Properties

Label 1336.2.a.e.1.1
Level $1336$
Weight $2$
Character 1336.1
Self dual yes
Analytic conductor $10.668$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1336,2,Mod(1,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, 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("1336.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.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: \(10.6680137100\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} - 2399 x^{3} + 148 x^{2} + 1224 x + 224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.31023\) of defining polynomial
Character \(\chi\) \(=\) 1336.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.31023 q^{3} -2.56191 q^{5} +1.56163 q^{7} +7.95764 q^{9} +O(q^{10})\) \(q-3.31023 q^{3} -2.56191 q^{5} +1.56163 q^{7} +7.95764 q^{9} +2.12161 q^{11} -6.14247 q^{13} +8.48052 q^{15} -7.33456 q^{17} -3.28452 q^{19} -5.16935 q^{21} -2.25065 q^{23} +1.56338 q^{25} -16.4109 q^{27} +3.64505 q^{29} +3.04298 q^{31} -7.02301 q^{33} -4.00075 q^{35} -7.09544 q^{37} +20.3330 q^{39} +8.18380 q^{41} +0.642462 q^{43} -20.3868 q^{45} +11.1274 q^{47} -4.56132 q^{49} +24.2791 q^{51} -6.50174 q^{53} -5.43536 q^{55} +10.8725 q^{57} +11.7360 q^{59} +5.63676 q^{61} +12.4269 q^{63} +15.7364 q^{65} +14.4572 q^{67} +7.45019 q^{69} -7.48926 q^{71} -9.35719 q^{73} -5.17517 q^{75} +3.31316 q^{77} -1.89017 q^{79} +30.4511 q^{81} +2.97961 q^{83} +18.7905 q^{85} -12.0660 q^{87} -6.79360 q^{89} -9.59225 q^{91} -10.0730 q^{93} +8.41464 q^{95} +13.9868 q^{97} +16.8830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.31023 −1.91116 −0.955582 0.294726i \(-0.904772\pi\)
−0.955582 + 0.294726i \(0.904772\pi\)
\(4\) 0 0
\(5\) −2.56191 −1.14572 −0.572861 0.819653i \(-0.694166\pi\)
−0.572861 + 0.819653i \(0.694166\pi\)
\(6\) 0 0
\(7\) 1.56163 0.590240 0.295120 0.955460i \(-0.404640\pi\)
0.295120 + 0.955460i \(0.404640\pi\)
\(8\) 0 0
\(9\) 7.95764 2.65255
\(10\) 0 0
\(11\) 2.12161 0.639688 0.319844 0.947470i \(-0.396369\pi\)
0.319844 + 0.947470i \(0.396369\pi\)
\(12\) 0 0
\(13\) −6.14247 −1.70361 −0.851807 0.523856i \(-0.824493\pi\)
−0.851807 + 0.523856i \(0.824493\pi\)
\(14\) 0 0
\(15\) 8.48052 2.18966
\(16\) 0 0
\(17\) −7.33456 −1.77889 −0.889446 0.457041i \(-0.848909\pi\)
−0.889446 + 0.457041i \(0.848909\pi\)
\(18\) 0 0
\(19\) −3.28452 −0.753520 −0.376760 0.926311i \(-0.622962\pi\)
−0.376760 + 0.926311i \(0.622962\pi\)
\(20\) 0 0
\(21\) −5.16935 −1.12805
\(22\) 0 0
\(23\) −2.25065 −0.469294 −0.234647 0.972081i \(-0.575393\pi\)
−0.234647 + 0.972081i \(0.575393\pi\)
\(24\) 0 0
\(25\) 1.56338 0.312677
\(26\) 0 0
\(27\) −16.4109 −3.15829
\(28\) 0 0
\(29\) 3.64505 0.676870 0.338435 0.940990i \(-0.390103\pi\)
0.338435 + 0.940990i \(0.390103\pi\)
\(30\) 0 0
\(31\) 3.04298 0.546535 0.273268 0.961938i \(-0.411896\pi\)
0.273268 + 0.961938i \(0.411896\pi\)
\(32\) 0 0
\(33\) −7.02301 −1.22255
\(34\) 0 0
\(35\) −4.00075 −0.676251
\(36\) 0 0
\(37\) −7.09544 −1.16648 −0.583242 0.812299i \(-0.698216\pi\)
−0.583242 + 0.812299i \(0.698216\pi\)
\(38\) 0 0
\(39\) 20.3330 3.25588
\(40\) 0 0
\(41\) 8.18380 1.27810 0.639048 0.769167i \(-0.279329\pi\)
0.639048 + 0.769167i \(0.279329\pi\)
\(42\) 0 0
\(43\) 0.642462 0.0979745 0.0489872 0.998799i \(-0.484401\pi\)
0.0489872 + 0.998799i \(0.484401\pi\)
\(44\) 0 0
\(45\) −20.3868 −3.03908
\(46\) 0 0
\(47\) 11.1274 1.62310 0.811552 0.584281i \(-0.198623\pi\)
0.811552 + 0.584281i \(0.198623\pi\)
\(48\) 0 0
\(49\) −4.56132 −0.651617
\(50\) 0 0
\(51\) 24.2791 3.39975
\(52\) 0 0
\(53\) −6.50174 −0.893083 −0.446541 0.894763i \(-0.647344\pi\)
−0.446541 + 0.894763i \(0.647344\pi\)
\(54\) 0 0
\(55\) −5.43536 −0.732904
\(56\) 0 0
\(57\) 10.8725 1.44010
\(58\) 0 0
\(59\) 11.7360 1.52790 0.763948 0.645278i \(-0.223259\pi\)
0.763948 + 0.645278i \(0.223259\pi\)
\(60\) 0 0
\(61\) 5.63676 0.721713 0.360857 0.932621i \(-0.382484\pi\)
0.360857 + 0.932621i \(0.382484\pi\)
\(62\) 0 0
\(63\) 12.4269 1.56564
\(64\) 0 0
\(65\) 15.7364 1.95187
\(66\) 0 0
\(67\) 14.4572 1.76623 0.883113 0.469161i \(-0.155443\pi\)
0.883113 + 0.469161i \(0.155443\pi\)
\(68\) 0 0
\(69\) 7.45019 0.896897
