Properties

Label 9075.2.a.dy.1.6
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $12$
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] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.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: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 18 x^{8} - 220 x^{7} + 61 x^{6} + 330 x^{5} - 145 x^{4} - 148 x^{3} + 71 x^{2} + 10 x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.488299\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.488299 q^{2} +1.00000 q^{3} -1.76156 q^{4} -0.488299 q^{6} -5.10895 q^{7} +1.83677 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.488299 q^{2} +1.00000 q^{3} -1.76156 q^{4} -0.488299 q^{6} -5.10895 q^{7} +1.83677 q^{8} +1.00000 q^{9} -1.76156 q^{12} -1.81774 q^{13} +2.49469 q^{14} +2.62624 q^{16} -0.639966 q^{17} -0.488299 q^{18} -4.39788 q^{19} -5.10895 q^{21} +4.37977 q^{23} +1.83677 q^{24} +0.887599 q^{26} +1.00000 q^{27} +8.99974 q^{28} +7.56018 q^{29} +2.20501 q^{31} -4.95592 q^{32} +0.312494 q^{34} -1.76156 q^{36} +5.97194 q^{37} +2.14748 q^{38} -1.81774 q^{39} -2.23182 q^{41} +2.49469 q^{42} -3.38068 q^{43} -2.13864 q^{46} -1.51581 q^{47} +2.62624 q^{48} +19.1013 q^{49} -0.639966 q^{51} +3.20206 q^{52} -9.17695 q^{53} -0.488299 q^{54} -9.38395 q^{56} -4.39788 q^{57} -3.69163 q^{58} +5.37174 q^{59} +7.26768 q^{61} -1.07670 q^{62} -5.10895 q^{63} -2.83250 q^{64} -7.25598 q^{67} +1.12734 q^{68} +4.37977 q^{69} +5.50864 q^{71} +1.83677 q^{72} +7.14322 q^{73} -2.91609 q^{74} +7.74714 q^{76} +0.887599 q^{78} +1.90516 q^{79} +1.00000 q^{81} +1.08979 q^{82} -0.161902 q^{83} +8.99974 q^{84} +1.65078 q^{86} +7.56018 q^{87} -1.19539 q^{89} +9.28673 q^{91} -7.71525 q^{92} +2.20501 q^{93} +0.740167 q^{94} -4.95592 q^{96} +14.2823 q^{97} -9.32716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9} + 12 q^{12} - 18 q^{13} + 6 q^{14} + 24 q^{16} - 18 q^{17} - 4 q^{18} - 16 q^{19} - 8 q^{21} - 12 q^{24} + 16 q^{26} + 12 q^{27} - 30 q^{28} - 28 q^{32} + 6 q^{34} + 12 q^{36} - 28 q^{38} - 18 q^{39} + 6 q^{42} - 32 q^{43} - 28 q^{46} - 4 q^{47} + 24 q^{48} + 12 q^{49} - 18 q^{51} - 48 q^{52} - 12 q^{53} - 4 q^{54} + 6 q^{56} - 16 q^{57} - 10 q^{58} + 20 q^{59} - 20 q^{61} - 20 q^{62} - 8 q^{63} - 6 q^{64} + 10 q^{67} - 26 q^{68} + 32 q^{71} - 12 q^{72} - 26 q^{73} + 68 q^{74} - 34 q^{76} + 16 q^{78} - 32 q^{79} + 12 q^{81} - 62 q^{82} - 26 q^{83} - 30 q^{84} - 36 q^{86} - 10 q^{89} + 8 q^{92} + 2 q^{94} - 28 q^{96} - 22 q^{97} - 60 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.488299 −0.345279 −0.172640 0.984985i \(-0.555230\pi\)
−0.172640 + 0.984985i \(0.555230\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.76156 −0.880782
\(5\) 0 0
\(6\) −0.488299 −0.199347
\(7\) −5.10895 −1.93100 −0.965500 0.260403i \(-0.916145\pi\)
−0.965500 + 0.260403i \(0.916145\pi\)
\(8\) 1.83677 0.649395
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −1.76156 −0.508520
\(13\) −1.81774 −0.504150 −0.252075 0.967708i \(-0.581113\pi\)
−0.252075 + 0.967708i \(0.581113\pi\)
\(14\) 2.49469 0.666735
\(15\) 0 0
\(16\) 2.62624 0.656559
\(17\) −0.639966 −0.155214 −0.0776072 0.996984i \(-0.524728\pi\)
−0.0776072 + 0.996984i \(0.524728\pi\)
\(18\) −0.488299 −0.115093
\(19\) −4.39788 −1.00894 −0.504471 0.863429i \(-0.668312\pi\)
−0.504471 + 0.863429i \(0.668312\pi\)
\(20\) 0 0
\(21\) −5.10895 −1.11486
\(22\) 0 0
\(23\) 4.37977 0.913246 0.456623 0.889660i \(-0.349059\pi\)
0.456623 + 0.889660i \(0.349059\pi\)
\(24\) 1.83677 0.374929
\(25\) 0 0
\(26\) 0.887599 0.174073
\(27\) 1.00000 0.192450
\(28\) 8.99974 1.70079
\(29\) 7.56018 1.40389 0.701945 0.712231i \(-0.252315\pi\)
0.701945 + 0.712231i \(0.252315\pi\)
\(30\) 0 0
\(31\) 2.20501 0.396032 0.198016 0.980199i \(-0.436550\pi\)
0.198016 + 0.980199i \(0.436550\pi\)
\(32\) −4.95592 −0.876092
\(33\) 0 0
\(34\) 0.312494 0.0535924
\(35\) 0 0
\(36\) −1.76156 −0.293594
\(37\) 5.97194 0.981780 0.490890 0.871221i \(-0.336672\pi\)
0.490890 + 0.871221i \(0.336672\pi\)
\(38\) 2.14748 0.348367
\(39\) −1.81774 −0.291071
\(40\) 0 0
\(41\) −2.23182 −0.348551 −0.174276 0.984697i \(-0.555758\pi\)
−0.174276 + 0.984697i \(0.555758\pi\)
\(42\) 2.49469 0.384939
\(43\) −3.38068 −0.515549 −0.257775 0.966205i \(-0.582989\pi\)
−0.257775 + 0.966205i \(0.582989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.13864 −0.315325
\(47\) −1.51581 −0.221103 −0.110552 0.993870i \(-0.535262\pi\)
−0.110552 + 0.993870i \(0.535262\pi\)
\(48\) 2.62624 0.379065
\(49\) 19.1013 2.72876
\(50\) 0 0
\(51\) −0.639966 −0.0896131
\(52\) 3.20206 0.444046
\(53\) −9.17695 −1.26055 −0.630275 0.776372i \(-0.717058\pi\)
−0.630275 + 0.776372i \(0.717058\pi\)
\(54\) −0.488299 −0.0664491
\(55\) 0 0
\(56\) −9.38395 −1.25398
\(57\) −4.39788 −0.582513
\(58\) −3.69163 −0.484734
\(59\) 5.37174 0.699341 0.349671 0.936873i \(-0.386294\pi\)
