Properties

Label 3344.2.o.a.1519.15
Level $3344$
Weight $2$
Character 3344.1519
Analytic conductor $26.702$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3344,2,Mod(1519,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3344.1519");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.o (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(26.7019744359\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1519.15
Character \(\chi\) \(=\) 3344.1519
Dual form 3344.2.o.a.1519.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.924782 q^{3} -2.57340 q^{5} +0.765529i q^{7} -2.14478 q^{9} +O(q^{10})\) \(q-0.924782 q^{3} -2.57340 q^{5} +0.765529i q^{7} -2.14478 q^{9} +1.00000i q^{11} -2.60853i q^{13} +2.37984 q^{15} -3.74816 q^{17} +(-2.28136 - 3.71421i) q^{19} -0.707948i q^{21} -3.60503i q^{23} +1.62240 q^{25} +4.75780 q^{27} +3.77031i q^{29} -7.30362 q^{31} -0.924782i q^{33} -1.97001i q^{35} +8.15874i q^{37} +2.41232i q^{39} -5.25952i q^{41} -0.574153i q^{43} +5.51937 q^{45} -1.77015i q^{47} +6.41396 q^{49} +3.46624 q^{51} -11.7573i q^{53} -2.57340i q^{55} +(2.10976 + 3.43484i) q^{57} -4.11799 q^{59} +9.17659 q^{61} -1.64189i q^{63} +6.71279i q^{65} -1.66350 q^{67} +3.33387i q^{69} -11.0522 q^{71} -2.32984 q^{73} -1.50036 q^{75} -0.765529 q^{77} -9.04543 q^{79} +2.03441 q^{81} +17.2218i q^{83} +9.64553 q^{85} -3.48672i q^{87} +3.39907i q^{89} +1.99691 q^{91} +6.75426 q^{93} +(5.87086 + 9.55816i) q^{95} +16.0585i q^{97} -2.14478i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q + 44 q^{9} - 16 q^{17} + 36 q^{25} - 32 q^{45} - 28 q^{49} + 24 q^{57} - 48 q^{61} - 24 q^{73} + 52 q^{81} + 24 q^{85} - 64 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3344\mathbb{Z}\right)^\times\).

\(n\) \(705\) \(837\) \(2433\) \(2927\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) −0.924782 −0.533923 −0.266962 0.963707i \(-0.586020\pi\)
−0.266962 + 0.963707i \(0.586020\pi\)
\(4\) 0 0
\(5\) −2.57340 −1.15086 −0.575430 0.817851i \(-0.695165\pi\)
−0.575430 + 0.817851i \(0.695165\pi\)
\(6\) 0 0
\(7\) 0.765529i 0.289343i 0.989480 + 0.144671i \(0.0462125\pi\)
−0.989480 + 0.144671i \(0.953787\pi\)
\(8\) 0 0
\(9\) −2.14478 −0.714926
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 2.60853i 0.723476i −0.932280 0.361738i \(-0.882184\pi\)
0.932280 0.361738i \(-0.117816\pi\)
\(14\) 0 0
\(15\) 2.37984 0.614471
\(16\) 0 0
\(17\) −3.74816 −0.909063 −0.454532 0.890731i \(-0.650193\pi\)
−0.454532 + 0.890731i \(0.650193\pi\)
\(18\) 0 0
\(19\) −2.28136 3.71421i −0.523381 0.852099i
\(20\) 0 0
\(21\) 0.707948i 0.154487i
\(22\) 0 0
\(23\) 3.60503i 0.751701i −0.926680 0.375850i \(-0.877351\pi\)
0.926680 0.375850i \(-0.122649\pi\)
\(24\) 0 0
\(25\) 1.62240 0.324479
\(26\) 0 0
\(27\) 4.75780 0.915639
\(28\) 0 0
\(29\) 3.77031i 0.700129i 0.936726 + 0.350065i \(0.113840\pi\)
−0.936726 + 0.350065i \(0.886160\pi\)
\(30\) 0 0
\(31\) −7.30362 −1.31177 −0.655885 0.754861i \(-0.727704\pi\)
−0.655885 + 0.754861i \(0.727704\pi\)
\(32\) 0 0
\(33\) 0.924782i 0.160984i
\(34\) 0 0
\(35\) 1.97001i 0.332993i
\(36\) 0 0
\(37\) 8.15874i 1.34129i 0.741779 + 0.670644i \(0.233982\pi\)
−0.741779 + 0.670644i \(0.766018\pi\)
\(38\) 0 0
\(39\) 2.41232i 0.386281i
\(40\) 0 0
\(41\) 5.25952i 0.821398i −0.911771 0.410699i \(-0.865285\pi\)
0.911771 0.410699i \(-0.134715\pi\)
\(42\) 0 0
\(43\) 0.574153i 0.0875575i −0.999041 0.0437788i \(-0.986060\pi\)
0.999041 0.0437788i \(-0.0139397\pi\)
\(44\) 0 0
\(45\) 5.51937 0.822780
\(46\) 0 0
\(47\) 1.77015i 0.258203i −0.991631 0.129101i \(-0.958791\pi\)
0.991631 0.129101i \(-0.0412092\pi\)
\(48\) 0 0
\(49\) 6.41396 0.916281
\(50\) 0 0
\(51\) 3.46624 0.485370
\(52\) 0 0
\(53\) 11.7573i 1.61500i −0.589871 0.807498i \(-0.700821\pi\)
0.589871 0.807498i \(-0.299179\pi\)
\(54\) 0 0
\(55\) 2.57340i 0.346997i
\(56\) 0 0
\(57\) 2.10976 + 3.43484i 0.279445 + 0.454955i
\(58\) 0 0
\(59\) −4.11799 −0.536117 −0.268059 0.963403i \(-0.586382\pi\)
−0.268059 + 0.963403i \(0.586382\pi\)
\(60\) 0 0
\(61\) 9.17659 1.17494 0.587471 0.809245i \(-0.300124\pi\)
0.587471 + 0.809245i \(0.300124\pi\)
\(62\) 0 0
\(63\) 1.64189i 0.206859i
\(64\) 0 0
\(65\) 6.71279i 0.832619i
\(66\) 0 0
\(67\) −1.66350 −0.203229 −0.101614 0.994824i \(-0.532401\pi\)
−0.101614 + 0.994824i \(0.532401\pi\)
\(68\) 0 0
\(69\) 3.33387i 0.401351i
