Properties

Label 531.4.a.f.1.2
Level $531$
Weight $4$
Character 531.1
Self dual yes
Analytic conductor $31.330$
Analytic rank $0$
Dimension $8$
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] = [531,4,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 531.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: \(31.3300142130\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.61734\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.61734 q^{2} +13.3198 q^{4} +3.21787 q^{5} +15.9864 q^{7} -24.5634 q^{8} +O(q^{10})\) \(q-4.61734 q^{2} +13.3198 q^{4} +3.21787 q^{5} +15.9864 q^{7} -24.5634 q^{8} -14.8580 q^{10} -54.2052 q^{11} +85.2841 q^{13} -73.8148 q^{14} +6.85884 q^{16} +48.1138 q^{17} +64.2348 q^{19} +42.8615 q^{20} +250.284 q^{22} +191.406 q^{23} -114.645 q^{25} -393.786 q^{26} +212.936 q^{28} +15.0020 q^{29} -209.724 q^{31} +164.837 q^{32} -222.158 q^{34} +51.4424 q^{35} +418.134 q^{37} -296.594 q^{38} -79.0418 q^{40} -226.787 q^{41} -207.102 q^{43} -722.003 q^{44} -883.784 q^{46} -330.575 q^{47} -87.4334 q^{49} +529.356 q^{50} +1135.97 q^{52} +449.141 q^{53} -174.426 q^{55} -392.681 q^{56} -69.2694 q^{58} +59.0000 q^{59} -393.282 q^{61} +968.368 q^{62} -815.980 q^{64} +274.434 q^{65} -67.7305 q^{67} +640.866 q^{68} -237.527 q^{70} +589.737 q^{71} -229.329 q^{73} -1930.67 q^{74} +855.595 q^{76} -866.549 q^{77} +563.476 q^{79} +22.0709 q^{80} +1047.15 q^{82} -1179.22 q^{83} +154.824 q^{85} +956.258 q^{86} +1331.46 q^{88} +1335.45 q^{89} +1363.39 q^{91} +2549.49 q^{92} +1526.37 q^{94} +206.699 q^{95} +1361.53 q^{97} +403.710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.61734 −1.63248 −0.816238 0.577716i \(-0.803944\pi\)
−0.816238 + 0.577716i \(0.803944\pi\)
\(3\) 0 0
\(4\) 13.3198 1.66498
\(5\) 3.21787 0.287815 0.143908 0.989591i \(-0.454033\pi\)
0.143908 + 0.989591i \(0.454033\pi\)
\(6\) 0 0
\(7\) 15.9864 0.863187 0.431594 0.902068i \(-0.357951\pi\)
0.431594 + 0.902068i \(0.357951\pi\)
\(8\) −24.5634 −1.08556
\(9\) 0 0
\(10\) −14.8580 −0.469852
\(11\) −54.2052 −1.48577 −0.742886 0.669418i \(-0.766544\pi\)
−0.742886 + 0.669418i \(0.766544\pi\)
\(12\) 0 0
\(13\) 85.2841 1.81951 0.909753 0.415151i \(-0.136271\pi\)
0.909753 + 0.415151i \(0.136271\pi\)
\(14\) −73.8148 −1.40913
\(15\) 0 0
\(16\) 6.85884 0.107169
\(17\) 48.1138 0.686430 0.343215 0.939257i \(-0.388484\pi\)
0.343215 + 0.939257i \(0.388484\pi\)
\(18\) 0 0
\(19\) 64.2348 0.775604 0.387802 0.921743i \(-0.373235\pi\)
0.387802 + 0.921743i \(0.373235\pi\)
\(20\) 42.8615 0.479206
\(21\) 0 0
\(22\) 250.284 2.42549
\(23\) 191.406 1.73525 0.867627 0.497216i \(-0.165645\pi\)
0.867627 + 0.497216i \(0.165645\pi\)
\(24\) 0 0
\(25\) −114.645 −0.917162
\(26\) −393.786 −2.97030
\(27\) 0 0
\(28\) 212.936 1.43719
\(29\) 15.0020 0.0960622 0.0480311 0.998846i \(-0.484705\pi\)
0.0480311 + 0.998846i \(0.484705\pi\)
\(30\) 0 0
\(31\) −209.724 −1.21508 −0.607542 0.794287i \(-0.707844\pi\)
−0.607542 + 0.794287i \(0.707844\pi\)
\(32\) 164.837 0.910606
\(33\) 0 0
\(34\) −222.158 −1.12058
\(35\) 51.4424 0.248439
\(36\) 0 0
\(37\) 418.134 1.85786 0.928931 0.370254i \(-0.120729\pi\)
0.928931 + 0.370254i \(0.120729\pi\)
\(38\) −296.594 −1.26615
\(39\) 0 0
\(40\) −79.0418 −0.312440
\(41\) −226.787 −0.863859 −0.431929 0.901907i \(-0.642167\pi\)
−0.431929 + 0.901907i \(0.642167\pi\)
\(42\) 0 0
\(43\) −207.102 −0.734481 −0.367241 0.930126i \(-0.619697\pi\)
−0.367241 + 0.930126i \(0.619697\pi\)
\(44\) −722.003 −2.47377
\(45\) 0 0
\(46\) −883.784 −2.83276
\(47\) −330.575 −1.02594 −0.512971 0.858406i \(-0.671455\pi\)
−0.512971 + 0.858406i \(0.671455\pi\)
\(48\) 0 0
\(49\) −87.4334 −0.254908
\(50\) 529.356 1.49724
\(51\) 0 0
\(52\) 1135.97 3.02943
\(53\) 449.141 1.16404 0.582021 0.813173i \(-0.302262\pi\)
0.582021 + 0.813173i \(0.302262\pi\)
\(54\) 0 0
\(55\) −174.426 −0.427628
\(56\) −392.681 −0.937039
\(57\) 0 0
\(58\) −69.2694 −0.156819
\(59\) 59.0000 0.130189
\(60\) 0 0
\(61\) −393.282 −0.825485 −0.412743 0.910848i \(-0.635429\pi\)
−0.412743 + 0.910848i \(0.635429\pi\)
\(62\) 968.368 1.98360
\(63\) 0 0
\(64\) −815.980 −1.59371
\(65\) 274.434 0.523682
\(66\) 0 0
\(67\) −67.7305 −0.123501 −0.0617507 0.998092i \(-0.519668\pi\)
−0.0617507 + 0.998092i \(0.519668\pi\)
\(68\) 640.866 1.14289
\(69\) 0 0
\(70\) −237.527 −0.405570
\(71\) 589.737 0.985759 0.492879 0.870098i \(-0.335944\pi\)
0.492879 + 0.870098i \(0.335944\pi\)
\(72\) 0 0