0.349671 + 0.936873i \(0.386294\pi\)
\(60\) 0 0
\(61\) 7.26768 0.930531 0.465266 0.885171i \(-0.345959\pi\)
0.465266 + 0.885171i \(0.345959\pi\)
\(62\) −1.07670 −0.136742
\(63\) −5.10895 −0.643667
\(64\) −2.83250 −0.354063
\(65\) 0 0
\(66\) 0 0
\(67\) −7.25598 −0.886459 −0.443230 0.896408i \(-0.646167\pi\)
−0.443230 + 0.896408i \(0.646167\pi\)
\(68\) 1.12734 0.136710
\(69\) 4.37977 0.527263
\(70\) 0 0
\(71\) 5.50864 0.653756 0.326878 0.945067i \(-0.394003\pi\)
0.326878 + 0.945067i \(0.394003\pi\)
\(72\) 1.83677 0.216465
\(73\) 7.14322 0.836051 0.418026 0.908435i \(-0.362722\pi\)
0.418026 + 0.908435i \(0.362722\pi\)
\(74\) −2.91609 −0.338988
\(75\) 0 0
\(76\) 7.74714 0.888658
\(77\) 0 0
\(78\) 0.887599 0.100501
\(79\) 1.90516 0.214347 0.107174 0.994240i \(-0.465820\pi\)
0.107174 + 0.994240i \(0.465820\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.08979 0.120348
\(83\) −0.161902 −0.0177710 −0.00888551 0.999961i \(-0.502828\pi\)
−0.00888551 + 0.999961i \(0.502828\pi\)
\(84\) 8.99974 0.981952
\(85\) 0 0
\(86\) 1.65078 0.178009
\(87\) 7.56018 0.810536
\(88\) 0 0
\(89\) −1.19539 −0.126711 −0.0633554 0.997991i \(-0.520180\pi\)
−0.0633554 + 0.997991i \(0.520180\pi\)
\(90\) 0 0
\(91\) 9.28673 0.973514
\(92\) −7.71525 −0.804371
\(93\) 2.20501 0.228649
\(94\) 0.740167 0.0763424
\(95\) 0 0
\(96\) −4.95592 −0.505812
\(97\) 14.2823 1.45015 0.725073 0.688672i \(-0.241806\pi\)
0.725073 + 0.688672i \(0.241806\pi\)
\(98\) −9.32716 −0.942185
\(99\) 0 0
\(100\) 0 0
\(101\) −4.29089 −0.426960 −0.213480 0.976947i \(-0.568480\pi\)
−0.213480 + 0.976947i \(0.568480\pi\)
\(102\) 0.312494 0.0309416
\(103\) 9.49802 0.935868 0.467934 0.883763i \(-0.344998\pi\)
0.467934 + 0.883763i \(0.344998\pi\)
\(104\) −3.33876 −0.327393
\(105\) 0 0
\(106\) 4.48109 0.435242
\(107\) 8.27470 0.799945 0.399973 0.916527i \(-0.369020\pi\)
0.399973 + 0.916527i \(0.369020\pi\)
\(108\) −1.76156 −0.169507
\(109\) −12.4980 −1.19709 −0.598546 0.801089i \(-0.704255\pi\)
−0.598546 + 0.801089i \(0.704255\pi\)
\(110\) 0 0
\(111\) 5.97194 0.566831
\(112\) −13.4173 −1.26782
\(113\) −16.2096 −1.52487 −0.762435 0.647065i \(-0.775996\pi\)
−0.762435 + 0.647065i \(0.775996\pi\)
\(114\) 2.14748 0.201130
\(115\) 0 0
\(116\) −13.3177 −1.23652
\(117\) −1.81774 −0.168050
\(118\) −2.62302 −0.241468
\(119\) 3.26955 0.299719
\(120\) 0 0
\(121\) 0 0
\(122\) −3.54880 −0.321293
\(123\) −2.23182 −0.201236
\(124\) −3.88427 −0.348817
\(125\) 0 0
\(126\) 2.49469 0.222245
\(127\) 9.88086 0.876785 0.438392 0.898784i \(-0.355548\pi\)
0.438392 + 0.898784i \(0.355548\pi\)
\(128\) 11.2950 0.998342
\(129\) −3.38068 −0.297653
\(130\) 0 0
\(131\) −12.7829 −1.11685 −0.558425 0.829555i \(-0.688594\pi\)
−0.558425 + 0.829555i \(0.688594\pi\)
\(132\) 0 0
\(133\) 22.4685 1.94827
\(134\) 3.54309 0.306076
\(135\) 0 0
\(136\) −1.17547 −0.100796
\(137\) −19.0589 −1.62831 −0.814157 0.580645i \(-0.802800\pi\)
−0.814157 + 0.580645i \(0.802800\pi\)
\(138\) −2.13864 −0.182053
\(139\) −11.7146 −0.993622 −0.496811 0.867859i \(-0.665496\pi\)
−0.496811 + 0.867859i \(0.665496\pi\)
\(140\) 0 0
\(141\) −1.51581 −0.127654
\(142\) −2.68986 −0.225728
\(143\) 0 0
\(144\) 2.62624 0.218853
\(145\) 0 0
\(146\) −3.48803 −0.288671
\(147\) 19.1013 1.57545
\(148\) −10.5199 −0.864735
\(149\) 7.45724 0.610921 0.305461 0.952205i \(-0.401190\pi\)
0.305461 + 0.952205i \(0.401190\pi\)
\(150\) 0 0
\(151\) −4.95905 −0.403562 −0.201781 0.979431i \(-0.564673\pi\)
−0.201781 + 0.979431i \(0.564673\pi\)
\(152\) −8.07787 −0.655202
\(153\) −0.639966 −0.0517382
\(154\) 0 0
\(155\) 0 0
\(156\) 3.20206 0.256370
\(157\) −20.5916 −1.64339 −0.821693 0.569930i \(-0.806970\pi\)
−0.821693 + 0.569930i \(0.806970\pi\)
\(158\) −0.930287 −0.0740096
\(159\) −9.17695 −0.727779
\(160\) 0 0
\(161\) −22.3760 −1.76348
\(162\) −0.488299 −0.0383644
\(163\) 7.22774 0.566120 0.283060 0.959102i \(-0.408650\pi\)
0.283060 + 0.959102i \(0.408650\pi\)
\(164\) 3.93149 0.306998
\(165\) 0 0
\(166\) 0.0790564 0.00613597
\(167\) 18.3130 1.41710 0.708550 0.705660i \(-0.249349\pi\)
0.708550 + 0.705660i \(0.249349\pi\)
\(168\) −9.38395 −0.723987
\(169\) −9.69583 −0.745833
\(170\) 0 0
\(171\) −4.39788 −0.336314
\(172\) 5.95529 0.454087
\(173\) −12.0157 −0.913536 −0.456768 0.889586i \(-0.650993\pi\)
−0.456768 + 0.889586i \(0.650993\pi\)
\(174\) −3.69163 −0.279862
\(175\) 0 0
\(176\) 0 0
\(177\) 5.37174 0.403765
\(178\) 0.583706 0.0437506
\(179\) −6.71004 −0.501532 −0.250766 0.968048i \(-0.580682\pi\)
−0.250766 + 0.968048i \(0.580682\pi\)
\(180\) 0 0
\(181\) −11.3348 −0.842508 −0.421254 0.906943i \(-0.638410\pi\)
−0.421254 + 0.906943i \(0.638410\pi\)
\(182\) −4.53470 −0.336134
\(183\) 7.26768 0.537242
\(184\) 8.04463 0.593058
\(185\) 0 0
\(186\) −1.07670 −0.0789478
\(187\) 0 0
\(188\) 2.67019 0.194744
\(189\) −5.10895 −0.371621
\(190\) 0 0
\(191\) 11.0864 0.802186 0.401093 0.916037i \(-0.368630\pi\)
