Properties

Label 1344.4.c.a.673.2
Level $1344$
Weight $4$
Character 1344.673
Analytic conductor $79.299$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,4,Mod(673,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.673");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.c (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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 722x^{3} + 11881x^{2} + 54936x + 127008 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 673.2
Root \(-2.93235 + 2.93235i\) of defining polynomial
Character \(\chi\) \(=\) 1344.673
Dual form 1344.4.c.a.673.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} -7.86469i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -7.86469i q^{5} -7.00000 q^{7} -9.00000 q^{9} -31.0937i q^{11} +48.9584i q^{13} -23.5941 q^{15} -43.0937 q^{17} +110.147i q^{19} +21.0000i q^{21} +170.469 q^{23} +63.1466 q^{25} +27.0000i q^{27} +78.6878i q^{29} -48.8753 q^{31} -93.2812 q^{33} +55.0529i q^{35} -6.95843i q^{37} +146.875 q^{39} +339.680 q^{41} +222.293i q^{43} +70.7822i q^{45} +314.918 q^{47} +49.0000 q^{49} +129.281i q^{51} -487.731i q^{53} -244.543 q^{55} +330.440 q^{57} -500.668i q^{59} +29.6040i q^{61} +63.0000 q^{63} +385.043 q^{65} +380.317i q^{67} -511.408i q^{69} -12.5976 q^{71} -857.735 q^{73} -189.440i q^{75} +217.656i q^{77} +799.481 q^{79} +81.0000 q^{81} +178.856i q^{83} +338.919i q^{85} +236.063 q^{87} -1029.18 q^{89} -342.709i q^{91} +146.626i q^{93} +866.269 q^{95} -360.904 q^{97} +279.844i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 42 q^{7} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 42 q^{7} - 54 q^{9} - 24 q^{15} - 16 q^{17} + 60 q^{23} - 138 q^{25} + 552 q^{31} + 168 q^{33} + 36 q^{39} - 272 q^{41} + 1576 q^{47} + 294 q^{49} + 1632 q^{55} + 432 q^{57} + 378 q^{63} - 664 q^{65} + 2548 q^{71} + 444 q^{73} + 3528 q^{79} + 486 q^{81} + 336 q^{87} + 16 q^{89} + 4776 q^{95} + 1548 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 3.00000i − 0.577350i
\(4\) 0 0
\(5\) − 7.86469i − 0.703440i −0.936105 0.351720i \(-0.885597\pi\)
0.936105 0.351720i \(-0.114403\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) − 31.0937i − 0.852283i −0.904657 0.426141i \(-0.859873\pi\)
0.904657 0.426141i \(-0.140127\pi\)
\(12\) 0 0
\(13\) 48.9584i 1.04451i 0.852790 + 0.522255i \(0.174909\pi\)
−0.852790 + 0.522255i \(0.825091\pi\)
\(14\) 0 0
\(15\) −23.5941 −0.406131
\(16\) 0 0
\(17\) −43.0937 −0.614810 −0.307405 0.951579i \(-0.599461\pi\)
−0.307405 + 0.951579i \(0.599461\pi\)
\(18\) 0 0
\(19\) 110.147i 1.32997i 0.746858 + 0.664983i \(0.231561\pi\)
−0.746858 + 0.664983i \(0.768439\pi\)
\(20\) 0 0
\(21\) 21.0000i 0.218218i
\(22\) 0 0
\(23\) 170.469 1.54545 0.772724 0.634742i \(-0.218894\pi\)
0.772724 + 0.634742i \(0.218894\pi\)
\(24\) 0 0
\(25\) 63.1466 0.505173
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 78.6878i 0.503861i 0.967745 + 0.251930i \(0.0810654\pi\)
−0.967745 + 0.251930i \(0.918935\pi\)
\(30\) 0 0
\(31\) −48.8753 −0.283170 −0.141585 0.989926i \(-0.545220\pi\)
−0.141585 + 0.989926i \(0.545220\pi\)
\(32\) 0 0
\(33\) −93.2812 −0.492066
\(34\) 0 0
\(35\) 55.0529i 0.265875i
\(36\) 0 0
\(37\) − 6.95843i − 0.0309178i −0.999881 0.0154589i \(-0.995079\pi\)
0.999881 0.0154589i \(-0.00492091\pi\)
\(38\) 0 0
\(39\) 146.875 0.603048
\(40\) 0 0
\(41\) 339.680 1.29388 0.646940 0.762541i \(-0.276048\pi\)
0.646940 + 0.762541i \(0.276048\pi\)
\(42\) 0 0
\(43\) 222.293i 0.788358i 0.919034 + 0.394179i \(0.128971\pi\)
−0.919034 + 0.394179i \(0.871029\pi\)
\(44\) 0 0
\(45\) 70.7822i 0.234480i
\(46\) 0 0
\(47\) 314.918 0.977350 0.488675 0.872466i \(-0.337480\pi\)
0.488675 + 0.872466i \(0.337480\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 129.281i 0.354961i
\(52\) 0 0
\(53\) − 487.731i − 1.26406i −0.774945 0.632028i \(-0.782223\pi\)
0.774945 0.632028i \(-0.217777\pi\)
\(54\) 0 0
\(55\) −244.543 −0.599530
\(56\) 0 0
\(57\) 330.440 0.767856
\(58\) 0 0
\(59\) − 500.668i − 1.10477i −0.833589 0.552385i \(-0.813718\pi\)
0.833589 0.552385i \(-0.186282\pi\)
\(60\) 0 0
\(61\) 29.6040i 0.0621377i 0.999517 + 0.0310688i \(0.00989111\pi\)
−0.999517 + 0.0310688i \(0.990109\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) 385.043 0.734749
\(66\) 0 0
\(67\) 380.317i 0.693480i 0.937961 + 0.346740i \(0.112711\pi\)
