Properties

Label 4527.2.a.k.1.3
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 1.3
Root \(0.208270\) 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)
 
\(f(q)\) \(=\) \(q-1.17266 q^{2} -0.624870 q^{4} -0.178789 q^{5} -0.0809018 q^{7} +3.07808 q^{8} +O(q^{10})\) \(q-1.17266 q^{2} -0.624870 q^{4} -0.178789 q^{5} -0.0809018 q^{7} +3.07808 q^{8} +0.209658 q^{10} +4.30391 q^{11} +1.10019 q^{13} +0.0948703 q^{14} -2.35980 q^{16} +3.06504 q^{17} -2.15853 q^{19} +0.111720 q^{20} -5.04702 q^{22} -9.15620 q^{23} -4.96803 q^{25} -1.29015 q^{26} +0.0505531 q^{28} +6.17849 q^{29} -9.82812 q^{31} -3.38892 q^{32} -3.59425 q^{34} +0.0144643 q^{35} +4.08321 q^{37} +2.53122 q^{38} -0.550326 q^{40} -4.58598 q^{41} +2.03126 q^{43} -2.68938 q^{44} +10.7371 q^{46} +5.12836 q^{47} -6.99345 q^{49} +5.82581 q^{50} -0.687477 q^{52} -2.69017 q^{53} -0.769491 q^{55} -0.249022 q^{56} -7.24527 q^{58} -9.21598 q^{59} -14.0554 q^{61} +11.5250 q^{62} +8.69364 q^{64} -0.196702 q^{65} +5.62801 q^{67} -1.91525 q^{68} -0.0169617 q^{70} +4.88191 q^{71} +10.9754 q^{73} -4.78821 q^{74} +1.34880 q^{76} -0.348194 q^{77} +17.4356 q^{79} +0.421905 q^{80} +5.37780 q^{82} +2.48372 q^{83} -0.547994 q^{85} -2.38198 q^{86} +13.2478 q^{88} -5.57230 q^{89} -0.0890076 q^{91} +5.72143 q^{92} -6.01382 q^{94} +0.385920 q^{95} -13.9731 q^{97} +8.20094 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.17266 −0.829195 −0.414598 0.910005i \(-0.636078\pi\)
−0.414598 + 0.910005i \(0.636078\pi\)
\(3\) 0 0
\(4\) −0.624870 −0.312435
\(5\) −0.178789 −0.0799568 −0.0399784 0.999201i \(-0.512729\pi\)
−0.0399784 + 0.999201i \(0.512729\pi\)
\(6\) 0 0
\(7\) −0.0809018 −0.0305780 −0.0152890 0.999883i \(-0.504867\pi\)
−0.0152890 + 0.999883i \(0.504867\pi\)
\(8\) 3.07808 1.08827
\(9\) 0 0
\(10\) 0.209658 0.0662998
\(11\) 4.30391 1.29768 0.648839 0.760926i \(-0.275255\pi\)
0.648839 + 0.760926i \(0.275255\pi\)
\(12\) 0 0
\(13\) 1.10019 0.305139 0.152569 0.988293i \(-0.451245\pi\)
0.152569 + 0.988293i \(0.451245\pi\)
\(14\) 0.0948703 0.0253551
\(15\) 0 0
\(16\) −2.35980 −0.589950
\(17\) 3.06504 0.743381 0.371690 0.928357i \(-0.378778\pi\)
0.371690 + 0.928357i \(0.378778\pi\)
\(18\) 0 0
\(19\) −2.15853 −0.495200 −0.247600 0.968862i \(-0.579642\pi\)
−0.247600 + 0.968862i \(0.579642\pi\)
\(20\) 0.111720 0.0249813
\(21\) 0 0
\(22\) −5.04702 −1.07603
\(23\) −9.15620 −1.90920 −0.954600 0.297892i \(-0.903716\pi\)
−0.954600 + 0.297892i \(0.903716\pi\)
\(24\) 0 0
\(25\) −4.96803 −0.993607
\(26\) −1.29015 −0.253020
\(27\) 0 0
\(28\) 0.0505531 0.00955364
\(29\) 6.17849 1.14732 0.573658 0.819095i \(-0.305524\pi\)
0.573658 + 0.819095i \(0.305524\pi\)
\(30\) 0 0
\(31\) −9.82812 −1.76518 −0.882591 0.470141i \(-0.844203\pi\)
−0.882591 + 0.470141i \(0.844203\pi\)
\(32\) −3.38892 −0.599082
\(33\) 0 0
\(34\) −3.59425 −0.616408
\(35\) 0.0144643 0.00244492
\(36\) 0 0
\(37\) 4.08321 0.671275 0.335637 0.941991i \(-0.391048\pi\)
0.335637 + 0.941991i \(0.391048\pi\)
\(38\) 2.53122 0.410618
\(39\) 0 0
\(40\) −0.550326 −0.0870141
\(41\) −4.58598 −0.716210 −0.358105 0.933681i \(-0.616577\pi\)
−0.358105 + 0.933681i \(0.616577\pi\)
\(42\) 0 0
\(43\) 2.03126 0.309764 0.154882 0.987933i \(-0.450500\pi\)
0.154882 + 0.987933i \(0.450500\pi\)
\(44\) −2.68938 −0.405440
\(45\) 0 0
\(46\) 10.7371 1.58310
\(47\) 5.12836 0.748049 0.374024 0.927419i \(-0.377978\pi\)
0.374024 + 0.927419i \(0.377978\pi\)
\(48\) 0 0
\(49\) −6.99345 −0.999065
\(50\) 5.82581 0.823894
\(51\) 0 0
\(52\) −0.687477 −0.0953359
\(53\) −2.69017 −0.369523 −0.184761 0.982783i \(-0.559151\pi\)
−0.184761 + 0.982783i \(0.559151\pi\)
\(54\) 0 0
\(55\) −0.769491 −0.103758
\(56\) −0.249022 −0.0332770
\(57\) 0 0
\(58\) −7.24527 −0.951350
\(59\) −9.21598 −1.19982 −0.599909 0.800068i \(-0.704797\pi\)
−0.599909 + 0.800068i \(0.704797\pi\)
\(60\) 0 0
\(61\) −14.0554 −1.79961 −0.899805 0.436291i \(-0.856292\pi\)
−0.899805 + 0.436291i \(0.856292\pi\)
\(62\) 11.5250 1.46368
\(63\) 0 0
\(64\) 8.69364 1.08671
\(65\) −0.196702 −0.0243979
\(66\) 0 0
\(67\) 5.62801 0.687570 0.343785 0.939048i \(-0.388291\pi\)
0.343785 + 0.939048i \(0.388291\pi\)
\(68\) −1.91525 −0.232258
\(69\) 0 0
\(70\) −0.0169617 −0.00202732
\(71\) 4.88191 0.579376 0.289688 0.957121i \(-0.406448\pi\)
0.289688 + 0.957121i \(0.406448\pi\)
\(72\) 0 0
\(73\) 10.9754 1.28457 0.642286 0.766465i \(-0.277986\pi\)
0.642286 + 0.766465i \(0.277986\pi\)
\(74\) −4.78821 −0.556618
\(75\) 0 0
\(76\) 1.34880 0.154718
\(77\) −0.348194 −0.0396804