0.401093 + 0.916037i \(0.368630\pi\)
\(192\) −2.83250 −0.204418
\(193\) −1.19205 −0.0858056 −0.0429028 0.999079i \(-0.513661\pi\)
−0.0429028 + 0.999079i \(0.513661\pi\)
\(194\) −6.97402 −0.500705
\(195\) 0 0
\(196\) −33.6482 −2.40344
\(197\) −18.3551 −1.30775 −0.653873 0.756604i \(-0.726857\pi\)
−0.653873 + 0.756604i \(0.726857\pi\)
\(198\) 0 0
\(199\) 22.1761 1.57202 0.786011 0.618213i \(-0.212143\pi\)
0.786011 + 0.618213i \(0.212143\pi\)
\(200\) 0 0
\(201\) −7.25598 −0.511798
\(202\) 2.09524 0.147420
\(203\) −38.6246 −2.71091
\(204\) 1.12734 0.0789296
\(205\) 0 0
\(206\) −4.63787 −0.323136
\(207\) 4.37977 0.304415
\(208\) −4.77381 −0.331004
\(209\) 0 0
\(210\) 0 0
\(211\) 8.30729 0.571897 0.285949 0.958245i \(-0.407691\pi\)
0.285949 + 0.958245i \(0.407691\pi\)
\(212\) 16.1658 1.11027
\(213\) 5.50864 0.377446
\(214\) −4.04052 −0.276205
\(215\) 0 0
\(216\) 1.83677 0.124976
\(217\) −11.2653 −0.764737
\(218\) 6.10276 0.413331
\(219\) 7.14322 0.482694
\(220\) 0 0
\(221\) 1.16329 0.0782514
\(222\) −2.91609 −0.195715
\(223\) 16.5497 1.10825 0.554124 0.832434i \(-0.313053\pi\)
0.554124 + 0.832434i \(0.313053\pi\)
\(224\) 25.3195 1.69173
\(225\) 0 0
\(226\) 7.91513 0.526506
\(227\) −8.48300 −0.563036 −0.281518 0.959556i \(-0.590838\pi\)
−0.281518 + 0.959556i \(0.590838\pi\)
\(228\) 7.74714 0.513067
\(229\) −3.39892 −0.224607 −0.112303 0.993674i \(-0.535823\pi\)
−0.112303 + 0.993674i \(0.535823\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 13.8863 0.911680
\(233\) −15.7123 −1.02935 −0.514675 0.857386i \(-0.672087\pi\)
−0.514675 + 0.857386i \(0.672087\pi\)
\(234\) 0.887599 0.0580242
\(235\) 0 0
\(236\) −9.46267 −0.615967
\(237\) 1.90516 0.123753
\(238\) −1.59652 −0.103487
\(239\) 30.6972 1.98564 0.992819 0.119624i \(-0.0381689\pi\)
0.992819 + 0.119624i \(0.0381689\pi\)
\(240\) 0 0
\(241\) 5.59097 0.360146 0.180073 0.983653i \(-0.442367\pi\)
0.180073 + 0.983653i \(0.442367\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −12.8025 −0.819595
\(245\) 0 0
\(246\) 1.08979 0.0694827
\(247\) 7.99419 0.508658
\(248\) 4.05009 0.257181
\(249\) −0.161902 −0.0102601
\(250\) 0 0
\(251\) 18.7237 1.18183 0.590915 0.806734i \(-0.298767\pi\)
0.590915 + 0.806734i \(0.298767\pi\)
\(252\) 8.99974 0.566930
\(253\) 0 0
\(254\) −4.82481 −0.302736
\(255\) 0 0
\(256\) 0.149694 0.00935585
\(257\) −4.98215 −0.310778 −0.155389 0.987853i \(-0.549663\pi\)
−0.155389 + 0.987853i \(0.549663\pi\)
\(258\) 1.65078 0.102773
\(259\) −30.5103 −1.89582
\(260\) 0 0
\(261\) 7.56018 0.467963
\(262\) 6.24190 0.385626
\(263\) −20.0788 −1.23811 −0.619055 0.785347i \(-0.712484\pi\)
−0.619055 + 0.785347i \(0.712484\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −10.9713 −0.672696
\(267\) −1.19539 −0.0731565
\(268\) 12.7819 0.780778
\(269\) −9.53027 −0.581071 −0.290535 0.956864i \(-0.593833\pi\)
−0.290535 + 0.956864i \(0.593833\pi\)
\(270\) 0 0
\(271\) −1.18095 −0.0717375 −0.0358688 0.999357i \(-0.511420\pi\)
−0.0358688 + 0.999357i \(0.511420\pi\)
\(272\) −1.68070 −0.101908
\(273\) 9.28673 0.562058
\(274\) 9.30645 0.562223
\(275\) 0 0
\(276\) −7.71525 −0.464404
\(277\) −8.22177 −0.493998 −0.246999 0.969016i \(-0.579444\pi\)
−0.246999 + 0.969016i \(0.579444\pi\)
\(278\) 5.72024 0.343077
\(279\) 2.20501 0.132011
\(280\) 0 0
\(281\) 10.4453 0.623116 0.311558 0.950227i \(-0.399149\pi\)
0.311558 + 0.950227i \(0.399149\pi\)
\(282\) 0.740167 0.0440763
\(283\) −24.8463 −1.47696 −0.738480 0.674275i \(-0.764456\pi\)
−0.738480 + 0.674275i \(0.764456\pi\)
\(284\) −9.70383 −0.575816
\(285\) 0 0
\(286\) 0 0
\(287\) 11.4022 0.673052
\(288\) −4.95592 −0.292031
\(289\) −16.5904 −0.975908
\(290\) 0 0
\(291\) 14.2823 0.837242
\(292\) −12.5832 −0.736379
\(293\) −24.8264 −1.45037 −0.725186 0.688553i \(-0.758246\pi\)
−0.725186 + 0.688553i \(0.758246\pi\)
\(294\) −9.32716 −0.543971
\(295\) 0 0
\(296\) 10.9691 0.637563
\(297\) 0 0
\(298\) −3.64136 −0.210939
\(299\) −7.96128 −0.460413
\(300\) 0 0
\(301\) 17.2717 0.995526
\(302\) 2.42150 0.139342
\(303\) −4.29089 −0.246505
\(304\) −11.5499 −0.662430
\(305\) 0 0
\(306\) 0.312494 0.0178641
\(307\) −12.4687 −0.711624 −0.355812 0.934557i \(-0.615796\pi\)
−0.355812 + 0.934557i \(0.615796\pi\)
\(308\) 0 0
\(309\) 9.49802 0.540324
\(310\) 0 0
\(311\) −3.15914 −0.179138 −0.0895692 0.995981i \(-0.528549\pi\)
−0.0895692 + 0.995981i \(0.528549\pi\)
\(312\) −3.33876 −0.189020
\(313\) −14.2948 −0.807988 −0.403994 0.914762i \(-0.632378\pi\)
−0.403994 + 0.914762i \(0.632378\pi\)
\(314\) 10.0548 0.567427
\(315\) 0 0
\(316\) −3.35606 −0.188793
\(317\) −26.8771 −1.50957 −0.754784 0.655974i \(-0.772258\pi\)
−0.754784 + 0.655974i \(0.772258\pi\)
\(318\) 4.48109 0.251287
\(319\) 0 0
\(320\) 0 0
\(321\) 8.27470 0.461849
\(322\) 10.9262 0.608893
\(323\) 2.81449 0.156602
\(324\) −1.76156 −0.0978647
\(325\) 0 0
\(326\) −3.52930 −0.195470