−0.937961 + 0.346740i \(0.887289\pi\)
\(68\) 0 0
\(69\) − 511.408i − 0.892265i
\(70\) 0 0
\(71\) −12.5976 −0.0210573 −0.0105286 0.999945i \(-0.503351\pi\)
−0.0105286 + 0.999945i \(0.503351\pi\)
\(72\) 0 0
\(73\) −857.735 −1.37521 −0.687605 0.726085i \(-0.741338\pi\)
−0.687605 + 0.726085i \(0.741338\pi\)
\(74\) 0 0
\(75\) − 189.440i − 0.291662i
\(76\) 0 0
\(77\) 217.656i 0.322133i
\(78\) 0 0
\(79\) 799.481 1.13859 0.569295 0.822133i \(-0.307216\pi\)
0.569295 + 0.822133i \(0.307216\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 178.856i 0.236530i 0.992982 + 0.118265i \(0.0377332\pi\)
−0.992982 + 0.118265i \(0.962267\pi\)
\(84\) 0 0
\(85\) 338.919i 0.432482i
\(86\) 0 0
\(87\) 236.063 0.290904
\(88\) 0 0
\(89\) −1029.18 −1.22576 −0.612881 0.790175i \(-0.709990\pi\)
−0.612881 + 0.790175i \(0.709990\pi\)
\(90\) 0 0
\(91\) − 342.709i − 0.394788i
\(92\) 0 0
\(93\) 146.626i 0.163488i
\(94\) 0 0
\(95\) 866.269 0.935551
\(96\) 0 0
\(97\) −360.904 −0.377775 −0.188888 0.981999i \(-0.560488\pi\)
−0.188888 + 0.981999i \(0.560488\pi\)
\(98\) 0 0
\(99\) 279.844i 0.284094i
\(100\) 0 0
\(101\) 768.144i 0.756764i 0.925650 + 0.378382i \(0.123519\pi\)
−0.925650 + 0.378382i \(0.876481\pi\)
\(102\) 0 0
\(103\) 1769.55 1.69280 0.846402 0.532544i \(-0.178764\pi\)
0.846402 + 0.532544i \(0.178764\pi\)
\(104\) 0 0
\(105\) 165.159 0.153503
\(106\) 0 0
\(107\) − 1013.38i − 0.915578i −0.889061 0.457789i \(-0.848641\pi\)
0.889061 0.457789i \(-0.151359\pi\)
\(108\) 0 0
\(109\) − 1221.05i − 1.07298i −0.843906 0.536491i \(-0.819750\pi\)
0.843906 0.536491i \(-0.180250\pi\)
\(110\) 0 0
\(111\) −20.8753 −0.0178504
\(112\) 0 0
\(113\) 612.481 0.509888 0.254944 0.966956i \(-0.417943\pi\)
0.254944 + 0.966956i \(0.417943\pi\)
\(114\) 0 0
\(115\) − 1340.69i − 1.08713i
\(116\) 0 0
\(117\) − 440.626i − 0.348170i
\(118\) 0 0
\(119\) 301.656 0.232376
\(120\) 0 0
\(121\) 364.180 0.273614
\(122\) 0 0
\(123\) − 1019.04i − 0.747022i
\(124\) 0 0
\(125\) − 1479.72i − 1.05880i
\(126\) 0 0
\(127\) 1373.22 0.959479 0.479740 0.877411i \(-0.340731\pi\)
0.479740 + 0.877411i \(0.340731\pi\)
\(128\) 0 0
\(129\) 666.880 0.455159
\(130\) 0 0
\(131\) 272.851i 0.181978i 0.995852 + 0.0909890i \(0.0290028\pi\)
−0.995852 + 0.0909890i \(0.970997\pi\)
\(132\) 0 0
\(133\) − 771.026i − 0.502680i
\(134\) 0 0
\(135\) 212.347 0.135377
\(136\) 0 0
\(137\) −1240.86 −0.773821 −0.386910 0.922117i \(-0.626458\pi\)
−0.386910 + 0.922117i \(0.626458\pi\)
\(138\) 0 0
\(139\) 1471.37i 0.897844i 0.893571 + 0.448922i \(0.148192\pi\)
−0.893571 + 0.448922i \(0.851808\pi\)
\(140\) 0 0
\(141\) − 944.753i − 0.564273i
\(142\) 0 0
\(143\) 1522.30 0.890218
\(144\) 0 0
\(145\) 618.856 0.354436
\(146\) 0 0
\(147\) − 147.000i − 0.0824786i
\(148\) 0 0
\(149\) 581.360i 0.319644i 0.987146 + 0.159822i \(0.0510920\pi\)
−0.987146 + 0.159822i \(0.948908\pi\)
\(150\) 0 0
\(151\) −2709.31 −1.46013 −0.730067 0.683375i \(-0.760511\pi\)
−0.730067 + 0.683375i \(0.760511\pi\)
\(152\) 0 0
\(153\) 387.844 0.204937
\(154\) 0 0
\(155\) 384.389i 0.199193i
\(156\) 0 0
\(157\) − 2885.43i − 1.46677i −0.679815 0.733383i \(-0.737940\pi\)
0.679815 0.733383i \(-0.262060\pi\)
\(158\) 0 0
\(159\) −1463.19 −0.729803
\(160\) 0 0
\(161\) −1193.29 −0.584125
\(162\) 0 0
\(163\) 3230.08i 1.55215i 0.630643 + 0.776073i \(0.282791\pi\)
−0.630643 + 0.776073i \(0.717209\pi\)
\(164\) 0 0
\(165\) 733.628i 0.346139i
\(166\) 0 0
\(167\) −234.777 −0.108788 −0.0543939 0.998520i \(-0.517323\pi\)
−0.0543939 + 0.998520i \(0.517323\pi\)
\(168\) 0 0
\(169\) −199.927 −0.0910002
\(170\) 0 0
\(171\) − 991.319i − 0.443322i
\(172\) 0 0
\(173\) − 1892.43i − 0.831667i −0.909441 0.415834i \(-0.863490\pi\)
0.909441 0.415834i \(-0.136510\pi\)
\(174\) 0 0
\(175\) −442.026 −0.190937
\(176\) 0 0
\(177\) −1502.00 −0.637839
\(178\) 0 0
\(179\) 340.579i 0.142213i 0.997469 + 0.0711064i \(0.0226530\pi\)
−0.997469 + 0.0711064i \(0.977347\pi\)
\(180\) 0 0
\(181\) − 3788.51i − 1.55579i −0.628396 0.777894i \(-0.716288\pi\)
0.628396 0.777894i \(-0.283712\pi\)
\(182\) 0 0
\(183\) 88.8119 0.0358752
\(184\) 0 0
\(185\) −54.7259 −0.0217488
\(186\) 0 0
\(187\) 1339.94i 0.523992i
\(188\) 0 0
\(189\) − 189.000i − 0.0727393i
\(190\) 0 0
\(191\) 5076.47 1.92314 0.961572 0.274554i \(-0.0885303\pi\)
0.961572 + 0.274554i \(0.0885303\pi\)
\(192\) 0 0