\(78\) 0 0
\(79\) 17.4356 1.96166 0.980828 0.194874i \(-0.0624299\pi\)
0.980828 + 0.194874i \(0.0624299\pi\)
\(80\) 0.421905 0.0471705
\(81\) 0 0
\(82\) 5.37780 0.593878
\(83\) 2.48372 0.272624 0.136312 0.990666i \(-0.456475\pi\)
0.136312 + 0.990666i \(0.456475\pi\)
\(84\) 0 0
\(85\) −0.547994 −0.0594383
\(86\) −2.38198 −0.256855
\(87\) 0 0
\(88\) 13.2478 1.41222
\(89\) −5.57230 −0.590662 −0.295331 0.955395i \(-0.595430\pi\)
−0.295331 + 0.955395i \(0.595430\pi\)
\(90\) 0 0
\(91\) −0.0890076 −0.00933053
\(92\) 5.72143 0.596500
\(93\) 0 0
\(94\) −6.01382 −0.620279
\(95\) 0.385920 0.0395946
\(96\) 0 0
\(97\) −13.9731 −1.41876 −0.709379 0.704828i \(-0.751024\pi\)
−0.709379 + 0.704828i \(0.751024\pi\)
\(98\) 8.20094 0.828420
\(99\) 0 0
\(100\) 3.10437 0.310437
\(101\) 7.98379 0.794417 0.397208 0.917728i \(-0.369979\pi\)
0.397208 + 0.917728i \(0.369979\pi\)
\(102\) 0 0
\(103\) 10.5314 1.03769 0.518846 0.854868i \(-0.326362\pi\)
0.518846 + 0.854868i \(0.326362\pi\)
\(104\) 3.38648 0.332072
\(105\) 0 0
\(106\) 3.15465 0.306407
\(107\) 3.31671 0.320638 0.160319 0.987065i \(-0.448748\pi\)
0.160319 + 0.987065i \(0.448748\pi\)
\(108\) 0 0
\(109\) −12.7956 −1.22559 −0.612797 0.790240i \(-0.709956\pi\)
−0.612797 + 0.790240i \(0.709956\pi\)
\(110\) 0.902350 0.0860357
\(111\) 0 0
\(112\) 0.190912 0.0180395
\(113\) 4.42086 0.415880 0.207940 0.978142i \(-0.433324\pi\)
0.207940 + 0.978142i \(0.433324\pi\)
\(114\) 0 0
\(115\) 1.63702 0.152653
\(116\) −3.86075 −0.358462
\(117\) 0 0
\(118\) 10.8072 0.994884
\(119\) −0.247967 −0.0227311
\(120\) 0 0
\(121\) 7.52364 0.683967
\(122\) 16.4822 1.49223
\(123\) 0 0
\(124\) 6.14130 0.551505
\(125\) 1.78217 0.159402
\(126\) 0 0
\(127\) −17.5071 −1.55351 −0.776753 0.629806i \(-0.783135\pi\)
−0.776753 + 0.629806i \(0.783135\pi\)
\(128\) −3.41685 −0.302010
\(129\) 0 0
\(130\) 0.230665 0.0202306
\(131\) −11.8336 −1.03390 −0.516952 0.856014i \(-0.672934\pi\)
−0.516952 + 0.856014i \(0.672934\pi\)
\(132\) 0 0
\(133\) 0.174629 0.0151422
\(134\) −6.59973 −0.570130
\(135\) 0 0
\(136\) 9.43443 0.808995
\(137\) 2.19499 0.187530 0.0937651 0.995594i \(-0.470110\pi\)
0.0937651 + 0.995594i \(0.470110\pi\)
\(138\) 0 0
\(139\) 6.11773 0.518899 0.259450 0.965757i \(-0.416459\pi\)
0.259450 + 0.965757i \(0.416459\pi\)
\(140\) −0.00903832 −0.000763878 0
\(141\) 0 0
\(142\) −5.72482 −0.480416
\(143\) 4.73513 0.395972
\(144\) 0 0
\(145\) −1.10464 −0.0917357
\(146\) −12.8704 −1.06516
\(147\) 0 0
\(148\) −2.55147 −0.209730
\(149\) 17.3950 1.42505 0.712526 0.701646i \(-0.247551\pi\)
0.712526 + 0.701646i \(0.247551\pi\)
\(150\) 0 0
\(151\) −21.9595 −1.78704 −0.893520 0.449024i \(-0.851772\pi\)
−0.893520 + 0.449024i \(0.851772\pi\)
\(152\) −6.64411 −0.538909
\(153\) 0 0
\(154\) 0.408313 0.0329028
\(155\) 1.75716 0.141138
\(156\) 0 0
\(157\) −19.5736 −1.56214 −0.781070 0.624444i \(-0.785326\pi\)
−0.781070 + 0.624444i \(0.785326\pi\)
\(158\) −20.4460 −1.62660
\(159\) 0 0
\(160\) 0.605900 0.0479006
\(161\) 0.740753 0.0583795
\(162\) 0 0
\(163\) −14.2549 −1.11653 −0.558265 0.829663i \(-0.688533\pi\)
−0.558265 + 0.829663i \(0.688533\pi\)
\(164\) 2.86564 0.223769
\(165\) 0 0
\(166\) −2.91256 −0.226058
\(167\) 19.3555 1.49777 0.748886 0.662698i \(-0.230589\pi\)
0.748886 + 0.662698i \(0.230589\pi\)
\(168\) 0 0
\(169\) −11.7896 −0.906890
\(170\) 0.642611 0.0492860
\(171\) 0 0
\(172\) −1.26927 −0.0967811
\(173\) −24.1751 −1.83800 −0.919000 0.394258i \(-0.871002\pi\)
−0.919000 + 0.394258i \(0.871002\pi\)
\(174\) 0 0
\(175\) 0.401923 0.0303825
\(176\) −10.1564 −0.765564
\(177\) 0 0
\(178\) 6.53441 0.489775
\(179\) −11.7844 −0.880807 −0.440403 0.897800i \(-0.645165\pi\)
−0.440403 + 0.897800i \(0.645165\pi\)
\(180\) 0 0
\(181\) 0.193747 0.0144011 0.00720054 0.999974i \(-0.497708\pi\)
0.00720054 + 0.999974i \(0.497708\pi\)
\(182\) 0.104376 0.00773683
\(183\) 0 0
\(184\) −28.1835 −2.07771
\(185\) −0.730031 −0.0536730
\(186\) 0 0
\(187\) 13.1916 0.964669
\(188\) −3.20456 −0.233716
\(189\) 0 0
\(190\) −0.452553 −0.0328317
\(191\) 20.0106 1.44791 0.723957 0.689845i \(-0.242321\pi\)
0.723957 + 0.689845i \(0.242321\pi\)
\(192\) 0 0
\(193\) −16.4429 −1.18358 −0.591792 0.806091i \(-0.701579\pi\)
−0.591792 + 0.806091i \(0.701579\pi\)
\(194\) 16.3857 1.17643
\(195\) 0 0
\(196\) 4.37000 0.312143
\(197\) −24.3311 −1.73352 −0.866761 0.498723i \(-0.833802\pi\)
−0.866761 + 0.498723i \(0.833802\pi\)
\(198\) 0 0
\(199\) −16.2467 −1.15170 −0.575851 0.817555i \(-0.695329\pi\)
