(-0.719709\pi\)
−0.636719 + 0.771096i \(0.719709\pi\)
\(434\) 11.1311 0.534308
\(435\) 0 0
\(436\) −1.45846 −0.0698477
\(437\) 2.70146 0.129228
\(438\) 0 0
\(439\) 19.2176 0.917208 0.458604 0.888641i \(-0.348350\pi\)
0.458604 + 0.888641i \(0.348350\pi\)
\(440\) −2.63537 −0.125636
\(441\) 0 0
\(442\) 19.4124 0.923355
\(443\) 5.97322 0.283796 0.141898 0.989881i \(-0.454679\pi\)
0.141898 + 0.989881i \(0.454679\pi\)
\(444\) 0 0
\(445\) 1.31035 0.0621164
\(446\) −11.3313 −0.536553
\(447\) 0 0
\(448\) 16.6500 0.786639
\(449\) −1.84304 −0.0869784 −0.0434892 0.999054i \(-0.513847\pi\)
−0.0434892 + 0.999054i \(0.513847\pi\)
\(450\) 0 0
\(451\) 7.36785 0.346938
\(452\) −1.39607 −0.0656655
\(453\) 0 0
\(454\) 8.65571 0.406233
\(455\) 5.40893 0.253575
\(456\) 0 0
\(457\) 27.3399 1.27891 0.639453 0.768830i \(-0.279161\pi\)
0.639453 + 0.768830i \(0.279161\pi\)
\(458\) −15.6012 −0.728998
\(459\) 0 0
\(460\) 0.694698 0.0323905
\(461\) −35.1434 −1.63679 −0.818395 0.574656i \(-0.805136\pi\)
−0.818395 + 0.574656i \(0.805136\pi\)
\(462\) 0 0
\(463\) −18.6765 −0.867969 −0.433984 0.900920i \(-0.642893\pi\)
−0.433984 + 0.900920i \(0.642893\pi\)
\(464\) 22.1681 1.02913
\(465\) 0 0
\(466\) 29.3857 1.36126
\(467\) 16.9925 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(468\) 0 0
\(469\) −20.0902 −0.927680
\(470\) −8.84588 −0.408030
\(471\) 0 0
\(472\) −7.70123 −0.354478
\(473\) 19.0130 0.874220
\(474\) 0 0
\(475\) −1.57296 −0.0721723
\(476\) −0.879940 −0.0403320
\(477\) 0 0
\(478\) −6.16006 −0.281755
\(479\) 11.4554 0.523409 0.261705 0.965148i \(-0.415715\pi\)
0.261705 + 0.965148i \(0.415715\pi\)
\(480\) 0 0
\(481\) 51.7570 2.35992
\(482\) 13.4069 0.610669
\(483\) 0 0
\(484\) 1.27675 0.0580340
\(485\) −8.43411 −0.382973
\(486\) 0 0
\(487\) −14.2183 −0.644295 −0.322147 0.946690i \(-0.604405\pi\)
−0.322147 + 0.946690i \(0.604405\pi\)
\(488\) 33.5852 1.52033
\(489\) 0 0
\(490\) −2.54087 −0.114785
\(491\) 4.53402 0.204617 0.102309 0.994753i \(-0.467377\pi\)
0.102309 + 0.994753i \(0.467377\pi\)
\(492\) 0 0
\(493\) −18.3495 −0.826419
\(494\) 2.15290 0.0968636
\(495\) 0 0
\(496\) 15.3774 0.690465
\(497\) −18.9421 −0.849671
\(498\) 0 0
\(499\) −38.1411 −1.70743 −0.853715 0.520741i \(-0.825656\pi\)
−0.853715 + 0.520741i \(0.825656\pi\)
\(500\) −0.839295 −0.0375344
\(501\) 0 0
\(502\) 30.0209 1.33990
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −0.371793 −0.0165446
\(506\) −16.6092 −0.738369
\(507\) 0 0
\(508\) 1.67310 0.0742319
\(509\) −34.3114 −1.52083 −0.760413 0.649440i \(-0.775003\pi\)
−0.760413 + 0.649440i \(0.775003\pi\)
\(510\) 0 0
\(511\) −12.4139 −0.549160
\(512\) 24.5965 1.08702
\(513\) 0 0
\(514\) 34.4654 1.52020