\(193\) −2335.52 −0.871059 −0.435530 0.900174i \(-0.643439\pi\)
−0.435530 + 0.900174i \(0.643439\pi\)
\(194\) 0 0
\(195\) − 1155.13i − 0.424208i
\(196\) 0 0
\(197\) − 3157.61i − 1.14198i −0.820957 0.570990i \(-0.806559\pi\)
0.820957 0.570990i \(-0.193441\pi\)
\(198\) 0 0
\(199\) 4559.29 1.62412 0.812059 0.583575i \(-0.198347\pi\)
0.812059 + 0.583575i \(0.198347\pi\)
\(200\) 0 0
\(201\) 1140.95 0.400381
\(202\) 0 0
\(203\) − 550.815i − 0.190441i
\(204\) 0 0
\(205\) − 2671.48i − 0.910167i
\(206\) 0 0
\(207\) −1534.22 −0.515150
\(208\) 0 0
\(209\) 3424.87 1.13351
\(210\) 0 0
\(211\) − 4066.41i − 1.32675i −0.748289 0.663373i \(-0.769124\pi\)
0.748289 0.663373i \(-0.230876\pi\)
\(212\) 0 0
\(213\) 37.7929i 0.0121574i
\(214\) 0 0
\(215\) 1748.27 0.554562
\(216\) 0 0
\(217\) 342.127 0.107028
\(218\) 0 0
\(219\) 2573.21i 0.793978i
\(220\) 0 0
\(221\) − 2109.80i − 0.642175i
\(222\) 0 0
\(223\) 3158.95 0.948604 0.474302 0.880362i \(-0.342700\pi\)
0.474302 + 0.880362i \(0.342700\pi\)
\(224\) 0 0
\(225\) −568.319 −0.168391
\(226\) 0 0
\(227\) 4076.18i 1.19183i 0.803047 + 0.595915i \(0.203211\pi\)
−0.803047 + 0.595915i \(0.796789\pi\)
\(228\) 0 0
\(229\) 2728.36i 0.787314i 0.919257 + 0.393657i \(0.128790\pi\)
−0.919257 + 0.393657i \(0.871210\pi\)
\(230\) 0 0
\(231\) 652.968 0.185983
\(232\) 0 0
\(233\) 1799.32 0.505910 0.252955 0.967478i \(-0.418597\pi\)
0.252955 + 0.967478i \(0.418597\pi\)
\(234\) 0 0
\(235\) − 2476.73i − 0.687507i
\(236\) 0 0
\(237\) − 2398.44i − 0.657366i
\(238\) 0 0
\(239\) 515.306 0.139466 0.0697330 0.997566i \(-0.477785\pi\)
0.0697330 + 0.997566i \(0.477785\pi\)
\(240\) 0 0
\(241\) 2837.68 0.758469 0.379235 0.925301i \(-0.376187\pi\)
0.379235 + 0.925301i \(0.376187\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) − 385.370i − 0.100491i
\(246\) 0 0
\(247\) −5392.60 −1.38916
\(248\) 0 0
\(249\) 536.567 0.136560
\(250\) 0 0
\(251\) − 3739.61i − 0.940406i −0.882558 0.470203i \(-0.844181\pi\)
0.882558 0.470203i \(-0.155819\pi\)
\(252\) 0 0
\(253\) − 5300.53i − 1.31716i
\(254\) 0 0
\(255\) 1016.76 0.249693
\(256\) 0 0
\(257\) 1082.08 0.262640 0.131320 0.991340i \(-0.458078\pi\)
0.131320 + 0.991340i \(0.458078\pi\)
\(258\) 0 0
\(259\) 48.7090i 0.0116858i
\(260\) 0 0
\(261\) − 708.190i − 0.167954i
\(262\) 0 0
\(263\) 6367.81 1.49299 0.746494 0.665392i \(-0.231736\pi\)
0.746494 + 0.665392i \(0.231736\pi\)
\(264\) 0 0
\(265\) −3835.85 −0.889187
\(266\) 0 0
\(267\) 3087.54i 0.707694i
\(268\) 0 0
\(269\) 4265.99i 0.966923i 0.875366 + 0.483461i \(0.160621\pi\)
−0.875366 + 0.483461i \(0.839379\pi\)
\(270\) 0 0
\(271\) 2158.77 0.483896 0.241948 0.970289i \(-0.422214\pi\)
0.241948 + 0.970289i \(0.422214\pi\)
\(272\) 0 0
\(273\) −1028.13 −0.227931
\(274\) 0 0
\(275\) − 1963.46i − 0.430550i
\(276\) 0 0
\(277\) − 4738.38i − 1.02780i −0.857849 0.513901i \(-0.828200\pi\)
0.857849 0.513901i \(-0.171800\pi\)
\(278\) 0 0
\(279\) 439.877 0.0943899
\(280\) 0 0
\(281\) 4054.46 0.860742 0.430371 0.902652i \(-0.358383\pi\)
0.430371 + 0.902652i \(0.358383\pi\)
\(282\) 0 0
\(283\) 583.193i 0.122499i 0.998122 + 0.0612495i \(0.0195085\pi\)
−0.998122 + 0.0612495i \(0.980491\pi\)
\(284\) 0 0
\(285\) − 2598.81i − 0.540141i
\(286\) 0 0
\(287\) −2377.76 −0.489041
\(288\) 0 0
\(289\) −3055.93 −0.622009
\(290\) 0 0
\(291\) 1082.71i 0.218109i
\(292\) 0 0
\(293\) 6986.16i 1.39295i 0.717579 + 0.696477i \(0.245250\pi\)
−0.717579 + 0.696477i \(0.754750\pi\)
\(294\) 0 0
\(295\) −3937.60 −0.777139
\(296\) 0 0
\(297\) 839.531 0.164022
\(298\) 0 0
\(299\) 8345.91i 1.61424i
\(300\) 0 0
\(301\) − 1556.05i − 0.297971i
\(302\) 0 0
\(303\) 2304.43 0.436918
\(304\) 0 0
\(305\) 232.826 0.0437101
\(306\) 0 0
\(307\) 7409.11i 1.37739i 0.725049 + 0.688697i \(0.241817\pi\)
−0.725049 + 0.688697i \(0.758183\pi\)
\(308\) 0 0
\(309\) − 5308.65i − 0.977341i
\(310\) 0 0
\(311\) 3756.12 0.684856 0.342428 0.939544i \(-0.388751\pi\)
0.342428 + 0.939544i \(0.388751\pi\)
\(312\) 0 0
\(313\) 7988.13 1.44254 0.721271 0.692653i \(-0.243558\pi\)
0.721271 + 0.692653i \(0.243558\pi\)
\(314\) 0 0
\(315\) − 495.476i − 0.0886251i
\(316\) 0 0
\(317\) − 968.306i − 0.171563i −0.996314 0.0857815i \(-0.972661\pi\)
0.996314 0.0857815i \(-0.0273387\pi\)
\(318\) 0 0
\(319\) 2446.70 0.429432
\(320\) 0 0
\(321\) −3040.13 −0.528609
\(322\) 0 0