\(73\) −229.329 −0.367684 −0.183842 0.982956i \(-0.558853\pi\)
−0.183842 + 0.982956i \(0.558853\pi\)
\(74\) −1930.67 −3.03291
\(75\) 0 0
\(76\) 855.595 1.29136
\(77\) −866.549 −1.28250
\(78\) 0 0
\(79\) 563.476 0.802482 0.401241 0.915973i \(-0.368579\pi\)
0.401241 + 0.915973i \(0.368579\pi\)
\(80\) 22.0709 0.0308450
\(81\) 0 0
\(82\) 1047.15 1.41023
\(83\) −1179.22 −1.55947 −0.779733 0.626112i \(-0.784645\pi\)
−0.779733 + 0.626112i \(0.784645\pi\)
\(84\) 0 0
\(85\) 154.824 0.197565
\(86\) 956.258 1.19902
\(87\) 0 0
\(88\) 1331.46 1.61289
\(89\) 1335.45 1.59053 0.795266 0.606261i \(-0.207331\pi\)
0.795266 + 0.606261i \(0.207331\pi\)
\(90\) 0 0
\(91\) 1363.39 1.57057
\(92\) 2549.49 2.88915
\(93\) 0 0
\(94\) 1526.37 1.67482
\(95\) 206.699 0.223231
\(96\) 0 0
\(97\) 1361.53 1.42518 0.712592 0.701578i \(-0.247521\pi\)
0.712592 + 0.701578i \(0.247521\pi\)
\(98\) 403.710 0.416131
\(99\) 0 0
\(100\) −1527.05 −1.52705
\(101\) −257.488 −0.253673 −0.126837 0.991924i \(-0.540482\pi\)
−0.126837 + 0.991924i \(0.540482\pi\)
\(102\) 0 0
\(103\) −343.256 −0.328370 −0.164185 0.986430i \(-0.552499\pi\)
−0.164185 + 0.986430i \(0.552499\pi\)
\(104\) −2094.86 −1.97518
\(105\) 0 0
\(106\) −2073.84 −1.90027
\(107\) 1319.34 1.19201 0.596007 0.802979i \(-0.296753\pi\)
0.596007 + 0.802979i \(0.296753\pi\)
\(108\) 0 0
\(109\) 282.298 0.248067 0.124033 0.992278i \(-0.460417\pi\)
0.124033 + 0.992278i \(0.460417\pi\)
\(110\) 805.382 0.698092
\(111\) 0 0
\(112\) 109.648 0.0925072
\(113\) 2326.67 1.93695 0.968473 0.249119i \(-0.0801411\pi\)
0.968473 + 0.249119i \(0.0801411\pi\)
\(114\) 0 0
\(115\) 615.919 0.499433
\(116\) 199.824 0.159941
\(117\) 0 0
\(118\) −272.423 −0.212530
\(119\) 769.169 0.592517
\(120\) 0 0
\(121\) 1607.21 1.20752
\(122\) 1815.92 1.34758
\(123\) 0 0
\(124\) −2793.49 −2.02309
\(125\) −771.149 −0.551789
\(126\) 0 0
\(127\) 660.219 0.461299 0.230650 0.973037i \(-0.425915\pi\)
0.230650 + 0.973037i \(0.425915\pi\)
\(128\) 2448.96 1.69109
\(129\) 0 0
\(130\) −1267.15 −0.854898
\(131\) −338.174 −0.225545 −0.112772 0.993621i \(-0.535973\pi\)
−0.112772 + 0.993621i \(0.535973\pi\)
\(132\) 0 0
\(133\) 1026.89 0.669491
\(134\) 312.734 0.201613
\(135\) 0 0
\(136\) −1181.84 −0.745159
\(137\) 1007.22 0.628121 0.314061 0.949403i \(-0.398311\pi\)
0.314061 + 0.949403i \(0.398311\pi\)
\(138\) 0 0
\(139\) 2107.75 1.28617 0.643084 0.765796i \(-0.277655\pi\)
0.643084 + 0.765796i \(0.277655\pi\)
\(140\) 685.203 0.413644
\(141\) 0 0
\(142\) −2723.01 −1.60923
\(143\) −4622.85 −2.70337
\(144\) 0 0
\(145\) 48.2746 0.0276482
\(146\) 1058.89 0.600235
\(147\) 0 0
\(148\) 5569.47 3.09329
\(149\) 2275.39 1.25105 0.625527 0.780203i \(-0.284884\pi\)
0.625527 + 0.780203i \(0.284884\pi\)
\(150\) 0 0
\(151\) 396.997 0.213955 0.106977 0.994261i \(-0.465883\pi\)
0.106977 + 0.994261i \(0.465883\pi\)
\(152\) −1577.82 −0.841962
\(153\) 0 0
\(154\) 4001.15 2.09365
\(155\) −674.867 −0.349720
\(156\) 0 0
\(157\) −2285.11 −1.16160 −0.580802 0.814045i \(-0.697261\pi\)
−0.580802 + 0.814045i \(0.697261\pi\)
\(158\) −2601.76 −1.31003
\(159\) 0 0
\(160\) 530.426 0.262086
\(161\) 3059.90 1.49785
\(162\) 0 0
\(163\) 1433.49 0.688831 0.344415 0.938817i \(-0.388077\pi\)
0.344415 + 0.938817i \(0.388077\pi\)
\(164\) −3020.76 −1.43830
\(165\) 0 0
\(166\) 5444.84 2.54579
\(167\) −270.302 −0.125249 −0.0626246 0.998037i \(-0.519947\pi\)
−0.0626246 + 0.998037i \(0.519947\pi\)
\(168\) 0 0
\(169\) 5076.39 2.31060
\(170\) −714.875 −0.322520
\(171\) 0 0
\(172\) −2758.55 −1.22289
\(173\) −1503.33 −0.660673 −0.330337 0.943863i \(-0.607162\pi\)
−0.330337 + 0.943863i \(0.607162\pi\)
\(174\) 0 0
\(175\) −1832.77 −0.791683
\(176\) −371.785 −0.159229
\(177\) 0 0
\(178\) −6166.22 −2.59650
\(179\) −1741.30 −0.727100 −0.363550 0.931575i \(-0.618435\pi\)
−0.363550 + 0.931575i \(0.618435\pi\)
\(180\) 0 0
\(181\) −3734.67 −1.53368 −0.766839 0.641839i \(-0.778172\pi\)
−0.766839 + 0.641839i \(0.778172\pi\)
\(182\) −6295.24 −2.56392
\(183\) 0 0
\(184\) −4701.56 −1.88372
\(185\) 1345.50 0.534721
\(186\) 0 0
\(187\) −2608.02 −1.01988
\(188\) −4403.19 −1.70817
\(189\) 0 0
\(190\) −954.401 −0.364419
\(191\) −1504.04 −0.569782 −0.284891 0.958560i \(-0.591957\pi\)
−0.284891 + 0.958560i \(0.591957\pi\)
\(192\) 0 0
\(193\) 2674.51 0.997488 0.498744 0.866749i \(-0.333795\pi\)
0.498744 + 0.866749i \(0.333795\pi\)
\(194\) −6286.67 −2.32658
\(195\) 0 0
\(196\) −1164.60 −0.424416
\(197\) 2496.55 0.902902 0.451451 0.892296i \(-0.350907\pi\)