\(515\) 2.66469 0.117420
\(516\) 0 0
\(517\) −16.8165 −0.739587
\(518\) 29.5054 1.29639
\(519\) 0 0
\(520\) 8.07001 0.353893
\(521\) −23.8175 −1.04346 −0.521732 0.853110i \(-0.674714\pi\)
−0.521732 + 0.853110i \(0.674714\pi\)
\(522\) 0 0
\(523\) −2.16718 −0.0947643 −0.0473821 0.998877i \(-0.515088\pi\)
−0.0473821 + 0.998877i \(0.515088\pi\)
\(524\) 2.34404 0.102400
\(525\) 0 0
\(526\) 14.5972 0.636469
\(527\) −12.7285 −0.554463
\(528\) 0 0
\(529\) 40.8200 1.77478
\(530\) 2.76144 0.119949
\(531\) 0 0
\(532\) −0.0975882 −0.00423099
\(533\) −22.5618 −0.977259
\(534\) 0 0
\(535\) 2.02805 0.0876804
\(536\) −29.9742 −1.29469
\(537\) 0 0
\(538\) −16.2803 −0.701895
\(539\) −4.83033 −0.208057
\(540\) 0 0
\(541\) 13.5019 0.580493 0.290246 0.956952i \(-0.406263\pi\)
0.290246 + 0.956952i \(0.406263\pi\)
\(542\) 15.9605 0.685560
\(543\) 0 0
\(544\) −2.53564 −0.108714
\(545\) −5.84423 −0.250339
\(546\) 0 0
\(547\) −1.05880 −0.0452710 −0.0226355 0.999744i \(-0.507206\pi\)
−0.0226355 + 0.999744i \(0.507206\pi\)
\(548\) −2.21544 −0.0946390
\(549\) 0 0
\(550\) 9.67092 0.412369
\(551\) −2.03502 −0.0866947
\(552\) 0 0
\(553\) −1.30728 −0.0555914
\(554\) 1.24781 0.0530145
\(555\) 0 0
\(556\) 2.45257 0.104012
\(557\) 9.20176 0.389891 0.194946 0.980814i \(-0.437547\pi\)
0.194946 + 0.980814i \(0.437547\pi\)
\(558\) 0 0
\(559\) −58.2216 −2.46251
\(560\) 4.25980 0.180010
\(561\) 0 0
\(562\) 29.5997 1.24859
\(563\) 29.6502 1.24961 0.624804 0.780782i \(-0.285179\pi\)
0.624804 + 0.780782i \(0.285179\pi\)
\(564\) 0 0
\(565\) −5.59420 −0.235350
\(566\) 10.9800 0.461526
\(567\) 0 0
\(568\) −28.2613 −1.18582
\(569\) −22.6715 −0.950439 −0.475220 0.879867i \(-0.657631\pi\)
−0.475220 + 0.879867i \(0.657631\pi\)
\(570\) 0 0
\(571\) −31.2013 −1.30573 −0.652866 0.757473i \(-0.726434\pi\)
−0.652866 + 0.757473i \(0.726434\pi\)
\(572\) 1.05248 0.0440065
\(573\) 0 0
\(574\) −12.8619 −0.536846
\(575\) −37.1600 −1.54968
\(576\) 0 0
\(577\) −3.98191 −0.165769 −0.0828845 0.996559i \(-0.526413\pi\)
−0.0828845 + 0.996559i \(0.526413\pi\)
\(578\) 10.4845 0.436099
\(579\) 0 0
\(580\) −0.523318 −0.0217296
\(581\) 4.43187 0.183865
\(582\) 0 0
\(583\) 5.24964 0.217418
\(584\) −18.5213 −0.766418
\(585\) 0 0
\(586\) −2.94834 −0.121795
\(587\) 12.7494 0.526225 0.263113 0.964765i \(-0.415251\pi\)
0.263113 + 0.964765i \(0.415251\pi\)
\(588\) 0 0
\(589\) −1.41163 −0.0581654
\(590\) −2.11709 −0.0871592
\(591\) 0 0
\(592\) 40.7612 1.67528
\(593\) 3.27771 0.134600 0.0672998 0.997733i \(-0.478562\pi\)
0.0672998 + 0.997733i \(0.478562\pi\)
\(594\) 0 0
\(595\) −3.52602 −0.144553
\(596\) −1.77745 −0.0728070
\(597\) 0 0
\(598\) 50.8606 2.07985
