Properties

Label 6003.2.a.p.1.1
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $0$
Dimension $14$
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] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, 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("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.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: \(47.9341963334\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 124 x^{10} - 216 x^{9} - 420 x^{8} + 647 x^{7} + 750 x^{6} - 939 x^{5} - 717 x^{4} + 604 x^{3} + 352 x^{2} - 128 x - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.55124\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.55124 q^{2} +4.50880 q^{4} +1.64740 q^{5} -2.91756 q^{7} -6.40055 q^{8} +O(q^{10})\) \(q-2.55124 q^{2} +4.50880 q^{4} +1.64740 q^{5} -2.91756 q^{7} -6.40055 q^{8} -4.20290 q^{10} +6.56108 q^{11} +5.20247 q^{13} +7.44337 q^{14} +7.31171 q^{16} +1.25844 q^{17} -4.89571 q^{19} +7.42779 q^{20} -16.7389 q^{22} +1.00000 q^{23} -2.28608 q^{25} -13.2727 q^{26} -13.1547 q^{28} +1.00000 q^{29} -2.43452 q^{31} -5.85279 q^{32} -3.21057 q^{34} -4.80637 q^{35} +2.46802 q^{37} +12.4901 q^{38} -10.5443 q^{40} +6.73385 q^{41} +0.485941 q^{43} +29.5826 q^{44} -2.55124 q^{46} +2.92199 q^{47} +1.51213 q^{49} +5.83233 q^{50} +23.4569 q^{52} +7.90067 q^{53} +10.8087 q^{55} +18.6740 q^{56} -2.55124 q^{58} +1.28036 q^{59} +11.7703 q^{61} +6.21104 q^{62} +0.308435 q^{64} +8.57053 q^{65} +3.52581 q^{67} +5.67404 q^{68} +12.2622 q^{70} -9.97469 q^{71} -5.56126 q^{73} -6.29651 q^{74} -22.0738 q^{76} -19.1423 q^{77} -5.84862 q^{79} +12.0453 q^{80} -17.1796 q^{82} +10.9870 q^{83} +2.07314 q^{85} -1.23975 q^{86} -41.9945 q^{88} -16.6318 q^{89} -15.1785 q^{91} +4.50880 q^{92} -7.45469 q^{94} -8.06517 q^{95} +6.97784 q^{97} -3.85780 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.55124 −1.80400 −0.901998 0.431740i \(-0.857900\pi\)
−0.901998 + 0.431740i \(0.857900\pi\)
\(3\) 0 0
\(4\) 4.50880 2.25440
\(5\) 1.64740 0.736739 0.368369 0.929680i \(-0.379916\pi\)
0.368369 + 0.929680i \(0.379916\pi\)
\(6\) 0 0
\(7\) −2.91756 −1.10273 −0.551366 0.834263i \(-0.685893\pi\)
−0.551366 + 0.834263i \(0.685893\pi\)
\(8\) −6.40055 −2.26294
\(9\) 0 0
\(10\) −4.20290 −1.32907
\(11\) 6.56108 1.97824 0.989120 0.147113i \(-0.0469980\pi\)
0.989120 + 0.147113i \(0.0469980\pi\)
\(12\) 0 0
\(13\) 5.20247 1.44290 0.721452 0.692464i \(-0.243475\pi\)
0.721452 + 0.692464i \(0.243475\pi\)
\(14\) 7.44337 1.98933
\(15\) 0 0
\(16\) 7.31171 1.82793
\(17\) 1.25844 0.305216 0.152608 0.988287i \(-0.451233\pi\)
0.152608 + 0.988287i \(0.451233\pi\)
\(18\) 0 0
\(19\) −4.89571 −1.12315 −0.561576 0.827425i \(-0.689805\pi\)
−0.561576 + 0.827425i \(0.689805\pi\)
\(20\) 7.42779 1.66091
\(21\) 0 0
\(22\) −16.7389 −3.56874
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −2.28608 −0.457216
\(26\) −13.2727 −2.60299
\(27\) 0 0
\(28\) −13.1547 −2.48600
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −2.43452 −0.437253 −0.218627 0.975809i \(-0.570158\pi\)
−0.218627 + 0.975809i \(0.570158\pi\)
\(32\) −5.85279 −1.03464
\(33\) 0 0
\(34\) −3.21057 −0.550608
\(35\) −4.80637 −0.812425
\(36\) 0 0
\(37\) 2.46802 0.405741 0.202870 0.979206i \(-0.434973\pi\)
0.202870 + 0.979206i \(0.434973\pi\)
\(38\) 12.4901 2.02616
\(39\) 0 0
\(40\) −10.5443 −1.66719
\(41\) 6.73385 1.05165 0.525825 0.850593i \(-0.323757\pi\)
0.525825 + 0.850593i \(0.323757\pi\)
\(42\) 0 0
\(43\) 0.485941 0.0741053 0.0370526 0.999313i \(-0.488203\pi\)
0.0370526 + 0.999313i \(0.488203\pi\)
\(44\) 29.5826 4.45975
\(45\) 0 0
\(46\) −2.55124 −0.376159
\(47\) 2.92199 0.426216 0.213108 0.977029i \(-0.431641\pi\)
0.213108 + 0.977029i \(0.431641\pi\)
\(48\) 0 0
\(49\) 1.51213 0.216019
\(50\) 5.83233 0.824816
\(51\) 0 0
\(52\) 23.4569 3.25289
\(53\) 7.90067 1.08524 0.542620 0.839978i \(-0.317432\pi\)
0.542620 + 0.839978i \(0.317432\pi\)
\(54\) 0 0
\(55\) 10.8087 1.45745
\(56\) 18.6740 2.49541
\(57\) 0 0
\(58\) −2.55124 −0.334994
\(59\) 1.28036 0.166688 0.0833441 0.996521i \(-0.473440\pi\)
0.0833441 + 0.996521i \(0.473440\pi\)
\(60\) 0 0
\(61\) 11.7703 1.50703 0.753515 0.657431i \(-0.228357\pi\)
0.753515 + 0.657431i \(0.228357\pi\)
\(62\) 6.21104 0.788803
\(63\) 0 0
\(64\) 0.308435 0.0385543
\(65\) 8.57053 1.06304
\(66\) 0 0
\(67\) 3.52581 0.430747 0.215373 0.976532i \(-0.430903\pi\)
0.215373 + 0.976532i \(0.430903\pi\)
\(68\) 5.67404 0.688079
\(69\) 0 0
\(70\) 12.2622 1.46561