0.451451 + 0.892296i \(0.350907\pi\)
\(198\) 0 0
\(199\) 4410.90 1.57126 0.785629 0.618698i \(-0.212339\pi\)
0.785629 + 0.618698i \(0.212339\pi\)
\(200\) 2816.07 0.995632
\(201\) 0 0
\(202\) 1188.91 0.414115
\(203\) 239.829 0.0829197
\(204\) 0 0
\(205\) −729.773 −0.248632
\(206\) 1584.93 0.536055
\(207\) 0 0
\(208\) 584.950 0.194995
\(209\) −3481.86 −1.15237
\(210\) 0 0
\(211\) −4046.61 −1.32029 −0.660143 0.751140i \(-0.729504\pi\)
−0.660143 + 0.751140i \(0.729504\pi\)
\(212\) 5982.47 1.93810
\(213\) 0 0
\(214\) −6091.84 −1.94593
\(215\) −666.427 −0.211395
\(216\) 0 0
\(217\) −3352.75 −1.04885
\(218\) −1303.47 −0.404963
\(219\) 0 0
\(220\) −2323.32 −0.711991
\(221\) 4103.34 1.24896
\(222\) 0 0
\(223\) 3533.34 1.06103 0.530516 0.847675i \(-0.321998\pi\)
0.530516 + 0.847675i \(0.321998\pi\)
\(224\) 2635.16 0.786023
\(225\) 0 0
\(226\) −10743.0 −3.16202
\(227\) −96.7822 −0.0282981 −0.0141490 0.999900i \(-0.504504\pi\)
−0.0141490 + 0.999900i \(0.504504\pi\)
\(228\) 0 0
\(229\) 1031.34 0.297611 0.148806 0.988866i \(-0.452457\pi\)
0.148806 + 0.988866i \(0.452457\pi\)
\(230\) −2843.91 −0.815312
\(231\) 0 0
\(232\) −368.500 −0.104281
\(233\) −4689.07 −1.31842 −0.659208 0.751961i \(-0.729108\pi\)
−0.659208 + 0.751961i \(0.729108\pi\)
\(234\) 0 0
\(235\) −1063.75 −0.295282
\(236\) 785.869 0.216761
\(237\) 0 0
\(238\) −3551.51 −0.967270
\(239\) −234.661 −0.0635103 −0.0317551 0.999496i \(-0.510110\pi\)
−0.0317551 + 0.999496i \(0.510110\pi\)
\(240\) 0 0
\(241\) −1334.65 −0.356731 −0.178365 0.983964i \(-0.557081\pi\)
−0.178365 + 0.983964i \(0.557081\pi\)
\(242\) −7421.01 −1.97124
\(243\) 0 0
\(244\) −5238.44 −1.37441
\(245\) −281.350 −0.0733664
\(246\) 0 0
\(247\) 5478.21 1.41121
\(248\) 5151.54 1.31904
\(249\) 0 0
\(250\) 3560.65 0.900782
\(251\) −1112.88 −0.279859 −0.139929 0.990161i \(-0.544688\pi\)
−0.139929 + 0.990161i \(0.544688\pi\)
\(252\) 0 0
\(253\) −10375.2 −2.57819
\(254\) −3048.46 −0.753060
\(255\) 0 0
\(256\) −4779.82 −1.16695
\(257\) 2171.13 0.526971 0.263486 0.964663i \(-0.415128\pi\)
0.263486 + 0.964663i \(0.415128\pi\)
\(258\) 0 0
\(259\) 6684.48 1.60368
\(260\) 3655.40 0.871918
\(261\) 0 0
\(262\) 1561.46 0.368196
\(263\) 5098.53 1.19540 0.597698 0.801722i \(-0.296082\pi\)
0.597698 + 0.801722i \(0.296082\pi\)
\(264\) 0 0
\(265\) 1445.28 0.335030
\(266\) −4741.48 −1.09293
\(267\) 0 0
\(268\) −902.157 −0.205627
\(269\) 2491.37 0.564689 0.282344 0.959313i \(-0.408888\pi\)
0.282344 + 0.959313i \(0.408888\pi\)
\(270\) 0 0
\(271\) −2614.02 −0.585943 −0.292972 0.956121i \(-0.594644\pi\)
−0.292972 + 0.956121i \(0.594644\pi\)
\(272\) 330.005 0.0735642
\(273\) 0 0
\(274\) −4650.67 −1.02539
\(275\) 6214.37 1.36269
\(276\) 0 0
\(277\) 4270.47 0.926309 0.463154 0.886278i \(-0.346718\pi\)
0.463154 + 0.886278i \(0.346718\pi\)
\(278\) −9732.21 −2.09964
\(279\) 0 0
\(280\) −1263.60 −0.269694
\(281\) 4081.99 0.866587 0.433294 0.901253i \(-0.357351\pi\)
0.433294 + 0.901253i \(0.357351\pi\)
\(282\) 0 0
\(283\) 1547.60 0.325071 0.162535 0.986703i \(-0.448033\pi\)
0.162535 + 0.986703i \(0.448033\pi\)
\(284\) 7855.18 1.64127
\(285\) 0 0
\(286\) 21345.2 4.41318
\(287\) −3625.52 −0.745672
\(288\) 0 0
\(289\) −2598.06 −0.528814
\(290\) −222.900 −0.0451350
\(291\) 0 0
\(292\) −3054.62 −0.612185
\(293\) −3046.36 −0.607407 −0.303703 0.952767i \(-0.598223\pi\)
−0.303703 + 0.952767i \(0.598223\pi\)
\(294\) 0 0
\(295\) 189.855 0.0374704
\(296\) −10270.8 −2.01681
\(297\) 0 0
\(298\) −10506.2 −2.04231
\(299\) 16323.9 3.15730
\(300\) 0 0
\(301\) −3310.82 −0.633995
\(302\) −1833.07 −0.349276
\(303\) 0 0
\(304\) 440.576 0.0831209
\(305\) −1265.53 −0.237587
\(306\) 0 0
\(307\) 5413.13 1.00633 0.503166 0.864190i \(-0.332168\pi\)
0.503166 + 0.864190i \(0.332168\pi\)
\(308\) −11542.3 −2.13533
\(309\) 0 0
\(310\) 3116.09 0.570909
\(311\) 945.000 0.172302 0.0861512 0.996282i \(-0.472543\pi\)
0.0861512 + 0.996282i \(0.472543\pi\)
\(312\) 0 0
\(313\) 4954.26 0.894669 0.447335 0.894367i \(-0.352373\pi\)
0.447335 + 0.894367i \(0.352373\pi\)
\(314\) 10551.1 1.89629
\(315\) 0 0
\(316\) 7505.40 1.33611
\(317\) 3881.82 0.687775 0.343887 0.939011i \(-0.388256\pi\)
0.343887 + 0.939011i \(0.388256\pi\)
\(318\) 0 0
\(319\) −813.188 −0.142727
\(320\) −2625.72 −0.458695
\(321\) 0 0
\(322\) −14128.6 −2.44520
\(323\) 3090.58 0.532397
\(324\) 0 0
\(325\) −9777.43 −1.66878
\(326\) −6618.90 −1.12450
\(327\) 0 0
\(328\) 5570.65 0.937768