\(323\) − 4746.63i − 0.817676i
\(324\) 0 0
\(325\) 3091.56i 0.527658i
\(326\) 0 0
\(327\) −3663.14 −0.619486
\(328\) 0 0
\(329\) −2204.42 −0.369404
\(330\) 0 0
\(331\) − 11428.0i − 1.89770i −0.315721 0.948852i \(-0.602246\pi\)
0.315721 0.948852i \(-0.397754\pi\)
\(332\) 0 0
\(333\) 62.6258i 0.0103059i
\(334\) 0 0
\(335\) 2991.08 0.487821
\(336\) 0 0
\(337\) 8189.96 1.32384 0.661922 0.749572i \(-0.269741\pi\)
0.661922 + 0.749572i \(0.269741\pi\)
\(338\) 0 0
\(339\) − 1837.44i − 0.294384i
\(340\) 0 0
\(341\) 1519.71i 0.241341i
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) −4022.07 −0.627655
\(346\) 0 0
\(347\) − 1519.83i − 0.235126i −0.993065 0.117563i \(-0.962492\pi\)
0.993065 0.117563i \(-0.0375081\pi\)
\(348\) 0 0
\(349\) 5156.67i 0.790918i 0.918484 + 0.395459i \(0.129414\pi\)
−0.918484 + 0.395459i \(0.870586\pi\)
\(350\) 0 0
\(351\) −1321.88 −0.201016
\(352\) 0 0
\(353\) −3904.42 −0.588700 −0.294350 0.955698i \(-0.595103\pi\)
−0.294350 + 0.955698i \(0.595103\pi\)
\(354\) 0 0
\(355\) 99.0766i 0.0148125i
\(356\) 0 0
\(357\) − 904.968i − 0.134162i
\(358\) 0 0
\(359\) 4589.29 0.674689 0.337345 0.941381i \(-0.390471\pi\)
0.337345 + 0.941381i \(0.390471\pi\)
\(360\) 0 0
\(361\) −5273.27 −0.768810
\(362\) 0 0
\(363\) − 1092.54i − 0.157971i
\(364\) 0 0
\(365\) 6745.82i 0.967377i
\(366\) 0 0
\(367\) 1951.21 0.277527 0.138763 0.990326i \(-0.455687\pi\)
0.138763 + 0.990326i \(0.455687\pi\)
\(368\) 0 0
\(369\) −3057.12 −0.431294
\(370\) 0 0
\(371\) 3414.12i 0.477768i
\(372\) 0 0
\(373\) 6271.39i 0.870564i 0.900294 + 0.435282i \(0.143351\pi\)
−0.900294 + 0.435282i \(0.856649\pi\)
\(374\) 0 0
\(375\) −4439.15 −0.611297
\(376\) 0 0
\(377\) −3852.43 −0.526287
\(378\) 0 0
\(379\) 461.952i 0.0626092i 0.999510 + 0.0313046i \(0.00996619\pi\)
−0.999510 + 0.0313046i \(0.990034\pi\)
\(380\) 0 0
\(381\) − 4119.67i − 0.553956i
\(382\) 0 0
\(383\) 10155.8 1.35493 0.677465 0.735555i \(-0.263078\pi\)
0.677465 + 0.735555i \(0.263078\pi\)
\(384\) 0 0
\(385\) 1711.80 0.226601
\(386\) 0 0
\(387\) − 2000.64i − 0.262786i
\(388\) 0 0
\(389\) − 4781.94i − 0.623276i −0.950201 0.311638i \(-0.899122\pi\)
0.950201 0.311638i \(-0.100878\pi\)
\(390\) 0 0
\(391\) −7346.16 −0.950157
\(392\) 0 0
\(393\) 818.554 0.105065
\(394\) 0 0
\(395\) − 6287.68i − 0.800930i
\(396\) 0 0
\(397\) − 10238.8i − 1.29438i −0.762329 0.647190i \(-0.775944\pi\)
0.762329 0.647190i \(-0.224056\pi\)
\(398\) 0 0
\(399\) −2313.08 −0.290222
\(400\) 0 0
\(401\) −12493.0 −1.55578 −0.777891 0.628399i \(-0.783710\pi\)
−0.777891 + 0.628399i \(0.783710\pi\)
\(402\) 0 0
\(403\) − 2392.86i − 0.295773i
\(404\) 0 0
\(405\) − 637.040i − 0.0781600i
\(406\) 0 0
\(407\) −216.363 −0.0263507
\(408\) 0 0
\(409\) −4755.09 −0.574876 −0.287438 0.957799i \(-0.592803\pi\)
−0.287438 + 0.957799i \(0.592803\pi\)
\(410\) 0 0
\(411\) 3722.57i 0.446766i
\(412\) 0 0
\(413\) 3504.68i 0.417564i
\(414\) 0 0
\(415\) 1406.64 0.166384
\(416\) 0 0
\(417\) 4414.12 0.518371
\(418\) 0 0
\(419\) 1948.68i 0.227205i 0.993526 + 0.113603i \(0.0362391\pi\)
−0.993526 + 0.113603i \(0.963761\pi\)
\(420\) 0 0
\(421\) − 1727.58i − 0.199993i −0.994988 0.0999963i \(-0.968117\pi\)
0.994988 0.0999963i \(-0.0318831\pi\)
\(422\) 0 0
\(423\) −2834.26 −0.325783
\(424\) 0 0
\(425\) −2721.22 −0.310585
\(426\) 0 0
\(427\) − 207.228i − 0.0234858i
\(428\) 0 0
\(429\) − 4566.90i − 0.513967i
\(430\) 0 0
\(431\) 9457.57 1.05697 0.528486 0.848942i \(-0.322760\pi\)
0.528486 + 0.848942i \(0.322760\pi\)
\(432\) 0 0
\(433\) 557.783 0.0619061 0.0309530 0.999521i \(-0.490146\pi\)
0.0309530 + 0.999521i \(0.490146\pi\)
\(434\) 0 0
\(435\) − 1856.57i − 0.204633i
\(436\) 0 0
\(437\) 18776.6i 2.05539i
\(438\) 0 0
\(439\) −7961.06 −0.865514 −0.432757 0.901511i \(-0.642459\pi\)
−0.432757 + 0.901511i \(0.642459\pi\)
\(440\) 0 0
\(441\) −441.000 −0.0476190
\(442\) 0 0
\(443\) − 778.834i − 0.0835294i −0.999127 0.0417647i \(-0.986702\pi\)
0.999127 0.0417647i \(-0.0132980\pi\)
\(444\) 0 0
\(445\) 8094.18i 0.862250i
\(446\) 0 0
\(447\) 1744.08 0.184546
\(448\) 0 0
\(449\) −1899.25 −0.199624 −0.0998118 0.995006i \(-0.531824\pi\)
−0.0998118 + 0.995006i \(0.531824\pi\)
\(450\) 0 0
\(451\) − 10561.9i − 1.10275i
\(452\) 0 0
\(453\) 8127.92i 0.843009i
\(454\) 0 0
\(455\) −2695.30 −0.277709
\(456\) 0 0
\(457\) 535.528 0.0548161 0.0274080 0.999624i \(-0.491275\pi\)