\(327\) −12.4980 −0.691141
\(328\) −4.09933 −0.226347
\(329\) 7.74418 0.426950
\(330\) 0 0
\(331\) 19.7763 1.08700 0.543502 0.839408i \(-0.317098\pi\)
0.543502 + 0.839408i \(0.317098\pi\)
\(332\) 0.285200 0.0156524
\(333\) 5.97194 0.327260
\(334\) −8.94221 −0.489296
\(335\) 0 0
\(336\) −13.4173 −0.731974
\(337\) 14.2580 0.776685 0.388343 0.921515i \(-0.373048\pi\)
0.388343 + 0.921515i \(0.373048\pi\)
\(338\) 4.73446 0.257521
\(339\) −16.2096 −0.880384
\(340\) 0 0
\(341\) 0 0
\(342\) 2.14748 0.116122
\(343\) −61.8251 −3.33824
\(344\) −6.20953 −0.334795
\(345\) 0 0
\(346\) 5.86725 0.315425
\(347\) 15.7954 0.847940 0.423970 0.905676i \(-0.360636\pi\)
0.423970 + 0.905676i \(0.360636\pi\)
\(348\) −13.3177 −0.713906
\(349\) 1.71442 0.0917705 0.0458853 0.998947i \(-0.485389\pi\)
0.0458853 + 0.998947i \(0.485389\pi\)
\(350\) 0 0
\(351\) −1.81774 −0.0970237
\(352\) 0 0
\(353\) 13.9610 0.743072 0.371536 0.928419i \(-0.378831\pi\)
0.371536 + 0.928419i \(0.378831\pi\)
\(354\) −2.62302 −0.139412
\(355\) 0 0
\(356\) 2.10575 0.111605
\(357\) 3.26955 0.173043
\(358\) 3.27650 0.173169
\(359\) −34.3062 −1.81061 −0.905306 0.424759i \(-0.860359\pi\)
−0.905306 + 0.424759i \(0.860359\pi\)
\(360\) 0 0
\(361\) 0.341312 0.0179638
\(362\) 5.53476 0.290901
\(363\) 0 0
\(364\) −16.3592 −0.857453
\(365\) 0 0
\(366\) −3.54880 −0.185499
\(367\) 3.61245 0.188569 0.0942843 0.995545i \(-0.469944\pi\)
0.0942843 + 0.995545i \(0.469944\pi\)
\(368\) 11.5023 0.599600
\(369\) −2.23182 −0.116184
\(370\) 0 0
\(371\) 46.8845 2.43412
\(372\) −3.88427 −0.201390
\(373\) −18.7064 −0.968579 −0.484290 0.874908i \(-0.660922\pi\)
−0.484290 + 0.874908i \(0.660922\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −2.78418 −0.143583
\(377\) −13.7424 −0.707771
\(378\) 2.49469 0.128313
\(379\) −9.37468 −0.481545 −0.240772 0.970582i \(-0.577401\pi\)
−0.240772 + 0.970582i \(0.577401\pi\)
\(380\) 0 0
\(381\) 9.88086 0.506212
\(382\) −5.41349 −0.276978
\(383\) −19.2155 −0.981865 −0.490932 0.871198i \(-0.663344\pi\)
−0.490932 + 0.871198i \(0.663344\pi\)
\(384\) 11.2950 0.576393
\(385\) 0 0
\(386\) 0.582076 0.0296269
\(387\) −3.38068 −0.171850
\(388\) −25.1591 −1.27726
\(389\) −9.29700 −0.471376 −0.235688 0.971829i \(-0.575734\pi\)
−0.235688 + 0.971829i \(0.575734\pi\)
\(390\) 0 0
\(391\) −2.80290 −0.141749
\(392\) 35.0847 1.77205
\(393\) −12.7829 −0.644814
\(394\) 8.96277 0.451538
\(395\) 0 0
\(396\) 0 0
\(397\) 13.8693 0.696078 0.348039 0.937480i \(-0.386848\pi\)
0.348039 + 0.937480i \(0.386848\pi\)
\(398\) −10.8286 −0.542787
\(399\) 22.4685 1.12483
\(400\) 0 0
\(401\) 10.6336 0.531015 0.265507 0.964109i \(-0.414461\pi\)
0.265507 + 0.964109i \(0.414461\pi\)
\(402\) 3.54309 0.176713
\(403\) −4.00813 −0.199659
\(404\) 7.55868 0.376058
\(405\) 0 0
\(406\) 18.8603 0.936022
\(407\) 0 0
\(408\) −1.17547 −0.0581943
\(409\) 3.71320 0.183606 0.0918030 0.995777i \(-0.470737\pi\)
0.0918030 + 0.995777i \(0.470737\pi\)
\(410\) 0 0
\(411\) −19.0589 −0.940107
\(412\) −16.7314 −0.824296
\(413\) −27.4439 −1.35043
\(414\) −2.13864 −0.105108
\(415\) 0 0
\(416\) 9.00857 0.441682
\(417\) −11.7146 −0.573668
\(418\) 0 0
\(419\) −15.8996 −0.776745 −0.388372 0.921503i \(-0.626963\pi\)
−0.388372 + 0.921503i \(0.626963\pi\)
\(420\) 0 0
\(421\) 4.36664 0.212817 0.106408 0.994323i \(-0.466065\pi\)
0.106408 + 0.994323i \(0.466065\pi\)
\(422\) −4.05644 −0.197464
\(423\) −1.51581 −0.0737011
\(424\) −16.8559 −0.818596
\(425\) 0 0
\(426\) −2.68986 −0.130324
\(427\) −37.1302 −1.79686
\(428\) −14.5764 −0.704577
\(429\) 0 0
\(430\) 0 0
\(431\) 20.1005 0.968209 0.484105 0.875010i \(-0.339145\pi\)
0.484105 + 0.875010i \(0.339145\pi\)
\(432\) 2.62624 0.126355
\(433\) −30.7232 −1.47646 −0.738230 0.674549i \(-0.764338\pi\)
−0.738230 + 0.674549i \(0.764338\pi\)
\(434\) 5.50082 0.264048
\(435\) 0 0
\(436\) 22.0160 1.05438
\(437\) −19.2617 −0.921412
\(438\) −3.48803 −0.166664
\(439\) −20.6104 −0.983679 −0.491840 0.870686i \(-0.663675\pi\)
−0.491840 + 0.870686i \(0.663675\pi\)
\(440\) 0 0
\(441\) 19.1013 0.909587
\(442\) −0.568033 −0.0270186
\(443\) 13.7997 0.655644 0.327822 0.944740i \(-0.393685\pi\)
0.327822 + 0.944740i \(0.393685\pi\)
\(444\) −10.5199 −0.499255
\(445\) 0 0
\(446\) −8.08118 −0.382655
\(447\) 7.45724 0.352716
\(448\) 14.4711 0.683696
\(449\) 27.2508 1.28605 0.643023 0.765847i \(-0.277680\pi\)
0.643023 + 0.765847i \(0.277680\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 28.5542 1.34308
\(453\) −4.95905 −0.232997
\(454\) 4.14224 0.194405
\(455\) 0 0
\(456\) −8.07787 −0.378281
\(457\) 4.62255 0.216234 0.108117 0.994138i \(-0.465518\pi\)
0.108117 + 0.994138i \(0.465518\pi\)
\(458\) 1.65969 0.0775521
\(459\) −0.639966 −0.0298710
\(460\) 0 0
\(461\) −25.3940 −1.18272 −0.591359 0.806409i \(-0.701408\pi\)
−0.591359 + 0.806409i \(0.701408\pi\)
\(462\) 0 0
\(463\) −12.8949 −0.599275 −0.299638 0.954053i \(-0.596866\pi\)