\(70\) 0 0
\(71\) −7.48926 −0.888812 −0.444406 0.895825i \(-0.646585\pi\)
−0.444406 + 0.895825i \(0.646585\pi\)
\(72\) 0 0
\(73\) −9.35719 −1.09518 −0.547588 0.836748i \(-0.684454\pi\)
−0.547588 + 0.836748i \(0.684454\pi\)
\(74\) 0 0
\(75\) −5.17517 −0.597577
\(76\) 0 0
\(77\) 3.31316 0.377570
\(78\) 0 0
\(79\) −1.89017 −0.212661 −0.106330 0.994331i \(-0.533910\pi\)
−0.106330 + 0.994331i \(0.533910\pi\)
\(80\) 0 0
\(81\) 30.4511 3.38346
\(82\) 0 0
\(83\) 2.97961 0.327055 0.163527 0.986539i \(-0.447713\pi\)
0.163527 + 0.986539i \(0.447713\pi\)
\(84\) 0 0
\(85\) 18.7905 2.03811
\(86\) 0 0
\(87\) −12.0660 −1.29361
\(88\) 0 0
\(89\) −6.79360 −0.720120 −0.360060 0.932929i \(-0.617244\pi\)
−0.360060 + 0.932929i \(0.617244\pi\)
\(90\) 0 0
\(91\) −9.59225 −1.00554
\(92\) 0 0
\(93\) −10.0730 −1.04452
\(94\) 0 0
\(95\) 8.41464 0.863323
\(96\) 0 0
\(97\) 13.9868 1.42015 0.710073 0.704128i \(-0.248662\pi\)
0.710073 + 0.704128i \(0.248662\pi\)
\(98\) 0 0
\(99\) 16.8830 1.69680
\(100\) 0 0
\(101\) 13.2281 1.31624 0.658122 0.752911i \(-0.271351\pi\)
0.658122 + 0.752911i \(0.271351\pi\)
\(102\) 0 0
\(103\) −9.28957 −0.915329 −0.457664 0.889125i \(-0.651314\pi\)
−0.457664 + 0.889125i \(0.651314\pi\)
\(104\) 0 0
\(105\) 13.2434 1.29243
\(106\) 0 0
\(107\) 6.51451 0.629781 0.314891 0.949128i \(-0.398032\pi\)
0.314891 + 0.949128i \(0.398032\pi\)
\(108\) 0 0
\(109\) −7.88317 −0.755071 −0.377536 0.925995i \(-0.623228\pi\)
−0.377536 + 0.925995i \(0.623228\pi\)
\(110\) 0 0
\(111\) 23.4876 2.22934
\(112\) 0 0
\(113\) 8.06423 0.758619 0.379309 0.925270i \(-0.376162\pi\)
0.379309 + 0.925270i \(0.376162\pi\)
\(114\) 0 0
\(115\) 5.76597 0.537680
\(116\) 0 0
\(117\) −48.8795 −4.51891
\(118\) 0 0
\(119\) −11.4539 −1.04997
\(120\) 0 0
\(121\) −6.49879 −0.590799
\(122\) 0 0
\(123\) −27.0903 −2.44265
\(124\) 0 0
\(125\) 8.80430 0.787481
\(126\) 0 0
\(127\) 8.06236 0.715418 0.357709 0.933833i \(-0.383558\pi\)
0.357709 + 0.933833i \(0.383558\pi\)
\(128\) 0 0
\(129\) −2.12670 −0.187245
\(130\) 0 0
\(131\) −16.2829 −1.42264 −0.711320 0.702868i \(-0.751902\pi\)
−0.711320 + 0.702868i \(0.751902\pi\)
\(132\) 0 0
\(133\) −5.12919 −0.444758
\(134\) 0 0
\(135\) 42.0434 3.61852
\(136\) 0 0
\(137\) 17.7692 1.51813 0.759064 0.651016i \(-0.225657\pi\)
0.759064 + 0.651016i \(0.225657\pi\)
\(138\) 0 0
\(139\) −4.92501 −0.417734 −0.208867 0.977944i \(-0.566978\pi\)
−0.208867 + 0.977944i \(0.566978\pi\)
\(140\) 0 0
\(141\) −36.8344 −3.10202
\(142\) 0 0
\(143\) −13.0319 −1.08978
\(144\) 0 0
\(145\) −9.33830 −0.775504
\(146\) 0 0
\(147\) 15.0990 1.24535
\(148\) 0 0
\(149\) −4.90152 −0.401548 −0.200774 0.979638i \(-0.564346\pi\)
−0.200774 + 0.979638i \(0.564346\pi\)
\(150\) 0 0
\(151\) 2.20868 0.179739 0.0898697 0.995954i \(-0.471355\pi\)
0.0898697 + 0.995954i \(0.471355\pi\)
\(152\) 0 0
\(153\) −58.3658 −4.71859
\(154\) 0 0
\(155\) −7.79584 −0.626177
\(156\) 0 0
\(157\) −15.8311 −1.26346 −0.631731 0.775188i \(-0.717655\pi\)
−0.631731 + 0.775188i \(0.717655\pi\)
\(158\) 0 0
\(159\) 21.5223 1.70683
\(160\) 0 0
\(161\) −3.51469 −0.276996
\(162\) 0 0
\(163\) 15.1837 1.18928 0.594640 0.803992i \(-0.297295\pi\)
0.594640 + 0.803992i \(0.297295\pi\)
\(164\) 0 0
\(165\) 17.9923 1.40070
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 24.7299 1.90230
\(170\) 0 0
\(171\) −26.1370 −1.99875
\(172\) 0 0
\(173\) 23.5401 1.78972 0.894859 0.446350i \(-0.147276\pi\)
0.894859 + 0.446350i \(0.147276\pi\)
\(174\) 0 0
\(175\) 2.44143 0.184554
\(176\) 0 0
\(177\) −38.8488 −2.92006
\(178\) 0 0
\(179\) 8.61437 0.643868 0.321934 0.946762i \(-0.395667\pi\)
0.321934 + 0.946762i \(0.395667\pi\)
\(180\) 0 0
\(181\) 8.10168 0.602193 0.301096 0.953594i \(-0.402647\pi\)
0.301096 + 0.953594i \(0.402647\pi\)
\(182\) 0 0
\(183\) −18.6590 −1.37931
\(184\) 0 0
\(185\) 18.1779 1.33646
\(186\) 0 0
\(187\) −15.5610 −1.13794
\(188\) 0 0
\(189\) −25.6278 −1.86415
\(190\) 0 0
\(191\) 8.41971 0.609229 0.304615 0.952476i \(-0.401472\pi\)
0.304615 + 0.952476i \(0.401472\pi\)
\(192\) 0 0
\(193\) 15.2603 1.09846 0.549228 0.835672i \(-0.314922\pi\)
0.549228 + 0.835672i \(0.314922\pi\)
\(194\) 0 0