\(70\) 0 0
\(71\) −11.0522 −1.31165 −0.655825 0.754913i \(-0.727679\pi\)
−0.655825 + 0.754913i \(0.727679\pi\)
\(72\) 0 0
\(73\) −2.32984 −0.272687 −0.136343 0.990662i \(-0.543535\pi\)
−0.136343 + 0.990662i \(0.543535\pi\)
\(74\) 0 0
\(75\) −1.50036 −0.173247
\(76\) 0 0
\(77\) −0.765529 −0.0872402
\(78\) 0 0
\(79\) −9.04543 −1.01769 −0.508845 0.860858i \(-0.669927\pi\)
−0.508845 + 0.860858i \(0.669927\pi\)
\(80\) 0 0
\(81\) 2.03441 0.226045
\(82\) 0 0
\(83\) 17.2218i 1.89034i 0.326576 + 0.945171i \(0.394105\pi\)
−0.326576 + 0.945171i \(0.605895\pi\)
\(84\) 0 0
\(85\) 9.64553 1.04620
\(86\) 0 0
\(87\) 3.48672i 0.373815i
\(88\) 0 0
\(89\) 3.39907i 0.360301i 0.983639 + 0.180150i \(0.0576584\pi\)
−0.983639 + 0.180150i \(0.942342\pi\)
\(90\) 0 0
\(91\) 1.99691 0.209333
\(92\) 0 0
\(93\) 6.75426 0.700384
\(94\) 0 0
\(95\) 5.87086 + 9.55816i 0.602338 + 0.980647i
\(96\) 0 0
\(97\) 16.0585i 1.63049i 0.579116 + 0.815245i \(0.303398\pi\)
−0.579116 + 0.815245i \(0.696602\pi\)
\(98\) 0 0
\(99\) 2.14478i 0.215558i
\(100\) 0 0
\(101\) 13.2482 1.31824 0.659121 0.752037i \(-0.270928\pi\)
0.659121 + 0.752037i \(0.270928\pi\)
\(102\) 0 0
\(103\) 13.9964 1.37911 0.689554 0.724234i \(-0.257806\pi\)
0.689554 + 0.724234i \(0.257806\pi\)
\(104\) 0 0
\(105\) 1.82183i 0.177793i
\(106\) 0 0
\(107\) 7.82819 0.756779 0.378390 0.925646i \(-0.376478\pi\)
0.378390 + 0.925646i \(0.376478\pi\)
\(108\) 0 0
\(109\) 19.8017i 1.89666i 0.317281 + 0.948332i \(0.397230\pi\)
−0.317281 + 0.948332i \(0.602770\pi\)
\(110\) 0 0
\(111\) 7.54506i 0.716145i
\(112\) 0 0
\(113\) 3.79125i 0.356651i −0.983972 0.178326i \(-0.942932\pi\)
0.983972 0.178326i \(-0.0570680\pi\)
\(114\) 0 0
\(115\) 9.27719i 0.865102i
\(116\) 0 0
\(117\) 5.59471i 0.517232i
\(118\) 0 0
\(119\) 2.86933i 0.263031i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 4.86391i 0.438564i
\(124\) 0 0
\(125\) 8.69193 0.777430
\(126\) 0 0
\(127\) −21.0742 −1.87003 −0.935017 0.354603i \(-0.884616\pi\)
−0.935017 + 0.354603i \(0.884616\pi\)
\(128\) 0 0
\(129\) 0.530966i 0.0467490i
\(130\) 0 0
\(131\) 17.3752i 1.51808i −0.651045 0.759039i \(-0.725669\pi\)
0.651045 0.759039i \(-0.274331\pi\)
\(132\) 0 0
\(133\) 2.84334 1.74645i 0.246549 0.151436i
\(134\) 0 0
\(135\) −12.2437 −1.05377
\(136\) 0 0
\(137\) −5.90625 −0.504605 −0.252302 0.967648i \(-0.581188\pi\)
−0.252302 + 0.967648i \(0.581188\pi\)
\(138\) 0 0
\(139\) 15.0930i 1.28017i −0.768305 0.640084i \(-0.778900\pi\)
0.768305 0.640084i \(-0.221100\pi\)
\(140\) 0 0
\(141\) 1.63700i 0.137860i
\(142\) 0 0
\(143\) 2.60853 0.218136
\(144\) 0 0
\(145\) 9.70253i 0.805751i
\(146\) 0 0
\(147\) −5.93152 −0.489224
\(148\) 0 0
\(149\) 7.75827 0.635582 0.317791 0.948161i \(-0.397059\pi\)
0.317791 + 0.948161i \(0.397059\pi\)
\(150\) 0 0
\(151\) 22.8250 1.85747 0.928737 0.370739i \(-0.120895\pi\)
0.928737 + 0.370739i \(0.120895\pi\)
\(152\) 0 0
\(153\) 8.03898 0.649913
\(154\) 0 0
\(155\) 18.7952 1.50966
\(156\) 0 0
\(157\) 24.5482 1.95916 0.979581 0.201048i \(-0.0644347\pi\)
0.979581 + 0.201048i \(0.0644347\pi\)
\(158\) 0 0
\(159\) 10.8730i 0.862284i
\(160\) 0 0
\(161\) 2.75976 0.217499
\(162\) 0 0
\(163\) 7.07574i 0.554215i 0.960839 + 0.277107i \(0.0893758\pi\)
−0.960839 + 0.277107i \(0.910624\pi\)
\(164\) 0 0
\(165\) 2.37984i 0.185270i
\(166\) 0 0
\(167\) 13.8729 1.07352 0.536758 0.843736i \(-0.319649\pi\)
0.536758 + 0.843736i \(0.319649\pi\)
\(168\) 0 0
\(169\) 6.19558 0.476583
\(170\) 0 0
\(171\) 4.89302 + 7.96616i 0.374178 + 0.609188i
\(172\) 0 0
\(173\) 17.1394i 1.30309i 0.758611 + 0.651543i \(0.225878\pi\)
−0.758611 + 0.651543i \(0.774122\pi\)
\(174\) 0 0
\(175\) 1.24199i 0.0938857i
\(176\) 0 0
\(177\) 3.80825 0.286245
\(178\) 0 0
\(179\) −18.7633 −1.40244 −0.701218 0.712947i \(-0.747360\pi\)
−0.701218 + 0.712947i \(0.747360\pi\)
\(180\) 0 0
\(181\) 22.8245i 1.69653i 0.529571 + 0.848265i \(0.322353\pi\)
−0.529571 + 0.848265i \(0.677647\pi\)
\(182\) 0 0
\(183\) −8.48635 −0.627329
\(184\) 0 0
\(185\) 20.9957i 1.54364i
\(186\) 0 0
\(187\) 3.74816i 0.274093i
\(188\) 0 0
\(189\) 3.64224i 0.264934i
\(190\) 0 0
\(191\) 23.5974i 1.70745i −0.520726 0.853724i \(-0.674339\pi\)
0.520726 0.853724i \(-0.325661\pi\)
\(192\) 0 0
\(193\) 11.0786i 0.797452i 0.917070 + 0.398726i \(0.130547\pi\)
−0.917070 + 0.398726i \(0.869453\pi\)
\(194\) 0 0