\(599\) −10.9237 −0.446332 −0.223166 0.974780i \(-0.571639\pi\)
−0.223166 + 0.974780i \(0.571639\pi\)
\(600\) 0 0
\(601\) −29.7446 −1.21331 −0.606653 0.794967i \(-0.707488\pi\)
−0.606653 + 0.794967i \(0.707488\pi\)
\(602\) −33.1907 −1.35275
\(603\) 0 0
\(604\) 2.42916 0.0988413
\(605\) 5.11608 0.207998
\(606\) 0 0
\(607\) −43.8414 −1.77947 −0.889735 0.456478i \(-0.849111\pi\)
−0.889735 + 0.456478i \(0.849111\pi\)
\(608\) −0.281210 −0.0114046
\(609\) 0 0
\(610\) 9.23267 0.373820
\(611\) 51.4953 2.08328
\(612\) 0 0
\(613\) −32.3195 −1.30537 −0.652687 0.757628i \(-0.726358\pi\)
−0.652687 + 0.757628i \(0.726358\pi\)
\(614\) −8.99061 −0.362831
\(615\) 0 0
\(616\) 8.74581 0.352379
\(617\) −40.9696 −1.64937 −0.824687 0.565589i \(-0.808649\pi\)
−0.824687 + 0.565589i \(0.808649\pi\)
\(618\) 0 0
\(619\) 22.5775 0.907466 0.453733 0.891138i \(-0.350092\pi\)
0.453733 + 0.891138i \(0.350092\pi\)
\(620\) −0.363011 −0.0145789
\(621\) 0 0
\(622\) 5.22926 0.209674
\(623\) −4.34856 −0.174221
\(624\) 0 0
\(625\) 19.8946 0.795782
\(626\) 34.6568 1.38517
\(627\) 0 0
\(628\) −0.604880 −0.0241373
\(629\) −33.7398 −1.34529
\(630\) 0 0
\(631\) 33.9719 1.35240 0.676199 0.736719i \(-0.263626\pi\)
0.676199 + 0.736719i \(0.263626\pi\)
\(632\) −1.95044 −0.0775844
\(633\) 0 0
\(634\) −11.2132 −0.445335
\(635\) 6.70432 0.266053
\(636\) 0 0
\(637\) 14.7914 0.586057
\(638\) 12.5118 0.495345
\(639\) 0 0
\(640\) 5.84721 0.231131
\(641\) 4.53431 0.179095 0.0895473 0.995983i \(-0.471458\pi\)
0.0895473 + 0.995983i \(0.471458\pi\)
\(642\) 0 0
\(643\) 38.3929 1.51407 0.757033 0.653376i \(-0.226648\pi\)
0.757033 + 0.653376i \(0.226648\pi\)
\(644\) −2.30545 −0.0908473
\(645\) 0 0
\(646\) −1.40345 −0.0552181
\(647\) 0.255028 0.0100262 0.00501309 0.999987i \(-0.498404\pi\)
0.00501309 + 0.999987i \(0.498404\pi\)
\(648\) 0 0
\(649\) −4.02470 −0.157983
\(650\) −29.6142 −1.16157
\(651\) 0 0
\(652\) −3.12981 −0.122573
\(653\) 2.04379 0.0799797 0.0399898 0.999200i \(-0.487267\pi\)
0.0399898 + 0.999200i \(0.487267\pi\)
\(654\) 0 0
\(655\) 9.39282 0.367008
\(656\) −17.7685 −0.693745
\(657\) 0 0
\(658\) 29.3562 1.14442
\(659\) −10.0768 −0.392534 −0.196267 0.980550i \(-0.562882\pi\)
−0.196267 + 0.980550i \(0.562882\pi\)
\(660\) 0 0
\(661\) −5.53010 −0.215096 −0.107548 0.994200i \(-0.534300\pi\)
−0.107548 + 0.994200i \(0.534300\pi\)
\(662\) 9.10050 0.353701
\(663\) 0 0
\(664\) 6.61225 0.256605
\(665\) −0.391047 −0.0151642
\(666\) 0 0
\(667\) −48.0757 −1.86150
\(668\) −0.878791 −0.0340015
\(669\) 0 0
\(670\) −8.23998 −0.318338
\(671\) 17.5518 0.677579
\(672\) 0 0
\(673\) −3.16280 −0.121917 −0.0609585 0.998140i \(-0.519416\pi\)
−0.0609585 + 0.998140i \(0.519416\pi\)
\(674\) 47.5704 1.83234