\(71\) −9.97469 −1.18378 −0.591889 0.806020i \(-0.701618\pi\)
−0.591889 + 0.806020i \(0.701618\pi\)
\(72\) 0 0
\(73\) −5.56126 −0.650896 −0.325448 0.945560i \(-0.605515\pi\)
−0.325448 + 0.945560i \(0.605515\pi\)
\(74\) −6.29651 −0.731954
\(75\) 0 0
\(76\) −22.0738 −2.53204
\(77\) −19.1423 −2.18147
\(78\) 0 0
\(79\) −5.84862 −0.658021 −0.329011 0.944326i \(-0.606715\pi\)
−0.329011 + 0.944326i \(0.606715\pi\)
\(80\) 12.0453 1.34670
\(81\) 0 0
\(82\) −17.1796 −1.89717
\(83\) 10.9870 1.20598 0.602991 0.797748i \(-0.293976\pi\)
0.602991 + 0.797748i \(0.293976\pi\)
\(84\) 0 0
\(85\) 2.07314 0.224864
\(86\) −1.23975 −0.133686
\(87\) 0 0
\(88\) −41.9945 −4.47663
\(89\) −16.6318 −1.76297 −0.881483 0.472216i \(-0.843454\pi\)
−0.881483 + 0.472216i \(0.843454\pi\)
\(90\) 0 0
\(91\) −15.1785 −1.59114
\(92\) 4.50880 0.470075
\(93\) 0 0
\(94\) −7.45469 −0.768893
\(95\) −8.06517 −0.827469
\(96\) 0 0
\(97\) 6.97784 0.708492 0.354246 0.935152i \(-0.384737\pi\)
0.354246 + 0.935152i \(0.384737\pi\)
\(98\) −3.85780 −0.389697
\(99\) 0 0
\(100\) −10.3075 −1.03075
\(101\) −4.13909 −0.411855 −0.205927 0.978567i \(-0.566021\pi\)
−0.205927 + 0.978567i \(0.566021\pi\)
\(102\) 0 0
\(103\) 12.8987 1.27095 0.635473 0.772123i \(-0.280805\pi\)
0.635473 + 0.772123i \(0.280805\pi\)
\(104\) −33.2987 −3.26520
\(105\) 0 0
\(106\) −20.1565 −1.95777
\(107\) 20.4276 1.97481 0.987406 0.158207i \(-0.0505714\pi\)
0.987406 + 0.158207i \(0.0505714\pi\)
\(108\) 0 0
\(109\) 3.37849 0.323601 0.161800 0.986824i \(-0.448270\pi\)
0.161800 + 0.986824i \(0.448270\pi\)
\(110\) −27.5756 −2.62923
\(111\) 0 0
\(112\) −21.3323 −2.01571
\(113\) 8.18175 0.769674 0.384837 0.922984i \(-0.374258\pi\)
0.384837 + 0.922984i \(0.374258\pi\)
\(114\) 0 0
\(115\) 1.64740 0.153621
\(116\) 4.50880 0.418632
\(117\) 0 0
\(118\) −3.26649 −0.300705
\(119\) −3.67156 −0.336571
\(120\) 0 0
\(121\) 32.0477 2.91343
\(122\) −30.0287 −2.71867
\(123\) 0 0
\(124\) −10.9768 −0.985744
\(125\) −12.0031 −1.07359
\(126\) 0 0
\(127\) 6.53978 0.580312 0.290156 0.956979i \(-0.406293\pi\)
0.290156 + 0.956979i \(0.406293\pi\)
\(128\) 10.9187 0.965086
\(129\) 0 0
\(130\) −21.8654 −1.91773
\(131\) −14.4084 −1.25887 −0.629435 0.777053i \(-0.716714\pi\)
−0.629435 + 0.777053i \(0.716714\pi\)
\(132\) 0 0
\(133\) 14.2835 1.23854
\(134\) −8.99518 −0.777065
\(135\) 0 0
\(136\) −8.05469 −0.690684
\(137\) 14.8821 1.27146 0.635730 0.771911i \(-0.280699\pi\)
0.635730 + 0.771911i \(0.280699\pi\)
\(138\) 0 0
\(139\) −13.3393 −1.13142 −0.565711 0.824603i \(-0.691398\pi\)
−0.565711 + 0.824603i \(0.691398\pi\)
\(140\) −21.6710 −1.83153
\(141\) 0 0
\(142\) 25.4478 2.13553
\(143\) 34.1338 2.85441
\(144\) 0 0
\(145\) 1.64740 0.136809
\(146\) 14.1881 1.17421
\(147\) 0 0
\(148\) 11.1278 0.914702
\(149\) −18.6630 −1.52893 −0.764465 0.644665i \(-0.776997\pi\)
−0.764465 + 0.644665i \(0.776997\pi\)
\(150\) 0 0
\(151\) −10.5101 −0.855300 −0.427650 0.903945i \(-0.640658\pi\)
−0.427650 + 0.903945i \(0.640658\pi\)
\(152\) 31.3352 2.54162
\(153\) 0 0
\(154\) 48.8365 3.93536
\(155\) −4.01063 −0.322141
\(156\) 0 0
\(157\) 9.85993 0.786908 0.393454 0.919344i \(-0.371280\pi\)
0.393454 + 0.919344i \(0.371280\pi\)
\(158\) 14.9212 1.18707
\(159\) 0 0
\(160\) −9.64188 −0.762257
\(161\) −2.91756 −0.229936
\(162\) 0 0
\(163\) −1.33183 −0.104317 −0.0521584 0.998639i \(-0.516610\pi\)
−0.0521584 + 0.998639i \(0.516610\pi\)
\(164\) 30.3616 2.37084
\(165\) 0 0
\(166\) −28.0305 −2.17559
\(167\) −15.9888 −1.23725 −0.618625 0.785687i \(-0.712310\pi\)
−0.618625 + 0.785687i \(0.712310\pi\)
\(168\) 0 0
\(169\) 14.0656 1.08197
\(170\) −5.28908 −0.405654
\(171\) 0 0
\(172\) 2.19101 0.167063
\(173\) 5.44324 0.413842 0.206921 0.978358i \(-0.433656\pi\)
0.206921 + 0.978358i \(0.433656\pi\)
\(174\) 0 0
\(175\) 6.66977 0.504187
\(176\) 47.9727 3.61608
\(177\) 0 0
\(178\) 42.4316 3.18038
\(179\) 5.56852 0.416210 0.208105 0.978106i \(-0.433270\pi\)
0.208105 + 0.978106i \(0.433270\pi\)
\(180\) 0 0
\(181\) 4.79673 0.356538 0.178269 0.983982i \(-0.442950\pi\)
0.178269 + 0.983982i \(0.442950\pi\)
\(182\) 38.7239 2.87041
\(183\) 0 0
\(184\) −6.40055 −0.471855
\(185\) 4.06582 0.298925
\(186\) 0 0
\(187\) 8.25670 0.603790
\(188\) 13.1747 0.960863
\(189\) 0 0
\(190\) 20.5762 1.49275
\(191\) 14.5446 1.05241 0.526206 0.850357i \(-0.323614\pi\)
0.526206 + 0.850357i \(0.323614\pi\)
\(192\) 0 0
\(193\) −24.2537 −1.74582 −0.872911 0.487880i \(-0.837771\pi\)
−0.872911 + 0.487880i \(0.837771\pi\)