−0.299638 + 0.954053i \(0.596866\pi\)
\(464\) 19.8548 0.921737
\(465\) 0 0
\(466\) 7.67231 0.355413
\(467\) −3.81025 −0.176317 −0.0881587 0.996106i \(-0.528098\pi\)
−0.0881587 + 0.996106i \(0.528098\pi\)
\(468\) 3.20206 0.148015
\(469\) 37.0704 1.71175
\(470\) 0 0
\(471\) −20.5916 −0.948809
\(472\) 9.86664 0.454149
\(473\) 0 0
\(474\) −0.930287 −0.0427295
\(475\) 0 0
\(476\) −5.75952 −0.263987
\(477\) −9.17695 −0.420184
\(478\) −14.9894 −0.685600
\(479\) 12.2918 0.561625 0.280812 0.959763i \(-0.409396\pi\)
0.280812 + 0.959763i \(0.409396\pi\)
\(480\) 0 0
\(481\) −10.8554 −0.494964
\(482\) −2.73006 −0.124351
\(483\) −22.3760 −1.01814
\(484\) 0 0
\(485\) 0 0
\(486\) −0.488299 −0.0221497
\(487\) −13.4389 −0.608975 −0.304488 0.952516i \(-0.598485\pi\)
−0.304488 + 0.952516i \(0.598485\pi\)
\(488\) 13.3490 0.604283
\(489\) 7.22774 0.326850
\(490\) 0 0
\(491\) −6.08182 −0.274469 −0.137234 0.990539i \(-0.543821\pi\)
−0.137234 + 0.990539i \(0.543821\pi\)
\(492\) 3.93149 0.177245
\(493\) −4.83826 −0.217904
\(494\) −3.90355 −0.175629
\(495\) 0 0
\(496\) 5.79088 0.260018
\(497\) −28.1434 −1.26240
\(498\) 0.0790564 0.00354260
\(499\) 18.7942 0.841344 0.420672 0.907213i \(-0.361794\pi\)
0.420672 + 0.907213i \(0.361794\pi\)
\(500\) 0 0
\(501\) 18.3130 0.818164
\(502\) −9.14276 −0.408061
\(503\) −6.06605 −0.270472 −0.135236 0.990813i \(-0.543179\pi\)
−0.135236 + 0.990813i \(0.543179\pi\)
\(504\) −9.38395 −0.417994
\(505\) 0 0
\(506\) 0 0
\(507\) −9.69583 −0.430607
\(508\) −17.4058 −0.772256
\(509\) −19.7444 −0.875154 −0.437577 0.899181i \(-0.644163\pi\)
−0.437577 + 0.899181i \(0.644163\pi\)
\(510\) 0 0
\(511\) −36.4943 −1.61441
\(512\) −22.6630 −1.00157
\(513\) −4.39788 −0.194171
\(514\) 2.43278 0.107305
\(515\) 0 0
\(516\) 5.95529 0.262167
\(517\) 0 0
\(518\) 14.8981 0.654587
\(519\) −12.0157 −0.527430
\(520\) 0 0
\(521\) −42.1434 −1.84634 −0.923168 0.384396i \(-0.874410\pi\)
−0.923168 + 0.384396i \(0.874410\pi\)
\(522\) −3.69163 −0.161578
\(523\) 6.37047 0.278561 0.139281 0.990253i \(-0.455521\pi\)
0.139281 + 0.990253i \(0.455521\pi\)
\(524\) 22.5180 0.983702
\(525\) 0 0
\(526\) 9.80445 0.427494
\(527\) −1.41113 −0.0614698
\(528\) 0 0
\(529\) −3.81758 −0.165982
\(530\) 0 0
\(531\) 5.37174 0.233114
\(532\) −39.5797 −1.71600
\(533\) 4.05686 0.175722
\(534\) 0.583706 0.0252594
\(535\) 0 0
\(536\) −13.3276 −0.575663
\(537\) −6.71004 −0.289560
\(538\) 4.65362 0.200632
\(539\) 0 0
\(540\) 0 0
\(541\) −12.3075 −0.529140 −0.264570 0.964367i \(-0.585230\pi\)
−0.264570 + 0.964367i \(0.585230\pi\)
\(542\) 0.576656 0.0247695
\(543\) −11.3348 −0.486422
\(544\) 3.17162 0.135982
\(545\) 0 0
\(546\) −4.53470 −0.194067
\(547\) −5.48939 −0.234709 −0.117355 0.993090i \(-0.537441\pi\)
−0.117355 + 0.993090i \(0.537441\pi\)
\(548\) 33.5735 1.43419
\(549\) 7.26768 0.310177
\(550\) 0 0
\(551\) −33.2487 −1.41644
\(552\) 8.04463 0.342402
\(553\) −9.73335 −0.413904
\(554\) 4.01468 0.170567
\(555\) 0 0
\(556\) 20.6361 0.875165
\(557\) −25.4636 −1.07893 −0.539464 0.842009i \(-0.681373\pi\)
−0.539464 + 0.842009i \(0.681373\pi\)
\(558\) −1.07670 −0.0455805
\(559\) 6.14520 0.259914
\(560\) 0 0
\(561\) 0 0
\(562\) −5.10044 −0.215149
\(563\) −38.3527 −1.61638 −0.808188 0.588925i \(-0.799551\pi\)
−0.808188 + 0.588925i \(0.799551\pi\)
\(564\) 2.67019 0.112435
\(565\) 0 0
\(566\) 12.1324 0.509964
\(567\) −5.10895 −0.214556
\(568\) 10.1181 0.424546
\(569\) 9.15245 0.383691 0.191845 0.981425i \(-0.438553\pi\)
0.191845 + 0.981425i \(0.438553\pi\)
\(570\) 0 0
\(571\) −18.3508 −0.767957 −0.383978 0.923342i \(-0.625446\pi\)
−0.383978 + 0.923342i \(0.625446\pi\)
\(572\) 0 0
\(573\) 11.0864 0.463142
\(574\) −5.56769 −0.232391
\(575\) 0 0
\(576\) −2.83250 −0.118021
\(577\) 19.9317 0.829770 0.414885 0.909874i \(-0.363822\pi\)
0.414885 + 0.909874i \(0.363822\pi\)
\(578\) 8.10109 0.336961
\(579\) −1.19205 −0.0495399
\(580\) 0 0
\(581\) 0.827147 0.0343158
\(582\) −6.97402 −0.289082
\(583\) 0 0
\(584\) 13.1204 0.542928
\(585\) 0 0
\(586\) 12.1227 0.500784
\(587\) 39.5465 1.63226 0.816129 0.577870i \(-0.196116\pi\)
0.816129 + 0.577870i \(0.196116\pi\)
\(588\) −33.6482 −1.38763
\(589\) −9.69736 −0.399573
\(590\) 0 0
\(591\) −18.3551 −0.755028
\(592\) 15.6837 0.644597
\(593\) 12.3314 0.506391 0.253195 0.967415i \(-0.418518\pi\)
0.253195 + 0.967415i \(0.418518\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −13.1364 −0.538088
\(597\) 22.1761 0.907607
\(598\) 3.88748 0.158971
\(599\) 34.7495 1.41983 0.709913 0.704289i \(-0.248734\pi\)
0.709913 + 0.704289i \(0.248734\pi\)
\(600\) 0 0
\(601\) 38.6097 1.57492 0.787462 0.616364i \(-0.211395\pi\)
0.787462 + 0.616364i \(0.211395\pi\)
\(602\) −8.43376 −0.343735
\(603\) −7.25598 −0.295486
\(604\) 8.73569 0.355450
\(605\) 0 0
\(606\) 2.09524 0.0851132
\(607\) −6.26341 −0.254224 −0.127112 0.991888i \(-0.540571\pi\)