0.0274080 + 0.999624i \(0.491275\pi\)
\(458\) 0 0
\(459\) − 1163.53i − 0.118320i
\(460\) 0 0
\(461\) 6361.75i 0.642725i 0.946956 + 0.321363i \(0.104141\pi\)
−0.946956 + 0.321363i \(0.895859\pi\)
\(462\) 0 0
\(463\) −4102.49 −0.411791 −0.205895 0.978574i \(-0.566011\pi\)
−0.205895 + 0.978574i \(0.566011\pi\)
\(464\) 0 0
\(465\) 1153.17 0.115004
\(466\) 0 0
\(467\) 11503.7i 1.13989i 0.821683 + 0.569945i \(0.193035\pi\)
−0.821683 + 0.569945i \(0.806965\pi\)
\(468\) 0 0
\(469\) − 2662.22i − 0.262111i
\(470\) 0 0
\(471\) −8656.29 −0.846838
\(472\) 0 0
\(473\) 6911.92 0.671904
\(474\) 0 0
\(475\) 6955.38i 0.671863i
\(476\) 0 0
\(477\) 4389.58i 0.421352i
\(478\) 0 0
\(479\) 5430.09 0.517969 0.258985 0.965881i \(-0.416612\pi\)
0.258985 + 0.965881i \(0.416612\pi\)
\(480\) 0 0
\(481\) 340.674 0.0322939
\(482\) 0 0
\(483\) 3579.86i 0.337245i
\(484\) 0 0
\(485\) 2838.40i 0.265742i
\(486\) 0 0
\(487\) −9563.99 −0.889909 −0.444955 0.895553i \(-0.646780\pi\)
−0.444955 + 0.895553i \(0.646780\pi\)
\(488\) 0 0
\(489\) 9690.25 0.896132
\(490\) 0 0
\(491\) 12301.3i 1.13065i 0.824868 + 0.565325i \(0.191249\pi\)
−0.824868 + 0.565325i \(0.808751\pi\)
\(492\) 0 0
\(493\) − 3390.95i − 0.309778i
\(494\) 0 0
\(495\) 2200.88 0.199843
\(496\) 0 0
\(497\) 88.1835 0.00795889
\(498\) 0 0
\(499\) − 4750.89i − 0.426211i −0.977029 0.213105i \(-0.931642\pi\)
0.977029 0.213105i \(-0.0683578\pi\)
\(500\) 0 0
\(501\) 704.330i 0.0628087i
\(502\) 0 0
\(503\) 2463.79 0.218400 0.109200 0.994020i \(-0.465171\pi\)
0.109200 + 0.994020i \(0.465171\pi\)
\(504\) 0 0
\(505\) 6041.22 0.532338
\(506\) 0 0
\(507\) 599.782i 0.0525390i
\(508\) 0 0
\(509\) 14836.9i 1.29201i 0.763334 + 0.646004i \(0.223561\pi\)
−0.763334 + 0.646004i \(0.776439\pi\)
\(510\) 0 0
\(511\) 6004.15 0.519780
\(512\) 0 0
\(513\) −2973.96 −0.255952
\(514\) 0 0
\(515\) − 13917.0i − 1.19079i
\(516\) 0 0
\(517\) − 9791.96i − 0.832979i
\(518\) 0 0
\(519\) −5677.28 −0.480163
\(520\) 0 0
\(521\) 13196.5 1.10969 0.554844 0.831955i \(-0.312778\pi\)
0.554844 + 0.831955i \(0.312778\pi\)
\(522\) 0 0
\(523\) 15562.4i 1.30114i 0.759447 + 0.650569i \(0.225470\pi\)
−0.759447 + 0.650569i \(0.774530\pi\)
\(524\) 0 0
\(525\) 1326.08i 0.110238i
\(526\) 0 0
\(527\) 2106.22 0.174095
\(528\) 0 0
\(529\) 16892.8 1.38841
\(530\) 0 0
\(531\) 4506.01i 0.368257i
\(532\) 0 0
\(533\) 16630.2i 1.35147i
\(534\) 0 0
\(535\) −7969.90 −0.644054
\(536\) 0 0
\(537\) 1021.74 0.0821066
\(538\) 0 0
\(539\) − 1523.59i − 0.121755i
\(540\) 0 0
\(541\) 13421.8i 1.06663i 0.845917 + 0.533315i \(0.179054\pi\)
−0.845917 + 0.533315i \(0.820946\pi\)
\(542\) 0 0
\(543\) −11365.5 −0.898234
\(544\) 0 0
\(545\) −9603.15 −0.754778
\(546\) 0 0
\(547\) − 12508.9i − 0.977770i −0.872348 0.488885i \(-0.837404\pi\)
0.872348 0.488885i \(-0.162596\pi\)
\(548\) 0 0
\(549\) − 266.436i − 0.0207126i
\(550\) 0 0
\(551\) −8667.19 −0.670118
\(552\) 0 0
\(553\) −5596.37 −0.430347
\(554\) 0 0
\(555\) 164.178i 0.0125567i
\(556\) 0 0
\(557\) 24112.0i 1.83421i 0.398643 + 0.917106i \(0.369481\pi\)
−0.398643 + 0.917106i \(0.630519\pi\)
\(558\) 0 0
\(559\) −10883.1 −0.823447
\(560\) 0 0
\(561\) 4019.83 0.302527
\(562\) 0 0
\(563\) − 10716.1i − 0.802186i −0.916037 0.401093i \(-0.868630\pi\)
0.916037 0.401093i \(-0.131370\pi\)
\(564\) 0 0
\(565\) − 4816.97i − 0.358675i
\(566\) 0 0
\(567\) −567.000 −0.0419961
\(568\) 0 0
\(569\) −8706.37 −0.641459 −0.320729 0.947171i \(-0.603928\pi\)
−0.320729 + 0.947171i \(0.603928\pi\)
\(570\) 0 0
\(571\) 2840.30i 0.208166i 0.994569 + 0.104083i \(0.0331908\pi\)
−0.994569 + 0.104083i \(0.966809\pi\)
\(572\) 0 0
\(573\) − 15229.4i − 1.11033i
\(574\) 0 0
\(575\) 10764.6 0.780718
\(576\) 0 0
\(577\) −2485.56 −0.179333 −0.0896665 0.995972i \(-0.528580\pi\)
−0.0896665 + 0.995972i \(0.528580\pi\)
\(578\) 0 0
\(579\) 7006.56i 0.502906i
\(580\) 0 0
\(581\) − 1251.99i − 0.0893998i
\(582\) 0 0
\(583\) −15165.4 −1.07733
\(584\) 0 0
\(585\) −3465.39 −0.244916
\(586\) 0 0
\(587\) 9350.79i 0.657493i 0.944418 + 0.328746i \(0.106626\pi\)
−0.944418 + 0.328746i \(0.893374\pi\)
\(588\) 0 0
\(589\) − 5383.45i − 0.376606i
\(590\) 0 0
\(591\) −9472.82 −0.659323
\(592\) 0 0
\(593\) 14248.9 0.986732 0.493366 0.869822i \(-0.335766\pi\)
0.493366 + 0.869822i \(0.335766\pi\)
\(594\) 0 0
\(595\) − 2372.43i − 0.163463i
\(596\) 0 0