\(195\) −52.0913 −3.73034
\(196\) 0 0
\(197\) 18.2933 1.30335 0.651673 0.758500i \(-0.274067\pi\)
0.651673 + 0.758500i \(0.274067\pi\)
\(198\) 0 0
\(199\) 7.83014 0.555064 0.277532 0.960716i \(-0.410484\pi\)
0.277532 + 0.960716i \(0.410484\pi\)
\(200\) 0 0
\(201\) −47.8566 −3.37555
\(202\) 0 0
\(203\) 5.69222 0.399516
\(204\) 0 0
\(205\) −20.9662 −1.46434
\(206\) 0 0
\(207\) −17.9099 −1.24482
\(208\) 0 0
\(209\) −6.96845 −0.482018
\(210\) 0 0
\(211\) −19.5183 −1.34369 −0.671847 0.740690i \(-0.734499\pi\)
−0.671847 + 0.740690i \(0.734499\pi\)
\(212\) 0 0
\(213\) 24.7912 1.69867
\(214\) 0 0
\(215\) −1.64593 −0.112251
\(216\) 0 0
\(217\) 4.75200 0.322587
\(218\) 0 0
\(219\) 30.9745 2.09306
\(220\) 0 0
\(221\) 45.0523 3.03054
\(222\) 0 0
\(223\) 2.92970 0.196187 0.0980935 0.995177i \(-0.468726\pi\)
0.0980935 + 0.995177i \(0.468726\pi\)
\(224\) 0 0
\(225\) 12.4408 0.829390
\(226\) 0 0
\(227\) 26.1969 1.73875 0.869375 0.494153i \(-0.164522\pi\)
0.869375 + 0.494153i \(0.164522\pi\)
\(228\) 0 0
\(229\) 9.16248 0.605474 0.302737 0.953074i \(-0.402100\pi\)
0.302737 + 0.953074i \(0.402100\pi\)
\(230\) 0 0
\(231\) −10.9673 −0.721597
\(232\) 0 0
\(233\) −3.32880 −0.218077 −0.109038 0.994038i \(-0.534777\pi\)
−0.109038 + 0.994038i \(0.534777\pi\)
\(234\) 0 0
\(235\) −28.5075 −1.85962
\(236\) 0 0
\(237\) 6.25691 0.406430
\(238\) 0 0
\(239\) 8.84311 0.572013 0.286007 0.958228i \(-0.407672\pi\)
0.286007 + 0.958228i \(0.407672\pi\)
\(240\) 0 0
\(241\) −2.90434 −0.187085 −0.0935424 0.995615i \(-0.529819\pi\)
−0.0935424 + 0.995615i \(0.529819\pi\)
\(242\) 0 0
\(243\) −51.5674 −3.30805
\(244\) 0 0
\(245\) 11.6857 0.746571
\(246\) 0 0
\(247\) 20.1750 1.28371
\(248\) 0 0
\(249\) −9.86320 −0.625055
\(250\) 0 0
\(251\) −10.3817 −0.655286 −0.327643 0.944802i \(-0.606254\pi\)
−0.327643 + 0.944802i \(0.606254\pi\)
\(252\) 0 0
\(253\) −4.77500 −0.300202
\(254\) 0 0
\(255\) −62.2009 −3.89517
\(256\) 0 0
\(257\) 7.74771 0.483289 0.241645 0.970365i \(-0.422313\pi\)
0.241645 + 0.970365i \(0.422313\pi\)
\(258\) 0 0
\(259\) −11.0804 −0.688505
\(260\) 0 0
\(261\) 29.0060 1.79543
\(262\) 0 0
\(263\) 4.69311 0.289389 0.144695 0.989476i \(-0.453780\pi\)
0.144695 + 0.989476i \(0.453780\pi\)
\(264\) 0 0
\(265\) 16.6569 1.02322
\(266\) 0 0
\(267\) 22.4884 1.37627
\(268\) 0 0
\(269\) −9.50641 −0.579616 −0.289808 0.957085i \(-0.593591\pi\)
−0.289808 + 0.957085i \(0.593591\pi\)
\(270\) 0 0
\(271\) −22.3745 −1.35915 −0.679576 0.733605i \(-0.737836\pi\)
−0.679576 + 0.733605i \(0.737836\pi\)
\(272\) 0 0
\(273\) 31.7526 1.92175
\(274\) 0 0
\(275\) 3.31689 0.200016
\(276\) 0 0
\(277\) −25.5847 −1.53724 −0.768618 0.639708i \(-0.779055\pi\)
−0.768618 + 0.639708i \(0.779055\pi\)
\(278\) 0 0
\(279\) 24.2149 1.44971
\(280\) 0 0
\(281\) 5.21507 0.311105 0.155553 0.987828i \(-0.450284\pi\)
0.155553 + 0.987828i \(0.450284\pi\)
\(282\) 0 0
\(283\) 19.2065 1.14171 0.570855 0.821051i \(-0.306612\pi\)
0.570855 + 0.821051i \(0.306612\pi\)
\(284\) 0 0
\(285\) −27.8544 −1.64995
\(286\) 0 0
\(287\) 12.7801 0.754383
\(288\) 0 0
\(289\) 36.7957 2.16446
\(290\) 0 0
\(291\) −46.2996 −2.71413
\(292\) 0 0
\(293\) 2.26658 0.132415 0.0662074 0.997806i \(-0.478910\pi\)
0.0662074 + 0.997806i \(0.478910\pi\)
\(294\) 0 0
\(295\) −30.0665 −1.75054
\(296\) 0 0
\(297\) −34.8175 −2.02032
\(298\) 0 0
\(299\) 13.8246 0.799495
\(300\) 0 0
\(301\) 1.00329 0.0578285
\(302\) 0 0
\(303\) −43.7880 −2.51556
\(304\) 0 0
\(305\) −14.4409 −0.826882
\(306\) 0 0
\(307\) 21.8192 1.24529 0.622644 0.782505i \(-0.286058\pi\)
0.622644 + 0.782505i \(0.286058\pi\)
\(308\) 0 0
\(309\) 30.7506 1.74934
\(310\) 0 0
\(311\) −34.6351 −1.96398 −0.981988 0.188944i \(-0.939494\pi\)
−0.981988 + 0.188944i \(0.939494\pi\)
\(312\) 0 0
\(313\) −3.39255 −0.191758 −0.0958792 0.995393i \(-0.530566\pi\)
−0.0958792 + 0.995393i \(0.530566\pi\)
\(314\) 0 0
\(315\) −31.8365 −1.79379
\(316\) 0 0
\(317\) −11.5365 −0.647956 −0.323978 0.946065i \(-0.605020\pi\)
−0.323978 + 0.946065i \(0.605020\pi\)
\(318\) 0 0
\(319\) 7.73337 0.432986
\(320\) 0 0
\(321\) −21.5645 −1.20361
\(322\) 0 0
\(323\) 24.0905 1.34043
\(324\) 0 0
\(325\) −9.60303 −0.532681