−0.127112 + 0.991888i \(0.540571\pi\)
\(608\) 21.7955 0.883926
\(609\) −38.6246 −1.56515
\(610\) 0 0
\(611\) 2.75534 0.111469
\(612\) 1.12734 0.0455700
\(613\) −18.9911 −0.767042 −0.383521 0.923532i \(-0.625289\pi\)
−0.383521 + 0.923532i \(0.625289\pi\)
\(614\) 6.08844 0.245709
\(615\) 0 0
\(616\) 0 0
\(617\) 7.58574 0.305390 0.152695 0.988273i \(-0.451205\pi\)
0.152695 + 0.988273i \(0.451205\pi\)
\(618\) −4.63787 −0.186563
\(619\) 27.9795 1.12459 0.562296 0.826936i \(-0.309918\pi\)
0.562296 + 0.826936i \(0.309918\pi\)
\(620\) 0 0
\(621\) 4.37977 0.175754
\(622\) 1.54260 0.0618528
\(623\) 6.10717 0.244679
\(624\) −4.77381 −0.191105
\(625\) 0 0
\(626\) 6.98012 0.278982
\(627\) 0 0
\(628\) 36.2734 1.44746
\(629\) −3.82183 −0.152387
\(630\) 0 0
\(631\) −25.4321 −1.01244 −0.506218 0.862406i \(-0.668957\pi\)
−0.506218 + 0.862406i \(0.668957\pi\)
\(632\) 3.49933 0.139196
\(633\) 8.30729 0.330185
\(634\) 13.1240 0.521223
\(635\) 0 0
\(636\) 16.1658 0.641015
\(637\) −34.7212 −1.37570
\(638\) 0 0
\(639\) 5.50864 0.217919
\(640\) 0 0
\(641\) −8.90150 −0.351588 −0.175794 0.984427i \(-0.556249\pi\)
−0.175794 + 0.984427i \(0.556249\pi\)
\(642\) −4.04052 −0.159467
\(643\) −19.4814 −0.768272 −0.384136 0.923276i \(-0.625501\pi\)
−0.384136 + 0.923276i \(0.625501\pi\)
\(644\) 39.4168 1.55324
\(645\) 0 0
\(646\) −1.37431 −0.0540716
\(647\) 32.8093 1.28987 0.644934 0.764238i \(-0.276885\pi\)
0.644934 + 0.764238i \(0.276885\pi\)
\(648\) 1.83677 0.0721550
\(649\) 0 0
\(650\) 0 0
\(651\) −11.2653 −0.441521
\(652\) −12.7321 −0.498629
\(653\) 14.9010 0.583122 0.291561 0.956552i \(-0.405825\pi\)
0.291561 + 0.956552i \(0.405825\pi\)
\(654\) 6.10276 0.238637
\(655\) 0 0
\(656\) −5.86128 −0.228844
\(657\) 7.14322 0.278684
\(658\) −3.78147 −0.147417
\(659\) −25.7059 −1.00136 −0.500680 0.865633i \(-0.666917\pi\)
−0.500680 + 0.865633i \(0.666917\pi\)
\(660\) 0 0
\(661\) −12.4887 −0.485753 −0.242876 0.970057i \(-0.578091\pi\)
−0.242876 + 0.970057i \(0.578091\pi\)
\(662\) −9.65675 −0.375320
\(663\) 1.16329 0.0451784
\(664\) −0.297376 −0.0115404
\(665\) 0 0
\(666\) −2.91609 −0.112996
\(667\) 33.1119 1.28210
\(668\) −32.2595 −1.24816
\(669\) 16.5497 0.639847
\(670\) 0 0
\(671\) 0 0
\(672\) 25.3195 0.976723
\(673\) 40.3440 1.55515 0.777573 0.628793i \(-0.216451\pi\)
0.777573 + 0.628793i \(0.216451\pi\)
\(674\) −6.96219 −0.268173
\(675\) 0 0
\(676\) 17.0798 0.656916
\(677\) −5.64803 −0.217071 −0.108536 0.994093i \(-0.534616\pi\)
−0.108536 + 0.994093i \(0.534616\pi\)
\(678\) 7.91513 0.303979
\(679\) −72.9674 −2.80023
\(680\) 0 0
\(681\) −8.48300 −0.325069
\(682\) 0 0
\(683\) 26.7349 1.02298 0.511492 0.859288i \(-0.329093\pi\)
0.511492 + 0.859288i \(0.329093\pi\)
\(684\) 7.74714 0.296219
\(685\) 0 0
\(686\) 30.1891 1.15263
\(687\) −3.39892 −0.129677
\(688\) −8.87848 −0.338489
\(689\) 16.6813 0.635506
\(690\) 0 0
\(691\) 39.2696 1.49389 0.746943 0.664888i \(-0.231521\pi\)
0.746943 + 0.664888i \(0.231521\pi\)
\(692\) 21.1664 0.804626
\(693\) 0 0
\(694\) −7.71286 −0.292776
\(695\) 0 0
\(696\) 13.8863 0.526359
\(697\) 1.42829 0.0541002
\(698\) −0.837147 −0.0316865
\(699\) −15.7123 −0.594295
\(700\) 0 0
\(701\) −7.23493 −0.273259 −0.136630 0.990622i \(-0.543627\pi\)
−0.136630 + 0.990622i \(0.543627\pi\)
\(702\) 0.887599 0.0335003
\(703\) −26.2638 −0.990559
\(704\) 0 0
\(705\) 0 0
\(706\) −6.81716 −0.256567
\(707\) 21.9219 0.824459
\(708\) −9.46267 −0.355629
\(709\) −9.96887 −0.374389 −0.187194 0.982323i \(-0.559939\pi\)
−0.187194 + 0.982323i \(0.559939\pi\)
\(710\) 0 0
\(711\) 1.90516 0.0714490
\(712\) −2.19565 −0.0822854
\(713\) 9.65745 0.361674
\(714\) −1.59652 −0.0597482
\(715\) 0 0
\(716\) 11.8202 0.441740
\(717\) 30.6972 1.14641
\(718\) 16.7517 0.625167
\(719\) 33.1006 1.23445 0.617223 0.786788i \(-0.288258\pi\)
0.617223 + 0.786788i \(0.288258\pi\)
\(720\) 0 0
\(721\) −48.5249 −1.80716
\(722\) −0.166662 −0.00620253
\(723\) 5.59097 0.207930
\(724\) 19.9669 0.742066
\(725\) 0 0
\(726\) 0 0
\(727\) −18.7123 −0.694000 −0.347000 0.937865i \(-0.612800\pi\)
−0.347000 + 0.937865i \(0.612800\pi\)
\(728\) 17.0576 0.632195
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.16352 0.0800207
\(732\) −12.8025 −0.473193
\(733\) 10.9499 0.404442 0.202221 0.979340i \(-0.435184\pi\)
0.202221 + 0.979340i \(0.435184\pi\)
\(734\) −1.76396 −0.0651089
\(735\) 0 0
\(736\) −21.7058 −0.800087
\(737\) 0 0
\(738\) 1.08979 0.0401158
\(739\) 38.0842 1.40095 0.700474 0.713677i \(-0.252972\pi\)
0.700474 + 0.713677i \(0.252972\pi\)
\(740\) 0 0
\(741\) 7.99419 0.293674
\(742\) −22.8937 −0.840453
\(743\) 16.5258 0.606275 0.303137 0.952947i \(-0.401966\pi\)
0.303137 + 0.952947i \(0.401966\pi\)
\(744\) 4.05009 0.148484
\(745\) 0 0
\(746\) 9.13430 0.334430
\(747\) −0.161902 −0.00592367
\(748\) 0 0
\(749\) −42.2750 −1.54469
\(750\) 0 0
\(751\) −17.9317 −0.654336 −0.327168 0.944966i \(-0.606094\pi\)