\(194\) −17.8021 −1.27812
\(195\) 0 0
\(196\) 6.81790 0.486993
\(197\) 13.2341 0.942892 0.471446 0.881895i \(-0.343732\pi\)
0.471446 + 0.881895i \(0.343732\pi\)
\(198\) 0 0
\(199\) −17.7549 −1.25861 −0.629307 0.777157i \(-0.716661\pi\)
−0.629307 + 0.777157i \(0.716661\pi\)
\(200\) 14.6322 1.03465
\(201\) 0 0
\(202\) 10.5598 0.742984
\(203\) −2.91756 −0.204772
\(204\) 0 0
\(205\) 11.0933 0.774792
\(206\) −32.9076 −2.29278
\(207\) 0 0
\(208\) 38.0389 2.63752
\(209\) −32.1211 −2.22186
\(210\) 0 0
\(211\) −8.27562 −0.569717 −0.284859 0.958570i \(-0.591947\pi\)
−0.284859 + 0.958570i \(0.591947\pi\)
\(212\) 35.6226 2.44657
\(213\) 0 0
\(214\) −52.1157 −3.56255
\(215\) 0.800538 0.0545962
\(216\) 0 0
\(217\) 7.10285 0.482173
\(218\) −8.61933 −0.583775
\(219\) 0 0
\(220\) 48.7343 3.28567
\(221\) 6.54697 0.440397
\(222\) 0 0
\(223\) 3.09231 0.207077 0.103538 0.994625i \(-0.466984\pi\)
0.103538 + 0.994625i \(0.466984\pi\)
\(224\) 17.0758 1.14093
\(225\) 0 0
\(226\) −20.8736 −1.38849
\(227\) −17.4806 −1.16023 −0.580115 0.814534i \(-0.696992\pi\)
−0.580115 + 0.814534i \(0.696992\pi\)
\(228\) 0 0
\(229\) 20.5863 1.36038 0.680191 0.733035i \(-0.261897\pi\)
0.680191 + 0.733035i \(0.261897\pi\)
\(230\) −4.20290 −0.277131
\(231\) 0 0
\(232\) −6.40055 −0.420217
\(233\) −7.89100 −0.516957 −0.258478 0.966017i \(-0.583221\pi\)
−0.258478 + 0.966017i \(0.583221\pi\)
\(234\) 0 0
\(235\) 4.81368 0.314010
\(236\) 5.77288 0.375782
\(237\) 0 0
\(238\) 9.36701 0.607173
\(239\) −19.9245 −1.28881 −0.644405 0.764684i \(-0.722895\pi\)
−0.644405 + 0.764684i \(0.722895\pi\)
\(240\) 0 0
\(241\) 10.8421 0.698399 0.349200 0.937048i \(-0.386454\pi\)
0.349200 + 0.937048i \(0.386454\pi\)
\(242\) −81.7614 −5.25582
\(243\) 0 0
\(244\) 53.0699 3.39745
\(245\) 2.49108 0.159149
\(246\) 0 0
\(247\) −25.4697 −1.62060
\(248\) 15.5823 0.989476
\(249\) 0 0
\(250\) 30.6227 1.93675
\(251\) 21.9174 1.38342 0.691708 0.722177i \(-0.256858\pi\)
0.691708 + 0.722177i \(0.256858\pi\)
\(252\) 0 0
\(253\) 6.56108 0.412491
\(254\) −16.6845 −1.04688
\(255\) 0 0
\(256\) −28.4730 −1.77956
\(257\) 3.65769 0.228161 0.114080 0.993472i \(-0.463608\pi\)
0.114080 + 0.993472i \(0.463608\pi\)
\(258\) 0 0
\(259\) −7.20059 −0.447423
\(260\) 38.6428 2.39653
\(261\) 0 0
\(262\) 36.7593 2.27100
\(263\) −26.5342 −1.63617 −0.818085 0.575098i \(-0.804964\pi\)
−0.818085 + 0.575098i \(0.804964\pi\)
\(264\) 0 0
\(265\) 13.0155 0.799538
\(266\) −36.4406 −2.23431
\(267\) 0 0
\(268\) 15.8972 0.971076
\(269\) 18.9277 1.15404 0.577022 0.816729i \(-0.304215\pi\)
0.577022 + 0.816729i \(0.304215\pi\)
\(270\) 0 0
\(271\) −1.52707 −0.0927629 −0.0463814 0.998924i \(-0.514769\pi\)
−0.0463814 + 0.998924i \(0.514769\pi\)
\(272\) 9.20132 0.557912
\(273\) 0 0
\(274\) −37.9677 −2.29371
\(275\) −14.9992 −0.904483
\(276\) 0 0
\(277\) −29.7373 −1.78674 −0.893370 0.449322i \(-0.851666\pi\)
−0.893370 + 0.449322i \(0.851666\pi\)
\(278\) 34.0316 2.04108
\(279\) 0 0
\(280\) 30.7634 1.83847
\(281\) 28.9096 1.72460 0.862301 0.506397i \(-0.169023\pi\)
0.862301 + 0.506397i \(0.169023\pi\)
\(282\) 0 0
\(283\) 10.7138 0.636871 0.318436 0.947944i \(-0.396843\pi\)
0.318436 + 0.947944i \(0.396843\pi\)
\(284\) −44.9739 −2.66871
\(285\) 0 0
\(286\) −87.0833 −5.14935
\(287\) −19.6464 −1.15969
\(288\) 0 0
\(289\) −15.4163 −0.906843
\(290\) −4.20290 −0.246803
\(291\) 0 0
\(292\) −25.0746 −1.46738
\(293\) 18.4098 1.07551 0.537757 0.843100i \(-0.319272\pi\)
0.537757 + 0.843100i \(0.319272\pi\)
\(294\) 0 0
\(295\) 2.10926 0.122806
\(296\) −15.7967 −0.918165
\(297\) 0 0
\(298\) 47.6136 2.75818
\(299\) 5.20247 0.300866
\(300\) 0 0
\(301\) −1.41776 −0.0817183
\(302\) 26.8137 1.54296
\(303\) 0 0
\(304\) −35.7960 −2.05304
\(305\) 19.3903 1.11029
\(306\) 0 0
\(307\) 8.79956 0.502217 0.251109 0.967959i \(-0.419205\pi\)
0.251109 + 0.967959i \(0.419205\pi\)
\(308\) −86.3089 −4.91791
\(309\) 0 0
\(310\) 10.2321 0.581142
\(311\) 15.0932 0.855856 0.427928 0.903813i \(-0.359244\pi\)
0.427928 + 0.903813i \(0.359244\pi\)
\(312\) 0 0
\(313\) −17.7920 −1.00567 −0.502833 0.864384i \(-0.667709\pi\)
−0.502833 + 0.864384i \(0.667709\pi\)
\(314\) −25.1550 −1.41958
\(315\) 0 0
\(316\) −26.3703 −1.48344
\(317\) 14.9547 0.839938 0.419969 0.907538i \(-0.362041\pi\)
0.419969 + 0.907538i \(0.362041\pi\)
\(318\) 0 0
\(319\) 6.56108 0.367350
\(320\) 0.508114 0.0284044
\(321\) 0 0
\(322\) 7.44337 0.414803
\(323\) −6.16093 −0.342804
\(324\) 0 0
\(325\) −11.8933 −0.659719