\(326\) 0 0
\(327\) 26.0951 1.44306
\(328\) 0 0
\(329\) 17.3769 0.958021
\(330\) 0 0
\(331\) −12.9843 −0.713684 −0.356842 0.934165i \(-0.616147\pi\)
−0.356842 + 0.934165i \(0.616147\pi\)
\(332\) 0 0
\(333\) −56.4630 −3.09415
\(334\) 0 0
\(335\) −37.0380 −2.02360
\(336\) 0 0
\(337\) −21.0996 −1.14937 −0.574685 0.818375i \(-0.694876\pi\)
−0.574685 + 0.818375i \(0.694876\pi\)
\(338\) 0 0
\(339\) −26.6945 −1.44984
\(340\) 0 0
\(341\) 6.45600 0.349612
\(342\) 0 0
\(343\) −18.0545 −0.974850
\(344\) 0 0
\(345\) −19.0867 −1.02759
\(346\) 0 0
\(347\) 12.0371 0.646184 0.323092 0.946368i \(-0.395278\pi\)
0.323092 + 0.946368i \(0.395278\pi\)
\(348\) 0 0
\(349\) −4.01866 −0.215114 −0.107557 0.994199i \(-0.534303\pi\)
−0.107557 + 0.994199i \(0.534303\pi\)
\(350\) 0 0
\(351\) 100.804 5.38050
\(352\) 0 0
\(353\) −33.2203 −1.76814 −0.884070 0.467354i \(-0.845207\pi\)
−0.884070 + 0.467354i \(0.845207\pi\)
\(354\) 0 0
\(355\) 19.1868 1.01833
\(356\) 0 0
\(357\) 37.9149 2.00667
\(358\) 0 0
\(359\) −12.9262 −0.682219 −0.341109 0.940024i \(-0.610803\pi\)
−0.341109 + 0.940024i \(0.610803\pi\)
\(360\) 0 0
\(361\) −8.21195 −0.432208
\(362\) 0 0
\(363\) 21.5125 1.12911
\(364\) 0 0
\(365\) 23.9723 1.25477
\(366\) 0 0
\(367\) 11.7877 0.615314 0.307657 0.951497i \(-0.400455\pi\)
0.307657 + 0.951497i \(0.400455\pi\)
\(368\) 0 0
\(369\) 65.1238 3.39021
\(370\) 0 0
\(371\) −10.1533 −0.527133
\(372\) 0 0
\(373\) −16.5556 −0.857218 −0.428609 0.903490i \(-0.640996\pi\)
−0.428609 + 0.903490i \(0.640996\pi\)
\(374\) 0 0
\(375\) −29.1443 −1.50500
\(376\) 0 0
\(377\) −22.3896 −1.15312
\(378\) 0 0
\(379\) 10.6357 0.546320 0.273160 0.961969i \(-0.411931\pi\)
0.273160 + 0.961969i \(0.411931\pi\)
\(380\) 0 0
\(381\) −26.6883 −1.36728
\(382\) 0 0
\(383\) −20.6397 −1.05464 −0.527320 0.849667i \(-0.676803\pi\)
−0.527320 + 0.849667i \(0.676803\pi\)
\(384\) 0 0
\(385\) −8.48802 −0.432590
\(386\) 0 0
\(387\) 5.11248 0.259882
\(388\) 0 0
\(389\) −11.5991 −0.588099 −0.294050 0.955790i \(-0.595003\pi\)
−0.294050 + 0.955790i \(0.595003\pi\)
\(390\) 0 0
\(391\) 16.5076 0.834823
\(392\) 0 0
\(393\) 53.9001 2.71890
\(394\) 0 0
\(395\) 4.84245 0.243650
\(396\) 0 0
\(397\) −2.23583 −0.112213 −0.0561066 0.998425i \(-0.517869\pi\)
−0.0561066 + 0.998425i \(0.517869\pi\)
\(398\) 0 0
\(399\) 16.9788 0.850004
\(400\) 0 0
\(401\) 3.98943 0.199223 0.0996114 0.995026i \(-0.468240\pi\)
0.0996114 + 0.995026i \(0.468240\pi\)
\(402\) 0 0
\(403\) −18.6914 −0.931085
\(404\) 0 0
\(405\) −78.0130 −3.87650
\(406\) 0 0
\(407\) −15.0537 −0.746186
\(408\) 0 0
\(409\) 12.8308 0.634441 0.317221 0.948352i \(-0.397250\pi\)
0.317221 + 0.948352i \(0.397250\pi\)
\(410\) 0 0
\(411\) −58.8203 −2.90139
\(412\) 0 0
\(413\) 18.3273 0.901825
\(414\) 0 0
\(415\) −7.63349 −0.374714
\(416\) 0 0
\(417\) 16.3029 0.798358
\(418\) 0 0
\(419\) −11.9249 −0.582571 −0.291286 0.956636i \(-0.594083\pi\)
−0.291286 + 0.956636i \(0.594083\pi\)
\(420\) 0 0
\(421\) −12.9454 −0.630919 −0.315459 0.948939i \(-0.602159\pi\)
−0.315459 + 0.948939i \(0.602159\pi\)
\(422\) 0 0
\(423\) 88.5481 4.30536
\(424\) 0 0
\(425\) −11.4667 −0.556218
\(426\) 0 0
\(427\) 8.80253 0.425984
\(428\) 0 0
\(429\) 43.1386 2.08275
\(430\) 0 0
\(431\) 30.8474 1.48587 0.742934 0.669365i \(-0.233434\pi\)
0.742934 + 0.669365i \(0.233434\pi\)
\(432\) 0 0
\(433\) −1.65891 −0.0797223 −0.0398612 0.999205i \(-0.512692\pi\)
−0.0398612 + 0.999205i \(0.512692\pi\)
\(434\) 0 0
\(435\) 30.9120 1.48211
\(436\) 0 0
\(437\) 7.39231 0.353622
\(438\) 0 0
\(439\) 27.8794 1.33061 0.665307 0.746570i \(-0.268301\pi\)
0.665307 + 0.746570i \(0.268301\pi\)
\(440\) 0 0
\(441\) −36.2973 −1.72844
\(442\) 0 0
\(443\) −5.25652 −0.249745 −0.124872 0.992173i \(-0.539852\pi\)
−0.124872 + 0.992173i \(0.539852\pi\)
\(444\) 0 0
\(445\) 17.4046 0.825057
\(446\) 0 0
\(447\) 16.2252 0.767424
\(448\) 0 0
\(449\) 14.0192 0.661609 0.330804 0.943699i \(-0.392680\pi\)
0.330804 + 0.943699i \(0.392680\pi\)
\(450\) 0 0
\(451\) 17.3628 0.817583
\(452\) 0 0
\(453\) −7.31123 −0.343512
\(454\) 0 0
\(455\) 24.5745 1.15207
\(456\) 0 0
\(457\) −13.6088 −0.636591 −0.318296 0.947991i \(-0.603110\pi\)