−0.327168 + 0.944966i \(0.606094\pi\)
\(752\) −3.98087 −0.145167
\(753\) 18.7237 0.682329
\(754\) 6.71041 0.244379
\(755\) 0 0
\(756\) 8.99974 0.327317
\(757\) −14.0679 −0.511307 −0.255653 0.966768i \(-0.582291\pi\)
−0.255653 + 0.966768i \(0.582291\pi\)
\(758\) 4.57764 0.166267
\(759\) 0 0
\(760\) 0 0
\(761\) 10.1182 0.366783 0.183392 0.983040i \(-0.441292\pi\)
0.183392 + 0.983040i \(0.441292\pi\)
\(762\) −4.82481 −0.174785
\(763\) 63.8516 2.31158
\(764\) −19.5295 −0.706551
\(765\) 0 0
\(766\) 9.38289 0.339018
\(767\) −9.76442 −0.352573
\(768\) 0.149694 0.00540160
\(769\) −16.0423 −0.578499 −0.289250 0.957254i \(-0.593406\pi\)
−0.289250 + 0.957254i \(0.593406\pi\)
\(770\) 0 0
\(771\) −4.98215 −0.179428
\(772\) 2.09987 0.0755760
\(773\) −40.4357 −1.45437 −0.727185 0.686442i \(-0.759172\pi\)
−0.727185 + 0.686442i \(0.759172\pi\)
\(774\) 1.65078 0.0593362
\(775\) 0 0
\(776\) 26.2332 0.941718
\(777\) −30.5103 −1.09455
\(778\) 4.53971 0.162757
\(779\) 9.81525 0.351668
\(780\) 0 0
\(781\) 0 0
\(782\) 1.36866 0.0489430
\(783\) 7.56018 0.270179
\(784\) 50.1646 1.79159
\(785\) 0 0
\(786\) 6.24190 0.222641
\(787\) −16.4314 −0.585716 −0.292858 0.956156i \(-0.594606\pi\)
−0.292858 + 0.956156i \(0.594606\pi\)
\(788\) 32.3337 1.15184
\(789\) −20.0788 −0.714824
\(790\) 0 0
\(791\) 82.8140 2.94453
\(792\) 0 0
\(793\) −13.2107 −0.469127
\(794\) −6.77235 −0.240342
\(795\) 0 0
\(796\) −39.0646 −1.38461
\(797\) −11.8928 −0.421266 −0.210633 0.977565i \(-0.567552\pi\)
−0.210633 + 0.977565i \(0.567552\pi\)
\(798\) −10.9713 −0.388381
\(799\) 0.970064 0.0343184
\(800\) 0 0
\(801\) −1.19539 −0.0422370
\(802\) −5.19236 −0.183349
\(803\) 0 0
\(804\) 12.7819 0.450782
\(805\) 0 0
\(806\) 1.95717 0.0689382
\(807\) −9.53027 −0.335481
\(808\) −7.88137 −0.277266
\(809\) 39.7032 1.39589 0.697944 0.716152i \(-0.254098\pi\)
0.697944 + 0.716152i \(0.254098\pi\)
\(810\) 0 0
\(811\) 29.1509 1.02363 0.511814 0.859096i \(-0.328974\pi\)
0.511814 + 0.859096i \(0.328974\pi\)
\(812\) 68.0396 2.38772
\(813\) −1.18095 −0.0414177
\(814\) 0 0
\(815\) 0 0
\(816\) −1.68070 −0.0588363
\(817\) 14.8678 0.520159
\(818\) −1.81315 −0.0633954
\(819\) 9.28673 0.324505
\(820\) 0 0
\(821\) 47.6040 1.66139 0.830696 0.556727i \(-0.187943\pi\)
0.830696 + 0.556727i \(0.187943\pi\)
\(822\) 9.30645 0.324600
\(823\) −19.2238 −0.670099 −0.335049 0.942201i \(-0.608753\pi\)
−0.335049 + 0.942201i \(0.608753\pi\)
\(824\) 17.4457 0.607748
\(825\) 0 0
\(826\) 13.4008 0.466275
\(827\) −30.6312 −1.06515 −0.532576 0.846382i \(-0.678776\pi\)
−0.532576 + 0.846382i \(0.678776\pi\)
\(828\) −7.71525 −0.268124
\(829\) −44.8613 −1.55810 −0.779048 0.626964i \(-0.784297\pi\)
−0.779048 + 0.626964i \(0.784297\pi\)
\(830\) 0 0
\(831\) −8.22177 −0.285210
\(832\) 5.14875 0.178501
\(833\) −12.2242 −0.423543
\(834\) 5.72024 0.198076
\(835\) 0 0
\(836\) 0 0
\(837\) 2.20501 0.0762163
\(838\) 7.76374 0.268194
\(839\) 4.34435 0.149984 0.0749919 0.997184i \(-0.476107\pi\)
0.0749919 + 0.997184i \(0.476107\pi\)
\(840\) 0 0
\(841\) 28.1563 0.970908
\(842\) −2.13222 −0.0734813
\(843\) 10.4453 0.359756
\(844\) −14.6338 −0.503717
\(845\) 0 0
\(846\) 0.740167 0.0254475
\(847\) 0 0
\(848\) −24.1008 −0.827626
\(849\) −24.8463 −0.852724
\(850\) 0 0
\(851\) 26.1557 0.896607
\(852\) −9.70383 −0.332448
\(853\) −40.3581 −1.38183 −0.690917 0.722934i \(-0.742793\pi\)
−0.690917 + 0.722934i \(0.742793\pi\)
\(854\) 18.1306 0.620417
\(855\) 0 0
\(856\) 15.1987 0.519481
\(857\) 36.6122 1.25065 0.625324 0.780365i \(-0.284967\pi\)
0.625324 + 0.780365i \(0.284967\pi\)
\(858\) 0 0
\(859\) 28.5710 0.974831 0.487415 0.873170i \(-0.337940\pi\)
0.487415 + 0.873170i \(0.337940\pi\)
\(860\) 0 0
\(861\) 11.4022 0.388587
\(862\) −9.81507 −0.334303
\(863\) −29.3094 −0.997704 −0.498852 0.866687i \(-0.666245\pi\)
−0.498852 + 0.866687i \(0.666245\pi\)
\(864\) −4.95592 −0.168604
\(865\) 0 0
\(866\) 15.0021 0.509791
\(867\) −16.5904 −0.563441
\(868\) 19.8445 0.673567
\(869\) 0 0
\(870\) 0 0
\(871\) 13.1895 0.446908
\(872\) −22.9559 −0.777386
\(873\) 14.2823 0.483382
\(874\) 9.40547 0.318145
\(875\) 0 0
\(876\) −12.5832 −0.425149
\(877\) −3.67475 −0.124087 −0.0620437 0.998073i \(-0.519762\pi\)
−0.0620437 + 0.998073i \(0.519762\pi\)
\(878\) 10.0640 0.339644
\(879\) −24.8264 −0.837373
\(880\) 0 0
\(881\) −11.1723 −0.376403 −0.188202 0.982130i \(-0.560266\pi\)
−0.188202 + 0.982130i \(0.560266\pi\)
\(882\) −9.32716 −0.314062
\(883\) −42.2570 −1.42206 −0.711030 0.703161i \(-0.751771\pi\)
−0.711030 + 0.703161i \(0.751771\pi\)
\(884\) −2.04921 −0.0689224
\(885\) 0 0
\(886\) −6.73838 −0.226380
\(887\) −55.8459 −1.87512 −0.937562 0.347819i \(-0.886922\pi\)
−0.937562 + 0.347819i \(0.886922\pi\)
\(888\) 10.9691 0.368097
\(889\) −50.4808 −1.69307
\(890\) 0 0
\(891\) 0 0
\(892\) −29.1533 −0.976125
\(893\) 6.66633 0.223080
\(894\) −3.64136 −0.121785