\(195\) 6.20787i 0.444555i
\(196\) 0 0
\(197\) −1.34017 −0.0954832 −0.0477416 0.998860i \(-0.515202\pi\)
−0.0477416 + 0.998860i \(0.515202\pi\)
\(198\) 0 0
\(199\) 15.1677i 1.07521i 0.843196 + 0.537606i \(0.180671\pi\)
−0.843196 + 0.537606i \(0.819329\pi\)
\(200\) 0 0
\(201\) 1.53838 0.108509
\(202\) 0 0
\(203\) −2.88629 −0.202578
\(204\) 0 0
\(205\) 13.5348i 0.945315i
\(206\) 0 0
\(207\) 7.73199i 0.537410i
\(208\) 0 0
\(209\) 3.71421 2.28136i 0.256918 0.157805i
\(210\) 0 0
\(211\) 22.0907 1.52078 0.760392 0.649464i \(-0.225007\pi\)
0.760392 + 0.649464i \(0.225007\pi\)
\(212\) 0 0
\(213\) 10.2208 0.700320
\(214\) 0 0
\(215\) 1.47753i 0.100766i
\(216\) 0 0
\(217\) 5.59114i 0.379551i
\(218\) 0 0
\(219\) 2.15459 0.145594
\(220\) 0 0
\(221\) 9.77720i 0.657685i
\(222\) 0 0
\(223\) 17.3476 1.16168 0.580841 0.814017i \(-0.302724\pi\)
0.580841 + 0.814017i \(0.302724\pi\)
\(224\) 0 0
\(225\) −3.47968 −0.231978
\(226\) 0 0
\(227\) −4.31237 −0.286222 −0.143111 0.989707i \(-0.545711\pi\)
−0.143111 + 0.989707i \(0.545711\pi\)
\(228\) 0 0
\(229\) 15.5294 1.02621 0.513106 0.858325i \(-0.328495\pi\)
0.513106 + 0.858325i \(0.328495\pi\)
\(230\) 0 0
\(231\) 0.707948 0.0465796
\(232\) 0 0
\(233\) −14.8624 −0.973666 −0.486833 0.873495i \(-0.661848\pi\)
−0.486833 + 0.873495i \(0.661848\pi\)
\(234\) 0 0
\(235\) 4.55530i 0.297155i
\(236\) 0 0
\(237\) 8.36505 0.543368
\(238\) 0 0
\(239\) 3.12277i 0.201995i 0.994887 + 0.100998i \(0.0322035\pi\)
−0.994887 + 0.100998i \(0.967797\pi\)
\(240\) 0 0
\(241\) 20.2566i 1.30484i −0.757858 0.652420i \(-0.773754\pi\)
0.757858 0.652420i \(-0.226246\pi\)
\(242\) 0 0
\(243\) −16.1548 −1.03633
\(244\) 0 0
\(245\) −16.5057 −1.05451
\(246\) 0 0
\(247\) −9.68863 + 5.95100i −0.616473 + 0.378653i
\(248\) 0 0
\(249\) 15.9264i 1.00930i
\(250\) 0 0
\(251\) 21.4793i 1.35576i 0.735172 + 0.677880i \(0.237101\pi\)
−0.735172 + 0.677880i \(0.762899\pi\)
\(252\) 0 0
\(253\) 3.60503 0.226646
\(254\) 0 0
\(255\) −8.92002 −0.558593
\(256\) 0 0
\(257\) 13.0893i 0.816489i −0.912873 0.408245i \(-0.866141\pi\)
0.912873 0.408245i \(-0.133859\pi\)
\(258\) 0 0
\(259\) −6.24576 −0.388092
\(260\) 0 0
\(261\) 8.08648i 0.500541i
\(262\) 0 0
\(263\) 23.0707i 1.42260i −0.702888 0.711300i \(-0.748107\pi\)
0.702888 0.711300i \(-0.251893\pi\)
\(264\) 0 0
\(265\) 30.2564i 1.85863i
\(266\) 0 0
\(267\) 3.14340i 0.192373i
\(268\) 0 0
\(269\) 5.99734i 0.365664i −0.983144 0.182832i \(-0.941474\pi\)
0.983144 0.182832i \(-0.0585265\pi\)
\(270\) 0 0
\(271\) 17.8489i 1.08424i 0.840300 + 0.542121i \(0.182379\pi\)
−0.840300 + 0.542121i \(0.817621\pi\)
\(272\) 0 0
\(273\) −1.84670 −0.111768
\(274\) 0 0
\(275\) 1.62240i 0.0978341i
\(276\) 0 0
\(277\) 16.3605 0.983010 0.491505 0.870875i \(-0.336447\pi\)
0.491505 + 0.870875i \(0.336447\pi\)
\(278\) 0 0
\(279\) 15.6647 0.937818
\(280\) 0 0
\(281\) 9.69477i 0.578342i −0.957278 0.289171i \(-0.906620\pi\)
0.957278 0.289171i \(-0.0933796\pi\)
\(282\) 0 0
\(283\) 20.4923i 1.21814i −0.793117 0.609069i \(-0.791543\pi\)
0.793117 0.609069i \(-0.208457\pi\)
\(284\) 0 0
\(285\) −5.42927 8.83922i −0.321602 0.523590i
\(286\) 0 0
\(287\) 4.02631 0.237666
\(288\) 0 0
\(289\) −2.95126 −0.173604
\(290\) 0 0
\(291\) 14.8506i 0.870557i
\(292\) 0 0
\(293\) 2.22695i 0.130100i −0.997882 0.0650500i \(-0.979279\pi\)
0.997882 0.0650500i \(-0.0207207\pi\)
\(294\) 0 0
\(295\) 10.5973 0.616996
\(296\) 0 0
\(297\) 4.75780i 0.276076i
\(298\) 0 0
\(299\) −9.40383 −0.543837
\(300\) 0 0
\(301\) 0.439531 0.0253341
\(302\) 0 0
\(303\) −12.2517 −0.703841
\(304\) 0 0
\(305\) −23.6151 −1.35219
\(306\) 0 0
\(307\) 21.5923 1.23233 0.616167 0.787615i \(-0.288684\pi\)
0.616167 + 0.787615i \(0.288684\pi\)
\(308\) 0 0
\(309\) −12.9436 −0.736338
\(310\) 0 0
\(311\) 17.4307i 0.988403i 0.869347 + 0.494202i \(0.164540\pi\)
−0.869347 + 0.494202i \(0.835460\pi\)
\(312\) 0 0
\(313\) −6.30327 −0.356282 −0.178141 0.984005i \(-0.557008\pi\)
−0.178141 + 0.984005i \(0.557008\pi\)
\(314\) 0 0
\(315\) 4.22524i 0.238066i
\(316\) 0 0
\(317\) 30.8999i 1.73551i 0.496989 + 0.867757i \(0.334439\pi\)
−0.496989 + 0.867757i \(0.665561\pi\)
\(318\) 0 0
\(319\) −3.77031 −0.211097
\(320\) 0 0
\(321\) −7.23937 −0.404062
\(322\) 0 0
\(323\) 8.55093 + 13.9215i 0.475786 + 0.774612i
\(324\) 0 0
\(325\) 4.23206i 0.234753i
\(326\) 0 0