−0.318296 + 0.947991i \(0.603110\pi\)
\(458\) 0 0
\(459\) 120.367 5.61825
\(460\) 0 0
\(461\) 20.1444 0.938217 0.469108 0.883141i \(-0.344575\pi\)
0.469108 + 0.883141i \(0.344575\pi\)
\(462\) 0 0
\(463\) 30.2893 1.40766 0.703831 0.710367i \(-0.251471\pi\)
0.703831 + 0.710367i \(0.251471\pi\)
\(464\) 0 0
\(465\) 25.8060 1.19673
\(466\) 0 0
\(467\) 12.9528 0.599383 0.299692 0.954036i \(-0.403116\pi\)
0.299692 + 0.954036i \(0.403116\pi\)
\(468\) 0 0
\(469\) 22.5767 1.04250
\(470\) 0 0
\(471\) 52.4047 2.41468
\(472\) 0 0
\(473\) 1.36305 0.0626731
\(474\) 0 0
\(475\) −5.13496 −0.235608
\(476\) 0 0
\(477\) −51.7385 −2.36894
\(478\) 0 0
\(479\) −28.5271 −1.30343 −0.651717 0.758462i \(-0.725951\pi\)
−0.651717 + 0.758462i \(0.725951\pi\)
\(480\) 0 0
\(481\) 43.5835 1.98724
\(482\) 0 0
\(483\) 11.6344 0.529385
\(484\) 0 0
\(485\) −35.8330 −1.62709
\(486\) 0 0
\(487\) −29.0787 −1.31768 −0.658842 0.752282i \(-0.728953\pi\)
−0.658842 + 0.752282i \(0.728953\pi\)
\(488\) 0 0
\(489\) −50.2616 −2.27291
\(490\) 0 0
\(491\) 16.9855 0.766545 0.383272 0.923635i \(-0.374797\pi\)
0.383272 + 0.923635i \(0.374797\pi\)
\(492\) 0 0
\(493\) −26.7349 −1.20408
\(494\) 0 0
\(495\) −43.2527 −1.94406
\(496\) 0 0
\(497\) −11.6954 −0.524613
\(498\) 0 0
\(499\) 30.4814 1.36454 0.682268 0.731102i \(-0.260994\pi\)
0.682268 + 0.731102i \(0.260994\pi\)
\(500\) 0 0
\(501\) −3.31023 −0.147890
\(502\) 0 0
\(503\) −3.93185 −0.175313 −0.0876564 0.996151i \(-0.527938\pi\)
−0.0876564 + 0.996151i \(0.527938\pi\)
\(504\) 0 0
\(505\) −33.8892 −1.50805
\(506\) 0 0
\(507\) −81.8617 −3.63561
\(508\) 0 0
\(509\) 15.4779 0.686045 0.343023 0.939327i \(-0.388549\pi\)
0.343023 + 0.939327i \(0.388549\pi\)
\(510\) 0 0
\(511\) −14.6124 −0.646417
\(512\) 0 0
\(513\) 53.9020 2.37983
\(514\) 0 0
\(515\) 23.7990 1.04871
\(516\) 0 0
\(517\) 23.6080 1.03828
\(518\) 0 0
\(519\) −77.9231 −3.42044
\(520\) 0 0
\(521\) −15.0406 −0.658939 −0.329469 0.944166i \(-0.606870\pi\)
−0.329469 + 0.944166i \(0.606870\pi\)
\(522\) 0 0
\(523\) −23.2973 −1.01872 −0.509359 0.860554i \(-0.670117\pi\)
−0.509359 + 0.860554i \(0.670117\pi\)
\(524\) 0 0
\(525\) −8.08169 −0.352714
\(526\) 0 0
\(527\) −22.3189 −0.972227
\(528\) 0 0
\(529\) −17.9346 −0.779763
\(530\) 0 0
\(531\) 93.3907 4.05281
\(532\) 0 0
\(533\) −50.2687 −2.17738
\(534\) 0 0
\(535\) −16.6896 −0.721553
\(536\) 0 0
\(537\) −28.5156 −1.23054
\(538\) 0 0
\(539\) −9.67731 −0.416831
\(540\) 0 0
\(541\) 28.0397 1.20552 0.602760 0.797922i \(-0.294068\pi\)
0.602760 + 0.797922i \(0.294068\pi\)
\(542\) 0 0
\(543\) −26.8184 −1.15089
\(544\) 0 0
\(545\) 20.1960 0.865101
\(546\) 0 0
\(547\) 6.99565 0.299112 0.149556 0.988753i \(-0.452215\pi\)
0.149556 + 0.988753i \(0.452215\pi\)
\(548\) 0 0
\(549\) 44.8553 1.91438
\(550\) 0 0
\(551\) −11.9722 −0.510035
\(552\) 0 0
\(553\) −2.95175 −0.125521
\(554\) 0 0
\(555\) −60.1730 −2.55420
\(556\) 0 0
\(557\) −0.213626 −0.00905163 −0.00452582 0.999990i \(-0.501441\pi\)
−0.00452582 + 0.999990i \(0.501441\pi\)
\(558\) 0 0
\(559\) −3.94630 −0.166911
\(560\) 0 0
\(561\) 51.5107 2.17478
\(562\) 0 0
\(563\) −17.8050 −0.750393 −0.375197 0.926945i \(-0.622425\pi\)
−0.375197 + 0.926945i \(0.622425\pi\)
\(564\) 0 0
\(565\) −20.6598 −0.869166
\(566\) 0 0
\(567\) 47.5533 1.99705
\(568\) 0 0
\(569\) −37.3874 −1.56736 −0.783681 0.621164i \(-0.786660\pi\)
−0.783681 + 0.621164i \(0.786660\pi\)
\(570\) 0 0
\(571\) −23.7937 −0.995737 −0.497869 0.867252i \(-0.665884\pi\)
−0.497869 + 0.867252i \(0.665884\pi\)
\(572\) 0 0
\(573\) −27.8712 −1.16434
\(574\) 0 0
\(575\) −3.51864 −0.146737
\(576\) 0 0
\(577\) 11.6525 0.485100 0.242550 0.970139i \(-0.422016\pi\)
0.242550 + 0.970139i \(0.422016\pi\)
\(578\) 0 0
\(579\) −50.5150 −2.09933
\(580\) 0 0
\(581\) 4.65305 0.193041
\(582\) 0 0
\(583\) −13.7941 −0.571294
\(584\) 0 0
\(585\) 125.225 5.17742
\(586\) 0 0
\(587\) 29.0583 1.19937 0.599683 0.800237i \(-0.295293\pi\)
0.599683 + 0.800237i \(0.295293\pi\)
\(588\) 0 0
\(589\) −9.99471 −0.411825
\(590\) 0 0
\(591\) −60.5552 −2.49091
\(592\) 0 0
\(593\) 29.6140 1.21610 0.608051 0.793898i \(-0.291952\pi\)
0.608051 + 0.793898i \(0.291952\pi\)
\(594\) 0 0