\(895\) 0 0
\(896\) −57.7053 −1.92780
\(897\) −7.96128 −0.265819
\(898\) −13.3066 −0.444045
\(899\) 16.6703 0.555985
\(900\) 0 0
\(901\) 5.87293 0.195656
\(902\) 0 0
\(903\) 17.2717 0.574767
\(904\) −29.7733 −0.990244
\(905\) 0 0
\(906\) 2.42150 0.0804490
\(907\) −12.0875 −0.401358 −0.200679 0.979657i \(-0.564315\pi\)
−0.200679 + 0.979657i \(0.564315\pi\)
\(908\) 14.9433 0.495912
\(909\) −4.29089 −0.142320
\(910\) 0 0
\(911\) −40.2306 −1.33290 −0.666450 0.745550i \(-0.732187\pi\)
−0.666450 + 0.745550i \(0.732187\pi\)
\(912\) −11.5499 −0.382454
\(913\) 0 0
\(914\) −2.25719 −0.0746611
\(915\) 0 0
\(916\) 5.98741 0.197830
\(917\) 65.3074 2.15664
\(918\) 0.312494 0.0103139
\(919\) 3.65013 0.120407 0.0602033 0.998186i \(-0.480825\pi\)
0.0602033 + 0.998186i \(0.480825\pi\)
\(920\) 0 0
\(921\) −12.4687 −0.410857
\(922\) 12.3999 0.408368
\(923\) −10.0133 −0.329591
\(924\) 0 0
\(925\) 0 0
\(926\) 6.29655 0.206917
\(927\) 9.49802 0.311956
\(928\) −37.4677 −1.22994
\(929\) 10.0510 0.329762 0.164881 0.986313i \(-0.447276\pi\)
0.164881 + 0.986313i \(0.447276\pi\)
\(930\) 0 0
\(931\) −84.0053 −2.75316
\(932\) 27.6783 0.906632
\(933\) −3.15914 −0.103426
\(934\) 1.86054 0.0608788
\(935\) 0 0
\(936\) −3.33876 −0.109131
\(937\) −28.2337 −0.922355 −0.461177 0.887308i \(-0.652573\pi\)
−0.461177 + 0.887308i \(0.652573\pi\)
\(938\) −18.1014 −0.591033
\(939\) −14.2948 −0.466492
\(940\) 0 0
\(941\) 32.6668 1.06491 0.532453 0.846459i \(-0.321270\pi\)
0.532453 + 0.846459i \(0.321270\pi\)
\(942\) 10.0548 0.327604
\(943\) −9.77485 −0.318313
\(944\) 14.1075 0.459159
\(945\) 0 0
\(946\) 0 0
\(947\) −57.1073 −1.85574 −0.927869 0.372906i \(-0.878361\pi\)
−0.927869 + 0.372906i \(0.878361\pi\)
\(948\) −3.35606 −0.109000
\(949\) −12.9845 −0.421495
\(950\) 0 0
\(951\) −26.8771 −0.871549
\(952\) 6.00540 0.194636
\(953\) −4.51422 −0.146230 −0.0731150 0.997324i \(-0.523294\pi\)
−0.0731150 + 0.997324i \(0.523294\pi\)
\(954\) 4.48109 0.145081
\(955\) 0 0
\(956\) −54.0751 −1.74891
\(957\) 0 0
\(958\) −6.00205 −0.193917
\(959\) 97.3710 3.14427
\(960\) 0 0
\(961\) −26.1379 −0.843159
\(962\) 5.30069 0.170901
\(963\) 8.27470 0.266648
\(964\) −9.84886 −0.317210
\(965\) 0 0
\(966\) 10.9262 0.351544
\(967\) 1.33608 0.0429654 0.0214827 0.999769i \(-0.493161\pi\)
0.0214827 + 0.999769i \(0.493161\pi\)
\(968\) 0 0
\(969\) 2.81449 0.0904144
\(970\) 0 0
\(971\) 2.71644 0.0871747 0.0435873 0.999050i \(-0.486121\pi\)
0.0435873 + 0.999050i \(0.486121\pi\)
\(972\) −1.76156 −0.0565022
\(973\) 59.8494 1.91868
\(974\) 6.56221 0.210267
\(975\) 0 0
\(976\) 19.0867 0.610949
\(977\) −16.6829 −0.533733 −0.266866 0.963734i \(-0.585988\pi\)
−0.266866 + 0.963734i \(0.585988\pi\)
\(978\) −3.52930 −0.112854
\(979\) 0 0
\(980\) 0 0
\(981\) −12.4980 −0.399031
\(982\) 2.96975 0.0947684
\(983\) 26.2517 0.837298 0.418649 0.908148i \(-0.362504\pi\)
0.418649 + 0.908148i \(0.362504\pi\)
\(984\) −4.09933 −0.130682
\(985\) 0 0
\(986\) 2.36251 0.0752378
\(987\) 7.74418 0.246500
\(988\) −14.0823 −0.448017
\(989\) −14.8066 −0.470823
\(990\) 0 0
\(991\) −50.3117 −1.59821 −0.799103 0.601195i \(-0.794692\pi\)
−0.799103 + 0.601195i \(0.794692\pi\)
\(992\) −10.9279 −0.346960
\(993\) 19.7763 0.627583
\(994\) 13.7424 0.435881
\(995\) 0 0
\(996\) 0.285200 0.00903691
\(997\) −45.2449 −1.43292 −0.716460 0.697628i \(-0.754239\pi\)
−0.716460 + 0.697628i \(0.754239\pi\)
\(998\) −9.17719 −0.290499
\(999\) 5.97194 0.188944
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 9075.2.a.dy.1.6 12
5.2 odd 4 1815.2.c.j.364.10 24
5.3 odd 4 1815.2.c.j.364.15 24
5.4 even 2 9075.2.a.dz.1.7 12
11.3 even 5 825.2.n.p.526.3 24
11.4 even 5 825.2.n.p.676.3 24
11.10 odd 2 9075.2.a.ea.1.7 12
55.3 odd 20 165.2.s.a.64.8 yes 48
55.4 even 10 825.2.n.o.676.4 24
55.14 even 10 825.2.n.o.526.4 24
55.32 even 4 1815.2.c.k.364.15 24
55.37 odd 20 165.2.s.a.49.8 yes 48
55.43 even 4 1815.2.c.k.364.10 24
55.47 odd 20 165.2.s.a.64.5 yes 48
55.48 odd 20 165.2.s.a.49.5 48
55.54 odd 2 9075.2.a.dx.1.6 12
165.47 even 20 495.2.ba.c.64.8 48
165.92 even 20 495.2.ba.c.379.5 48
165.113 even 20 495.2.ba.c.64.5 48
165.158 even 20 495.2.ba.c.379.8 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.s.a.49.5 48 55.48 odd 20
165.2.s.a.49.8 yes 48 55.37 odd 20
165.2.s.a.64.5 yes 48 55.47 odd 20
165.2.s.a.64.8 yes 48 55.3 odd 20
495.2.ba.c.64.5 48 165.113 even 20
495.2.ba.c.64.8 48 165.47 even 20
495.2.ba.c.379.5 48 165.92 even 20
495.2.ba.c.379.8 48 165.158 even 20
825.2.n.o.526.4 24 55.14 even 10
825.2.n.o.676.4 24 55.4 even 10
825.2.n.p.526.3 24 11.3 even 5
825.2.n.p.676.3 24 11.4 even 5
1815.2.c.j.364.10 24 5.2 odd 4
1815.2.c.j.364.15 24 5.3 odd 4
1815.2.c.k.364.10 24 55.43 even 4
1815.2.c.k.364.15 24 55.32 even 4
9075.2.a.dx.1.6 12 55.54 odd 2
9075.2.a.dy.1.6 12 1.1 even 1 trivial
9075.2.a.dz.1.7 12 5.4 even 2
9075.2.a.ea.1.7 12 11.10 odd 2