\(326\) 3.39781 0.188187
\(327\) 0 0
\(328\) −43.1004 −2.37982
\(329\) −8.52507 −0.470002
\(330\) 0 0
\(331\) 14.5635 0.800480 0.400240 0.916410i \(-0.368927\pi\)
0.400240 + 0.916410i \(0.368927\pi\)
\(332\) 49.5383 2.71877
\(333\) 0 0
\(334\) 40.7912 2.23199
\(335\) 5.80842 0.317348
\(336\) 0 0
\(337\) −24.2334 −1.32008 −0.660039 0.751231i \(-0.729460\pi\)
−0.660039 + 0.751231i \(0.729460\pi\)
\(338\) −35.8848 −1.95188
\(339\) 0 0
\(340\) 9.34740 0.506934
\(341\) −15.9731 −0.864991
\(342\) 0 0
\(343\) 16.0112 0.864522
\(344\) −3.11029 −0.167696
\(345\) 0 0
\(346\) −13.8870 −0.746569
\(347\) −8.45970 −0.454140 −0.227070 0.973878i \(-0.572915\pi\)
−0.227070 + 0.973878i \(0.572915\pi\)
\(348\) 0 0
\(349\) −1.28519 −0.0687946 −0.0343973 0.999408i \(-0.510951\pi\)
−0.0343973 + 0.999408i \(0.510951\pi\)
\(350\) −17.0162 −0.909552
\(351\) 0 0
\(352\) −38.4006 −2.04676
\(353\) 13.9545 0.742725 0.371363 0.928488i \(-0.378891\pi\)
0.371363 + 0.928488i \(0.378891\pi\)
\(354\) 0 0
\(355\) −16.4323 −0.872135
\(356\) −74.9895 −3.97443
\(357\) 0 0
\(358\) −14.2066 −0.750842
\(359\) 30.4675 1.60801 0.804007 0.594620i \(-0.202697\pi\)
0.804007 + 0.594620i \(0.202697\pi\)
\(360\) 0 0
\(361\) 4.96794 0.261471
\(362\) −12.2376 −0.643194
\(363\) 0 0
\(364\) −68.4368 −3.58706
\(365\) −9.16161 −0.479540
\(366\) 0 0
\(367\) 5.18438 0.270623 0.135311 0.990803i \(-0.456797\pi\)
0.135311 + 0.990803i \(0.456797\pi\)
\(368\) 7.31171 0.381149
\(369\) 0 0
\(370\) −10.3729 −0.539259
\(371\) −23.0506 −1.19673
\(372\) 0 0
\(373\) 17.9429 0.929048 0.464524 0.885561i \(-0.346225\pi\)
0.464524 + 0.885561i \(0.346225\pi\)
\(374\) −21.0648 −1.08923
\(375\) 0 0
\(376\) −18.7024 −0.964501
\(377\) 5.20247 0.267941
\(378\) 0 0
\(379\) −22.6550 −1.16371 −0.581853 0.813294i \(-0.697672\pi\)
−0.581853 + 0.813294i \(0.697672\pi\)
\(380\) −36.3643 −1.86545
\(381\) 0 0
\(382\) −37.1068 −1.89855
\(383\) 11.7444 0.600113 0.300056 0.953921i \(-0.402994\pi\)
0.300056 + 0.953921i \(0.402994\pi\)
\(384\) 0 0
\(385\) −31.5350 −1.60717
\(386\) 61.8770 3.14946
\(387\) 0 0
\(388\) 31.4617 1.59723
\(389\) −10.7623 −0.545671 −0.272835 0.962061i \(-0.587961\pi\)
−0.272835 + 0.962061i \(0.587961\pi\)
\(390\) 0 0
\(391\) 1.25844 0.0636419
\(392\) −9.67847 −0.488837
\(393\) 0 0
\(394\) −33.7634 −1.70097
\(395\) −9.63500 −0.484790
\(396\) 0 0
\(397\) 5.03714 0.252807 0.126403 0.991979i \(-0.459657\pi\)
0.126403 + 0.991979i \(0.459657\pi\)
\(398\) 45.2970 2.27053
\(399\) 0 0
\(400\) −16.7152 −0.835758
\(401\) −28.6191 −1.42917 −0.714586 0.699548i \(-0.753385\pi\)
−0.714586 + 0.699548i \(0.753385\pi\)
\(402\) 0 0
\(403\) −12.6655 −0.630914
\(404\) −18.6623 −0.928486
\(405\) 0 0
\(406\) 7.44337 0.369408
\(407\) 16.1929 0.802652
\(408\) 0 0
\(409\) 7.44849 0.368304 0.184152 0.982898i \(-0.441046\pi\)
0.184152 + 0.982898i \(0.441046\pi\)
\(410\) −28.3017 −1.39772
\(411\) 0 0
\(412\) 58.1577 2.86523
\(413\) −3.73551 −0.183813
\(414\) 0 0
\(415\) 18.1000 0.888493
\(416\) −30.4490 −1.49288
\(417\) 0 0
\(418\) 81.9485 4.00823
\(419\) −2.54093 −0.124133 −0.0620663 0.998072i \(-0.519769\pi\)
−0.0620663 + 0.998072i \(0.519769\pi\)
\(420\) 0 0
\(421\) −8.17670 −0.398508 −0.199254 0.979948i \(-0.563852\pi\)
−0.199254 + 0.979948i \(0.563852\pi\)
\(422\) 21.1131 1.02777
\(423\) 0 0
\(424\) −50.5686 −2.45583
\(425\) −2.87689 −0.139550
\(426\) 0 0
\(427\) −34.3404 −1.66185
\(428\) 92.1041 4.45202
\(429\) 0 0
\(430\) −2.04236 −0.0984914
\(431\) −23.9124 −1.15182 −0.575910 0.817513i \(-0.695352\pi\)
−0.575910 + 0.817513i \(0.695352\pi\)
\(432\) 0 0
\(433\) −11.1094 −0.533884 −0.266942 0.963713i \(-0.586013\pi\)
−0.266942 + 0.963713i \(0.586013\pi\)
\(434\) −18.1211 −0.869839
\(435\) 0 0
\(436\) 15.2330 0.729526
\(437\) −4.89571 −0.234193
\(438\) 0 0
\(439\) −18.6475 −0.889997 −0.444998 0.895531i \(-0.646796\pi\)
−0.444998 + 0.895531i \(0.646796\pi\)
\(440\) −69.1817 −3.29811
\(441\) 0 0
\(442\) −16.7029 −0.794474
\(443\) 4.40464 0.209271 0.104635 0.994511i \(-0.466632\pi\)
0.104635 + 0.994511i \(0.466632\pi\)
\(444\) 0 0
\(445\) −27.3992 −1.29884
\(446\) −7.88922 −0.373565
\(447\) 0 0
\(448\) −0.899875 −0.0425151
\(449\) 22.6058 1.06684 0.533418 0.845852i \(-0.320907\pi\)
0.533418 + 0.845852i \(0.320907\pi\)
\(450\) 0 0
\(451\) 44.1813 2.08042
\(452\) 36.8899 1.73516
\(453\) 0 0
\(454\) 44.5972 2.09305
\(455\) −25.0050 −1.17225
\(456\) 0 0
\(457\) 36.8455 1.72356 0.861781 0.507281i \(-0.169349\pi\)