−0.575851 + 0.817555i \(0.695329\pi\)
\(200\) −15.2920 −1.08131
\(201\) 0 0
\(202\) −9.36226 −0.658727
\(203\) −0.499851 −0.0350827
\(204\) 0 0
\(205\) 0.819922 0.0572659
\(206\) −12.3498 −0.860449
\(207\) 0 0
\(208\) −2.59623 −0.180016
\(209\) −9.29010 −0.642610
\(210\) 0 0
\(211\) 13.5844 0.935189 0.467594 0.883943i \(-0.345121\pi\)
0.467594 + 0.883943i \(0.345121\pi\)
\(212\) 1.68100 0.115452
\(213\) 0 0
\(214\) −3.88937 −0.265872
\(215\) −0.363166 −0.0247677
\(216\) 0 0
\(217\) 0.795113 0.0539758
\(218\) 15.0049 1.01626
\(219\) 0 0
\(220\) 0.480831 0.0324176
\(221\) 3.37213 0.226834
\(222\) 0 0
\(223\) 24.6725 1.65219 0.826096 0.563530i \(-0.190557\pi\)
0.826096 + 0.563530i \(0.190557\pi\)
\(224\) 0.274169 0.0183187
\(225\) 0 0
\(226\) −5.18417 −0.344846
\(227\) −17.9854 −1.19373 −0.596866 0.802341i \(-0.703588\pi\)
−0.596866 + 0.802341i \(0.703588\pi\)
\(228\) 0 0
\(229\) 15.8558 1.04778 0.523891 0.851786i \(-0.324480\pi\)
0.523891 + 0.851786i \(0.324480\pi\)
\(230\) −1.91967 −0.126579
\(231\) 0 0
\(232\) 19.0179 1.24858
\(233\) −2.33607 −0.153041 −0.0765205 0.997068i \(-0.524381\pi\)
−0.0765205 + 0.997068i \(0.524381\pi\)
\(234\) 0 0
\(235\) −0.916894 −0.0598115
\(236\) 5.75879 0.374865
\(237\) 0 0
\(238\) 0.290781 0.0188485
\(239\) −4.84852 −0.313625 −0.156812 0.987628i \(-0.550122\pi\)
−0.156812 + 0.987628i \(0.550122\pi\)
\(240\) 0 0
\(241\) −3.42679 −0.220739 −0.110369 0.993891i \(-0.535203\pi\)
−0.110369 + 0.993891i \(0.535203\pi\)
\(242\) −8.82267 −0.567143
\(243\) 0 0
\(244\) 8.78280 0.562261
\(245\) 1.25035 0.0798820
\(246\) 0 0
\(247\) −2.37480 −0.151105
\(248\) −30.2517 −1.92099
\(249\) 0 0
\(250\) −2.08988 −0.132176
\(251\) 25.7166 1.62321 0.811607 0.584204i \(-0.198593\pi\)
0.811607 + 0.584204i \(0.198593\pi\)
\(252\) 0 0
\(253\) −39.4074 −2.47753
\(254\) 20.5299 1.28816
\(255\) 0 0
\(256\) −13.3805 −0.836280
\(257\) 24.1897 1.50891 0.754457 0.656350i \(-0.227900\pi\)
0.754457 + 0.656350i \(0.227900\pi\)
\(258\) 0 0
\(259\) −0.330339 −0.0205262
\(260\) 0.122913 0.00762275
\(261\) 0 0
\(262\) 13.8768 0.857309
\(263\) −18.1779 −1.12089 −0.560447 0.828190i \(-0.689371\pi\)
−0.560447 + 0.828190i \(0.689371\pi\)
\(264\) 0 0
\(265\) 0.480972 0.0295458
\(266\) −0.204780 −0.0125559
\(267\) 0 0
\(268\) −3.51677 −0.214821
\(269\) 5.69010 0.346931 0.173466 0.984840i \(-0.444503\pi\)
0.173466 + 0.984840i \(0.444503\pi\)
\(270\) 0 0
\(271\) 16.4138 0.997069 0.498535 0.866870i \(-0.333872\pi\)
0.498535 + 0.866870i \(0.333872\pi\)
\(272\) −7.23287 −0.438557
\(273\) 0 0
\(274\) −2.57397 −0.155499
\(275\) −21.3820 −1.28938
\(276\) 0 0
\(277\) −26.4497 −1.58921 −0.794605 0.607127i \(-0.792322\pi\)
−0.794605 + 0.607127i \(0.792322\pi\)
\(278\) −7.17402 −0.430269
\(279\) 0 0
\(280\) 0.0445223 0.00266072
\(281\) −3.32607 −0.198417 −0.0992084 0.995067i \(-0.531631\pi\)
−0.0992084 + 0.995067i \(0.531631\pi\)
\(282\) 0 0
\(283\) −21.9817 −1.30668 −0.653339 0.757065i \(-0.726633\pi\)
−0.653339 + 0.757065i \(0.726633\pi\)
\(284\) −3.05056 −0.181017
\(285\) 0 0
\(286\) −5.55270 −0.328338
\(287\) 0.371014 0.0219003
\(288\) 0 0
\(289\) −7.60555 −0.447385
\(290\) 1.29537 0.0760669
\(291\) 0 0
\(292\) −6.85819 −0.401345
\(293\) 6.54118 0.382140 0.191070 0.981576i \(-0.438804\pi\)
0.191070 + 0.981576i \(0.438804\pi\)
\(294\) 0 0
\(295\) 1.64771 0.0959336
\(296\) 12.5684 0.730525
\(297\) 0 0
\(298\) −20.3984 −1.18165
\(299\) −10.0736 −0.582570
\(300\) 0 0
\(301\) −0.164333 −0.00947197
\(302\) 25.7510 1.48181
\(303\) 0 0
\(304\) 5.09369 0.292143
\(305\) 2.51295 0.143891
\(306\) 0 0
\(307\) 15.6985 0.895959 0.447980 0.894044i \(-0.352144\pi\)
0.447980 + 0.894044i \(0.352144\pi\)
\(308\) 0.217576 0.0123975
\(309\) 0 0
\(310\) −2.06055 −0.117031
\(311\) −25.6166 −1.45258 −0.726292 0.687386i \(-0.758758\pi\)
−0.726292 + 0.687386i \(0.758758\pi\)
\(312\) 0 0
\(313\) 3.69793 0.209019 0.104510 0.994524i \(-0.466673\pi\)
0.104510 + 0.994524i \(0.466673\pi\)
\(314\) 22.9531 1.29532
\(315\) 0 0
\(316\) −10.8950 −0.612890
\(317\) −19.0196 −1.06825 −0.534124 0.845406i \(-0.679359\pi\)
−0.534124 + 0.845406i \(0.679359\pi\)
\(318\) 0 0
\(319\) 26.5917 1.48885
\(320\) −1.55433 −0.0868894
\(321\) 0 0
\(322\) −0.868651 −0.0484080
\(323\) −6.61597 −0.368122
\(324\) 0 0
\(325\) −5.46580 −0.303188
\(326\) 16.7161 0.925822
\(327\) 0 0
\(328\) −14.1160 −0.779427
\(329\) −0.414894 −0.0228738
\(330\) 0 0
\(331\) −4.74934 −0.261048 −0.130524 0.991445i \(-0.541666\pi\)