\(327\) 18.3123i 1.01267i
\(328\) 0 0
\(329\) 1.35510 0.0747091
\(330\) 0 0
\(331\) −26.5527 −1.45947 −0.729734 0.683731i \(-0.760356\pi\)
−0.729734 + 0.683731i \(0.760356\pi\)
\(332\) 0 0
\(333\) 17.4987i 0.958922i
\(334\) 0 0
\(335\) 4.28085 0.233888
\(336\) 0 0
\(337\) 11.0220i 0.600407i −0.953875 0.300204i \(-0.902945\pi\)
0.953875 0.300204i \(-0.0970547\pi\)
\(338\) 0 0
\(339\) 3.50608i 0.190424i
\(340\) 0 0
\(341\) 7.30362i 0.395513i
\(342\) 0 0
\(343\) 10.2688i 0.554462i
\(344\) 0 0
\(345\) 8.57938i 0.461898i
\(346\) 0 0
\(347\) 12.2891i 0.659711i 0.944031 + 0.329856i \(0.107000\pi\)
−0.944031 + 0.329856i \(0.893000\pi\)
\(348\) 0 0
\(349\) 22.1394 1.18509 0.592546 0.805536i \(-0.298123\pi\)
0.592546 + 0.805536i \(0.298123\pi\)
\(350\) 0 0
\(351\) 12.4109i 0.662442i
\(352\) 0 0
\(353\) −25.6400 −1.36468 −0.682339 0.731035i \(-0.739037\pi\)
−0.682339 + 0.731035i \(0.739037\pi\)
\(354\) 0 0
\(355\) 28.4416 1.50952
\(356\) 0 0
\(357\) 2.65351i 0.140438i
\(358\) 0 0
\(359\) 8.13933i 0.429577i 0.976661 + 0.214789i \(0.0689063\pi\)
−0.976661 + 0.214789i \(0.931094\pi\)
\(360\) 0 0
\(361\) −8.59076 + 16.9469i −0.452145 + 0.891944i
\(362\) 0 0
\(363\) 0.924782 0.0485385
\(364\) 0 0
\(365\) 5.99561 0.313825
\(366\) 0 0
\(367\) 30.2059i 1.57674i −0.615203 0.788369i \(-0.710926\pi\)
0.615203 0.788369i \(-0.289074\pi\)
\(368\) 0 0
\(369\) 11.2805i 0.587239i
\(370\) 0 0
\(371\) 9.00059 0.467288
\(372\) 0 0
\(373\) 5.16531i 0.267450i −0.991018 0.133725i \(-0.957306\pi\)
0.991018 0.133725i \(-0.0426939\pi\)
\(374\) 0 0
\(375\) −8.03815 −0.415088
\(376\) 0 0
\(377\) 9.83497 0.506527
\(378\) 0 0
\(379\) 15.0822 0.774722 0.387361 0.921928i \(-0.373387\pi\)
0.387361 + 0.921928i \(0.373387\pi\)
\(380\) 0 0
\(381\) 19.4891 0.998454
\(382\) 0 0
\(383\) −5.75172 −0.293899 −0.146949 0.989144i \(-0.546945\pi\)
−0.146949 + 0.989144i \(0.546945\pi\)
\(384\) 0 0
\(385\) 1.97001 0.100401
\(386\) 0 0
\(387\) 1.23143i 0.0625971i
\(388\) 0 0
\(389\) −24.8324 −1.25905 −0.629527 0.776978i \(-0.716751\pi\)
−0.629527 + 0.776978i \(0.716751\pi\)
\(390\) 0 0
\(391\) 13.5122i 0.683344i
\(392\) 0 0
\(393\) 16.0683i 0.810537i
\(394\) 0 0
\(395\) 23.2775 1.17122
\(396\) 0 0
\(397\) 10.0021 0.501991 0.250995 0.967988i \(-0.419242\pi\)
0.250995 + 0.967988i \(0.419242\pi\)
\(398\) 0 0
\(399\) −2.62947 + 1.61509i −0.131638 + 0.0808555i
\(400\) 0 0
\(401\) 13.6318i 0.680739i 0.940292 + 0.340369i \(0.110552\pi\)
−0.940292 + 0.340369i \(0.889448\pi\)
\(402\) 0 0
\(403\) 19.0517i 0.949033i
\(404\) 0 0
\(405\) −5.23534 −0.260146
\(406\) 0 0
\(407\) −8.15874 −0.404414
\(408\) 0 0
\(409\) 32.7709i 1.62042i −0.586142 0.810208i \(-0.699354\pi\)
0.586142 0.810208i \(-0.300646\pi\)
\(410\) 0 0
\(411\) 5.46199 0.269420
\(412\) 0 0
\(413\) 3.15245i 0.155122i
\(414\) 0 0
\(415\) 44.3187i 2.17552i
\(416\) 0 0
\(417\) 13.9577i 0.683512i
\(418\) 0 0
\(419\) 21.6469i 1.05752i 0.848771 + 0.528760i \(0.177343\pi\)
−0.848771 + 0.528760i \(0.822657\pi\)
\(420\) 0 0
\(421\) 16.5251i 0.805385i −0.915335 0.402692i \(-0.868074\pi\)
0.915335 0.402692i \(-0.131926\pi\)
\(422\) 0 0
\(423\) 3.79657i 0.184596i
\(424\) 0 0
\(425\) −6.08100 −0.294972
\(426\) 0 0
\(427\) 7.02495i 0.339961i
\(428\) 0 0
\(429\) −2.41232 −0.116468
\(430\) 0 0
\(431\) −8.49914 −0.409389 −0.204694 0.978826i \(-0.565620\pi\)
−0.204694 + 0.978826i \(0.565620\pi\)
\(432\) 0 0
\(433\) 8.14216i 0.391287i −0.980675 0.195644i \(-0.937320\pi\)
0.980675 0.195644i \(-0.0626796\pi\)
\(434\) 0 0
\(435\) 8.97273i 0.430209i
\(436\) 0 0
\(437\) −13.3899 + 8.22438i −0.640523 + 0.393426i
\(438\) 0 0
\(439\) −9.56561 −0.456541 −0.228271 0.973598i \(-0.573307\pi\)
−0.228271 + 0.973598i \(0.573307\pi\)
\(440\) 0 0
\(441\) −13.7565 −0.655073
\(442\) 0 0
\(443\) 31.8773i 1.51454i −0.653104 0.757269i \(-0.726533\pi\)
0.653104 0.757269i \(-0.273467\pi\)
\(444\) 0 0
\(445\) 8.74717i 0.414656i
\(446\) 0 0
\(447\) −7.17471 −0.339352
\(448\) 0 0
\(449\) 17.9188i 0.845640i 0.906214 + 0.422820i \(0.138960\pi\)
−0.906214 + 0.422820i \(0.861040\pi\)
\(450\) 0 0
\(451\) 5.25952 0.247661
\(452\) 0 0
\(453\) −21.1082 −0.991749
\(454\) 0 0
\(455\) −5.13884 −0.240913
\(456\) 0 0
\(457\) 4.14113 0.193714 0.0968569 0.995298i \(-0.469121\pi\)
0.0968569 + 0.995298i \(0.469121\pi\)
\(458\) 0 0
\(459\) −17.8330 −0.832374
\(460\) 0 0