\(595\) 29.3438 1.20298
\(596\) 0 0
\(597\) −25.9196 −1.06082
\(598\) 0 0
\(599\) −9.97837 −0.407705 −0.203853 0.979002i \(-0.565346\pi\)
−0.203853 + 0.979002i \(0.565346\pi\)
\(600\) 0 0
\(601\) 15.4179 0.628910 0.314455 0.949272i \(-0.398178\pi\)
0.314455 + 0.949272i \(0.398178\pi\)
\(602\) 0 0
\(603\) 115.045 4.68500
\(604\) 0 0
\(605\) 16.6493 0.676891
\(606\) 0 0
\(607\) −16.9572 −0.688271 −0.344136 0.938920i \(-0.611828\pi\)
−0.344136 + 0.938920i \(0.611828\pi\)
\(608\) 0 0
\(609\) −18.8426 −0.763540
\(610\) 0 0
\(611\) −68.3499 −2.76514
\(612\) 0 0
\(613\) 7.20743 0.291105 0.145553 0.989351i \(-0.453504\pi\)
0.145553 + 0.989351i \(0.453504\pi\)
\(614\) 0 0
\(615\) 69.4029 2.79860
\(616\) 0 0
\(617\) 41.4731 1.66964 0.834822 0.550520i \(-0.185570\pi\)
0.834822 + 0.550520i \(0.185570\pi\)
\(618\) 0 0
\(619\) 42.9537 1.72645 0.863226 0.504817i \(-0.168440\pi\)
0.863226 + 0.504817i \(0.168440\pi\)
\(620\) 0 0
\(621\) 36.9353 1.48216
\(622\) 0 0
\(623\) −10.6091 −0.425044
\(624\) 0 0
\(625\) −30.3728 −1.21491
\(626\) 0 0
\(627\) 23.0672 0.921215
\(628\) 0 0
\(629\) 52.0419 2.07505
\(630\) 0 0
\(631\) 30.2821 1.20551 0.602756 0.797926i \(-0.294069\pi\)
0.602756 + 0.797926i \(0.294069\pi\)
\(632\) 0 0
\(633\) 64.6101 2.56802
\(634\) 0 0
\(635\) −20.6550 −0.819670
\(636\) 0 0
\(637\) 28.0177 1.11010
\(638\) 0 0
\(639\) −59.5969 −2.35762
\(640\) 0 0
\(641\) 39.0327 1.54170 0.770849 0.637018i \(-0.219832\pi\)
0.770849 + 0.637018i \(0.219832\pi\)
\(642\) 0 0
\(643\) −42.4273 −1.67317 −0.836585 0.547837i \(-0.815451\pi\)
−0.836585 + 0.547837i \(0.815451\pi\)
\(644\) 0 0
\(645\) 5.44841 0.214531
\(646\) 0 0
\(647\) −36.2379 −1.42466 −0.712330 0.701845i \(-0.752360\pi\)
−0.712330 + 0.701845i \(0.752360\pi\)
\(648\) 0 0
\(649\) 24.8991 0.977377
\(650\) 0 0
\(651\) −15.7302 −0.616516
\(652\) 0 0
\(653\) −45.0559 −1.76317 −0.881587 0.472022i \(-0.843524\pi\)
−0.881587 + 0.472022i \(0.843524\pi\)
\(654\) 0 0
\(655\) 41.7152 1.62995
\(656\) 0 0
\(657\) −74.4611 −2.90500
\(658\) 0 0
\(659\) 21.3965 0.833487 0.416744 0.909024i \(-0.363171\pi\)
0.416744 + 0.909024i \(0.363171\pi\)
\(660\) 0 0
\(661\) −35.2681 −1.37177 −0.685885 0.727710i \(-0.740585\pi\)
−0.685885 + 0.727710i \(0.740585\pi\)
\(662\) 0 0
\(663\) −149.134 −5.79187
\(664\) 0 0
\(665\) 13.1405 0.509568
\(666\) 0 0
\(667\) −8.20376 −0.317651
\(668\) 0 0
\(669\) −9.69798 −0.374945
\(670\) 0 0
\(671\) 11.9590 0.461671
\(672\) 0 0
\(673\) 37.7417 1.45484 0.727418 0.686195i \(-0.240720\pi\)
0.727418 + 0.686195i \(0.240720\pi\)
\(674\) 0 0
\(675\) −25.6566 −0.987523
\(676\) 0 0
\(677\) 31.3028 1.20307 0.601533 0.798848i \(-0.294557\pi\)
0.601533 + 0.798848i \(0.294557\pi\)
\(678\) 0 0
\(679\) 21.8422 0.838227
\(680\) 0 0
\(681\) −86.7178 −3.32303
\(682\) 0 0
\(683\) 35.4903 1.35800 0.679000 0.734138i \(-0.262414\pi\)
0.679000 + 0.734138i \(0.262414\pi\)
\(684\) 0 0
\(685\) −45.5232 −1.73935
\(686\) 0 0
\(687\) −30.3299 −1.15716
\(688\) 0 0
\(689\) 39.9367 1.52147
\(690\) 0 0
\(691\) 6.36456 0.242119 0.121060 0.992645i \(-0.461371\pi\)
0.121060 + 0.992645i \(0.461371\pi\)
\(692\) 0 0
\(693\) 26.3649 1.00152
\(694\) 0 0
\(695\) 12.6174 0.478607
\(696\) 0 0
\(697\) −60.0246 −2.27359
\(698\) 0 0
\(699\) 11.0191 0.416780
\(700\) 0 0
\(701\) −24.2577 −0.916200 −0.458100 0.888901i \(-0.651470\pi\)
−0.458100 + 0.888901i \(0.651470\pi\)
\(702\) 0 0
\(703\) 23.3051 0.878968
\(704\) 0 0
\(705\) 94.3664 3.55404
\(706\) 0 0
\(707\) 20.6574 0.776900
\(708\) 0 0
\(709\) −35.4415 −1.33103 −0.665517 0.746383i \(-0.731789\pi\)
−0.665517 + 0.746383i \(0.731789\pi\)
\(710\) 0 0
\(711\) −15.0413 −0.564093
\(712\) 0 0
\(713\) −6.84869 −0.256486
\(714\) 0 0
\(715\) 33.3865 1.24859
\(716\) 0 0
\(717\) −29.2727 −1.09321
\(718\) 0 0
\(719\) −7.95811 −0.296787 −0.148394 0.988928i \(-0.547410\pi\)
−0.148394 + 0.988928i \(0.547410\pi\)
\(720\) 0 0
\(721\) −14.5069 −0.540264
\(722\) 0 0
\(723\) 9.61404 0.357550
\(724\) 0 0
\(725\) 5.69862 0.211641
\(726\) 0 0
\(727\) 34.7441 1.28859 0.644293 0.764778i \(-0.277152\pi\)
0.644293 + 0.764778i \(0.277152\pi\)
\(728\) 0 0
\(729\) 79.3469 2.93877
\(730\) 0 0