−0.130524 + 0.991445i \(0.541666\pi\)
\(332\) −1.55200 −0.0851771
\(333\) 0 0
\(334\) −22.6974 −1.24195
\(335\) −1.00622 −0.0549759
\(336\) 0 0
\(337\) −29.8451 −1.62576 −0.812882 0.582428i \(-0.802103\pi\)
−0.812882 + 0.582428i \(0.802103\pi\)
\(338\) 13.8252 0.751989
\(339\) 0 0
\(340\) 0.342425 0.0185706
\(341\) −42.2993 −2.29064
\(342\) 0 0
\(343\) 1.13210 0.0611274
\(344\) 6.25238 0.337105
\(345\) 0 0
\(346\) 28.3492 1.52406
\(347\) −15.3781 −0.825540 −0.412770 0.910835i \(-0.635439\pi\)
−0.412770 + 0.910835i \(0.635439\pi\)
\(348\) 0 0
\(349\) 6.64130 0.355501 0.177750 0.984076i \(-0.443118\pi\)
0.177750 + 0.984076i \(0.443118\pi\)
\(350\) −0.471319 −0.0251930
\(351\) 0 0
\(352\) −14.5856 −0.777415
\(353\) 33.7272 1.79512 0.897558 0.440896i \(-0.145339\pi\)
0.897558 + 0.440896i \(0.145339\pi\)
\(354\) 0 0
\(355\) −0.872830 −0.0463250
\(356\) 3.48196 0.184544
\(357\) 0 0
\(358\) 13.8191 0.730361
\(359\) 27.2110 1.43614 0.718071 0.695970i \(-0.245025\pi\)
0.718071 + 0.695970i \(0.245025\pi\)
\(360\) 0 0
\(361\) −14.3408 −0.754777
\(362\) −0.227199 −0.0119413
\(363\) 0 0
\(364\) 0.0556181 0.00291518
\(365\) −1.96228 −0.102710
\(366\) 0 0
\(367\) 14.0523 0.733525 0.366763 0.930315i \(-0.380466\pi\)
0.366763 + 0.930315i \(0.380466\pi\)
\(368\) 21.6068 1.12633
\(369\) 0 0
\(370\) 0.856078 0.0445054
\(371\) 0.217639 0.0112993
\(372\) 0 0
\(373\) 19.8673 1.02869 0.514345 0.857584i \(-0.328035\pi\)
0.514345 + 0.857584i \(0.328035\pi\)
\(374\) −15.4693 −0.799899
\(375\) 0 0
\(376\) 15.7855 0.814075
\(377\) 6.79753 0.350091
\(378\) 0 0
\(379\) −16.4863 −0.846844 −0.423422 0.905933i \(-0.639171\pi\)
−0.423422 + 0.905933i \(0.639171\pi\)
\(380\) −0.241150 −0.0123707
\(381\) 0 0
\(382\) −23.4656 −1.20060
\(383\) −22.2392 −1.13637 −0.568186 0.822900i \(-0.692355\pi\)
−0.568186 + 0.822900i \(0.692355\pi\)
\(384\) 0 0
\(385\) 0.0622532 0.00317272
\(386\) 19.2819 0.981422
\(387\) 0 0
\(388\) 8.73139 0.443269
\(389\) 10.3650 0.525528 0.262764 0.964860i \(-0.415366\pi\)
0.262764 + 0.964860i \(0.415366\pi\)
\(390\) 0 0
\(391\) −28.0641 −1.41926
\(392\) −21.5264 −1.08725
\(393\) 0 0
\(394\) 28.5322 1.43743
\(395\) −3.11729 −0.156848
\(396\) 0 0
\(397\) −2.54363 −0.127661 −0.0638306 0.997961i \(-0.520332\pi\)
−0.0638306 + 0.997961i \(0.520332\pi\)
\(398\) 19.0519 0.954985
\(399\) 0 0
\(400\) 11.7236 0.586178
\(401\) −12.4246 −0.620455 −0.310227 0.950662i \(-0.600405\pi\)
−0.310227 + 0.950662i \(0.600405\pi\)
\(402\) 0 0
\(403\) −10.8128 −0.538625
\(404\) −4.98883 −0.248203
\(405\) 0 0
\(406\) 0.586155 0.0290904
\(407\) 17.5738 0.871098
\(408\) 0 0
\(409\) 14.3575 0.709932 0.354966 0.934879i \(-0.384492\pi\)
0.354966 + 0.934879i \(0.384492\pi\)
\(410\) −0.961490 −0.0474846
\(411\) 0 0
\(412\) −6.58076 −0.324211
\(413\) 0.745589 0.0366880
\(414\) 0 0
\(415\) −0.444061 −0.0217981
\(416\) −3.72846 −0.182803
\(417\) 0 0
\(418\) 10.8941 0.532849
\(419\) −15.6860 −0.766309 −0.383155 0.923684i \(-0.625162\pi\)
−0.383155 + 0.923684i \(0.625162\pi\)
\(420\) 0 0
\(421\) 0.689920 0.0336247 0.0168123 0.999859i \(-0.494648\pi\)
0.0168123 + 0.999859i \(0.494648\pi\)
\(422\) −15.9299 −0.775454
\(423\) 0 0
\(424\) −8.28054 −0.402139
\(425\) −15.2272 −0.738628
\(426\) 0 0
\(427\) 1.13711 0.0550285
\(428\) −2.07251 −0.100179
\(429\) 0 0
\(430\) 0.425870 0.0205373
\(431\) −29.0858 −1.40102 −0.700508 0.713645i \(-0.747043\pi\)
−0.700508 + 0.713645i \(0.747043\pi\)
\(432\) 0 0
\(433\) 18.1169 0.870643 0.435321 0.900275i \(-0.356635\pi\)
0.435321 + 0.900275i \(0.356635\pi\)
\(434\) −0.932396 −0.0447565
\(435\) 0 0
\(436\) 7.99557 0.382918
\(437\) 19.7639 0.945435
\(438\) 0 0
\(439\) 3.25712 0.155454 0.0777270 0.996975i \(-0.475234\pi\)
0.0777270 + 0.996975i \(0.475234\pi\)
\(440\) −2.36855 −0.112916
\(441\) 0 0
\(442\) −3.95436 −0.188090
\(443\) 21.5717 1.02490 0.512451 0.858717i \(-0.328738\pi\)
0.512451 + 0.858717i \(0.328738\pi\)
\(444\) 0 0
\(445\) 0.996264 0.0472274
\(446\) −28.9324 −1.36999
\(447\) 0 0
\(448\) −0.703331 −0.0332293
\(449\) −18.9612 −0.894834 −0.447417 0.894325i \(-0.647656\pi\)
−0.447417 + 0.894325i \(0.647656\pi\)
\(450\) 0 0
\(451\) −19.7377 −0.929410
\(452\) −2.76246 −0.129935
\(453\) 0 0
\(454\) 21.0907 0.989837
\(455\) 0.0159136 0.000746039 0
\(456\) 0 0
\(457\) 4.18789 0.195901 0.0979506 0.995191i \(-0.468771\pi\)
0.0979506 + 0.995191i \(0.468771\pi\)
\(458\) −18.5935 −0.868815
\(459\) 0 0
\(460\) −1.02293 −0.0476942
\(461\) −13.7101 −0.638543 −0.319271 0.947663i \(-0.603438\pi\)