0.861781 + 0.507281i \(0.169349\pi\)
\(458\) −52.5206 −2.45413
\(459\) 0 0
\(460\) 7.42779 0.346323
\(461\) −8.77617 −0.408747 −0.204373 0.978893i \(-0.565516\pi\)
−0.204373 + 0.978893i \(0.565516\pi\)
\(462\) 0 0
\(463\) −5.87863 −0.273203 −0.136602 0.990626i \(-0.543618\pi\)
−0.136602 + 0.990626i \(0.543618\pi\)
\(464\) 7.31171 0.339438
\(465\) 0 0
\(466\) 20.1318 0.932588
\(467\) −4.25825 −0.197048 −0.0985241 0.995135i \(-0.531412\pi\)
−0.0985241 + 0.995135i \(0.531412\pi\)
\(468\) 0 0
\(469\) −10.2868 −0.474998
\(470\) −12.2808 −0.566473
\(471\) 0 0
\(472\) −8.19499 −0.377205
\(473\) 3.18830 0.146598
\(474\) 0 0
\(475\) 11.1920 0.513523
\(476\) −16.5543 −0.758767
\(477\) 0 0
\(478\) 50.8322 2.32501
\(479\) −33.3895 −1.52561 −0.762803 0.646631i \(-0.776177\pi\)
−0.762803 + 0.646631i \(0.776177\pi\)
\(480\) 0 0
\(481\) 12.8398 0.585445
\(482\) −27.6607 −1.25991
\(483\) 0 0
\(484\) 144.497 6.56805
\(485\) 11.4953 0.521974
\(486\) 0 0
\(487\) −7.38462 −0.334629 −0.167314 0.985904i \(-0.553510\pi\)
−0.167314 + 0.985904i \(0.553510\pi\)
\(488\) −75.3363 −3.41031
\(489\) 0 0
\(490\) −6.35533 −0.287105
\(491\) 42.4415 1.91536 0.957679 0.287838i \(-0.0929364\pi\)
0.957679 + 0.287838i \(0.0929364\pi\)
\(492\) 0 0
\(493\) 1.25844 0.0566771
\(494\) 64.9793 2.92356
\(495\) 0 0
\(496\) −17.8005 −0.799267
\(497\) 29.1017 1.30539
\(498\) 0 0
\(499\) 40.0787 1.79417 0.897086 0.441857i \(-0.145680\pi\)
0.897086 + 0.441857i \(0.145680\pi\)
\(500\) −54.1195 −2.42030
\(501\) 0 0
\(502\) −55.9166 −2.49568
\(503\) 0.118814 0.00529764 0.00264882 0.999996i \(-0.499157\pi\)
0.00264882 + 0.999996i \(0.499157\pi\)
\(504\) 0 0
\(505\) −6.81872 −0.303429
\(506\) −16.7389 −0.744133
\(507\) 0 0
\(508\) 29.4866 1.30826
\(509\) 3.69720 0.163876 0.0819378 0.996637i \(-0.473889\pi\)
0.0819378 + 0.996637i \(0.473889\pi\)
\(510\) 0 0
\(511\) 16.2253 0.717764
\(512\) 50.8040 2.24524
\(513\) 0 0
\(514\) −9.33164 −0.411601
\(515\) 21.2493 0.936356
\(516\) 0 0
\(517\) 19.1714 0.843158
\(518\) 18.3704 0.807150
\(519\) 0 0
\(520\) −54.8561 −2.40560
\(521\) −9.03211 −0.395704 −0.197852 0.980232i \(-0.563397\pi\)
−0.197852 + 0.980232i \(0.563397\pi\)
\(522\) 0 0
\(523\) −11.2806 −0.493267 −0.246634 0.969109i \(-0.579324\pi\)
−0.246634 + 0.969109i \(0.579324\pi\)
\(524\) −64.9648 −2.83800
\(525\) 0 0
\(526\) 67.6950 2.95164
\(527\) −3.06369 −0.133456
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −33.2057 −1.44236
\(531\) 0 0
\(532\) 64.4015 2.79216
\(533\) 35.0326 1.51743
\(534\) 0 0
\(535\) 33.6524 1.45492
\(536\) −22.5672 −0.974752
\(537\) 0 0
\(538\) −48.2891 −2.08189
\(539\) 9.92121 0.427337
\(540\) 0 0
\(541\) 12.8519 0.552548 0.276274 0.961079i \(-0.410900\pi\)
0.276274 + 0.961079i \(0.410900\pi\)
\(542\) 3.89592 0.167344
\(543\) 0 0
\(544\) −7.36537 −0.315787
\(545\) 5.56572 0.238409
\(546\) 0 0
\(547\) −36.1104 −1.54397 −0.771985 0.635641i \(-0.780736\pi\)
−0.771985 + 0.635641i \(0.780736\pi\)
\(548\) 67.1003 2.86638
\(549\) 0 0
\(550\) 38.2664 1.63168
\(551\) −4.89571 −0.208564
\(552\) 0 0
\(553\) 17.0637 0.725621
\(554\) 75.8668 3.22327
\(555\) 0 0
\(556\) −60.1442 −2.55068
\(557\) −25.5611 −1.08306 −0.541529 0.840682i \(-0.682155\pi\)
−0.541529 + 0.840682i \(0.682155\pi\)
\(558\) 0 0
\(559\) 2.52809 0.106927
\(560\) −35.1428 −1.48505
\(561\) 0 0
\(562\) −73.7552 −3.11117
\(563\) 17.3413 0.730850 0.365425 0.930841i \(-0.380924\pi\)
0.365425 + 0.930841i \(0.380924\pi\)
\(564\) 0 0
\(565\) 13.4786 0.567049
\(566\) −27.3335 −1.14891
\(567\) 0 0
\(568\) 63.8435 2.67881
\(569\) 42.8650 1.79699 0.898497 0.438980i \(-0.144660\pi\)
0.898497 + 0.438980i \(0.144660\pi\)
\(570\) 0 0
\(571\) 33.5508 1.40406 0.702028 0.712149i \(-0.252278\pi\)
0.702028 + 0.712149i \(0.252278\pi\)
\(572\) 153.903 6.43499
\(573\) 0 0
\(574\) 50.1226 2.09208
\(575\) −2.28608 −0.0953362
\(576\) 0 0
\(577\) 32.4048 1.34903 0.674514 0.738262i \(-0.264353\pi\)
0.674514 + 0.738262i \(0.264353\pi\)
\(578\) 39.3307 1.63594
\(579\) 0 0
\(580\) 7.42779 0.308422
\(581\) −32.0552 −1.32987
\(582\) 0 0
\(583\) 51.8369 2.14686
\(584\) 35.5951 1.47294
\(585\) 0 0
\(586\) −46.9678 −1.94022
\(587\) −34.5477 −1.42593 −0.712967 0.701197i \(-0.752649\pi\)
−0.712967 + 0.701197i \(0.752649\pi\)
\(588\) 0 0
\(589\) 11.9187 0.491102
\(590\) −5.38121 −0.221541
\(591\) 0 0
\(592\) 18.0455 0.741664
\(593\) 17.2013 0.706371 0.353186 0.935553i \(-0.385098\pi\)
0.353186 + 0.935553i \(0.385098\pi\)