\(461\) 8.48067 0.394984 0.197492 0.980304i \(-0.436720\pi\)
0.197492 + 0.980304i \(0.436720\pi\)
\(462\) 0 0
\(463\) 21.0728i 0.979336i 0.871909 + 0.489668i \(0.162882\pi\)
−0.871909 + 0.489668i \(0.837118\pi\)
\(464\) 0 0
\(465\) −17.3814 −0.806044
\(466\) 0 0
\(467\) 11.6322i 0.538273i −0.963102 0.269137i \(-0.913262\pi\)
0.963102 0.269137i \(-0.0867383\pi\)
\(468\) 0 0
\(469\) 1.27346i 0.0588029i
\(470\) 0 0
\(471\) −22.7018 −1.04604
\(472\) 0 0
\(473\) 0.574153 0.0263996
\(474\) 0 0
\(475\) −3.70127 6.02592i −0.169826 0.276488i
\(476\) 0 0
\(477\) 25.2169i 1.15460i
\(478\) 0 0
\(479\) 21.0362i 0.961167i 0.876949 + 0.480583i \(0.159575\pi\)
−0.876949 + 0.480583i \(0.840425\pi\)
\(480\) 0 0
\(481\) 21.2823 0.970390
\(482\) 0 0
\(483\) −2.55217 −0.116128
\(484\) 0 0
\(485\) 41.3249i 1.87647i
\(486\) 0 0
\(487\) −5.23979 −0.237438 −0.118719 0.992928i \(-0.537879\pi\)
−0.118719 + 0.992928i \(0.537879\pi\)
\(488\) 0 0
\(489\) 6.54352i 0.295908i
\(490\) 0 0
\(491\) 8.71576i 0.393337i −0.980470 0.196668i \(-0.936988\pi\)
0.980470 0.196668i \(-0.0630122\pi\)
\(492\) 0 0
\(493\) 14.1318i 0.636462i
\(494\) 0 0
\(495\) 5.51937i 0.248077i
\(496\) 0 0
\(497\) 8.46075i 0.379516i
\(498\) 0 0
\(499\) 24.0516i 1.07670i −0.842722 0.538349i \(-0.819048\pi\)
0.842722 0.538349i \(-0.180952\pi\)
\(500\) 0 0
\(501\) −12.8294 −0.573175
\(502\) 0 0
\(503\) 8.14499i 0.363167i 0.983376 + 0.181583i \(0.0581223\pi\)
−0.983376 + 0.181583i \(0.941878\pi\)
\(504\) 0 0
\(505\) −34.0929 −1.51711
\(506\) 0 0
\(507\) −5.72956 −0.254459
\(508\) 0 0
\(509\) 24.8864i 1.10307i 0.834151 + 0.551536i \(0.185958\pi\)
−0.834151 + 0.551536i \(0.814042\pi\)
\(510\) 0 0
\(511\) 1.78356i 0.0789000i
\(512\) 0 0
\(513\) −10.8543 17.6715i −0.479228 0.780215i
\(514\) 0 0
\(515\) −36.0184 −1.58716
\(516\) 0 0
\(517\) 1.77015 0.0778510
\(518\) 0 0
\(519\) 15.8502i 0.695748i
\(520\) 0 0
\(521\) 4.50065i 0.197177i 0.995128 + 0.0985886i \(0.0314328\pi\)
−0.995128 + 0.0985886i \(0.968567\pi\)
\(522\) 0 0
\(523\) −28.4924 −1.24588 −0.622942 0.782268i \(-0.714063\pi\)
−0.622942 + 0.782268i \(0.714063\pi\)
\(524\) 0 0
\(525\) 1.14857i 0.0501278i
\(526\) 0 0
\(527\) 27.3752 1.19248
\(528\) 0 0
\(529\) 10.0038 0.434946
\(530\) 0 0
\(531\) 8.83218 0.383284
\(532\) 0 0
\(533\) −13.7196 −0.594262
\(534\) 0 0
\(535\) −20.1451 −0.870947
\(536\) 0 0
\(537\) 17.3520 0.748793
\(538\) 0 0
\(539\) 6.41396i 0.276269i
\(540\) 0 0
\(541\) 18.5155 0.796044 0.398022 0.917376i \(-0.369697\pi\)
0.398022 + 0.917376i \(0.369697\pi\)
\(542\) 0 0
\(543\) 21.1077i 0.905817i
\(544\) 0 0
\(545\) 50.9578i 2.18279i
\(546\) 0 0
\(547\) −5.58928 −0.238980 −0.119490 0.992835i \(-0.538126\pi\)
−0.119490 + 0.992835i \(0.538126\pi\)
\(548\) 0 0
\(549\) −19.6817 −0.839997
\(550\) 0 0
\(551\) 14.0037 8.60145i 0.596580 0.366434i
\(552\) 0 0
\(553\) 6.92454i 0.294461i
\(554\) 0 0
\(555\) 19.4165i 0.824183i
\(556\) 0 0
\(557\) −11.1418 −0.472092 −0.236046 0.971742i \(-0.575852\pi\)
−0.236046 + 0.971742i \(0.575852\pi\)
\(558\) 0 0
\(559\) −1.49769 −0.0633457
\(560\) 0 0
\(561\) 3.46624i 0.146345i
\(562\) 0 0
\(563\) −14.5096 −0.611505 −0.305752 0.952111i \(-0.598908\pi\)
−0.305752 + 0.952111i \(0.598908\pi\)
\(564\) 0 0
\(565\) 9.75642i 0.410456i
\(566\) 0 0
\(567\) 1.55740i 0.0654045i
\(568\) 0 0
\(569\) 40.4383i 1.69526i −0.530587 0.847630i \(-0.678029\pi\)
0.530587 0.847630i \(-0.321971\pi\)
\(570\) 0 0
\(571\) 11.6196i 0.486265i 0.969993 + 0.243132i \(0.0781750\pi\)
−0.969993 + 0.243132i \(0.921825\pi\)
\(572\) 0 0
\(573\) 21.8225i 0.911646i
\(574\) 0 0
\(575\) 5.84878i 0.243911i
\(576\) 0 0
\(577\) 16.9948 0.707501 0.353750 0.935340i \(-0.384906\pi\)
0.353750 + 0.935340i \(0.384906\pi\)
\(578\) 0 0
\(579\) 10.2452i 0.425778i
\(580\) 0 0
\(581\) −13.1838 −0.546957
\(582\) 0 0
\(583\) 11.7573 0.486940
\(584\) 0 0
\(585\) 14.3974i 0.595261i
\(586\) 0 0
\(587\) 9.92060i 0.409467i −0.978818 0.204734i \(-0.934367\pi\)
0.978818 0.204734i \(-0.0656328\pi\)
\(588\) 0 0
\(589\) 16.6622 + 27.1272i 0.686555 + 1.11776i
\(590\) 0 0
\(591\) 1.23937 0.0509807
\(592\) 0 0
\(593\) −8.16956 −0.335483 −0.167742 0.985831i \(-0.553647\pi\)
−0.167742 + 0.985831i \(0.553647\pi\)
\(594\) 0 0
\(595\) 7.38394i 0.302712i
\(596\) 0 0
\(597\) 14.0268i 0.574080i
\(598\) 0 0