−0.319271 + 0.947663i \(0.603438\pi\)
\(462\) 0 0
\(463\) 4.87808 0.226703 0.113352 0.993555i \(-0.463841\pi\)
0.113352 + 0.993555i \(0.463841\pi\)
\(464\) −14.5800 −0.676859
\(465\) 0 0
\(466\) 2.73941 0.126901
\(467\) −12.3373 −0.570904 −0.285452 0.958393i \(-0.592144\pi\)
−0.285452 + 0.958393i \(0.592144\pi\)
\(468\) 0 0
\(469\) −0.455316 −0.0210245
\(470\) 1.07520 0.0495955
\(471\) 0 0
\(472\) −28.3675 −1.30572
\(473\) 8.74236 0.401974
\(474\) 0 0
\(475\) 10.7236 0.492034
\(476\) 0.154947 0.00710199
\(477\) 0 0
\(478\) 5.68566 0.260056
\(479\) 3.56361 0.162826 0.0814128 0.996680i \(-0.474057\pi\)
0.0814128 + 0.996680i \(0.474057\pi\)
\(480\) 0 0
\(481\) 4.49231 0.204832
\(482\) 4.01846 0.183036
\(483\) 0 0
\(484\) −4.70130 −0.213695
\(485\) 2.49824 0.113439
\(486\) 0 0
\(487\) −5.09262 −0.230769 −0.115384 0.993321i \(-0.536810\pi\)
−0.115384 + 0.993321i \(0.536810\pi\)
\(488\) −43.2637 −1.95845
\(489\) 0 0
\(490\) −1.46624 −0.0662378
\(491\) −27.1049 −1.22323 −0.611614 0.791156i \(-0.709479\pi\)
−0.611614 + 0.791156i \(0.709479\pi\)
\(492\) 0 0
\(493\) 18.9373 0.852893
\(494\) 2.78483 0.125295
\(495\) 0 0
\(496\) 23.1924 1.04137
\(497\) −0.394955 −0.0177162
\(498\) 0 0
\(499\) −4.49778 −0.201348 −0.100674 0.994919i \(-0.532100\pi\)
−0.100674 + 0.994919i \(0.532100\pi\)
\(500\) −1.11363 −0.0498028
\(501\) 0 0
\(502\) −30.1568 −1.34596
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −1.42741 −0.0635190
\(506\) 46.2115 2.05435
\(507\) 0 0
\(508\) 10.9397 0.485369
\(509\) 9.92656 0.439987 0.219993 0.975501i \(-0.429396\pi\)
0.219993 + 0.975501i \(0.429396\pi\)
\(510\) 0 0
\(511\) −0.887929 −0.0392797
\(512\) 22.5244 0.995449
\(513\) 0 0
\(514\) −28.3663 −1.25118
\(515\) −1.88290 −0.0829704
\(516\) 0 0
\(517\) 22.0720 0.970726
\(518\) 0.387375 0.0170203
\(519\) 0 0
\(520\) −0.605464 −0.0265514
\(521\) 8.65790 0.379309 0.189655 0.981851i \(-0.439263\pi\)
0.189655 + 0.981851i \(0.439263\pi\)
\(522\) 0 0
\(523\) 17.7111 0.774451 0.387226 0.921985i \(-0.373433\pi\)
0.387226 + 0.921985i \(0.373433\pi\)
\(524\) 7.39445 0.323028
\(525\) 0 0
\(526\) 21.3164 0.929441
\(527\) −30.1236 −1.31220
\(528\) 0 0
\(529\) 60.8360 2.64504
\(530\) −0.564016 −0.0244993
\(531\) 0 0
\(532\) −0.109120 −0.00473096
\(533\) −5.04547 −0.218543
\(534\) 0 0
\(535\) −0.592990 −0.0256372
\(536\) 17.3234 0.748259
\(537\) 0 0
\(538\) −6.67255 −0.287674
\(539\) −30.0992 −1.29646
\(540\) 0 0
\(541\) −10.8187 −0.465131 −0.232565 0.972581i \(-0.574712\pi\)
−0.232565 + 0.972581i \(0.574712\pi\)
\(542\) −19.2478 −0.826765
\(543\) 0 0
\(544\) −10.3872 −0.445346
\(545\) 2.28771 0.0979945
\(546\) 0 0
\(547\) −33.1188 −1.41606 −0.708028 0.706184i \(-0.750415\pi\)
−0.708028 + 0.706184i \(0.750415\pi\)
\(548\) −1.37158 −0.0585910
\(549\) 0 0
\(550\) 25.0738 1.06915
\(551\) −13.3364 −0.568151
\(552\) 0 0
\(553\) −1.41057 −0.0599835
\(554\) 31.0165 1.31777
\(555\) 0 0
\(556\) −3.82279 −0.162122
\(557\) −12.3324 −0.522541 −0.261270 0.965266i \(-0.584141\pi\)
−0.261270 + 0.965266i \(0.584141\pi\)
\(558\) 0 0
\(559\) 2.23478 0.0945210
\(560\) −0.0341329 −0.00144238
\(561\) 0 0
\(562\) 3.90035 0.164526
\(563\) −29.3356 −1.23635 −0.618174 0.786041i \(-0.712127\pi\)
−0.618174 + 0.786041i \(0.712127\pi\)
\(564\) 0 0
\(565\) −0.790400 −0.0332524
\(566\) 25.7771 1.08349
\(567\) 0 0
\(568\) 15.0269 0.630514
\(569\) −0.339405 −0.0142286 −0.00711431 0.999975i \(-0.502265\pi\)
−0.00711431 + 0.999975i \(0.502265\pi\)
\(570\) 0 0
\(571\) −16.5993 −0.694658 −0.347329 0.937743i \(-0.612911\pi\)
−0.347329 + 0.937743i \(0.612911\pi\)
\(572\) −2.95884 −0.123715
\(573\) 0 0
\(574\) −0.435073 −0.0181596
\(575\) 45.4883 1.89699
\(576\) 0 0
\(577\) −19.0207 −0.791840 −0.395920 0.918285i \(-0.629574\pi\)
−0.395920 + 0.918285i \(0.629574\pi\)
\(578\) 8.91872 0.370970
\(579\) 0 0
\(580\) 0.690259 0.0286614
\(581\) −0.200937 −0.00833629
\(582\) 0 0
\(583\) −11.5782 −0.479522
\(584\) 33.7831 1.39796
\(585\) 0 0
\(586\) −7.67058 −0.316869
\(587\) −29.3939 −1.21322 −0.606609 0.795000i \(-0.707471\pi\)
−0.606609 + 0.795000i \(0.707471\pi\)
\(588\) 0 0
\(589\) 21.2143 0.874118
\(590\) −1.93221 −0.0795477
\(591\) 0 0
\(592\) −9.63554 −0.396018
\(593\) 21.6941 0.890871 0.445436 0.895314i \(-0.353049\pi\)
0.445436 + 0.895314i \(0.353049\pi\)
\(594\) 0 0
\(595\) 0.0443337 0.00181751
\(596\) −10.8696 −0.445236
\(597\) 0 0
\(598\) 11.8129 0.483065
\(599\) 7.41054 0.302786 0.151393 0.988474i \(-0.451624\pi\)