\(594\) 0 0
\(595\) −6.04851 −0.247965
\(596\) −84.1477 −3.44682
\(597\) 0 0
\(598\) −13.2727 −0.542762
\(599\) 32.8586 1.34257 0.671283 0.741201i \(-0.265743\pi\)
0.671283 + 0.741201i \(0.265743\pi\)
\(600\) 0 0
\(601\) 1.47346 0.0601038 0.0300519 0.999548i \(-0.490433\pi\)
0.0300519 + 0.999548i \(0.490433\pi\)
\(602\) 3.61704 0.147420
\(603\) 0 0
\(604\) −47.3880 −1.92819
\(605\) 52.7954 2.14644
\(606\) 0 0
\(607\) −13.3362 −0.541300 −0.270650 0.962678i \(-0.587239\pi\)
−0.270650 + 0.962678i \(0.587239\pi\)
\(608\) 28.6536 1.16206
\(609\) 0 0
\(610\) −49.4693 −2.00295
\(611\) 15.2016 0.614989
\(612\) 0 0
\(613\) 13.1307 0.530345 0.265173 0.964201i \(-0.414571\pi\)
0.265173 + 0.964201i \(0.414571\pi\)
\(614\) −22.4498 −0.905998
\(615\) 0 0
\(616\) 122.521 4.93653
\(617\) 41.4164 1.66736 0.833681 0.552247i \(-0.186229\pi\)
0.833681 + 0.552247i \(0.186229\pi\)
\(618\) 0 0
\(619\) −34.0601 −1.36899 −0.684496 0.729017i \(-0.739978\pi\)
−0.684496 + 0.729017i \(0.739978\pi\)
\(620\) −18.0831 −0.726236
\(621\) 0 0
\(622\) −38.5063 −1.54396
\(623\) 48.5241 1.94408
\(624\) 0 0
\(625\) −8.34343 −0.333737
\(626\) 45.3917 1.81422
\(627\) 0 0
\(628\) 44.4565 1.77401
\(629\) 3.10585 0.123838
\(630\) 0 0
\(631\) −5.31051 −0.211408 −0.105704 0.994398i \(-0.533710\pi\)
−0.105704 + 0.994398i \(0.533710\pi\)
\(632\) 37.4344 1.48906
\(633\) 0 0
\(634\) −38.1529 −1.51524
\(635\) 10.7736 0.427538
\(636\) 0 0
\(637\) 7.86681 0.311694
\(638\) −16.7389 −0.662698
\(639\) 0 0
\(640\) 17.9874 0.711016
\(641\) 27.3092 1.07865 0.539323 0.842099i \(-0.318680\pi\)
0.539323 + 0.842099i \(0.318680\pi\)
\(642\) 0 0
\(643\) −35.1059 −1.38444 −0.692220 0.721686i \(-0.743367\pi\)
−0.692220 + 0.721686i \(0.743367\pi\)
\(644\) −13.1547 −0.518367
\(645\) 0 0
\(646\) 15.7180 0.618416
\(647\) 28.1684 1.10741 0.553706 0.832712i \(-0.313213\pi\)
0.553706 + 0.832712i \(0.313213\pi\)
\(648\) 0 0
\(649\) 8.40052 0.329749
\(650\) 30.3425 1.19013
\(651\) 0 0
\(652\) −6.00495 −0.235172
\(653\) −27.6829 −1.08331 −0.541657 0.840599i \(-0.682203\pi\)
−0.541657 + 0.840599i \(0.682203\pi\)
\(654\) 0 0
\(655\) −23.7364 −0.927458
\(656\) 49.2360 1.92234
\(657\) 0 0
\(658\) 21.7495 0.847883
\(659\) 32.3726 1.26106 0.630530 0.776165i \(-0.282838\pi\)
0.630530 + 0.776165i \(0.282838\pi\)
\(660\) 0 0
\(661\) −13.6507 −0.530950 −0.265475 0.964118i \(-0.585529\pi\)
−0.265475 + 0.964118i \(0.585529\pi\)
\(662\) −37.1548 −1.44406
\(663\) 0 0
\(664\) −70.3229 −2.72906
\(665\) 23.5306 0.912477
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) −72.0903 −2.78926
\(669\) 0 0
\(670\) −14.8186 −0.572494
\(671\) 77.2257 2.98126
\(672\) 0 0
\(673\) −35.5130 −1.36892 −0.684462 0.729048i \(-0.739963\pi\)
−0.684462 + 0.729048i \(0.739963\pi\)
\(674\) 61.8252 2.38142
\(675\) 0 0
\(676\) 63.4193 2.43920
\(677\) 3.45875 0.132930 0.0664652 0.997789i \(-0.478828\pi\)
0.0664652 + 0.997789i \(0.478828\pi\)
\(678\) 0 0
\(679\) −20.3582 −0.781277
\(680\) −13.2693 −0.508853
\(681\) 0 0
\(682\) 40.7511 1.56044
\(683\) −30.8279 −1.17960 −0.589798 0.807551i \(-0.700792\pi\)
−0.589798 + 0.807551i \(0.700792\pi\)
\(684\) 0 0
\(685\) 24.5167 0.936734
\(686\) −40.8483 −1.55959
\(687\) 0 0
\(688\) 3.55306 0.135459
\(689\) 41.1029 1.56590
\(690\) 0 0
\(691\) 30.5507 1.16220 0.581102 0.813831i \(-0.302622\pi\)
0.581102 + 0.813831i \(0.302622\pi\)
\(692\) 24.5425 0.932966
\(693\) 0 0
\(694\) 21.5827 0.819268
\(695\) −21.9751 −0.833563
\(696\) 0 0
\(697\) 8.47412 0.320980
\(698\) 3.27882 0.124105
\(699\) 0 0
\(700\) 30.0727 1.13664
\(701\) −23.9520 −0.904656 −0.452328 0.891852i \(-0.649406\pi\)
−0.452328 + 0.891852i \(0.649406\pi\)
\(702\) 0 0
\(703\) −12.0827 −0.455708
\(704\) 2.02366 0.0762697
\(705\) 0 0
\(706\) −35.6013 −1.33987
\(707\) 12.0760 0.454165
\(708\) 0 0
\(709\) 10.3049 0.387007 0.193503 0.981100i \(-0.438015\pi\)
0.193503 + 0.981100i \(0.438015\pi\)
\(710\) 41.9226 1.57333
\(711\) 0 0
\(712\) 106.453 3.98948
\(713\) −2.43452 −0.0911736
\(714\) 0 0
\(715\) 56.2319 2.10295
\(716\) 25.1074 0.938306
\(717\) 0 0
\(718\) −77.7298 −2.90085
\(719\) 47.4660 1.77018 0.885092 0.465416i \(-0.154095\pi\)
0.885092 + 0.465416i \(0.154095\pi\)
\(720\) 0 0
\(721\) −37.6327 −1.40151
\(722\) −12.6744 −0.471692
\(723\) 0 0
\(724\) 21.6275 0.803781
\(725\) −2.28608 −0.0849029
\(726\) 0 0
\(727\) 41.6344 1.54414 0.772068 0.635540i \(-0.219222\pi\)
0.772068 + 0.635540i \(0.219222\pi\)
\(728\) 97.1507 3.60064