\(329\) −5284.71 −0.885580
\(330\) 0 0
\(331\) 5526.01 0.917634 0.458817 0.888531i \(-0.348273\pi\)
0.458817 + 0.888531i \(0.348273\pi\)
\(332\) −15706.9 −2.59647
\(333\) 0 0
\(334\) 1248.08 0.204466
\(335\) −217.948 −0.0355456
\(336\) 0 0
\(337\) 3200.62 0.517356 0.258678 0.965964i \(-0.416713\pi\)
0.258678 + 0.965964i \(0.416713\pi\)
\(338\) −23439.4 −3.77200
\(339\) 0 0
\(340\) 2062.23 0.328941
\(341\) 11368.2 1.80534
\(342\) 0 0
\(343\) −6881.10 −1.08322
\(344\) 5087.11 0.797321
\(345\) 0 0
\(346\) 6941.40 1.07853
\(347\) 3533.99 0.546728 0.273364 0.961911i \(-0.411864\pi\)
0.273364 + 0.961911i \(0.411864\pi\)
\(348\) 0 0
\(349\) −2116.90 −0.324684 −0.162342 0.986735i \(-0.551905\pi\)
−0.162342 + 0.986735i \(0.551905\pi\)
\(350\) 8462.52 1.29240
\(351\) 0 0
\(352\) −8935.04 −1.35295
\(353\) 1961.72 0.295784 0.147892 0.989004i \(-0.452751\pi\)
0.147892 + 0.989004i \(0.452751\pi\)
\(354\) 0 0
\(355\) 1897.70 0.283717
\(356\) 17787.9 2.64820
\(357\) 0 0
\(358\) 8040.17 1.18697
\(359\) −4288.50 −0.630469 −0.315234 0.949014i \(-0.602083\pi\)
−0.315234 + 0.949014i \(0.602083\pi\)
\(360\) 0 0
\(361\) −2732.89 −0.398439
\(362\) 17244.2 2.50369
\(363\) 0 0
\(364\) 18160.1 2.61497
\(365\) −737.952 −0.105825
\(366\) 0 0
\(367\) −1246.33 −0.177269 −0.0886345 0.996064i \(-0.528250\pi\)
−0.0886345 + 0.996064i \(0.528250\pi\)
\(368\) 1312.82 0.185966
\(369\) 0 0
\(370\) −6212.65 −0.872919
\(371\) 7180.17 1.00479
\(372\) 0 0
\(373\) 11151.5 1.54799 0.773996 0.633190i \(-0.218255\pi\)
0.773996 + 0.633190i \(0.218255\pi\)
\(374\) 12042.1 1.66493
\(375\) 0 0
\(376\) 8120.02 1.11372
\(377\) 1279.43 0.174786
\(378\) 0 0
\(379\) 6741.73 0.913719 0.456860 0.889539i \(-0.348974\pi\)
0.456860 + 0.889539i \(0.348974\pi\)
\(380\) 2753.20 0.371674
\(381\) 0 0
\(382\) 6944.65 0.930155
\(383\) −5172.09 −0.690030 −0.345015 0.938597i \(-0.612126\pi\)
−0.345015 + 0.938597i \(0.612126\pi\)
\(384\) 0 0
\(385\) −2788.45 −0.369123
\(386\) −12349.1 −1.62837
\(387\) 0 0
\(388\) 18135.4 2.37290
\(389\) −10739.9 −1.39983 −0.699914 0.714228i \(-0.746778\pi\)
−0.699914 + 0.714228i \(0.746778\pi\)
\(390\) 0 0
\(391\) 9209.25 1.19113
\(392\) 2147.66 0.276717
\(393\) 0 0
\(394\) −11527.4 −1.47397
\(395\) 1813.20 0.230967
\(396\) 0 0
\(397\) 3755.94 0.474825 0.237412 0.971409i \(-0.423701\pi\)
0.237412 + 0.971409i \(0.423701\pi\)
\(398\) −20366.6 −2.56504
\(399\) 0 0
\(400\) −786.333 −0.0982916
\(401\) −10542.2 −1.31285 −0.656424 0.754392i \(-0.727932\pi\)
−0.656424 + 0.754392i \(0.727932\pi\)
\(402\) 0 0
\(403\) −17886.2 −2.21085
\(404\) −3429.69 −0.422360
\(405\) 0 0
\(406\) −1107.37 −0.135364
\(407\) −22665.1 −2.76036
\(408\) 0 0
\(409\) −14298.9 −1.72870 −0.864348 0.502894i \(-0.832269\pi\)
−0.864348 + 0.502894i \(0.832269\pi\)
\(410\) 3369.61 0.405885
\(411\) 0 0
\(412\) −4572.11 −0.546727
\(413\) 943.201 0.112377
\(414\) 0 0
\(415\) −3794.57 −0.448839
\(416\) 14058.0 1.65685
\(417\) 0 0
\(418\) 16076.9 1.88122
\(419\) −2605.91 −0.303835 −0.151917 0.988393i \(-0.548545\pi\)
−0.151917 + 0.988393i \(0.548545\pi\)
\(420\) 0 0
\(421\) −11765.6 −1.36204 −0.681022 0.732263i \(-0.738464\pi\)
−0.681022 + 0.732263i \(0.738464\pi\)
\(422\) 18684.6 2.15534
\(423\) 0 0
\(424\) −11032.4 −1.26364
\(425\) −5516.02 −0.629568
\(426\) 0 0
\(427\) −6287.18 −0.712548
\(428\) 17573.4 1.98467
\(429\) 0 0
\(430\) 3077.12 0.345097
\(431\) −11883.8 −1.32813 −0.664063 0.747677i \(-0.731169\pi\)
−0.664063 + 0.747677i \(0.731169\pi\)
\(432\) 0 0
\(433\) 3187.43 0.353760 0.176880 0.984232i \(-0.443400\pi\)
0.176880 + 0.984232i \(0.443400\pi\)
\(434\) 15480.8 1.71221
\(435\) 0 0
\(436\) 3760.16 0.413025
\(437\) 12294.9 1.34587
\(438\) 0 0
\(439\) 16420.7 1.78524 0.892618 0.450814i \(-0.148866\pi\)
0.892618 + 0.450814i \(0.148866\pi\)
\(440\) 4284.48 0.464215
\(441\) 0 0
\(442\) −18946.5 −2.03890
\(443\) −632.306 −0.0678143 −0.0339072 0.999425i \(-0.510795\pi\)
−0.0339072 + 0.999425i \(0.510795\pi\)
\(444\) 0 0
\(445\) 4297.31 0.457780
\(446\) −16314.6 −1.73211
\(447\) 0 0
\(448\) −13044.6 −1.37567
\(449\) −6771.85 −0.711767 −0.355884 0.934530i \(-0.615820\pi\)
−0.355884 + 0.934530i \(0.615820\pi\)
\(450\) 0 0
\(451\) 12293.0 1.28350
\(452\) 30990.8 3.22497
\(453\) 0 0
\(454\) 446.876 0.0461959
\(455\) 4387.22 0.452035
\(456\) 0 0
\(457\) −12278.5 −1.25681 −0.628406 0.777886i \(-0.716292\pi\)
−0.628406 + 0.777886i \(0.716292\pi\)
\(458\) −4762.06 −0.485843
\(459\) 0 0