0.151393 + 0.988474i \(0.451624\pi\)
\(600\) 0 0
\(601\) −17.7208 −0.722848 −0.361424 0.932402i \(-0.617709\pi\)
−0.361424 + 0.932402i \(0.617709\pi\)
\(602\) 0.192706 0.00785411
\(603\) 0 0
\(604\) 13.7218 0.558333
\(605\) −1.34514 −0.0546878
\(606\) 0 0
\(607\) 37.0318 1.50307 0.751536 0.659692i \(-0.229313\pi\)
0.751536 + 0.659692i \(0.229313\pi\)
\(608\) 7.31507 0.296665
\(609\) 0 0
\(610\) −2.94683 −0.119314
\(611\) 5.64219 0.228259
\(612\) 0 0
\(613\) 37.6348 1.52005 0.760027 0.649891i \(-0.225185\pi\)
0.760027 + 0.649891i \(0.225185\pi\)
\(614\) −18.4090 −0.742925
\(615\) 0 0
\(616\) −1.07177 −0.0431828
\(617\) 8.73601 0.351699 0.175849 0.984417i \(-0.443733\pi\)
0.175849 + 0.984417i \(0.443733\pi\)
\(618\) 0 0
\(619\) 17.0846 0.686687 0.343343 0.939210i \(-0.388441\pi\)
0.343343 + 0.939210i \(0.388441\pi\)
\(620\) −1.09799 −0.0440965
\(621\) 0 0
\(622\) 30.0396 1.20448
\(623\) 0.450809 0.0180613
\(624\) 0 0
\(625\) 24.5215 0.980862
\(626\) −4.33641 −0.173318
\(627\) 0 0
\(628\) 12.2309 0.488067
\(629\) 12.5152 0.499013
\(630\) 0 0
\(631\) −38.0472 −1.51464 −0.757318 0.653046i \(-0.773491\pi\)
−0.757318 + 0.653046i \(0.773491\pi\)
\(632\) 53.6681 2.13480
\(633\) 0 0
\(634\) 22.3035 0.885787
\(635\) 3.13008 0.124213
\(636\) 0 0
\(637\) −7.69415 −0.304853
\(638\) −31.1830 −1.23455
\(639\) 0 0
\(640\) 0.610894 0.0241477
\(641\) −27.5905 −1.08976 −0.544879 0.838514i \(-0.683425\pi\)
−0.544879 + 0.838514i \(0.683425\pi\)
\(642\) 0 0
\(643\) 6.89268 0.271821 0.135910 0.990721i \(-0.456604\pi\)
0.135910 + 0.990721i \(0.456604\pi\)
\(644\) −0.462874 −0.0182398
\(645\) 0 0
\(646\) 7.75827 0.305245
\(647\) −8.60822 −0.338424 −0.169212 0.985580i \(-0.554122\pi\)
−0.169212 + 0.985580i \(0.554122\pi\)
\(648\) 0 0
\(649\) −39.6647 −1.55698
\(650\) 6.40952 0.251402
\(651\) 0 0
\(652\) 8.90746 0.348843
\(653\) −3.73378 −0.146114 −0.0730571 0.997328i \(-0.523276\pi\)
−0.0730571 + 0.997328i \(0.523276\pi\)
\(654\) 0 0
\(655\) 2.11571 0.0826677
\(656\) 10.8220 0.422528
\(657\) 0 0
\(658\) 0.486529 0.0189669
\(659\) −10.1373 −0.394893 −0.197447 0.980314i \(-0.563265\pi\)
−0.197447 + 0.980314i \(0.563265\pi\)
\(660\) 0 0
\(661\) −11.0003 −0.427863 −0.213932 0.976849i \(-0.568627\pi\)
−0.213932 + 0.976849i \(0.568627\pi\)
\(662\) 5.56936 0.216459
\(663\) 0 0
\(664\) 7.64508 0.296687
\(665\) −0.0312216 −0.00121072
\(666\) 0 0
\(667\) −56.5715 −2.19046
\(668\) −12.0947 −0.467957
\(669\) 0 0
\(670\) 1.17996 0.0455858
\(671\) −60.4932 −2.33531
\(672\) 0 0
\(673\) −44.9940 −1.73439 −0.867196 0.497967i \(-0.834080\pi\)
−0.867196 + 0.497967i \(0.834080\pi\)
\(674\) 34.9981 1.34808
\(675\) 0 0
\(676\) 7.36695 0.283344
\(677\) −15.4870 −0.595213 −0.297607 0.954689i \(-0.596188\pi\)
−0.297607 + 0.954689i \(0.596188\pi\)
\(678\) 0 0
\(679\) 1.13045 0.0433828
\(680\) −1.68677 −0.0646846
\(681\) 0 0
\(682\) 49.6027 1.89939
\(683\) −33.7255 −1.29047 −0.645235 0.763984i \(-0.723241\pi\)
−0.645235 + 0.763984i \(0.723241\pi\)
\(684\) 0 0
\(685\) −0.392439 −0.0149943
\(686\) −1.32756 −0.0506866
\(687\) 0 0
\(688\) −4.79336 −0.182745
\(689\) −2.95970 −0.112756
\(690\) 0 0
\(691\) 13.5728 0.516333 0.258166 0.966100i \(-0.416882\pi\)
0.258166 + 0.966100i \(0.416882\pi\)
\(692\) 15.1063 0.574255
\(693\) 0 0
\(694\) 18.0333 0.684534
\(695\) −1.09378 −0.0414895
\(696\) 0 0
\(697\) −14.0562 −0.532417
\(698\) −7.78799 −0.294780
\(699\) 0 0
\(700\) −0.251149 −0.00949256
\(701\) 34.7789 1.31358 0.656790 0.754074i \(-0.271914\pi\)
0.656790 + 0.754074i \(0.271914\pi\)
\(702\) 0 0
\(703\) −8.81371 −0.332415
\(704\) 37.4167 1.41019
\(705\) 0 0
\(706\) −39.5505 −1.48850
\(707\) −0.645903 −0.0242917
\(708\) 0 0
\(709\) −23.1752 −0.870362 −0.435181 0.900343i \(-0.643316\pi\)
−0.435181 + 0.900343i \(0.643316\pi\)
\(710\) 1.02353 0.0384125
\(711\) 0 0
\(712\) −17.1520 −0.642797
\(713\) 89.9882 3.37009
\(714\) 0 0
\(715\) −0.846588 −0.0316606
\(716\) 7.36371 0.275195
\(717\) 0 0
\(718\) −31.9092 −1.19084
\(719\) 39.5082 1.47341 0.736704 0.676215i \(-0.236381\pi\)
0.736704 + 0.676215i \(0.236381\pi\)
\(720\) 0 0
\(721\) −0.852011 −0.0317305
\(722\) 16.8168 0.625858
\(723\) 0 0
\(724\) −0.121067 −0.00449940
\(725\) −30.6950 −1.13998
\(726\) 0 0
\(727\) −5.75881 −0.213583 −0.106791 0.994281i \(-0.534058\pi\)
−0.106791 + 0.994281i \(0.534058\pi\)
\(728\) −0.273972 −0.0101541
\(729\) 0 0
\(730\) 2.30108 0.0851669
\(731\) 6.22589 0.230273
\(732\) 0 0
\(733\) −41.9036 −1.54774 −0.773872 0.633343i \(-0.781682\pi\)