\(731\) −4.71217 −0.174286
\(732\) 0 0
\(733\) 1.93098 0.0713226 0.0356613 0.999364i \(-0.488646\pi\)
0.0356613 + 0.999364i \(0.488646\pi\)
\(734\) 0 0
\(735\) −38.6823 −1.42682
\(736\) 0 0
\(737\) 30.6724 1.12983
\(738\) 0 0
\(739\) −19.7931 −0.728102 −0.364051 0.931379i \(-0.618607\pi\)
−0.364051 + 0.931379i \(0.618607\pi\)
\(740\) 0 0
\(741\) −66.7840 −2.45337
\(742\) 0 0
\(743\) 33.3838 1.22473 0.612366 0.790575i \(-0.290218\pi\)
0.612366 + 0.790575i \(0.290218\pi\)
\(744\) 0 0
\(745\) 12.5572 0.460062
\(746\) 0 0
\(747\) 23.7107 0.867528
\(748\) 0 0
\(749\) 10.1732 0.371722
\(750\) 0 0
\(751\) 6.85107 0.249999 0.125000 0.992157i \(-0.460107\pi\)
0.125000 + 0.992157i \(0.460107\pi\)
\(752\) 0 0
\(753\) 34.3658 1.25236
\(754\) 0 0
\(755\) −5.65843 −0.205931
\(756\) 0 0
\(757\) −23.6656 −0.860141 −0.430070 0.902795i \(-0.641511\pi\)
−0.430070 + 0.902795i \(0.641511\pi\)
\(758\) 0 0
\(759\) 15.8064 0.573735
\(760\) 0 0
\(761\) −2.41790 −0.0876488 −0.0438244 0.999039i \(-0.513954\pi\)
−0.0438244 + 0.999039i \(0.513954\pi\)
\(762\) 0 0
\(763\) −12.3106 −0.445673
\(764\) 0 0
\(765\) 149.528 5.40619
\(766\) 0 0
\(767\) −72.0879 −2.60294
\(768\) 0 0
\(769\) −18.0804 −0.651995 −0.325997 0.945371i \(-0.605700\pi\)
−0.325997 + 0.945371i \(0.605700\pi\)
\(770\) 0 0
\(771\) −25.6467 −0.923645
\(772\) 0 0
\(773\) 11.0927 0.398976 0.199488 0.979900i \(-0.436072\pi\)
0.199488 + 0.979900i \(0.436072\pi\)
\(774\) 0 0
\(775\) 4.75734 0.170889
\(776\) 0 0
\(777\) 36.6789 1.31585
\(778\) 0 0
\(779\) −26.8798 −0.963070
\(780\) 0 0
\(781\) −15.8893 −0.568563
\(782\) 0 0
\(783\) −59.8188 −2.13775
\(784\) 0 0
\(785\) 40.5579 1.44757
\(786\) 0 0
\(787\) 27.8184 0.991620 0.495810 0.868431i \(-0.334871\pi\)
0.495810 + 0.868431i \(0.334871\pi\)
\(788\) 0 0
\(789\) −15.5353 −0.553070
\(790\) 0 0
\(791\) 12.5933 0.447767
\(792\) 0 0
\(793\) −34.6236 −1.22952
\(794\) 0 0
\(795\) −55.1381 −1.95555
\(796\) 0 0
\(797\) −21.3973 −0.757931 −0.378966 0.925411i \(-0.623720\pi\)
−0.378966 + 0.925411i \(0.623720\pi\)
\(798\) 0 0
\(799\) −81.6148 −2.88732
\(800\) 0 0
\(801\) −54.0610 −1.91015
\(802\) 0 0
\(803\) −19.8523 −0.700571
\(804\) 0 0
\(805\) 9.00431 0.317360
\(806\) 0 0
\(807\) 31.4684 1.10774
\(808\) 0 0
\(809\) −46.5325 −1.63600 −0.817998 0.575220i \(-0.804916\pi\)
−0.817998 + 0.575220i \(0.804916\pi\)
\(810\) 0 0
\(811\) 54.1372 1.90101 0.950506 0.310705i \(-0.100565\pi\)
0.950506 + 0.310705i \(0.100565\pi\)
\(812\) 0 0
\(813\) 74.0647 2.59756
\(814\) 0 0
\(815\) −38.8993 −1.36258
\(816\) 0 0
\(817\) −2.11018 −0.0738257
\(818\) 0 0
\(819\) −76.3317 −2.66724
\(820\) 0 0
\(821\) −19.6157 −0.684592 −0.342296 0.939592i \(-0.611205\pi\)
−0.342296 + 0.939592i \(0.611205\pi\)
\(822\) 0 0
\(823\) 28.4516 0.991759 0.495879 0.868391i \(-0.334846\pi\)
0.495879 + 0.868391i \(0.334846\pi\)
\(824\) 0 0
\(825\) −10.9797 −0.382263
\(826\) 0 0
\(827\) 32.1268 1.11716 0.558579 0.829451i \(-0.311347\pi\)
0.558579 + 0.829451i \(0.311347\pi\)
\(828\) 0 0
\(829\) −29.8622 −1.03716 −0.518579 0.855030i \(-0.673539\pi\)
−0.518579 + 0.855030i \(0.673539\pi\)
\(830\) 0 0
\(831\) 84.6913 2.93791
\(832\) 0 0
\(833\) 33.4552 1.15916
\(834\) 0 0
\(835\) −2.56191 −0.0886586
\(836\) 0 0
\(837\) −49.9381 −1.72611
\(838\) 0 0
\(839\) 53.8590 1.85942 0.929710 0.368292i \(-0.120057\pi\)
0.929710 + 0.368292i \(0.120057\pi\)
\(840\) 0 0
\(841\) −15.7136 −0.541848
\(842\) 0 0
\(843\) −17.2631 −0.594573
\(844\) 0 0
\(845\) −63.3558 −2.17950
\(846\) 0 0
\(847\) −10.1487 −0.348713
\(848\) 0 0
\(849\) −63.5781 −2.18199
\(850\) 0 0
\(851\) 15.9694 0.547424
\(852\) 0 0
\(853\) 5.78694 0.198141 0.0990705 0.995080i \(-0.468413\pi\)
0.0990705 + 0.995080i \(0.468413\pi\)
\(854\) 0 0
\(855\) 66.9606 2.29001
\(856\) 0 0
\(857\) 45.3296 1.54843 0.774215 0.632923i \(-0.218145\pi\)
0.774215 + 0.632923i \(0.218145\pi\)
\(858\) 0 0
\(859\) 12.2246 0.417097 0.208548 0.978012i \(-0.433126\pi\)
0.208548 + 0.978012i \(0.433126\pi\)
\(860\) 0 0
\(861\) −42.3050 −1.44175
\(862\) 0 0
\(863\) 0.466464 0.0158786 0.00793931 0.999968i \(-0.497473\pi\)
0.00793931 + 0.999968i \(0.497473\pi\)
\(864\) 0 0