\(675\) 0 0
\(676\) −1.30783 −0.0503012
\(677\) 47.8219 1.83794 0.918972 0.394323i \(-0.129021\pi\)
0.918972 + 0.394323i \(0.129021\pi\)
\(678\) 0 0
\(679\) 27.9897 1.07415
\(680\) −5.26075 −0.201740
\(681\) 0 0
\(682\) 8.67906 0.332338
\(683\) −13.2468 −0.506873 −0.253437 0.967352i \(-0.581561\pi\)
−0.253437 + 0.967352i \(0.581561\pi\)
\(684\) 0 0
\(685\) −8.87754 −0.339193
\(686\) 27.0974 1.03458
\(687\) 0 0
\(688\) −45.8524 −1.74811
\(689\) −16.0754 −0.612425
\(690\) 0 0
\(691\) −7.80850 −0.297049 −0.148525 0.988909i \(-0.547452\pi\)
−0.148525 + 0.988909i \(0.547452\pi\)
\(692\) −2.39563 −0.0910681
\(693\) 0 0
\(694\) −21.9575 −0.833494
\(695\) 9.82774 0.372787
\(696\) 0 0
\(697\) 14.7078 0.557097
\(698\) 22.1390 0.837975
\(699\) 0 0
\(700\) 1.34237 0.0507370
\(701\) −7.68972 −0.290437 −0.145218 0.989400i \(-0.546388\pi\)
−0.145218 + 0.989400i \(0.546388\pi\)
\(702\) 0 0
\(703\) −3.74185 −0.141127
\(704\) 12.9823 0.489288
\(705\) 0 0
\(706\) 21.1557 0.796205
\(707\) 1.23384 0.0464034
\(708\) 0 0
\(709\) 31.0522 1.16619 0.583096 0.812403i \(-0.301841\pi\)
0.583096 + 0.812403i \(0.301841\pi\)
\(710\) −7.76911 −0.291569
\(711\) 0 0
\(712\) −6.48796 −0.243146
\(713\) −33.3488 −1.24892
\(714\) 0 0
\(715\) 4.21742 0.157723
\(716\) 1.37992 0.0515701
\(717\) 0 0
\(718\) −22.5048 −0.839873
\(719\) 30.4494 1.13557 0.567786 0.823176i \(-0.307800\pi\)
0.567786 + 0.823176i \(0.307800\pi\)
\(720\) 0 0
\(721\) −8.84312 −0.329335
\(722\) 25.7059 0.956675
\(723\) 0 0
\(724\) −0.239952 −0.00891774
\(725\) 27.9927 1.03962
\(726\) 0 0
\(727\) −6.14587 −0.227938 −0.113969 0.993484i \(-0.536356\pi\)
−0.113969 + 0.993484i \(0.536356\pi\)
\(728\) −26.7814 −0.992584
\(729\) 0 0
\(730\) −5.09156 −0.188447
\(731\) 37.9540 1.40378
\(732\) 0 0
\(733\) −25.4237 −0.939046 −0.469523 0.882920i \(-0.655574\pi\)
−0.469523 + 0.882920i \(0.655574\pi\)
\(734\) −7.65638 −0.282602
\(735\) 0 0
\(736\) −6.64338 −0.244878
\(737\) −15.6646 −0.577014
\(738\) 0 0
\(739\) −19.8464 −0.730063 −0.365032 0.930995i \(-0.618942\pi\)
−0.365032 + 0.930995i \(0.618942\pi\)
\(740\) −0.962242 −0.0353727
\(741\) 0 0
\(742\) −9.16420 −0.336428
\(743\) 28.7154 1.05346 0.526732 0.850031i \(-0.323417\pi\)
0.526732 + 0.850031i \(0.323417\pi\)
\(744\) 0 0
\(745\) −7.12243 −0.260946
\(746\) 24.5824 0.900025
\(747\) 0 0
\(748\) −0.686102 −0.0250864
\(749\) −6.73036 −0.245922
\(750\) 0 0
\(751\) 24.0374 0.877139 0.438569 0.898697i \(-0.355485\pi\)
0.438569 + 0.898697i \(0.355485\pi\)
\(752\) 40.5551 1.47889
\(753\) 0 0
\(754\) −38.3134 −1.39529
\(755\) 9.73394 0.354254
\(756\) 0 0
\(757\) −23.2463 −0.844902 −0.422451 0.906386i \(-0.638830\pi\)