\(599\) 13.6191 0.556463 0.278231 0.960514i \(-0.410252\pi\)
0.278231 + 0.960514i \(0.410252\pi\)
\(600\) 0 0
\(601\) 45.5092i 1.85636i 0.372132 + 0.928180i \(0.378627\pi\)
−0.372132 + 0.928180i \(0.621373\pi\)
\(602\) 0 0
\(603\) 3.56784 0.145294
\(604\) 0 0
\(605\) 2.57340 0.104624
\(606\) 0 0
\(607\) −12.3767 −0.502356 −0.251178 0.967941i \(-0.580818\pi\)
−0.251178 + 0.967941i \(0.580818\pi\)
\(608\) 0 0
\(609\) 2.66919 0.108161
\(610\) 0 0
\(611\) −4.61748 −0.186803
\(612\) 0 0
\(613\) −33.7650 −1.36375 −0.681877 0.731467i \(-0.738836\pi\)
−0.681877 + 0.731467i \(0.738836\pi\)
\(614\) 0 0
\(615\) 12.5168i 0.504725i
\(616\) 0 0
\(617\) 8.86095 0.356728 0.178364 0.983965i \(-0.442919\pi\)
0.178364 + 0.983965i \(0.442919\pi\)
\(618\) 0 0
\(619\) 9.50658i 0.382102i −0.981580 0.191051i \(-0.938810\pi\)
0.981580 0.191051i \(-0.0611895\pi\)
\(620\) 0 0
\(621\) 17.1520i 0.688286i
\(622\) 0 0
\(623\) −2.60209 −0.104250
\(624\) 0 0
\(625\) −30.4798 −1.21919
\(626\) 0 0
\(627\) −3.43484 + 2.10976i −0.137174 + 0.0842559i
\(628\) 0 0
\(629\) 30.5803i 1.21932i
\(630\) 0 0
\(631\) 5.28201i 0.210273i 0.994458 + 0.105137i \(0.0335280\pi\)
−0.994458 + 0.105137i \(0.966472\pi\)
\(632\) 0 0
\(633\) −20.4291 −0.811982
\(634\) 0 0
\(635\) 54.2324 2.15215
\(636\) 0 0
\(637\) 16.7310i 0.662907i
\(638\) 0 0
\(639\) 23.7044 0.937732
\(640\) 0 0
\(641\) 24.0110i 0.948377i −0.880423 0.474189i \(-0.842741\pi\)
0.880423 0.474189i \(-0.157259\pi\)
\(642\) 0 0
\(643\) 35.1370i 1.38567i −0.721096 0.692835i \(-0.756362\pi\)
0.721096 0.692835i \(-0.243638\pi\)
\(644\) 0 0
\(645\) 1.36639i 0.0538015i
\(646\) 0 0
\(647\) 6.91965i 0.272039i 0.990706 + 0.136020i \(0.0434311\pi\)
−0.990706 + 0.136020i \(0.956569\pi\)
\(648\) 0 0
\(649\) 4.11799i 0.161645i
\(650\) 0 0
\(651\) 5.17059i 0.202651i
\(652\) 0 0
\(653\) −7.56654 −0.296102 −0.148051 0.988980i \(-0.547300\pi\)
−0.148051 + 0.988980i \(0.547300\pi\)
\(654\) 0 0
\(655\) 44.7133i 1.74709i
\(656\) 0 0
\(657\) 4.99699 0.194951
\(658\) 0 0
\(659\) −13.1971 −0.514084 −0.257042 0.966400i \(-0.582748\pi\)
−0.257042 + 0.966400i \(0.582748\pi\)
\(660\) 0 0
\(661\) 15.4346i 0.600338i 0.953886 + 0.300169i \(0.0970430\pi\)
−0.953886 + 0.300169i \(0.902957\pi\)
\(662\) 0 0
\(663\) 9.04178i 0.351154i
\(664\) 0 0
\(665\) −7.31705 + 4.49432i −0.283743 + 0.174282i
\(666\) 0 0
\(667\) 13.5921 0.526288
\(668\) 0 0
\(669\) −16.0428 −0.620249
\(670\) 0 0
\(671\) 9.17659i 0.354258i
\(672\) 0 0
\(673\) 36.8348i 1.41988i 0.704263 + 0.709939i \(0.251277\pi\)
−0.704263 + 0.709939i \(0.748723\pi\)
\(674\) 0 0
\(675\) 7.71903 0.297106
\(676\) 0 0
\(677\) 39.0494i 1.50079i −0.660989 0.750396i \(-0.729863\pi\)
0.660989 0.750396i \(-0.270137\pi\)
\(678\) 0 0
\(679\) −12.2932 −0.471771
\(680\) 0 0
\(681\) 3.98800 0.152821
\(682\) 0 0
\(683\) −29.4734 −1.12777 −0.563885 0.825853i \(-0.690694\pi\)
−0.563885 + 0.825853i \(0.690694\pi\)
\(684\) 0 0
\(685\) 15.1992 0.580730
\(686\) 0 0
\(687\) −14.3613 −0.547918
\(688\) 0 0
\(689\) −30.6694 −1.16841
\(690\) 0 0
\(691\) 20.7405i 0.789008i 0.918894 + 0.394504i \(0.129084\pi\)
−0.918894 + 0.394504i \(0.870916\pi\)
\(692\) 0 0
\(693\) 1.64189 0.0623703
\(694\) 0 0
\(695\) 38.8402i 1.47329i
\(696\) 0 0
\(697\) 19.7135i 0.746703i
\(698\) 0 0
\(699\) 13.7444 0.519863
\(700\) 0 0
\(701\) 49.8681 1.88349 0.941745 0.336326i \(-0.109185\pi\)
0.941745 + 0.336326i \(0.109185\pi\)
\(702\) 0 0
\(703\) 30.3033 18.6131i 1.14291 0.702005i
\(704\) 0 0
\(705\) 4.21266i 0.158658i
\(706\) 0 0
\(707\) 10.1419i 0.381424i
\(708\) 0 0
\(709\) −5.33476 −0.200351 −0.100176 0.994970i \(-0.531940\pi\)
−0.100176 + 0.994970i \(0.531940\pi\)
\(710\) 0 0
\(711\) 19.4004 0.727573
\(712\) 0 0
\(713\) 26.3298i 0.986058i
\(714\) 0 0
\(715\) −6.71279 −0.251044
\(716\) 0 0
\(717\) 2.88789i 0.107850i
\(718\) 0 0
\(719\) 51.1019i 1.90578i −0.303319 0.952889i \(-0.598095\pi\)
0.303319 0.952889i \(-0.401905\pi\)
\(720\) 0 0
\(721\) 10.7147i 0.399035i
\(722\) 0 0
\(723\) 18.7329i 0.696684i
\(724\) 0 0
\(725\) 6.11694i 0.227177i
\(726\) 0 0
\(727\) 1.04368i 0.0387080i −0.999813 0.0193540i \(-0.993839\pi\)
0.999813 0.0193540i \(-0.00616095\pi\)
\(728\) 0 0
\(729\) 8.83644 0.327275
\(730\) 0 0
\(731\) 2.15202i 0.0795953i
\(732\) 0 0
\(733\) 35.7389 1.32005 0.660023 0.751246i \(-0.270547\pi\)
0.660023 + 0.751246i \(0.270547\pi\)