\(460\) 8203.93 0.831543
\(461\) 10989.3 1.11024 0.555120 0.831770i \(-0.312672\pi\)
0.555120 + 0.831770i \(0.312672\pi\)
\(462\) 0 0
\(463\) −12034.6 −1.20798 −0.603990 0.796992i \(-0.706423\pi\)
−0.603990 + 0.796992i \(0.706423\pi\)
\(464\) 102.896 0.0102949
\(465\) 0 0
\(466\) 21651.0 2.15228
\(467\) 9494.20 0.940769 0.470384 0.882462i \(-0.344115\pi\)
0.470384 + 0.882462i \(0.344115\pi\)
\(468\) 0 0
\(469\) −1082.77 −0.106605
\(470\) 4911.68 0.482040
\(471\) 0 0
\(472\) −1449.24 −0.141328
\(473\) 11226.0 1.09127
\(474\) 0 0
\(475\) −7364.21 −0.711354
\(476\) 10245.2 0.986527
\(477\) 0 0
\(478\) 1083.51 0.103679
\(479\) −1700.69 −0.162227 −0.0811135 0.996705i \(-0.525848\pi\)
−0.0811135 + 0.996705i \(0.525848\pi\)
\(480\) 0 0
\(481\) 35660.2 3.38039
\(482\) 6162.51 0.582354
\(483\) 0 0
\(484\) 21407.7 2.01049
\(485\) 4381.25 0.410190
\(486\) 0 0
\(487\) 3687.27 0.343093 0.171546 0.985176i \(-0.445124\pi\)
0.171546 + 0.985176i \(0.445124\pi\)
\(488\) 9660.33 0.896112
\(489\) 0 0
\(490\) 1299.09 0.119769
\(491\) −14020.6 −1.28867 −0.644337 0.764742i \(-0.722866\pi\)
−0.644337 + 0.764742i \(0.722866\pi\)
\(492\) 0 0
\(493\) 721.804 0.0659400
\(494\) −25294.7 −2.30377
\(495\) 0 0
\(496\) −1438.47 −0.130220
\(497\) 9427.80 0.850894
\(498\) 0 0
\(499\) 831.548 0.0745996 0.0372998 0.999304i \(-0.488124\pi\)
0.0372998 + 0.999304i \(0.488124\pi\)
\(500\) −10271.6 −0.918715
\(501\) 0 0
\(502\) 5138.56 0.456862
\(503\) −14503.5 −1.28565 −0.642824 0.766014i \(-0.722237\pi\)
−0.642824 + 0.766014i \(0.722237\pi\)
\(504\) 0 0
\(505\) −828.563 −0.0730110
\(506\) 47905.7 4.20883
\(507\) 0 0
\(508\) 8794.00 0.768052
\(509\) −3554.46 −0.309526 −0.154763 0.987952i \(-0.549461\pi\)
−0.154763 + 0.987952i \(0.549461\pi\)
\(510\) 0 0
\(511\) −3666.16 −0.317380
\(512\) 2478.40 0.213927
\(513\) 0 0
\(514\) −10024.9 −0.860267
\(515\) −1104.56 −0.0945098
\(516\) 0 0
\(517\) 17918.9 1.52431
\(518\) −30864.5 −2.61797
\(519\) 0 0
\(520\) −6741.01 −0.568486
\(521\) −3095.33 −0.260286 −0.130143 0.991495i \(-0.541544\pi\)
−0.130143 + 0.991495i \(0.541544\pi\)
\(522\) 0 0
\(523\) −10451.2 −0.873802 −0.436901 0.899510i \(-0.643924\pi\)
−0.436901 + 0.899510i \(0.643924\pi\)
\(524\) −4504.41 −0.375527
\(525\) 0 0
\(526\) −23541.6 −1.95145
\(527\) −10090.6 −0.834070
\(528\) 0 0
\(529\) 24469.1 2.01110
\(530\) −6673.35 −0.546928
\(531\) 0 0
\(532\) 13677.9 1.11469
\(533\) −19341.3 −1.57179
\(534\) 0 0
\(535\) 4245.47 0.343080
\(536\) 1663.69 0.134068
\(537\) 0 0
\(538\) −11503.5 −0.921841
\(539\) 4739.35 0.378735
\(540\) 0 0
\(541\) −8714.59 −0.692550 −0.346275 0.938133i \(-0.612554\pi\)
−0.346275 + 0.938133i \(0.612554\pi\)
\(542\) 12069.8 0.956538
\(543\) 0 0
\(544\) 7930.95 0.625067
\(545\) 908.401 0.0713975
\(546\) 0 0
\(547\) −7782.31 −0.608314 −0.304157 0.952622i \(-0.598375\pi\)
−0.304157 + 0.952622i \(0.598375\pi\)
\(548\) 13416.0 1.04581
\(549\) 0 0
\(550\) −28693.9 −2.22456
\(551\) 963.651 0.0745062
\(552\) 0 0
\(553\) 9007.99 0.692692
\(554\) −19718.2 −1.51218
\(555\) 0 0
\(556\) 28074.9 2.14144
\(557\) 12895.7 0.980982 0.490491 0.871446i \(-0.336817\pi\)
0.490491 + 0.871446i \(0.336817\pi\)
\(558\) 0 0
\(559\) −17662.5 −1.33639
\(560\) 352.835 0.0266250
\(561\) 0 0
\(562\) −18847.9 −1.41468
\(563\) −10805.3 −0.808863 −0.404431 0.914568i \(-0.632531\pi\)
−0.404431 + 0.914568i \(0.632531\pi\)
\(564\) 0 0
\(565\) 7486.94 0.557483
\(566\) −7145.78 −0.530670
\(567\) 0 0
\(568\) −14485.9 −1.07010
\(569\) 11897.7 0.876589 0.438294 0.898831i \(-0.355583\pi\)
0.438294 + 0.898831i \(0.355583\pi\)
\(570\) 0 0
\(571\) 20402.8 1.49532 0.747662 0.664080i \(-0.231176\pi\)
0.747662 + 0.664080i \(0.231176\pi\)
\(572\) −61575.4 −4.50105
\(573\) 0 0
\(574\) 16740.3 1.21729
\(575\) −21943.7 −1.59151
\(576\) 0 0
\(577\) 21770.5 1.57074 0.785370 0.619027i \(-0.212473\pi\)
0.785370 + 0.619027i \(0.212473\pi\)
\(578\) 11996.1 0.863276
\(579\) 0 0
\(580\) 643.009 0.0460336
\(581\) −18851.5 −1.34611
\(582\) 0 0
\(583\) −24345.8 −1.72950
\(584\) 5633.09 0.399142
\(585\) 0 0
\(586\) 14066.1 0.991576
\(587\) −21616.4 −1.51994 −0.759970 0.649959i \(-0.774786\pi\)
−0.759970 + 0.649959i \(0.774786\pi\)
\(588\) 0 0
\(589\) −13471.6 −0.942424
\(590\) −876.623 −0.0611695
\(591\) 0 0
\(592\) 2867.91 0.199106
\(593\) 18257.7 1.26434 0.632172 0.774828i \(-0.282164\pi\)
0.632172 + 0.774828i \(0.282164\pi\)
\(594\) 0 0
\(595\) 2475.09 0.170536
\(596\) 30307.7 2.08297