\(729\) 0 0
\(730\) 23.3734 0.865089
\(731\) 0.611526 0.0226181
\(732\) 0 0
\(733\) 27.0455 0.998948 0.499474 0.866329i \(-0.333527\pi\)
0.499474 + 0.866329i \(0.333527\pi\)
\(734\) −13.2266 −0.488202
\(735\) 0 0
\(736\) −5.85279 −0.215737
\(737\) 23.1331 0.852120
\(738\) 0 0
\(739\) 39.7406 1.46188 0.730942 0.682440i \(-0.239081\pi\)
0.730942 + 0.682440i \(0.239081\pi\)
\(740\) 18.3320 0.673897
\(741\) 0 0
\(742\) 58.8076 2.15889
\(743\) −35.1971 −1.29126 −0.645629 0.763651i \(-0.723405\pi\)
−0.645629 + 0.763651i \(0.723405\pi\)
\(744\) 0 0
\(745\) −30.7453 −1.12642
\(746\) −45.7766 −1.67600
\(747\) 0 0
\(748\) 37.2278 1.36118
\(749\) −59.5987 −2.17769
\(750\) 0 0
\(751\) −12.2429 −0.446750 −0.223375 0.974733i \(-0.571707\pi\)
−0.223375 + 0.974733i \(0.571707\pi\)
\(752\) 21.3648 0.779092
\(753\) 0 0
\(754\) −13.2727 −0.483364
\(755\) −17.3143 −0.630132
\(756\) 0 0
\(757\) −18.7925 −0.683025 −0.341513 0.939877i \(-0.610939\pi\)
−0.341513 + 0.939877i \(0.610939\pi\)
\(758\) 57.7981 2.09932
\(759\) 0 0
\(760\) 51.6216 1.87251
\(761\) 39.8907 1.44604 0.723019 0.690828i \(-0.242754\pi\)
0.723019 + 0.690828i \(0.242754\pi\)
\(762\) 0 0
\(763\) −9.85694 −0.356845
\(764\) 65.5789 2.37256
\(765\) 0 0
\(766\) −29.9628 −1.08260
\(767\) 6.66101 0.240515
\(768\) 0 0
\(769\) −32.6742 −1.17826 −0.589131 0.808038i \(-0.700530\pi\)
−0.589131 + 0.808038i \(0.700530\pi\)
\(770\) 80.4532 2.89933
\(771\) 0 0
\(772\) −109.355 −3.93578
\(773\) 0.742529 0.0267069 0.0133535 0.999911i \(-0.495749\pi\)
0.0133535 + 0.999911i \(0.495749\pi\)
\(774\) 0 0
\(775\) 5.56552 0.199919
\(776\) −44.6620 −1.60327
\(777\) 0 0
\(778\) 27.4572 0.984388
\(779\) −32.9670 −1.18116
\(780\) 0 0
\(781\) −65.4447 −2.34180
\(782\) −3.21057 −0.114810
\(783\) 0 0
\(784\) 11.0563 0.394866
\(785\) 16.2432 0.579746
\(786\) 0 0
\(787\) −17.6413 −0.628846 −0.314423 0.949283i \(-0.601811\pi\)
−0.314423 + 0.949283i \(0.601811\pi\)
\(788\) 59.6701 2.12566
\(789\) 0 0
\(790\) 24.5812 0.874558
\(791\) −23.8707 −0.848745
\(792\) 0 0
\(793\) 61.2344 2.17450
\(794\) −12.8509 −0.456062
\(795\) 0 0
\(796\) −80.0535 −2.83742
\(797\) 9.68090 0.342915 0.171458 0.985192i \(-0.445152\pi\)
0.171458 + 0.985192i \(0.445152\pi\)
\(798\) 0 0
\(799\) 3.67714 0.130088
\(800\) 13.3800 0.473053
\(801\) 0 0
\(802\) 73.0142 2.57822
\(803\) −36.4879 −1.28763
\(804\) 0 0
\(805\) −4.80637 −0.169402
\(806\) 32.3127 1.13817
\(807\) 0 0
\(808\) 26.4925 0.932001
\(809\) 29.2901 1.02978 0.514892 0.857255i \(-0.327832\pi\)
0.514892 + 0.857255i \(0.327832\pi\)
\(810\) 0 0
\(811\) 4.53571 0.159270 0.0796352 0.996824i \(-0.474624\pi\)
0.0796352 + 0.996824i \(0.474624\pi\)
\(812\) −13.1547 −0.461639
\(813\) 0 0
\(814\) −41.3119 −1.44798
\(815\) −2.19405 −0.0768542
\(816\) 0 0
\(817\) −2.37902 −0.0832315
\(818\) −19.0029 −0.664419
\(819\) 0 0
\(820\) 50.0176 1.74669
\(821\) −6.70897 −0.234145 −0.117072 0.993123i \(-0.537351\pi\)
−0.117072 + 0.993123i \(0.537351\pi\)
\(822\) 0 0
\(823\) −7.24785 −0.252644 −0.126322 0.991989i \(-0.540317\pi\)
−0.126322 + 0.991989i \(0.540317\pi\)
\(824\) −82.5588 −2.87607
\(825\) 0 0
\(826\) 9.53017 0.331597
\(827\) −14.4279 −0.501707 −0.250853 0.968025i \(-0.580711\pi\)
−0.250853 + 0.968025i \(0.580711\pi\)
\(828\) 0 0
\(829\) −43.7318 −1.51887 −0.759433 0.650585i \(-0.774524\pi\)
−0.759433 + 0.650585i \(0.774524\pi\)
\(830\) −46.1773 −1.60284
\(831\) 0 0
\(832\) 1.60462 0.0556302
\(833\) 1.90292 0.0659323
\(834\) 0 0
\(835\) −26.3399 −0.911529
\(836\) −144.828 −5.00897
\(837\) 0 0
\(838\) 6.48251 0.223935
\(839\) 35.6685 1.23141 0.615706 0.787976i \(-0.288871\pi\)
0.615706 + 0.787976i \(0.288871\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 20.8607 0.718907
\(843\) 0 0
\(844\) −37.3132 −1.28437
\(845\) 23.1717 0.797131
\(846\) 0 0
\(847\) −93.5011 −3.21274
\(848\) 57.7674 1.98374
\(849\) 0 0
\(850\) 7.33962 0.251747
\(851\) 2.46802 0.0846027
\(852\) 0 0
\(853\) 37.4978 1.28390 0.641950 0.766746i \(-0.278126\pi\)
0.641950 + 0.766746i \(0.278126\pi\)
\(854\) 87.6105 2.99797
\(855\) 0 0
\(856\) −130.748 −4.46888
\(857\) −42.6622 −1.45731 −0.728656 0.684879i \(-0.759855\pi\)
−0.728656 + 0.684879i \(0.759855\pi\)
\(858\) 0 0
\(859\) 17.7446 0.605439 0.302720 0.953080i \(-0.402105\pi\)
0.302720 + 0.953080i \(0.402105\pi\)
\(860\) 3.60947 0.123082
\(861\) 0 0
\(862\) 61.0062 2.07788
\(863\) 12.6765 0.431512 0.215756 0.976447i \(-0.430778\pi\)
0.215756 + 0.976447i \(0.430778\pi\)