−0.422451 + 0.906386i \(0.638830\pi\)
\(758\) 3.30447 0.120024
\(759\) 0 0
\(760\) −0.583434 −0.0211634
\(761\) −44.3555 −1.60789 −0.803943 0.594706i \(-0.797268\pi\)
−0.803943 + 0.594706i \(0.797268\pi\)
\(762\) 0 0
\(763\) 19.3948 0.702140
\(764\) −1.24031 −0.0448728
\(765\) 0 0
\(766\) −22.1563 −0.800539
\(767\) 12.3244 0.445009
\(768\) 0 0
\(769\) −17.4534 −0.629385 −0.314693 0.949194i \(-0.601901\pi\)
−0.314693 + 0.949194i \(0.601901\pi\)
\(770\) 2.40425 0.0866431
\(771\) 0 0
\(772\) 3.67569 0.132291
\(773\) 1.73607 0.0624422 0.0312211 0.999513i \(-0.490060\pi\)
0.0312211 + 0.999513i \(0.490060\pi\)
\(774\) 0 0
\(775\) 19.4177 0.697506
\(776\) 41.7600 1.49910
\(777\) 0 0
\(778\) −20.5343 −0.736190
\(779\) 1.63114 0.0584417
\(780\) 0 0
\(781\) −14.7695 −0.528493
\(782\) −33.1555 −1.18564
\(783\) 0 0
\(784\) 11.6490 0.416035
\(785\) −2.42383 −0.0865100
\(786\) 0 0
\(787\) −16.1636 −0.576169 −0.288085 0.957605i \(-0.593018\pi\)
−0.288085 + 0.957605i \(0.593018\pi\)
\(788\) 0.348483 0.0124142
\(789\) 0 0
\(790\) −0.536182 −0.0190765
\(791\) 18.5651 0.660099
\(792\) 0 0
\(793\) −53.7469 −1.90861
\(794\) 36.4134 1.29226
\(795\) 0 0
\(796\) −1.73560 −0.0615167
\(797\) 26.8208 0.950040 0.475020 0.879975i \(-0.342441\pi\)
0.475020 + 0.879975i \(0.342441\pi\)
\(798\) 0 0
\(799\) −33.5692 −1.18759
\(800\) 3.86819 0.136761
\(801\) 0 0
\(802\) −28.8466 −1.01861
\(803\) −9.67933 −0.341576
\(804\) 0 0
\(805\) −9.23819 −0.325603
\(806\) −26.5770 −0.936133
\(807\) 0 0
\(808\) 1.84087 0.0647615
\(809\) −33.4065 −1.17451 −0.587255 0.809402i \(-0.699791\pi\)
−0.587255 + 0.809402i \(0.699791\pi\)
\(810\) 0 0
\(811\) 34.8499 1.22375 0.611873 0.790956i \(-0.290416\pi\)
0.611873 + 0.790956i \(0.290416\pi\)
\(812\) 1.73670 0.0609462
\(813\) 0 0
\(814\) 23.0058 0.806352
\(815\) −12.5415 −0.439310
\(816\) 0 0
\(817\) 4.20922 0.147262
\(818\) −19.3523 −0.676636
\(819\) 0 0
\(820\) 0.419458 0.0146481
\(821\) 24.4182 0.852201 0.426101 0.904676i \(-0.359887\pi\)
0.426101 + 0.904676i \(0.359887\pi\)
\(822\) 0 0
\(823\) 4.51039 0.157222 0.0786112 0.996905i \(-0.474951\pi\)
0.0786112 + 0.996905i \(0.474951\pi\)
\(824\) −13.1937 −0.459626
\(825\) 0 0
\(826\) 7.02584 0.244460
\(827\) 2.91499 0.101364 0.0506821 0.998715i \(-0.483860\pi\)
0.0506821 + 0.998715i \(0.483860\pi\)
\(828\) 0 0
\(829\) 35.2382 1.22387 0.611937 0.790906i \(-0.290390\pi\)
0.611937 + 0.790906i \(0.290390\pi\)
\(830\) 1.81773 0.0630943
\(831\) 0 0
\(832\) −39.7542 −1.37823
\(833\) −9.64236 −0.334088
\(834\) 0 0
\(835\) −3.52142 −0.121864
\(836\) −0.0760910 −0.00263166
\(837\) 0 0
\(838\) 55.4281 1.91473
\(839\) 14.4876 0.500166 0.250083 0.968224i \(-0.419542\pi\)