\(597\) − 13677.9i − 0.937685i
\(598\) 0 0
\(599\) −27065.7 −1.84620 −0.923102 0.384555i \(-0.874355\pi\)
−0.923102 + 0.384555i \(0.874355\pi\)
\(600\) 0 0
\(601\) −16949.1 −1.15036 −0.575181 0.818026i \(-0.695068\pi\)
−0.575181 + 0.818026i \(0.695068\pi\)
\(602\) 0 0
\(603\) − 3422.85i − 0.231160i
\(604\) 0 0
\(605\) − 2864.16i − 0.192471i
\(606\) 0 0
\(607\) −16657.8 −1.11387 −0.556937 0.830555i \(-0.688023\pi\)
−0.556937 + 0.830555i \(0.688023\pi\)
\(608\) 0 0
\(609\) −1652.44 −0.109951
\(610\) 0 0
\(611\) 15417.9i 1.02085i
\(612\) 0 0
\(613\) − 3871.55i − 0.255091i −0.991833 0.127545i \(-0.959290\pi\)
0.991833 0.127545i \(-0.0407098\pi\)
\(614\) 0 0
\(615\) −8014.44 −0.525485
\(616\) 0 0
\(617\) 27607.7 1.80137 0.900683 0.434476i \(-0.143067\pi\)
0.900683 + 0.434476i \(0.143067\pi\)
\(618\) 0 0
\(619\) 11637.0i 0.755625i 0.925882 + 0.377813i \(0.123324\pi\)
−0.925882 + 0.377813i \(0.876676\pi\)
\(620\) 0 0
\(621\) 4602.67i 0.297422i
\(622\) 0 0
\(623\) 7204.26 0.463295
\(624\) 0 0
\(625\) −3744.18 −0.239628
\(626\) 0 0
\(627\) − 10274.6i − 0.654431i
\(628\) 0 0
\(629\) 299.865i 0.0190086i
\(630\) 0 0
\(631\) −14402.0 −0.908613 −0.454306 0.890846i \(-0.650113\pi\)
−0.454306 + 0.890846i \(0.650113\pi\)
\(632\) 0 0
\(633\) −12199.2 −0.765997
\(634\) 0 0
\(635\) − 10800.0i − 0.674936i
\(636\) 0 0
\(637\) 2398.96i 0.149216i
\(638\) 0 0
\(639\) 113.379 0.00701908
\(640\) 0 0
\(641\) −20510.6 −1.26384 −0.631918 0.775035i \(-0.717732\pi\)
−0.631918 + 0.775035i \(0.717732\pi\)
\(642\) 0 0
\(643\) 11283.3i 0.692020i 0.938231 + 0.346010i \(0.112464\pi\)
−0.938231 + 0.346010i \(0.887536\pi\)
\(644\) 0 0
\(645\) − 5244.80i − 0.320177i
\(646\) 0 0
\(647\) −26999.9 −1.64061 −0.820306 0.571925i \(-0.806197\pi\)
−0.820306 + 0.571925i \(0.806197\pi\)
\(648\) 0 0
\(649\) −15567.6 −0.941577
\(650\) 0 0
\(651\) − 1026.38i − 0.0617927i
\(652\) 0 0
\(653\) 18706.7i 1.12105i 0.828136 + 0.560527i \(0.189401\pi\)
−0.828136 + 0.560527i \(0.810599\pi\)
\(654\) 0 0
\(655\) 2145.89 0.128011
\(656\) 0 0
\(657\) 7719.62 0.458403
\(658\) 0 0
\(659\) − 26475.4i − 1.56500i −0.622651 0.782500i \(-0.713944\pi\)
0.622651 0.782500i \(-0.286056\pi\)
\(660\) 0 0
\(661\) − 14782.9i − 0.869876i −0.900460 0.434938i \(-0.856770\pi\)
0.900460 0.434938i \(-0.143230\pi\)
\(662\) 0 0
\(663\) −6329.40 −0.370760
\(664\) 0 0
\(665\) −6063.88 −0.353605
\(666\) 0 0
\(667\) 13413.9i 0.778691i
\(668\) 0 0
\(669\) − 9476.84i − 0.547677i
\(670\) 0 0
\(671\) 920.498 0.0529589
\(672\) 0 0
\(673\) 30451.4 1.74416 0.872078 0.489367i \(-0.162772\pi\)
0.872078 + 0.489367i \(0.162772\pi\)
\(674\) 0 0
\(675\) 1704.96i 0.0972205i
\(676\) 0 0
\(677\) 29136.3i 1.65406i 0.562158 + 0.827030i \(0.309971\pi\)
−0.562158 + 0.827030i \(0.690029\pi\)
\(678\) 0 0
\(679\) 2526.32 0.142786
\(680\) 0 0
\(681\) 12228.5 0.688104
\(682\) 0 0
\(683\) − 22580.3i − 1.26502i −0.774552 0.632510i \(-0.782025\pi\)
0.774552 0.632510i \(-0.217975\pi\)
\(684\) 0 0
\(685\) 9758.95i 0.544336i
\(686\) 0 0
\(687\) 8185.08 0.454556
\(688\) 0 0
\(689\) 23878.5 1.32032
\(690\) 0 0
\(691\) 17484.1i 0.962558i 0.876567 + 0.481279i \(0.159828\pi\)
−0.876567 + 0.481279i \(0.840172\pi\)
\(692\) 0 0
\(693\) − 1958.91i − 0.107378i
\(694\) 0 0
\(695\) 11571.9 0.631579
\(696\) 0 0
\(697\) −14638.1 −0.795491
\(698\) 0 0
\(699\) − 5397.95i − 0.292088i
\(700\) 0 0
\(701\) 16469.3i 0.887355i 0.896186 + 0.443678i \(0.146326\pi\)
−0.896186 + 0.443678i \(0.853674\pi\)
\(702\) 0 0
\(703\) 766.447 0.0411196
\(704\) 0 0
\(705\) −7430.19 −0.396932
\(706\) 0 0
\(707\) − 5377.01i − 0.286030i
\(708\) 0 0
\(709\) 21999.8i 1.16533i 0.812712 + 0.582666i \(0.197990\pi\)
−0.812712 + 0.582666i \(0.802010\pi\)
\(710\) 0 0
\(711\) −7195.33 −0.379530
\(712\) 0 0
\(713\) −8331.74 −0.437624
\(714\) 0 0
\(715\) − 11972.4i − 0.626214i
\(716\) 0 0
\(717\) − 1545.92i − 0.0805208i
\(718\) 0 0
\(719\) 24993.7 1.29639 0.648197 0.761472i \(-0.275523\pi\)
0.648197 + 0.761472i \(0.275523\pi\)
\(720\) 0 0
\(721\) −12386.8 −0.639820
\(722\) 0 0
\(723\) − 8513.04i − 0.437903i
\(724\) 0 0
\(725\) 4968.87i 0.254537i
\(726\) 0 0
\(727\) −22235.5 −1.13435 −0.567173 0.823598i \(-0.691963\pi\)
−0.567173 + 0.823598i \(0.691963\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) − 9579.44i − 0.484690i
\(732\) 0 0