\(597\) 0 0
\(598\) −75372.8 −5.15422
\(599\) 2679.25 0.182756 0.0913781 0.995816i \(-0.470873\pi\)
0.0913781 + 0.995816i \(0.470873\pi\)
\(600\) 0 0
\(601\) −19874.2 −1.34889 −0.674447 0.738324i \(-0.735618\pi\)
−0.674447 + 0.738324i \(0.735618\pi\)
\(602\) 15287.2 1.03498
\(603\) 0 0
\(604\) 5287.93 0.356230
\(605\) 5171.79 0.347542
\(606\) 0 0
\(607\) −20437.2 −1.36659 −0.683297 0.730141i \(-0.739454\pi\)
−0.683297 + 0.730141i \(0.739454\pi\)
\(608\) 10588.3 0.706269
\(609\) 0 0
\(610\) 5843.39 0.387856
\(611\) −28192.8 −1.86671
\(612\) 0 0
\(613\) 19812.1 1.30539 0.652695 0.757621i \(-0.273638\pi\)
0.652695 + 0.757621i \(0.273638\pi\)
\(614\) −24994.3 −1.64281
\(615\) 0 0
\(616\) 21285.4 1.39223
\(617\) −27960.5 −1.82439 −0.912194 0.409758i \(-0.865613\pi\)
−0.912194 + 0.409758i \(0.865613\pi\)
\(618\) 0 0
\(619\) −28522.1 −1.85202 −0.926008 0.377503i \(-0.876783\pi\)
−0.926008 + 0.377503i \(0.876783\pi\)
\(620\) −8989.10 −0.582276
\(621\) 0 0
\(622\) −4363.38 −0.281279
\(623\) 21349.1 1.37293
\(624\) 0 0
\(625\) 11849.2 0.758349
\(626\) −22875.5 −1.46053
\(627\) 0 0
\(628\) −30437.2 −1.93404
\(629\) 20118.0 1.27529
\(630\) 0 0
\(631\) −13610.9 −0.858706 −0.429353 0.903137i \(-0.641258\pi\)
−0.429353 + 0.903137i \(0.641258\pi\)
\(632\) −13840.9 −0.871140
\(633\) 0 0
\(634\) −17923.7 −1.12278
\(635\) 2124.50 0.132769
\(636\) 0 0
\(637\) −7456.68 −0.463806
\(638\) 3754.76 0.232998
\(639\) 0 0
\(640\) 7880.44 0.486721
\(641\) 19792.9 1.21961 0.609807 0.792550i \(-0.291247\pi\)
0.609807 + 0.792550i \(0.291247\pi\)
\(642\) 0 0
\(643\) 14995.9 0.919719 0.459859 0.887992i \(-0.347900\pi\)
0.459859 + 0.887992i \(0.347900\pi\)
\(644\) 40757.2 2.49388
\(645\) 0 0
\(646\) −14270.2 −0.869126
\(647\) −29860.2 −1.81441 −0.907207 0.420684i \(-0.861790\pi\)
−0.907207 + 0.420684i \(0.861790\pi\)
\(648\) 0 0
\(649\) −3198.11 −0.193431
\(650\) 45145.7 2.72424
\(651\) 0 0
\(652\) 19093.8 1.14689
\(653\) −17572.2 −1.05307 −0.526533 0.850155i \(-0.676508\pi\)
−0.526533 + 0.850155i \(0.676508\pi\)
\(654\) 0 0
\(655\) −1088.20 −0.0649153
\(656\) −1555.50 −0.0925791
\(657\) 0 0
\(658\) 24401.3 1.44569
\(659\) −14231.1 −0.841222 −0.420611 0.907241i \(-0.638184\pi\)
−0.420611 + 0.907241i \(0.638184\pi\)
\(660\) 0 0
\(661\) −28811.1 −1.69535 −0.847673 0.530520i \(-0.821997\pi\)
−0.847673 + 0.530520i \(0.821997\pi\)
\(662\) −25515.4 −1.49802
\(663\) 0 0
\(664\) 28965.5 1.69289
\(665\) 3304.39 0.192690
\(666\) 0 0
\(667\) 2871.47 0.166692
\(668\) −3600.38 −0.208537
\(669\) 0 0
\(670\) 1006.34 0.0580273
\(671\) 21317.9 1.22648
\(672\) 0 0
\(673\) 16148.1 0.924909 0.462455 0.886643i \(-0.346969\pi\)
0.462455 + 0.886643i \(0.346969\pi\)
\(674\) −14778.4 −0.844572
\(675\) 0 0
\(676\) 67616.5 3.84709
\(677\) −20624.3 −1.17084 −0.585418 0.810731i \(-0.699070\pi\)
−0.585418 + 0.810731i \(0.699070\pi\)
\(678\) 0 0
\(679\) 21766.1 1.23020
\(680\) −3803.00 −0.214468
\(681\) 0 0
\(682\) −52490.6 −2.94717
\(683\) 26607.3 1.49063 0.745314 0.666714i \(-0.232300\pi\)
0.745314 + 0.666714i \(0.232300\pi\)
\(684\) 0 0
\(685\) 3241.11 0.180783
\(686\) 31772.4 1.76833
\(687\) 0 0
\(688\) −1420.48 −0.0787138
\(689\) 38304.6 2.11798
\(690\) 0 0
\(691\) 31832.9 1.75251 0.876254 0.481850i \(-0.160035\pi\)
0.876254 + 0.481850i \(0.160035\pi\)
\(692\) −20024.1 −1.10000
\(693\) 0 0
\(694\) −16317.6 −0.892520
\(695\) 6782.49 0.370179
\(696\) 0 0
\(697\) −10911.6 −0.592978
\(698\) 9774.42 0.530039
\(699\) 0 0
\(700\) −24412.2 −1.31813
\(701\) 26303.0 1.41719 0.708594 0.705616i \(-0.249330\pi\)
0.708594 + 0.705616i \(0.249330\pi\)
\(702\) 0 0
\(703\) 26858.8 1.44096
\(704\) 44230.4 2.36789
\(705\) 0 0
\(706\) −9057.92 −0.482860
\(707\) −4116.31 −0.218967
\(708\) 0 0
\(709\) 31256.4 1.65565 0.827826 0.560984i \(-0.189577\pi\)
0.827826 + 0.560984i \(0.189577\pi\)
\(710\) −8762.32 −0.463160
\(711\) 0 0
\(712\) −32803.1 −1.72661
\(713\) −40142.4 −2.10848
\(714\) 0 0
\(715\) −14875.7 −0.778072
\(716\) −23193.8 −1.21060
\(717\) 0 0
\(718\) 19801.4 1.02922
\(719\) −20206.1 −1.04807 −0.524034 0.851697i \(-0.675574\pi\)
−0.524034 + 0.851697i \(0.675574\pi\)
\(720\) 0 0
\(721\) −5487.45 −0.283444
\(722\) 12618.7 0.650442
\(723\) 0 0
\(724\) −49745.1 −2.55354
\(725\) −1719.91 −0.0881047
\(726\) 0 0
\(727\) −15044.8 −0.767510 −0.383755 0.923435i \(-0.625369\pi\)
−0.383755 + 0.923435i \(0.625369\pi\)
\(728\) −33489.5 −1.70495
\(729\) 0 0
\(730\) 3407.37 0.172757