\(864\) 0 0
\(865\) 8.96718 0.304893
\(866\) 28.3427 0.963124
\(867\) 0 0
\(868\) 32.0254 1.08701
\(869\) −38.3732 −1.30172
\(870\) 0 0
\(871\) 18.3429 0.621526
\(872\) −21.6242 −0.732288
\(873\) 0 0
\(874\) 12.4901 0.422484
\(875\) 35.0196 1.18388
\(876\) 0 0
\(877\) 0.131207 0.00443054 0.00221527 0.999998i \(-0.499295\pi\)
0.00221527 + 0.999998i \(0.499295\pi\)
\(878\) 47.5742 1.60555
\(879\) 0 0
\(880\) 79.0301 2.66410
\(881\) 52.2114 1.75905 0.879523 0.475856i \(-0.157862\pi\)
0.879523 + 0.475856i \(0.157862\pi\)
\(882\) 0 0
\(883\) 49.7337 1.67367 0.836836 0.547453i \(-0.184403\pi\)
0.836836 + 0.547453i \(0.184403\pi\)
\(884\) 29.5190 0.992832
\(885\) 0 0
\(886\) −11.2373 −0.377523
\(887\) −1.15616 −0.0388200 −0.0194100 0.999812i \(-0.506179\pi\)
−0.0194100 + 0.999812i \(0.506179\pi\)
\(888\) 0 0
\(889\) −19.0802 −0.639929
\(890\) 69.9017 2.34311
\(891\) 0 0
\(892\) 13.9426 0.466834
\(893\) −14.3052 −0.478706
\(894\) 0 0
\(895\) 9.17356 0.306638
\(896\) −31.8559 −1.06423
\(897\) 0 0
\(898\) −57.6728 −1.92457
\(899\) −2.43452 −0.0811959
\(900\) 0 0
\(901\) 9.94248 0.331232
\(902\) −112.717 −3.75306
\(903\) 0 0
\(904\) −52.3677 −1.74172
\(905\) 7.90212 0.262676
\(906\) 0 0
\(907\) −34.3077 −1.13917 −0.569585 0.821933i \(-0.692896\pi\)
−0.569585 + 0.821933i \(0.692896\pi\)
\(908\) −78.8168 −2.61563
\(909\) 0 0
\(910\) 63.7936 2.11474
\(911\) 41.4208 1.37233 0.686167 0.727444i \(-0.259292\pi\)
0.686167 + 0.727444i \(0.259292\pi\)
\(912\) 0 0
\(913\) 72.0866 2.38572
\(914\) −94.0016 −3.10930
\(915\) 0 0
\(916\) 92.8197 3.06685
\(917\) 42.0374 1.38820
\(918\) 0 0
\(919\) −6.42221 −0.211849 −0.105925 0.994374i \(-0.533780\pi\)
−0.105925 + 0.994374i \(0.533780\pi\)
\(920\) −10.5443 −0.347634
\(921\) 0 0
\(922\) 22.3901 0.737378
\(923\) −51.8930 −1.70808
\(924\) 0 0
\(925\) −5.64210 −0.185511
\(926\) 14.9978 0.492857
\(927\) 0 0
\(928\) −5.85279 −0.192127
\(929\) −26.4758 −0.868641 −0.434321 0.900758i \(-0.643012\pi\)
−0.434321 + 0.900758i \(0.643012\pi\)
\(930\) 0 0
\(931\) −7.40295 −0.242622
\(932\) −35.5790 −1.16543
\(933\) 0 0
\(934\) 10.8638 0.355474
\(935\) 13.6021 0.444835
\(936\) 0 0
\(937\) 5.44658 0.177932 0.0889660 0.996035i \(-0.471644\pi\)
0.0889660 + 0.996035i \(0.471644\pi\)
\(938\) 26.2439 0.856895
\(939\) 0 0
\(940\) 21.7040 0.707905
\(941\) −9.09767 −0.296576 −0.148288 0.988944i \(-0.547376\pi\)
−0.148288 + 0.988944i \(0.547376\pi\)
\(942\) 0 0
\(943\) 6.73385 0.219284
\(944\) 9.36160 0.304694
\(945\) 0 0
\(946\) −8.13410 −0.264462
\(947\) 42.5878 1.38392 0.691959 0.721937i \(-0.256748\pi\)
0.691959 + 0.721937i \(0.256748\pi\)
\(948\) 0 0
\(949\) −28.9323 −0.939181
\(950\) −28.5534 −0.926394
\(951\) 0 0
\(952\) 23.5000 0.761639
\(953\) −11.7665 −0.381155 −0.190577 0.981672i \(-0.561036\pi\)
−0.190577 + 0.981672i \(0.561036\pi\)
\(954\) 0 0
\(955\) 23.9608 0.775353
\(956\) −89.8358 −2.90550
\(957\) 0 0
\(958\) 85.1845 2.75219
\(959\) −43.4192 −1.40208
\(960\) 0 0
\(961\) −25.0731 −0.808810
\(962\) −32.7574 −1.05614
\(963\) 0 0
\(964\) 48.8848 1.57447
\(965\) −39.9555 −1.28621
\(966\) 0 0
\(967\) 30.0505 0.966360 0.483180 0.875521i \(-0.339482\pi\)
0.483180 + 0.875521i \(0.339482\pi\)
\(968\) −205.123 −6.59291
\(969\) 0 0
\(970\) −29.3272 −0.941638
\(971\) −35.1648 −1.12849 −0.564246 0.825607i \(-0.690833\pi\)
−0.564246 + 0.825607i \(0.690833\pi\)
\(972\) 0 0
\(973\) 38.9181 1.24766
\(974\) 18.8399 0.603669
\(975\) 0 0
\(976\) 86.0608 2.75474
\(977\) 26.7076 0.854452 0.427226 0.904145i \(-0.359491\pi\)
0.427226 + 0.904145i \(0.359491\pi\)
\(978\) 0 0
\(979\) −109.122 −3.48757
\(980\) 11.2318 0.358786
\(981\) 0 0
\(982\) −108.278 −3.45530
\(983\) −20.0805 −0.640468 −0.320234 0.947338i \(-0.603762\pi\)
−0.320234 + 0.947338i \(0.603762\pi\)
\(984\) 0 0
\(985\) 21.8019 0.694665
\(986\) −3.21057 −0.102245
\(987\) 0 0
\(988\) −114.838 −3.65349
\(989\) 0.485941 0.0154520
\(990\) 0 0
\(991\) 26.1961 0.832146 0.416073 0.909331i \(-0.363406\pi\)
0.416073 + 0.909331i \(0.363406\pi\)
\(992\) 14.2488 0.452398
\(993\) 0 0
\(994\) −74.2453 −2.35492
\(995\) −29.2494 −0.927269
\(996\) 0 0
\(997\) 39.6800 1.25668 0.628340 0.777939i \(-0.283735\pi\)
0.628340 + 0.777939i \(0.283735\pi\)
\(998\) −102.250 −3.23668
\(999\) 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 6003.2.a.p.1.1 14
3.2 odd 2 2001.2.a.m.1.14 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.m.1.14 14 3.2 odd 2
6003.2.a.p.1.1 14 1.1 even 1 trivial