0.250083 + 0.968224i \(0.419542\pi\)
\(840\) 0 0
\(841\) 7.21557 0.248813
\(842\) −41.1994 −1.41983
\(843\) 0 0
\(844\) 2.47670 0.0852515
\(845\) −5.24063 −0.180283
\(846\) 0 0
\(847\) −16.9784 −0.583384
\(848\) −12.6602 −0.434753
\(849\) 0 0
\(850\) 19.3052 0.662163
\(851\) −88.3985 −3.03026
\(852\) 0 0
\(853\) 4.92917 0.168772 0.0843858 0.996433i \(-0.473107\pi\)
0.0843858 + 0.996433i \(0.473107\pi\)
\(854\) −30.6398 −1.04847
\(855\) 0 0
\(856\) −10.0416 −0.343213
\(857\) −52.5344 −1.79454 −0.897270 0.441482i \(-0.854453\pi\)
−0.897270 + 0.441482i \(0.854453\pi\)
\(858\) 0 0
\(859\) 32.5431 1.11036 0.555179 0.831731i \(-0.312650\pi\)
0.555179 + 0.831731i \(0.312650\pi\)
\(860\) 1.08243 0.0369105
\(861\) 0 0
\(862\) 19.5141 0.664653
\(863\) −51.5604 −1.75514 −0.877568 0.479451i \(-0.840836\pi\)
−0.877568 + 0.479451i \(0.840836\pi\)
\(864\) 0 0
\(865\) −9.59956 −0.326395
\(866\) 36.0681 1.22564
\(867\) 0 0
\(868\) 1.20470 0.0408901
\(869\) −1.01931 −0.0345777
\(870\) 0 0
\(871\) 47.9682 1.62534
\(872\) 28.9367 0.979919
\(873\) 0 0
\(874\) −3.67705 −0.124378
\(875\) 11.1611 0.377313
\(876\) 0 0
\(877\) 35.8616 1.21096 0.605481 0.795860i \(-0.292981\pi\)
0.605481 + 0.795860i \(0.292981\pi\)
\(878\) −26.1578 −0.882783
\(879\) 0 0
\(880\) 3.32143 0.111965
\(881\) 41.2432 1.38952 0.694759 0.719243i \(-0.255511\pi\)
0.694759 + 0.719243i \(0.255511\pi\)
\(882\) 0 0
\(883\) 24.3041 0.817898 0.408949 0.912557i \(-0.365895\pi\)
0.408949 + 0.912557i \(0.365895\pi\)
\(884\) 2.10098 0.0706636
\(885\) 0 0
\(886\) −8.13035 −0.273144
\(887\) −51.7010 −1.73595 −0.867975 0.496608i \(-0.834579\pi\)
−0.867975 + 0.496608i \(0.834579\pi\)
\(888\) 0 0
\(889\) −22.2492 −0.746213
\(890\) −1.78356 −0.0597850
\(891\) 0 0
\(892\) −1.22637 −0.0410619
\(893\) −3.72294 −0.124583
\(894\) 0 0
\(895\) 5.52951 0.184831
\(896\) −19.4047 −0.648266
\(897\) 0 0
\(898\) 2.50862 0.0837138
\(899\) 25.1217 0.837856
\(900\) 0 0
\(901\) 10.4794 0.349119
\(902\) −10.0286 −0.333917
\(903\) 0 0
\(904\) 27.6987 0.921246
\(905\) −0.961515 −0.0319618
\(906\) 0 0
\(907\) −5.06194 −0.168079 −0.0840395 0.996462i \(-0.526782\pi\)
−0.0840395 + 0.996462i \(0.526782\pi\)
\(908\) 0.936796 0.0310887
\(909\) 0 0
\(910\) −7.36228 −0.244057
\(911\) 40.4144 1.33899 0.669495 0.742817i \(-0.266511\pi\)
0.669495 + 0.742817i \(0.266511\pi\)
\(912\) 0 0
\(913\) 3.45559 0.114363
\(914\) −37.2133 −1.23091
\(915\) 0 0
\(916\) −1.68850 −0.0557896
\(917\) −31.1713 −1.02937
\(918\) 0 0
\(919\) 31.2276 1.03010 0.515052 0.857159i \(-0.327773\pi\)
0.515052 + 0.857159i \(0.327773\pi\)
\(920\) −13.7832 −0.454418
\(921\) 0 0
\(922\) 47.8349 1.57536
\(923\) 45.2270 1.48867
\(924\) 0 0
\(925\) 51.4711 1.69236