\(731\) −9964.44 −0.504170
\(732\) 0 0
\(733\) 30276.0 1.52560 0.762802 0.646632i \(-0.223823\pi\)
0.762802 + 0.646632i \(0.223823\pi\)
\(734\) 5754.71 0.289387
\(735\) 0 0
\(736\) 31550.8 1.58013
\(737\) 3671.35 0.183495
\(738\) 0 0
\(739\) −7587.02 −0.377663 −0.188832 0.982009i \(-0.560470\pi\)
−0.188832 + 0.982009i \(0.560470\pi\)
\(740\) 17921.9 0.890298
\(741\) 0 0
\(742\) −33153.3 −1.64029
\(743\) 12898.6 0.636882 0.318441 0.947943i \(-0.396841\pi\)
0.318441 + 0.947943i \(0.396841\pi\)
\(744\) 0 0
\(745\) 7321.91 0.360072
\(746\) −51490.1 −2.52706
\(747\) 0 0
\(748\) −34738.3 −1.69807
\(749\) 21091.6 1.02893
\(750\) 0 0
\(751\) −18857.3 −0.916260 −0.458130 0.888885i \(-0.651481\pi\)
−0.458130 + 0.888885i \(0.651481\pi\)
\(752\) −2267.36 −0.109949
\(753\) 0 0
\(754\) −5907.58 −0.285333
\(755\) 1277.49 0.0615795
\(756\) 0 0
\(757\) −25156.1 −1.20781 −0.603905 0.797056i \(-0.706390\pi\)
−0.603905 + 0.797056i \(0.706390\pi\)
\(758\) −31128.9 −1.49162
\(759\) 0 0
\(760\) −5077.23 −0.242330
\(761\) −2380.10 −0.113375 −0.0566875 0.998392i \(-0.518054\pi\)
−0.0566875 + 0.998392i \(0.518054\pi\)
\(762\) 0 0
\(763\) 4512.95 0.214128
\(764\) −20033.5 −0.948673
\(765\) 0 0
\(766\) 23881.3 1.12646
\(767\) 5031.76 0.236879
\(768\) 0 0
\(769\) −27770.6 −1.30226 −0.651128 0.758968i \(-0.725704\pi\)
−0.651128 + 0.758968i \(0.725704\pi\)
\(770\) 12875.2 0.602584
\(771\) 0 0
\(772\) 35623.9 1.66079
\(773\) 15649.4 0.728165 0.364082 0.931367i \(-0.381383\pi\)
0.364082 + 0.931367i \(0.381383\pi\)
\(774\) 0 0
\(775\) 24043.9 1.11443
\(776\) −33443.9 −1.54712
\(777\) 0 0
\(778\) 49589.6 2.28518
\(779\) −14567.6 −0.670012
\(780\) 0 0
\(781\) −31966.8 −1.46461
\(782\) −42522.2 −1.94449
\(783\) 0 0
\(784\) −599.691 −0.0273183
\(785\) −7353.20 −0.334327
\(786\) 0 0
\(787\) 12216.7 0.553338 0.276669 0.960965i \(-0.410769\pi\)
0.276669 + 0.960965i \(0.410769\pi\)
\(788\) 33253.6 1.50331
\(789\) 0 0
\(790\) −8372.14 −0.377047
\(791\) 37195.2 1.67195
\(792\) 0 0
\(793\) −33540.7 −1.50197
\(794\) −17342.5 −0.775140
\(795\) 0 0
\(796\) 58752.3 2.61611
\(797\) −31212.6 −1.38721 −0.693605 0.720356i \(-0.743979\pi\)
−0.693605 + 0.720356i \(0.743979\pi\)
\(798\) 0 0
\(799\) −15905.2 −0.704237
\(800\) −18897.8 −0.835173
\(801\) 0 0
\(802\) 48676.9 2.14319
\(803\) 12430.8 0.546295
\(804\) 0 0
\(805\) 9846.36 0.431104
\(806\) 82586.5 3.60916
\(807\) 0 0
\(808\) 6324.76 0.275377
\(809\) 27090.0 1.17730 0.588650 0.808388i \(-0.299660\pi\)
0.588650 + 0.808388i \(0.299660\pi\)
\(810\) 0 0
\(811\) 2136.04 0.0924863 0.0462431 0.998930i \(-0.485275\pi\)
0.0462431 + 0.998930i \(0.485275\pi\)
\(812\) 3194.48 0.138059
\(813\) 0 0
\(814\) 104652. 4.50622
\(815\) 4612.78 0.198256
\(816\) 0 0
\(817\) −13303.1 −0.569666
\(818\) 66023.0 2.82205
\(819\) 0 0
\(820\) −9720.43 −0.413966
\(821\) 13041.2 0.554373 0.277187 0.960816i \(-0.410598\pi\)
0.277187 + 0.960816i \(0.410598\pi\)
\(822\) 0 0
\(823\) −15741.7 −0.666734 −0.333367 0.942797i \(-0.608185\pi\)
−0.333367 + 0.942797i \(0.608185\pi\)
\(824\) 8431.53 0.356464
\(825\) 0 0
\(826\) −4355.08 −0.183453
\(827\) −29188.2 −1.22730 −0.613648 0.789580i \(-0.710299\pi\)
−0.613648 + 0.789580i \(0.710299\pi\)
\(828\) 0 0
\(829\) −23688.9 −0.992461 −0.496230 0.868191i \(-0.665283\pi\)
−0.496230 + 0.868191i \(0.665283\pi\)
\(830\) 17520.8 0.732718
\(831\) 0 0
\(832\) −69590.2 −2.89977
\(833\) −4206.75 −0.174976
\(834\) 0 0
\(835\) −869.799 −0.0360487
\(836\) −46377.7 −1.91867
\(837\) 0 0
\(838\) 12032.3 0.496003
\(839\) −46609.6 −1.91793 −0.958965 0.283525i \(-0.908496\pi\)
−0.958965 + 0.283525i \(0.908496\pi\)
\(840\) 0 0
\(841\) −24163.9 −0.990772
\(842\) 54325.8 2.22350
\(843\) 0 0
\(844\) −53900.1 −2.19825
\(845\) 16335.2 0.665026
\(846\) 0 0
\(847\) 25693.5 1.04231
\(848\) 3080.59 0.124750
\(849\) 0 0
\(850\) 25469.3 1.02775
\(851\) 80033.2 3.22386
\(852\) 0 0
\(853\) 15069.2 0.604877 0.302439 0.953169i \(-0.402199\pi\)
0.302439 + 0.953169i \(0.402199\pi\)
\(854\) 29030.1 1.16322
\(855\) 0 0
\(856\) −32407.4 −1.29400
\(857\) 32618.8 1.30016 0.650081 0.759865i \(-0.274735\pi\)
0.650081 + 0.759865i \(0.274735\pi\)
\(858\) 0 0
\(859\) −6931.99 −0.275339 −0.137670 0.990478i \(-0.543961\pi\)
−0.137670 + 0.990478i \(0.543961\pi\)
\(860\) −8876.68 −0.351968
\(861\) 0 0
\(862\) 54871.5 2.16813
\(863\) −6692.94 −0.263998 −0.131999 0.991250i \(-0.542140\pi\)
−0.131999 + 0.991250i \(0.542140\pi\)
\(864\) 0 0