\(734\) 0 0
\(735\) 15.2642 0.563028
\(736\) 0 0
\(737\) 1.66350i 0.0612758i
\(738\) 0 0
\(739\) 23.9647i 0.881555i −0.897616 0.440778i \(-0.854703\pi\)
0.897616 0.440778i \(-0.145297\pi\)
\(740\) 0 0
\(741\) 8.95988 5.50338i 0.329149 0.202172i
\(742\) 0 0
\(743\) −3.59899 −0.132034 −0.0660170 0.997818i \(-0.521029\pi\)
−0.0660170 + 0.997818i \(0.521029\pi\)
\(744\) 0 0
\(745\) −19.9651 −0.731466
\(746\) 0 0
\(747\) 36.9370i 1.35145i
\(748\) 0 0
\(749\) 5.99271i 0.218969i
\(750\) 0 0
\(751\) 26.7342 0.975545 0.487773 0.872971i \(-0.337810\pi\)
0.487773 + 0.872971i \(0.337810\pi\)
\(752\) 0 0
\(753\) 19.8637i 0.723872i
\(754\) 0 0
\(755\) −58.7380 −2.13769
\(756\) 0 0
\(757\) −2.86207 −0.104024 −0.0520119 0.998646i \(-0.516563\pi\)
−0.0520119 + 0.998646i \(0.516563\pi\)
\(758\) 0 0
\(759\) −3.33387 −0.121012
\(760\) 0 0
\(761\) 5.93442 0.215123 0.107561 0.994198i \(-0.465696\pi\)
0.107561 + 0.994198i \(0.465696\pi\)
\(762\) 0 0
\(763\) −15.1588 −0.548786
\(764\) 0 0
\(765\) −20.6875 −0.747959
\(766\) 0 0
\(767\) 10.7419i 0.387868i
\(768\) 0 0
\(769\) 19.3655 0.698339 0.349169 0.937060i \(-0.386464\pi\)
0.349169 + 0.937060i \(0.386464\pi\)
\(770\) 0 0
\(771\) 12.1048i 0.435943i
\(772\) 0 0
\(773\) 34.7544i 1.25003i 0.780613 + 0.625015i \(0.214907\pi\)
−0.780613 + 0.625015i \(0.785093\pi\)
\(774\) 0 0
\(775\) −11.8494 −0.425642
\(776\) 0 0
\(777\) 5.77597 0.207212
\(778\) 0 0
\(779\) −19.5350 + 11.9989i −0.699913 + 0.429904i
\(780\) 0 0
\(781\) 11.0522i 0.395477i
\(782\) 0 0
\(783\) 17.9384i 0.641066i
\(784\) 0 0
\(785\) −63.1725 −2.25472
\(786\) 0 0
\(787\) −10.5553 −0.376254 −0.188127 0.982145i \(-0.560242\pi\)
−0.188127 + 0.982145i \(0.560242\pi\)
\(788\) 0 0
\(789\) 21.3354i 0.759560i
\(790\) 0 0
\(791\) 2.90232 0.103194
\(792\) 0 0
\(793\) 23.9374i 0.850042i
\(794\) 0 0
\(795\) 27.9806i 0.992368i
\(796\) 0 0
\(797\) 15.8878i 0.562776i 0.959594 + 0.281388i \(0.0907947\pi\)
−0.959594 + 0.281388i \(0.909205\pi\)
\(798\) 0 0
\(799\) 6.63480i 0.234723i
\(800\) 0 0
\(801\) 7.29025i 0.257588i
\(802\) 0 0
\(803\) 2.32984i 0.0822182i
\(804\) 0 0
\(805\) −7.10196 −0.250311
\(806\) 0 0
\(807\) 5.54623i 0.195237i
\(808\) 0 0
\(809\) −20.0958 −0.706532 −0.353266 0.935523i \(-0.614929\pi\)
−0.353266 + 0.935523i \(0.614929\pi\)
\(810\) 0 0
\(811\) −0.698862 −0.0245404 −0.0122702 0.999925i \(-0.503906\pi\)
−0.0122702 + 0.999925i \(0.503906\pi\)
\(812\) 0 0
\(813\) 16.5063i 0.578903i
\(814\) 0 0
\(815\) 18.2087i 0.637824i
\(816\) 0 0
\(817\) −2.13253 + 1.30985i −0.0746077 + 0.0458259i
\(818\) 0 0
\(819\) −4.28292 −0.149657
\(820\) 0 0
\(821\) 2.92951 0.102241 0.0511203 0.998693i \(-0.483721\pi\)
0.0511203 + 0.998693i \(0.483721\pi\)
\(822\) 0 0
\(823\) 29.9843i 1.04519i −0.852582 0.522593i \(-0.824965\pi\)
0.852582 0.522593i \(-0.175035\pi\)
\(824\) 0 0
\(825\) 1.50036i 0.0522359i
\(826\) 0 0
\(827\) −32.2067 −1.11994 −0.559969 0.828514i \(-0.689187\pi\)
−0.559969 + 0.828514i \(0.689187\pi\)
\(828\) 0 0
\(829\) 5.21195i 0.181019i −0.995896 0.0905093i \(-0.971151\pi\)
0.995896 0.0905093i \(-0.0288495\pi\)
\(830\) 0 0
\(831\) −15.1299 −0.524852
\(832\) 0 0
\(833\) −24.0406 −0.832957
\(834\) 0 0
\(835\) −35.7005 −1.23547
\(836\) 0 0
\(837\) −34.7492 −1.20111
\(838\) 0 0
\(839\) 43.1109 1.48835 0.744177 0.667982i \(-0.232842\pi\)
0.744177 + 0.667982i \(0.232842\pi\)
\(840\) 0 0
\(841\) 14.7847 0.509819
\(842\) 0 0
\(843\) 8.96555i 0.308790i
\(844\) 0 0
\(845\) −15.9437 −0.548480
\(846\) 0 0
\(847\) 0.765529i 0.0263039i
\(848\) 0 0
\(849\) 18.9509i 0.650393i
\(850\) 0 0
\(851\) 29.4125 1.00825
\(852\) 0 0
\(853\) 17.0560 0.583985 0.291993 0.956421i \(-0.405682\pi\)
0.291993 + 0.956421i \(0.405682\pi\)
\(854\) 0 0
\(855\) −12.5917 20.5001i −0.430627 0.701090i
\(856\) 0 0
\(857\) 47.6403i 1.62736i 0.581312 + 0.813680i \(0.302539\pi\)
−0.581312 + 0.813680i \(0.697461\pi\)
\(858\) 0 0
\(859\) 54.4089i 1.85641i 0.372072 + 0.928204i \(0.378648\pi\)
−0.372072 + 0.928204i \(0.621352\pi\)
\(860\) 0 0
\(861\) −3.72346 −0.126895
\(862\) 0 0
\(863\) −0.0365416 −0.00124389 −0.000621945 1.00000i \(-0.500198\pi\)
−0.000621945 1.00000i \(0.500198\pi\)
\(864\) 0 0
\(865\) 44.1066i 1.49967i
\(866\) 0 0
\(867\) 2.72927 0.0926910
\(868\) 0 0
\(869\) 9.04543i 0.306845i
\(870\) 0 0
\(871\) 4.33929i 0.147031i
\(872\) 0 0