−0.773872 + 0.633343i \(0.781682\pi\)
\(734\) −16.4786 −0.608236
\(735\) 0 0
\(736\) 31.0296 1.14377
\(737\) 24.2224 0.892245
\(738\) 0 0
\(739\) −14.5650 −0.535780 −0.267890 0.963449i \(-0.586326\pi\)
−0.267890 + 0.963449i \(0.586326\pi\)
\(740\) 0.456174 0.0167693
\(741\) 0 0
\(742\) −0.255217 −0.00936931
\(743\) 22.9633 0.842440 0.421220 0.906958i \(-0.361602\pi\)
0.421220 + 0.906958i \(0.361602\pi\)
\(744\) 0 0
\(745\) −3.11003 −0.113943
\(746\) −23.2976 −0.852984
\(747\) 0 0
\(748\) −8.24306 −0.301396
\(749\) −0.268328 −0.00980448
\(750\) 0 0
\(751\) −8.52609 −0.311121 −0.155561 0.987826i \(-0.549718\pi\)
−0.155561 + 0.987826i \(0.549718\pi\)
\(752\) −12.1019 −0.441311
\(753\) 0 0
\(754\) −7.97119 −0.290294
\(755\) 3.92611 0.142886
\(756\) 0 0
\(757\) −19.7364 −0.717331 −0.358665 0.933466i \(-0.616768\pi\)
−0.358665 + 0.933466i \(0.616768\pi\)
\(758\) 19.3328 0.702199
\(759\) 0 0
\(760\) 1.18789 0.0430894
\(761\) −44.7253 −1.62129 −0.810646 0.585537i \(-0.800884\pi\)
−0.810646 + 0.585537i \(0.800884\pi\)
\(762\) 0 0
\(763\) 1.03519 0.0374762
\(764\) −12.5040 −0.452379
\(765\) 0 0
\(766\) 26.0791 0.942275
\(767\) −10.1394 −0.366111
\(768\) 0 0
\(769\) −28.1170 −1.01393 −0.506963 0.861968i \(-0.669232\pi\)
−0.506963 + 0.861968i \(0.669232\pi\)
\(770\) −0.0730018 −0.00263080
\(771\) 0 0
\(772\) 10.2746 0.369793
\(773\) −34.5978 −1.24440 −0.622199 0.782859i \(-0.713760\pi\)
−0.622199 + 0.782859i \(0.713760\pi\)
\(774\) 0 0
\(775\) 48.8264 1.75390
\(776\) −43.0104 −1.54398
\(777\) 0 0
\(778\) −12.1546 −0.435765
\(779\) 9.89897 0.354667
\(780\) 0 0
\(781\) 21.0113 0.751843
\(782\) 32.9096 1.17685
\(783\) 0 0
\(784\) 16.5031 0.589398
\(785\) 3.49953 0.124904
\(786\) 0 0
\(787\) −46.1869 −1.64639 −0.823193 0.567762i \(-0.807810\pi\)
−0.823193 + 0.567762i \(0.807810\pi\)
\(788\) 15.2038 0.541613
\(789\) 0 0
\(790\) 3.65552 0.130057
\(791\) −0.357656 −0.0127168
\(792\) 0 0
\(793\) −15.4637 −0.549131
\(794\) 2.98282 0.105856
\(795\) 0 0
\(796\) 10.1521 0.359832
\(797\) −31.0936 −1.10139 −0.550695 0.834707i \(-0.685637\pi\)
−0.550695 + 0.834707i \(0.685637\pi\)
\(798\) 0 0
\(799\) 15.7186 0.556085
\(800\) 16.8363 0.595252
\(801\) 0 0
\(802\) 14.5698 0.514478
\(803\) 47.2371 1.66696
\(804\) 0 0
\(805\) −0.132438 −0.00466784
\(806\) 12.6798 0.446626
\(807\) 0 0
\(808\) 24.5747 0.864536
\(809\) 22.7451 0.799676 0.399838 0.916586i \(-0.369066\pi\)
0.399838 + 0.916586i \(0.369066\pi\)
\(810\) 0 0
\(811\) −20.2514 −0.711121 −0.355561 0.934653i \(-0.615710\pi\)
−0.355561 + 0.934653i \(0.615710\pi\)
\(812\) 0.312342 0.0109610
\(813\) 0 0
\(814\) −20.6080 −0.722311
\(815\) 2.54862 0.0892741
\(816\) 0 0
\(817\) −4.38453 −0.153395
\(818\) −16.8364 −0.588673
\(819\) 0 0
\(820\) −0.512345 −0.0178918
\(821\) −47.0787 −1.64306 −0.821529 0.570167i \(-0.806879\pi\)
−0.821529 + 0.570167i \(0.806879\pi\)
\(822\) 0 0
\(823\) −3.97277 −0.138482 −0.0692411 0.997600i \(-0.522058\pi\)
−0.0692411 + 0.997600i \(0.522058\pi\)
\(824\) 32.4165 1.12928
\(825\) 0 0
\(826\) −0.874322 −0.0304216
\(827\) 12.1175 0.421366 0.210683 0.977554i \(-0.432431\pi\)
0.210683 + 0.977554i \(0.432431\pi\)
\(828\) 0 0
\(829\) 27.0304 0.938804 0.469402 0.882985i \(-0.344470\pi\)
0.469402 + 0.882985i \(0.344470\pi\)
\(830\) 0.520732 0.0180749
\(831\) 0 0
\(832\) 9.56468 0.331596
\(833\) −21.4352 −0.742686
\(834\) 0 0
\(835\) −3.46054 −0.119757
\(836\) 5.80511 0.200774
\(837\) 0 0
\(838\) 18.3943 0.635420
\(839\) 21.4624 0.740964 0.370482 0.928840i \(-0.379193\pi\)
0.370482 + 0.928840i \(0.379193\pi\)
\(840\) 0 0
\(841\) 9.17374 0.316336
\(842\) −0.809042 −0.0278814
\(843\) 0 0
\(844\) −8.48848 −0.292186
\(845\) 2.10784 0.0725120
\(846\) 0 0
\(847\) −0.608676 −0.0209144
\(848\) 6.34825 0.218000
\(849\) 0 0
\(850\) 17.8563 0.612467
\(851\) −37.3866 −1.28160
\(852\) 0 0
\(853\) −20.5093 −0.702226 −0.351113 0.936333i \(-0.614197\pi\)
−0.351113 + 0.936333i \(0.614197\pi\)
\(854\) −1.33344 −0.0456294
\(855\) 0 0
\(856\) 10.2091 0.348939
\(857\) 23.2772 0.795135 0.397567 0.917573i \(-0.369854\pi\)
0.397567 + 0.917573i \(0.369854\pi\)
\(858\) 0 0
\(859\) −3.80551 −0.129842 −0.0649212 0.997890i \(-0.520680\pi\)
−0.0649212 + 0.997890i \(0.520680\pi\)
\(860\) 0.226932 0.00773830
\(861\) 0 0
\(862\) 34.1078 1.16172
\(863\) 35.9633 1.22420 0.612102 0.790779i \(-0.290324\pi\)
0.612102 + 0.790779i \(0.290324\pi\)
\(864\) 0 0
\(865\) 4.32224 0.146960
\(866\) −21.2450 −0.721933
\(867\) 0 0