\(865\) −4837.54 −0.190152
\(866\) −14717.4 −0.577505
\(867\) 0 0
\(868\) −44658.0 −1.74630
\(869\) −30543.4 −1.19230
\(870\) 0 0
\(871\) −5776.34 −0.224711
\(872\) −6934.20 −0.269291
\(873\) 0 0
\(874\) −56769.7 −2.19710
\(875\) −12327.9 −0.476297
\(876\) 0 0
\(877\) 7773.39 0.299303 0.149652 0.988739i \(-0.452185\pi\)
0.149652 + 0.988739i \(0.452185\pi\)
\(878\) −75820.0 −2.91435
\(879\) 0 0
\(880\) −1196.36 −0.0458286
\(881\) 8298.43 0.317345 0.158673 0.987331i \(-0.449279\pi\)
0.158673 + 0.987331i \(0.449279\pi\)
\(882\) 0 0
\(883\) −30480.9 −1.16168 −0.580841 0.814017i \(-0.697276\pi\)
−0.580841 + 0.814017i \(0.697276\pi\)
\(884\) 54655.7 2.07949
\(885\) 0 0
\(886\) 2919.57 0.110705
\(887\) −7695.04 −0.291290 −0.145645 0.989337i \(-0.546526\pi\)
−0.145645 + 0.989337i \(0.546526\pi\)
\(888\) 0 0
\(889\) 10554.6 0.398188
\(890\) −19842.1 −0.747314
\(891\) 0 0
\(892\) 47063.4 1.76659
\(893\) −21234.4 −0.795724
\(894\) 0 0
\(895\) −5603.29 −0.209271
\(896\) 39150.1 1.45973
\(897\) 0 0
\(898\) 31267.9 1.16194
\(899\) −3146.29 −0.116724
\(900\) 0 0
\(901\) 21609.9 0.799034
\(902\) −56761.1 −2.09528
\(903\) 0 0
\(904\) −57150.9 −2.10267
\(905\) −12017.7 −0.441416
\(906\) 0 0
\(907\) 20682.7 0.757176 0.378588 0.925565i \(-0.376410\pi\)
0.378588 + 0.925565i \(0.376410\pi\)
\(908\) −1289.12 −0.0471156
\(909\) 0 0
\(910\) −20257.3 −0.737937
\(911\) 27515.9 1.00071 0.500353 0.865822i \(-0.333204\pi\)
0.500353 + 0.865822i \(0.333204\pi\)
\(912\) 0 0
\(913\) 63919.6 2.31701
\(914\) 56693.9 2.05171
\(915\) 0 0
\(916\) 13737.3 0.495516
\(917\) −5406.19 −0.194687
\(918\) 0 0
\(919\) −4136.30 −0.148470 −0.0742350 0.997241i \(-0.523651\pi\)
−0.0742350 + 0.997241i \(0.523651\pi\)
\(920\) −15129.0 −0.542163
\(921\) 0 0
\(922\) −50741.1 −1.81244
\(923\) 50295.2 1.79359
\(924\) 0 0
\(925\) −47937.1 −1.70396
\(926\) 55567.7 1.97200
\(927\) 0 0
\(928\) 2472.89 0.0874748
\(929\) 41730.5 1.47377 0.736886 0.676017i \(-0.236295\pi\)
0.736886 + 0.676017i \(0.236295\pi\)
\(930\) 0 0
\(931\) −5616.27 −0.197708
\(932\) −62457.5 −2.19513
\(933\) 0 0
\(934\) −43837.9 −1.53578
\(935\) −8392.28 −0.293537
\(936\) 0 0
\(937\) 21380.2 0.745422 0.372711 0.927947i \(-0.378428\pi\)
0.372711 + 0.927947i \(0.378428\pi\)
\(938\) 4999.51 0.174030
\(939\) 0 0
\(940\) −14168.9 −0.491637
\(941\) 21613.7 0.748764 0.374382 0.927275i \(-0.377855\pi\)
0.374382 + 0.927275i \(0.377855\pi\)
\(942\) 0 0
\(943\) −43408.3 −1.49901
\(944\) 404.671 0.0139523
\(945\) 0 0
\(946\) −51834.2 −1.78147
\(947\) 6008.44 0.206175 0.103088 0.994672i \(-0.467128\pi\)
0.103088 + 0.994672i \(0.467128\pi\)
\(948\) 0 0
\(949\) −19558.1 −0.669003
\(950\) 34003.1 1.16127
\(951\) 0 0
\(952\) −18893.4 −0.643212
\(953\) 5173.79 0.175861 0.0879305 0.996127i \(-0.471975\pi\)
0.0879305 + 0.996127i \(0.471975\pi\)
\(954\) 0 0
\(955\) −4839.80 −0.163992
\(956\) −3125.64 −0.105743
\(957\) 0 0
\(958\) 7852.68 0.264832
\(959\) 16101.9 0.542186
\(960\) 0 0
\(961\) 14193.3 0.476430
\(962\) −164655. −5.51840
\(963\) 0 0
\(964\) −17777.2 −0.593948
\(965\) 8606.22 0.287092
\(966\) 0 0
\(967\) 26508.3 0.881540 0.440770 0.897620i \(-0.354705\pi\)
0.440770 + 0.897620i \(0.354705\pi\)
\(968\) −39478.4 −1.31083
\(969\) 0 0
\(970\) −20229.7 −0.669625
\(971\) 22770.6 0.752569 0.376285 0.926504i \(-0.377202\pi\)
0.376285 + 0.926504i \(0.377202\pi\)
\(972\) 0 0
\(973\) 33695.5 1.11020
\(974\) −17025.4 −0.560091
\(975\) 0 0
\(976\) −2697.46 −0.0884667
\(977\) 10137.9 0.331974 0.165987 0.986128i \(-0.446919\pi\)
0.165987 + 0.986128i \(0.446919\pi\)
\(978\) 0 0
\(979\) −72388.3 −2.36317
\(980\) −3747.53 −0.122153
\(981\) 0 0
\(982\) 64737.6 2.10373
\(983\) 19066.2 0.618635 0.309317 0.950959i \(-0.399899\pi\)
0.309317 + 0.950959i \(0.399899\pi\)
\(984\) 0 0
\(985\) 8033.58 0.259869
\(986\) −3332.81 −0.107645
\(987\) 0 0
\(988\) 72968.7 2.34964
\(989\) −39640.4 −1.27451
\(990\) 0 0
\(991\) −17491.5 −0.560681 −0.280341 0.959901i \(-0.590447\pi\)
−0.280341 + 0.959901i \(0.590447\pi\)
\(992\) −34570.4 −1.10646
\(993\) 0 0
\(994\) −43531.3 −1.38906
\(995\) 14193.7 0.452232
\(996\) 0 0
\(997\) −7297.85 −0.231821 −0.115910 0.993260i \(-0.536979\pi\)
−0.115910 + 0.993260i \(0.536979\pi\)
\(998\) −3839.54 −0.121782
\(999\) 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 531.4.a.f.1.2 8
3.2 odd 2 177.4.a.c.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.c.1.7 8 3.2 odd 2
531.4.a.f.1.2 8 1.1 even 1 trivial