\(733\) − 23498.3i − 1.18408i −0.805910 0.592038i \(-0.798324\pi\)
0.805910 0.592038i \(-0.201676\pi\)
\(734\) 0 0
\(735\) −1156.11 −0.0580187
\(736\) 0 0
\(737\) 11825.5 0.591041
\(738\) 0 0
\(739\) 8773.06i 0.436701i 0.975870 + 0.218351i \(0.0700677\pi\)
−0.975870 + 0.218351i \(0.929932\pi\)
\(740\) 0 0
\(741\) 16177.8i 0.802033i
\(742\) 0 0
\(743\) 5806.02 0.286679 0.143339 0.989674i \(-0.454216\pi\)
0.143339 + 0.989674i \(0.454216\pi\)
\(744\) 0 0
\(745\) 4572.22 0.224850
\(746\) 0 0
\(747\) − 1609.70i − 0.0788432i
\(748\) 0 0
\(749\) 7093.64i 0.346056i
\(750\) 0 0
\(751\) −18504.6 −0.899125 −0.449562 0.893249i \(-0.648420\pi\)
−0.449562 + 0.893249i \(0.648420\pi\)
\(752\) 0 0
\(753\) −11218.8 −0.542944
\(754\) 0 0
\(755\) 21307.9i 1.02712i
\(756\) 0 0
\(757\) 35271.9i 1.69350i 0.531991 + 0.846750i \(0.321444\pi\)
−0.531991 + 0.846750i \(0.678556\pi\)
\(758\) 0 0
\(759\) −15901.6 −0.760462
\(760\) 0 0
\(761\) −6989.26 −0.332931 −0.166465 0.986047i \(-0.553235\pi\)
−0.166465 + 0.986047i \(0.553235\pi\)
\(762\) 0 0
\(763\) 8547.32i 0.405549i
\(764\) 0 0
\(765\) − 3050.27i − 0.144161i
\(766\) 0 0
\(767\) 24511.9 1.15394
\(768\) 0 0
\(769\) −14248.4 −0.668152 −0.334076 0.942546i \(-0.608424\pi\)
−0.334076 + 0.942546i \(0.608424\pi\)
\(770\) 0 0
\(771\) − 3246.25i − 0.151635i
\(772\) 0 0
\(773\) − 11637.6i − 0.541496i −0.962650 0.270748i \(-0.912729\pi\)
0.962650 0.270748i \(-0.0872711\pi\)
\(774\) 0 0
\(775\) −3086.31 −0.143050
\(776\) 0 0
\(777\) 146.127 0.00674681
\(778\) 0 0
\(779\) 37414.6i 1.72082i
\(780\) 0 0
\(781\) 391.708i 0.0179467i
\(782\) 0 0
\(783\) −2124.57 −0.0969680
\(784\) 0 0
\(785\) −22693.0 −1.03178
\(786\) 0 0
\(787\) − 14596.1i − 0.661112i −0.943786 0.330556i \(-0.892764\pi\)
0.943786 0.330556i \(-0.107236\pi\)
\(788\) 0 0
\(789\) − 19103.4i − 0.861977i
\(790\) 0 0
\(791\) −4287.36 −0.192720
\(792\) 0 0
\(793\) −1449.36 −0.0649034
\(794\) 0 0
\(795\) 11507.6i 0.513373i
\(796\) 0 0
\(797\) 8911.71i 0.396072i 0.980195 + 0.198036i \(0.0634562\pi\)
−0.980195 + 0.198036i \(0.936544\pi\)
\(798\) 0 0
\(799\) −13571.0 −0.600884
\(800\) 0 0
\(801\) 9262.62 0.408587
\(802\) 0 0
\(803\) 26670.2i 1.17207i
\(804\) 0 0
\(805\) 9384.83i 0.410896i
\(806\) 0 0
\(807\) 12798.0 0.558253
\(808\) 0 0
\(809\) 14318.4 0.622260 0.311130 0.950367i \(-0.399292\pi\)
0.311130 + 0.950367i \(0.399292\pi\)
\(810\) 0 0
\(811\) − 17454.9i − 0.755762i −0.925854 0.377881i \(-0.876653\pi\)
0.925854 0.377881i \(-0.123347\pi\)
\(812\) 0 0
\(813\) − 6476.31i − 0.279378i
\(814\) 0 0
\(815\) 25403.6 1.09184
\(816\) 0 0
\(817\) −24484.8 −1.04849
\(818\) 0 0
\(819\) 3084.38i 0.131596i
\(820\) 0 0
\(821\) − 29192.7i − 1.24096i −0.784220 0.620482i \(-0.786937\pi\)
0.784220 0.620482i \(-0.213063\pi\)
\(822\) 0 0
\(823\) 13828.3 0.585693 0.292846 0.956159i \(-0.405398\pi\)
0.292846 + 0.956159i \(0.405398\pi\)
\(824\) 0 0
\(825\) −5890.39 −0.248578
\(826\) 0 0
\(827\) − 37841.9i − 1.59116i −0.605846 0.795582i \(-0.707165\pi\)
0.605846 0.795582i \(-0.292835\pi\)
\(828\) 0 0
\(829\) − 4910.72i − 0.205738i −0.994695 0.102869i \(-0.967198\pi\)
0.994695 0.102869i \(-0.0328022\pi\)
\(830\) 0 0
\(831\) −14215.1 −0.593402
\(832\) 0 0
\(833\) −2111.59 −0.0878300
\(834\) 0 0
\(835\) 1846.45i 0.0765257i
\(836\) 0 0
\(837\) − 1319.63i − 0.0544960i
\(838\) 0 0
\(839\) 19949.5 0.820899 0.410450 0.911883i \(-0.365372\pi\)
0.410450 + 0.911883i \(0.365372\pi\)
\(840\) 0 0
\(841\) 18197.2 0.746124
\(842\) 0 0
\(843\) − 12163.4i − 0.496950i
\(844\) 0 0
\(845\) 1572.37i 0.0640132i
\(846\) 0 0
\(847\) −2549.26 −0.103416
\(848\) 0 0
\(849\) 1749.58 0.0707248
\(850\) 0 0
\(851\) − 1186.20i − 0.0477819i
\(852\) 0 0
\(853\) 3800.95i 0.152570i 0.997086 + 0.0762849i \(0.0243059\pi\)
−0.997086 + 0.0762849i \(0.975694\pi\)
\(854\) 0 0
\(855\) −7796.42 −0.311850
\(856\) 0 0
\(857\) −3277.49 −0.130638 −0.0653192 0.997864i \(-0.520807\pi\)
−0.0653192 + 0.997864i \(0.520807\pi\)
\(858\) 0 0
\(859\) − 16774.9i − 0.666300i −0.942874 0.333150i \(-0.891888\pi\)
0.942874 0.333150i \(-0.108112\pi\)
\(860\) 0 0
\(861\) 7133.28i 0.282348i
\(862\) 0 0
\(863\) −5411.95 −0.213470 −0.106735 0.994287i \(-0.534040\pi\)
−0.106735 + 0.994287i \(0.534040\pi\)
\(864\) 0 0
\(865\) −14883.3 −0.585028
\(866\) 0 0
\(867\) 9167.79i 0.359117i
\(868\) 0 0
\(869\) − 24858.9i − 0.970402i