\(865\) −60.3075 −2.05052
\(866\) 0 0
\(867\) −121.802 −4.13663
\(868\) 0 0
\(869\) −4.01020 −0.136037
\(870\) 0 0
\(871\) −88.8027 −3.00897
\(872\) 0 0
\(873\) 111.302 3.76700
\(874\) 0 0
\(875\) 13.7490 0.464803
\(876\) 0 0
\(877\) 46.3905 1.56649 0.783247 0.621710i \(-0.213562\pi\)
0.783247 + 0.621710i \(0.213562\pi\)
\(878\) 0 0
\(879\) −7.50289 −0.253066
\(880\) 0 0
\(881\) 54.2865 1.82896 0.914480 0.404632i \(-0.132600\pi\)
0.914480 + 0.404632i \(0.132600\pi\)
\(882\) 0 0
\(883\) 18.3626 0.617952 0.308976 0.951070i \(-0.400014\pi\)
0.308976 + 0.951070i \(0.400014\pi\)
\(884\) 0 0
\(885\) 99.5272 3.34557
\(886\) 0 0
\(887\) −3.35576 −0.112675 −0.0563377 0.998412i \(-0.517942\pi\)
−0.0563377 + 0.998412i \(0.517942\pi\)
\(888\) 0 0
\(889\) 12.5904 0.422269
\(890\) 0 0
\(891\) 64.6052 2.16436
\(892\) 0 0
\(893\) −36.5482 −1.22304
\(894\) 0 0
\(895\) −22.0692 −0.737693
\(896\) 0 0
\(897\) −45.7625 −1.52797
\(898\) 0 0
\(899\) 11.0918 0.369933
\(900\) 0 0
\(901\) 47.6874 1.58870
\(902\) 0 0
\(903\) −3.32111 −0.110520
\(904\) 0 0
\(905\) −20.7558 −0.689945
\(906\) 0 0
\(907\) 3.01069 0.0999684 0.0499842 0.998750i \(-0.484083\pi\)
0.0499842 + 0.998750i \(0.484083\pi\)
\(908\) 0 0
\(909\) 105.264 3.49140
\(910\) 0 0
\(911\) 3.18018 0.105364 0.0526820 0.998611i \(-0.483223\pi\)
0.0526820 + 0.998611i \(0.483223\pi\)
\(912\) 0 0
\(913\) 6.32156 0.209213
\(914\) 0 0
\(915\) 47.8027 1.58031
\(916\) 0 0
\(917\) −25.4278 −0.839700
\(918\) 0 0
\(919\) 36.3109 1.19779 0.598893 0.800829i \(-0.295608\pi\)
0.598893 + 0.800829i \(0.295608\pi\)
\(920\) 0 0
\(921\) −72.2267 −2.37995
\(922\) 0 0
\(923\) 46.0026 1.51419
\(924\) 0 0
\(925\) −11.0929 −0.364732
\(926\) 0 0
\(927\) −73.9231 −2.42795
\(928\) 0 0
\(929\) 5.21107 0.170970 0.0854849 0.996339i \(-0.472756\pi\)
0.0854849 + 0.996339i \(0.472756\pi\)
\(930\) 0 0
\(931\) 14.9817 0.491006
\(932\) 0 0
\(933\) 114.650 3.75348
\(934\) 0 0
\(935\) 39.8660 1.30376
\(936\) 0 0
\(937\) 24.0899 0.786984 0.393492 0.919328i \(-0.371267\pi\)
0.393492 + 0.919328i \(0.371267\pi\)
\(938\) 0 0
\(939\) 11.2301 0.366482
\(940\) 0 0
\(941\) 45.5653 1.48539 0.742694 0.669631i \(-0.233548\pi\)
0.742694 + 0.669631i \(0.233548\pi\)
\(942\) 0 0
\(943\) −18.4189 −0.599802
\(944\) 0 0
\(945\) 65.6561 2.13579
\(946\) 0 0
\(947\) −18.5248 −0.601974 −0.300987 0.953628i \(-0.597316\pi\)
−0.300987 + 0.953628i \(0.597316\pi\)
\(948\) 0 0
\(949\) 57.4762 1.86576
\(950\) 0 0
\(951\) 38.1886 1.23835
\(952\) 0 0
\(953\) 14.1796 0.459323 0.229662 0.973271i \(-0.426238\pi\)
0.229662 + 0.973271i \(0.426238\pi\)
\(954\) 0 0
\(955\) −21.5705 −0.698007
\(956\) 0 0
\(957\) −25.5992 −0.827506
\(958\) 0 0
\(959\) 27.7489 0.896060
\(960\) 0 0
\(961\) −21.7403 −0.701299
\(962\) 0 0
\(963\) 51.8401 1.67052
\(964\) 0 0
\(965\) −39.0954 −1.25853
\(966\) 0 0
\(967\) −24.4050 −0.784811 −0.392405 0.919792i \(-0.628357\pi\)
−0.392405 + 0.919792i \(0.628357\pi\)
\(968\) 0 0
\(969\) −79.7451 −2.56178
\(970\) 0 0
\(971\) 24.0560 0.771994 0.385997 0.922500i \(-0.373857\pi\)
0.385997 + 0.922500i \(0.373857\pi\)
\(972\) 0 0
\(973\) −7.69104 −0.246564
\(974\) 0 0
\(975\) 31.7883 1.01804
\(976\) 0 0
\(977\) −37.8248 −1.21012 −0.605062 0.796179i \(-0.706852\pi\)
−0.605062 + 0.796179i \(0.706852\pi\)
\(978\) 0 0
\(979\) −14.4133 −0.460653
\(980\) 0 0
\(981\) −62.7315 −2.00286
\(982\) 0 0
\(983\) 17.8078 0.567980 0.283990 0.958827i \(-0.408342\pi\)
0.283990 + 0.958827i \(0.408342\pi\)
\(984\) 0 0
\(985\) −46.8659 −1.49327
\(986\) 0 0
\(987\) −57.5217 −1.83093
\(988\) 0 0
\(989\) −1.44596 −0.0459788
\(990\) 0 0
\(991\) −45.2086 −1.43610 −0.718050 0.695992i \(-0.754965\pi\)
−0.718050 + 0.695992i \(0.754965\pi\)
\(992\) 0 0
\(993\) 42.9812 1.36397
\(994\) 0 0
\(995\) −20.0601 −0.635948
\(996\) 0 0
\(997\) −16.8763 −0.534477 −0.267239 0.963630i \(-0.586111\pi\)
−0.267239 + 0.963630i \(0.586111\pi\)
\(998\) 0 0
\(999\) 116.443 3.68409
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 1336.2.a.e.1.1 12
4.3 odd 2 2672.2.a.n.1.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1336.2.a.e.1.1 12 1.1 even 1 trivial
2672.2.a.n.1.12 12 4.3 odd 2