\(873\) 34.4418i 1.16568i
\(874\) 0 0
\(875\) 6.65393i 0.224944i
\(876\) 0 0
\(877\) 30.3011i 1.02319i −0.859225 0.511597i \(-0.829054\pi\)
0.859225 0.511597i \(-0.170946\pi\)
\(878\) 0 0
\(879\) 2.05945i 0.0694635i
\(880\) 0 0
\(881\) −25.0140 −0.842743 −0.421372 0.906888i \(-0.638451\pi\)
−0.421372 + 0.906888i \(0.638451\pi\)
\(882\) 0 0
\(883\) 34.0342i 1.14534i −0.819785 0.572671i \(-0.805907\pi\)
0.819785 0.572671i \(-0.194093\pi\)
\(884\) 0 0
\(885\) −9.80015 −0.329428
\(886\) 0 0
\(887\) −17.5033 −0.587702 −0.293851 0.955851i \(-0.594937\pi\)
−0.293851 + 0.955851i \(0.594937\pi\)
\(888\) 0 0
\(889\) 16.1329i 0.541081i
\(890\) 0 0
\(891\) 2.03441i 0.0681551i
\(892\) 0 0
\(893\) −6.57471 + 4.03835i −0.220014 + 0.135138i
\(894\) 0 0
\(895\) 48.2855 1.61401
\(896\) 0 0
\(897\) 8.69649 0.290367
\(898\) 0 0
\(899\) 27.5369i 0.918409i
\(900\) 0 0
\(901\) 44.0685i 1.46813i
\(902\) 0 0
\(903\) −0.406470 −0.0135265
\(904\) 0 0
\(905\) 58.7366i 1.95247i
\(906\) 0 0
\(907\) 55.3675 1.83845 0.919224 0.393735i \(-0.128817\pi\)
0.919224 + 0.393735i \(0.128817\pi\)
\(908\) 0 0
\(909\) −28.4144 −0.942446
\(910\) 0 0
\(911\) 45.0058 1.49111 0.745554 0.666445i \(-0.232185\pi\)
0.745554 + 0.666445i \(0.232185\pi\)
\(912\) 0 0
\(913\) −17.2218 −0.569959
\(914\) 0 0
\(915\) 21.8388 0.721968
\(916\) 0 0
\(917\) 13.3012 0.439245
\(918\) 0 0
\(919\) 33.8325i 1.11603i 0.829830 + 0.558016i \(0.188437\pi\)
−0.829830 + 0.558016i \(0.811563\pi\)
\(920\) 0 0
\(921\) −19.9681 −0.657972
\(922\) 0 0
\(923\) 28.8299i 0.948946i
\(924\) 0 0
\(925\) 13.2367i 0.435220i
\(926\) 0 0
\(927\) −30.0192 −0.985961
\(928\) 0 0
\(929\) −40.9987 −1.34512 −0.672562 0.740040i \(-0.734806\pi\)
−0.672562 + 0.740040i \(0.734806\pi\)
\(930\) 0 0
\(931\) −14.6326 23.8228i −0.479564 0.780762i
\(932\) 0 0
\(933\) 16.1196i 0.527731i
\(934\) 0 0
\(935\) 9.64553i 0.315443i
\(936\) 0 0
\(937\) 24.5155 0.800886 0.400443 0.916322i \(-0.368856\pi\)
0.400443 + 0.916322i \(0.368856\pi\)
\(938\) 0 0
\(939\) 5.82915 0.190227
\(940\) 0 0
\(941\) 3.64399i 0.118791i 0.998235 + 0.0593954i \(0.0189173\pi\)
−0.998235 + 0.0593954i \(0.981083\pi\)
\(942\) 0 0
\(943\) −18.9607 −0.617446
\(944\) 0 0
\(945\) 9.37293i 0.304902i
\(946\) 0 0
\(947\) 40.8600i 1.32777i −0.747834 0.663886i \(-0.768906\pi\)
0.747834 0.663886i \(-0.231094\pi\)
\(948\) 0 0
\(949\) 6.07745i 0.197282i
\(950\) 0 0
\(951\) 28.5757i 0.926631i
\(952\) 0 0
\(953\) 27.7349i 0.898422i −0.893426 0.449211i \(-0.851705\pi\)
0.893426 0.449211i \(-0.148295\pi\)
\(954\) 0 0
\(955\) 60.7256i 1.96503i
\(956\) 0 0
\(957\) 3.48672 0.112710
\(958\) 0 0
\(959\) 4.52141i 0.146004i
\(960\) 0 0
\(961\) 22.3429 0.720739
\(962\) 0 0
\(963\) −16.7897 −0.541041
\(964\) 0 0
\(965\) 28.5096i 0.917755i
\(966\) 0 0
\(967\) 38.9264i 1.25179i 0.779908 + 0.625894i \(0.215266\pi\)
−0.779908 + 0.625894i \(0.784734\pi\)
\(968\) 0 0
\(969\) −7.90774 12.8743i −0.254033 0.413583i
\(970\) 0 0
\(971\) 52.9327 1.69869 0.849346 0.527836i \(-0.176997\pi\)
0.849346 + 0.527836i \(0.176997\pi\)
\(972\) 0 0
\(973\) 11.5541 0.370408
\(974\) 0 0
\(975\) 3.91374i 0.125340i
\(976\) 0 0
\(977\) 50.7846i 1.62474i −0.583140 0.812372i \(-0.698176\pi\)
0.583140 0.812372i \(-0.301824\pi\)
\(978\) 0 0
\(979\) −3.39907 −0.108635
\(980\) 0 0
\(981\) 42.4703i 1.35597i
\(982\) 0 0
\(983\) 23.7078 0.756161 0.378081 0.925773i \(-0.376584\pi\)
0.378081 + 0.925773i \(0.376584\pi\)
\(984\) 0 0
\(985\) 3.44880 0.109888
\(986\) 0 0
\(987\) −1.25317 −0.0398889
\(988\) 0 0
\(989\) −2.06984 −0.0658170
\(990\) 0 0
\(991\) −39.1036 −1.24217 −0.621084 0.783744i \(-0.713307\pi\)
−0.621084 + 0.783744i \(0.713307\pi\)
\(992\) 0 0
\(993\) 24.5555 0.779244
\(994\) 0 0
\(995\) 39.0326i 1.23742i
\(996\) 0 0
\(997\) −53.6755 −1.69992 −0.849960 0.526847i \(-0.823374\pi\)
−0.849960 + 0.526847i \(0.823374\pi\)
\(998\) 0 0
\(999\) 38.8177i 1.22814i
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 3344.2.o.a.1519.15 36
4.3 odd 2 inner 3344.2.o.a.1519.21 yes 36
19.18 odd 2 inner 3344.2.o.a.1519.22 yes 36
76.75 even 2 inner 3344.2.o.a.1519.16 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3344.2.o.a.1519.15 36 1.1 even 1 trivial
3344.2.o.a.1519.16 yes 36 76.75 even 2 inner
3344.2.o.a.1519.21 yes 36 4.3 odd 2 inner
3344.2.o.a.1519.22 yes 36 19.18 odd 2 inner