\(870\) 0 0
\(871\) −18619.7 −0.724346
\(872\) 0 0
\(873\) 3248.13 0.125925
\(874\) 0 0
\(875\) 10358.0i 0.400188i
\(876\) 0 0
\(877\) − 29953.6i − 1.15332i −0.816984 0.576661i \(-0.804355\pi\)
0.816984 0.576661i \(-0.195645\pi\)
\(878\) 0 0
\(879\) 20958.5 0.804223
\(880\) 0 0
\(881\) 8509.06 0.325400 0.162700 0.986676i \(-0.447980\pi\)
0.162700 + 0.986676i \(0.447980\pi\)
\(882\) 0 0
\(883\) − 3532.20i − 0.134618i −0.997732 0.0673091i \(-0.978559\pi\)
0.997732 0.0673091i \(-0.0214414\pi\)
\(884\) 0 0
\(885\) 11812.8i 0.448681i
\(886\) 0 0
\(887\) −30349.5 −1.14886 −0.574429 0.818555i \(-0.694776\pi\)
−0.574429 + 0.818555i \(0.694776\pi\)
\(888\) 0 0
\(889\) −9612.56 −0.362649
\(890\) 0 0
\(891\) − 2518.59i − 0.0946981i
\(892\) 0 0
\(893\) 34687.1i 1.29984i
\(894\) 0 0
\(895\) 2678.55 0.100038
\(896\) 0 0
\(897\) 25037.7 0.931980
\(898\) 0 0
\(899\) − 3845.89i − 0.142678i
\(900\) 0 0
\(901\) 21018.1i 0.777154i
\(902\) 0 0
\(903\) −4668.16 −0.172034
\(904\) 0 0
\(905\) −29795.5 −1.09440
\(906\) 0 0
\(907\) − 36207.2i − 1.32551i −0.748835 0.662756i \(-0.769387\pi\)
0.748835 0.662756i \(-0.230613\pi\)
\(908\) 0 0
\(909\) − 6913.29i − 0.252255i
\(910\) 0 0
\(911\) 40684.8 1.47964 0.739818 0.672808i \(-0.234912\pi\)
0.739818 + 0.672808i \(0.234912\pi\)
\(912\) 0 0
\(913\) 5561.29 0.201590
\(914\) 0 0
\(915\) − 698.478i − 0.0252360i
\(916\) 0 0
\(917\) − 1909.96i − 0.0687812i
\(918\) 0 0
\(919\) −46932.9 −1.68463 −0.842314 0.538986i \(-0.818807\pi\)
−0.842314 + 0.538986i \(0.818807\pi\)
\(920\) 0 0
\(921\) 22227.3 0.795239
\(922\) 0 0
\(923\) − 616.761i − 0.0219945i
\(924\) 0 0
\(925\) − 439.401i − 0.0156188i
\(926\) 0 0
\(927\) −15925.9 −0.564268
\(928\) 0 0
\(929\) −9916.16 −0.350203 −0.175101 0.984550i \(-0.556025\pi\)
−0.175101 + 0.984550i \(0.556025\pi\)
\(930\) 0 0
\(931\) 5397.18i 0.189995i
\(932\) 0 0
\(933\) − 11268.4i − 0.395402i
\(934\) 0 0
\(935\) 10538.3 0.368597
\(936\) 0 0
\(937\) 38298.5 1.33528 0.667640 0.744484i \(-0.267305\pi\)
0.667640 + 0.744484i \(0.267305\pi\)
\(938\) 0 0
\(939\) − 23964.4i − 0.832852i
\(940\) 0 0
\(941\) 37798.8i 1.30947i 0.755861 + 0.654733i \(0.227219\pi\)
−0.755861 + 0.654733i \(0.772781\pi\)
\(942\) 0 0
\(943\) 57905.0 1.99963
\(944\) 0 0
\(945\) −1486.43 −0.0511677
\(946\) 0 0
\(947\) 7106.74i 0.243863i 0.992539 + 0.121931i \(0.0389088\pi\)
−0.992539 + 0.121931i \(0.961091\pi\)
\(948\) 0 0
\(949\) − 41993.4i − 1.43642i
\(950\) 0 0
\(951\) −2904.92 −0.0990520
\(952\) 0 0
\(953\) 50709.2 1.72364 0.861821 0.507212i \(-0.169324\pi\)
0.861821 + 0.507212i \(0.169324\pi\)
\(954\) 0 0
\(955\) − 39924.9i − 1.35282i
\(956\) 0 0
\(957\) − 7340.09i − 0.247933i
\(958\) 0 0
\(959\) 8685.99 0.292477
\(960\) 0 0
\(961\) −27402.2 −0.919815
\(962\) 0 0
\(963\) 9120.39i 0.305193i
\(964\) 0 0
\(965\) 18368.2i 0.612738i
\(966\) 0 0
\(967\) 26572.3 0.883667 0.441833 0.897097i \(-0.354328\pi\)
0.441833 + 0.897097i \(0.354328\pi\)
\(968\) 0 0
\(969\) −14239.9 −0.472086
\(970\) 0 0
\(971\) 24027.0i 0.794091i 0.917799 + 0.397045i \(0.129964\pi\)
−0.917799 + 0.397045i \(0.870036\pi\)
\(972\) 0 0
\(973\) − 10299.6i − 0.339353i
\(974\) 0 0
\(975\) 9274.67 0.304643
\(976\) 0 0
\(977\) 9671.08 0.316689 0.158345 0.987384i \(-0.449384\pi\)
0.158345 + 0.987384i \(0.449384\pi\)
\(978\) 0 0
\(979\) 32001.0i 1.04470i
\(980\) 0 0
\(981\) 10989.4i 0.357660i
\(982\) 0 0
\(983\) −24568.3 −0.797159 −0.398579 0.917134i \(-0.630497\pi\)
−0.398579 + 0.917134i \(0.630497\pi\)
\(984\) 0 0
\(985\) −24833.6 −0.803314
\(986\) 0 0
\(987\) 6613.27i 0.213275i
\(988\) 0 0
\(989\) 37894.2i 1.21837i
\(990\) 0 0
\(991\) 33699.2 1.08021 0.540106 0.841597i \(-0.318384\pi\)
0.540106 + 0.841597i \(0.318384\pi\)
\(992\) 0 0
\(993\) −34284.0 −1.09564
\(994\) 0 0
\(995\) − 35857.4i − 1.14247i
\(996\) 0 0
\(997\) − 43096.4i − 1.36898i −0.729020 0.684492i \(-0.760024\pi\)
0.729020 0.684492i \(-0.239976\pi\)
\(998\) 0 0
\(999\) 187.877 0.00595013
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 1344.4.c.a.673.2 6
4.3 odd 2 1344.4.c.d.673.5 yes 6
8.3 odd 2 1344.4.c.d.673.2 yes 6
8.5 even 2 inner 1344.4.c.a.673.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1344.4.c.a.673.2 6 1.1 even 1 trivial
1344.4.c.a.673.5 yes 6 8.5 even 2 inner
1344.4.c.d.673.2 yes 6 8.3 odd 2
1344.4.c.d.673.5 yes 6 4.3 odd 2