\(868\) −0.496842 −0.0168639
\(869\) 75.0412 2.54560
\(870\) 0 0
\(871\) 6.19189 0.209804
\(872\) −39.3858 −1.33377
\(873\) 0 0
\(874\) −23.1763 −0.783951
\(875\) −0.144181 −0.00487421
\(876\) 0 0
\(877\) 14.2388 0.480810 0.240405 0.970673i \(-0.422720\pi\)
0.240405 + 0.970673i \(0.422720\pi\)
\(878\) −3.81949 −0.128902
\(879\) 0 0
\(880\) 1.81584 0.0612120
\(881\) −27.4845 −0.925976 −0.462988 0.886365i \(-0.653223\pi\)
−0.462988 + 0.886365i \(0.653223\pi\)
\(882\) 0 0
\(883\) 24.3360 0.818973 0.409487 0.912316i \(-0.365708\pi\)
0.409487 + 0.912316i \(0.365708\pi\)
\(884\) −2.10714 −0.0708709
\(885\) 0 0
\(886\) −25.2962 −0.849844
\(887\) 26.2023 0.879787 0.439893 0.898050i \(-0.355016\pi\)
0.439893 + 0.898050i \(0.355016\pi\)
\(888\) 0 0
\(889\) 1.41636 0.0475031
\(890\) −1.16828 −0.0391608
\(891\) 0 0
\(892\) −15.4171 −0.516202
\(893\) −11.0697 −0.370434
\(894\) 0 0
\(895\) 2.10692 0.0704265
\(896\) 0.276429 0.00923485
\(897\) 0 0
\(898\) 22.2350 0.741992
\(899\) −60.7230 −2.02522
\(900\) 0 0
\(901\) −8.24546 −0.274696
\(902\) 23.1456 0.770663
\(903\) 0 0
\(904\) 13.6078 0.452587
\(905\) −0.0346397 −0.00115146
\(906\) 0 0
\(907\) 58.2340 1.93363 0.966814 0.255482i \(-0.0822342\pi\)
0.966814 + 0.255482i \(0.0822342\pi\)
\(908\) 11.2385 0.372963
\(909\) 0 0
\(910\) −0.0186612 −0.000618612 0
\(911\) 30.4851 1.01002 0.505008 0.863114i \(-0.331489\pi\)
0.505008 + 0.863114i \(0.331489\pi\)
\(912\) 0 0
\(913\) 10.6897 0.353778
\(914\) −4.91097 −0.162440
\(915\) 0 0
\(916\) −9.90781 −0.327363
\(917\) 0.957358 0.0316148
\(918\) 0 0
\(919\) −2.94915 −0.0972833 −0.0486417 0.998816i \(-0.515489\pi\)
−0.0486417 + 0.998816i \(0.515489\pi\)
\(920\) 5.03889 0.166127
\(921\) 0 0
\(922\) 16.0773 0.529477
\(923\) 5.37104 0.176790
\(924\) 0 0
\(925\) −20.2855 −0.666983
\(926\) −5.72032 −0.187982
\(927\) 0 0
\(928\) −20.9384 −0.687336
\(929\) −28.9881 −0.951070 −0.475535 0.879697i \(-0.657745\pi\)
−0.475535 + 0.879697i \(0.657745\pi\)
\(930\) 0 0
\(931\) 15.0956 0.494737
\(932\) 1.45974 0.0478153
\(933\) 0 0
\(934\) 14.4675 0.473391
\(935\) −2.35852 −0.0771318
\(936\) 0 0
\(937\) −27.5952 −0.901497 −0.450748 0.892651i \(-0.648843\pi\)
−0.450748 + 0.892651i \(0.648843\pi\)
\(938\) 0.533930 0.0174334
\(939\) 0 0
\(940\) 0.572939 0.0186872
\(941\) −19.2064 −0.626112 −0.313056 0.949735i \(-0.601353\pi\)
−0.313056 + 0.949735i \(0.601353\pi\)
\(942\) 0 0
\(943\) 41.9902 1.36739
\(944\) 21.7478 0.707832
\(945\) 0 0
\(946\) −10.2518 −0.333315
\(947\) 41.2922 1.34182 0.670908 0.741541i \(-0.265905\pi\)
0.670908 + 0.741541i \(0.265905\pi\)
\(948\) 0 0
\(949\) 12.0750 0.391973
\(950\) −12.5752 −0.407992
\(951\) 0 0
\(952\) −0.763262 −0.0247375
\(953\) 24.0272 0.778317 0.389158 0.921171i \(-0.372766\pi\)
0.389158 + 0.921171i \(0.372766\pi\)
\(954\) 0 0
\(955\) −3.57767 −0.115771
\(956\) 3.02969 0.0979872
\(957\) 0 0
\(958\) −4.17890 −0.135014
\(959\) −0.177578 −0.00573430
\(960\) 0 0
\(961\) 65.5920 2.11587
\(962\) −5.26795 −0.169846
\(963\) 0 0
\(964\) 2.14130 0.0689665
\(965\) 2.93980 0.0946355
\(966\) 0 0
\(967\) −10.8553 −0.349084 −0.174542 0.984650i \(-0.555845\pi\)
−0.174542 + 0.984650i \(0.555845\pi\)
\(968\) 23.1584 0.744338
\(969\) 0 0
\(970\) −2.92958 −0.0940633
\(971\) 17.1111 0.549122 0.274561 0.961570i \(-0.411467\pi\)
0.274561 + 0.961570i \(0.411467\pi\)
\(972\) 0 0
\(973\) −0.494936 −0.0158669
\(974\) 5.97191 0.191352
\(975\) 0 0
\(976\) 33.1679 1.06168
\(977\) −22.5277 −0.720726 −0.360363 0.932812i \(-0.617347\pi\)
−0.360363 + 0.932812i \(0.617347\pi\)
\(978\) 0 0
\(979\) −23.9827 −0.766489
\(980\) −0.781306 −0.0249579
\(981\) 0 0
\(982\) 31.7849 1.01430
\(983\) 27.0618 0.863138 0.431569 0.902080i \(-0.357960\pi\)
0.431569 + 0.902080i \(0.357960\pi\)
\(984\) 0 0
\(985\) 4.35014 0.138607
\(986\) −22.2070 −0.707215
\(987\) 0 0
\(988\) 1.48394 0.0472104
\(989\) −18.5986 −0.591401
\(990\) 0 0
\(991\) −19.8216 −0.629655 −0.314827 0.949149i \(-0.601947\pi\)
−0.314827 + 0.949149i \(0.601947\pi\)
\(992\) 33.3067 1.05749
\(993\) 0 0
\(994\) 0.463148 0.0146902
\(995\) 2.90473 0.0920863
\(996\) 0 0
\(997\) −23.1120 −0.731966 −0.365983 0.930622i \(-0.619267\pi\)
−0.365983 + 0.930622i \(0.619267\pi\)
\(998\) 5.27436 0.166957
\(999\) 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.1.3 10
3.2 odd 2 503.2.a.e.1.8 10
12.11 even 2 8048.2.a.p.1.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.8 10 3.2 odd 2
4527.2.a.k.1.3 10 1.1 even 1 trivial
8048.2.a.p.1.5 10 12.11 even 2