\(926\) 25.4212 0.835391
\(927\) 0 0
\(928\) 5.00447 0.164280
\(929\) 51.3773 1.68563 0.842817 0.538201i \(-0.180896\pi\)
0.842817 + 0.538201i \(0.180896\pi\)
\(930\) 0 0
\(931\) −1.06937 −0.0350472
\(932\) 3.18037 0.104176
\(933\) 0 0
\(934\) −23.1290 −0.756805
\(935\) −2.74929 −0.0899114
\(936\) 0 0
\(937\) 22.6093 0.738614 0.369307 0.929307i \(-0.379595\pi\)
0.369307 + 0.929307i \(0.379595\pi\)
\(938\) 27.3455 0.892861
\(939\) 0 0
\(940\) −0.957377 −0.0312262
\(941\) 51.2485 1.67065 0.835326 0.549754i \(-0.185279\pi\)
0.835326 + 0.549754i \(0.185279\pi\)
\(942\) 0 0
\(943\) 38.5344 1.25485
\(944\) 9.70610 0.315906
\(945\) 0 0
\(946\) −25.8793 −0.841408
\(947\) 54.1474 1.75955 0.879777 0.475387i \(-0.157692\pi\)
0.879777 + 0.475387i \(0.157692\pi\)
\(948\) 0 0
\(949\) 29.6400 0.962155
\(950\) 2.14101 0.0694635
\(951\) 0 0
\(952\) 17.4585 0.565833
\(953\) 34.0986 1.10456 0.552281 0.833658i \(-0.313758\pi\)
0.552281 + 0.833658i \(0.313758\pi\)
\(954\) 0 0
\(955\) −4.97006 −0.160827
\(956\) −0.666695 −0.0215624
\(957\) 0 0
\(958\) −15.5923 −0.503764
\(959\) 29.4613 0.951354
\(960\) 0 0
\(961\) −13.5738 −0.437863
\(962\) −70.4482 −2.27134
\(963\) 0 0
\(964\) 1.45101 0.0467340
\(965\) 14.7289 0.474140
\(966\) 0 0
\(967\) 20.9385 0.673336 0.336668 0.941623i \(-0.390700\pi\)
0.336668 + 0.941623i \(0.390700\pi\)
\(968\) −25.3314 −0.814181
\(969\) 0 0
\(970\) 11.4800 0.368599
\(971\) 5.00722 0.160689 0.0803447 0.996767i \(-0.474398\pi\)
0.0803447 + 0.996767i \(0.474398\pi\)
\(972\) 0 0
\(973\) −32.6146 −1.04558
\(974\) 19.3531 0.620113
\(975\) 0 0
\(976\) −42.3284 −1.35490
\(977\) −12.5547 −0.401662 −0.200831 0.979626i \(-0.564364\pi\)
−0.200831 + 0.979626i \(0.564364\pi\)
\(978\) 0 0
\(979\) −3.39064 −0.108365
\(980\) −0.274995 −0.00878440
\(981\) 0 0
\(982\) −6.17140 −0.196937
\(983\) −1.43810 −0.0458683 −0.0229341 0.999737i \(-0.507301\pi\)
−0.0229341 + 0.999737i \(0.507301\pi\)
\(984\) 0 0
\(985\) 1.39641 0.0444934
\(986\) 24.9761 0.795401
\(987\) 0 0
\(988\) 0.233005 0.00741289
\(989\) 99.4397 3.16200
\(990\) 0 0
\(991\) −41.6595 −1.32336 −0.661678 0.749788i \(-0.730155\pi\)
−0.661678 + 0.749788i \(0.730155\pi\)
\(992\) 3.47146 0.110219
\(993\) 0 0
\(994\) 25.7828 0.817781
\(995\) −6.95475 −0.220480
\(996\) 0 0
\(997\) 46.5145 1.47313 0.736564 0.676367i \(-0.236447\pi\)
0.736564 + 0.676367i \(0.236447\pi\)
\(998\) 51.9151 1.64334
\(999\) 0 0
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 4527.2.a.k.1.2 10
3.2 odd 2 503.2.a.e.1.9 10
12.11 even 2 8048.2.a.p.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.9 10 3.2 odd 2
4527.2.a.k.1.2 10 1.1 even 1 trivial
8048.2.a.p.1.9 10 12.11 even 2