Properties

Label 1344.4.c.c.673.1
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: 6.0.14024243776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 17x^{4} - 164x^{3} + 299x^{2} + 2466x + 13042 \) 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 673.1
Root \(-3.08965 - 2.87659i\) of defining polynomial
Character \(\chi\) \(=\) 1344.673
Dual form 1344.4.c.c.673.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} -13.9325i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -13.9325i q^{5} +7.00000 q^{7} -9.00000 q^{9} +44.5866i q^{11} -81.2363i q^{13} -41.7974 q^{15} -136.899 q^{17} +114.510i q^{19} -21.0000i q^{21} +67.0337 q^{23} -69.1139 q^{25} +27.0000i q^{27} +222.366i q^{29} +135.161 q^{31} +133.760 q^{33} -97.5273i q^{35} +298.444i q^{37} -243.709 q^{39} +88.8987 q^{41} +241.435i q^{43} +125.392i q^{45} -377.065 q^{47} +49.0000 q^{49} +410.696i q^{51} +174.621i q^{53} +621.202 q^{55} +343.531 q^{57} -26.8799i q^{59} -51.2106i q^{61} -63.0000 q^{63} -1131.82 q^{65} -1011.62i q^{67} -201.101i q^{69} -502.872 q^{71} -709.769 q^{73} +207.342i q^{75} +312.106i q^{77} -716.903 q^{79} +81.0000 q^{81} +882.618i q^{83} +1907.34i q^{85} +667.099 q^{87} +1294.29 q^{89} -568.654i q^{91} -405.483i q^{93} +1595.41 q^{95} -386.914 q^{97} -401.279i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 42 q^{7} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 42 q^{7} - 54 q^{9} - 24 q^{15} - 72 q^{17} - 196 q^{23} + 246 q^{25} + 104 q^{31} + 408 q^{33} - 348 q^{39} - 216 q^{41} - 408 q^{47} + 294 q^{49} + 720 q^{55} + 336 q^{57} - 378 q^{63} - 2472 q^{65} - 1164 q^{71} - 276 q^{73} + 1352 q^{79} + 486 q^{81} + 720 q^{87} + 2520 q^{89} + 4648 q^{95} - 2340 q^{97}+O(q^{100}) \) 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
\(5\) − 13.9325i − 1.24616i −0.782159 0.623079i \(-0.785881\pi\)
0.782159 0.623079i \(-0.214119\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 44.5866i 1.22212i 0.791583 + 0.611062i \(0.209257\pi\)
−0.791583 + 0.611062i \(0.790743\pi\)
\(12\) 0 0
\(13\) − 81.2363i − 1.73315i −0.499051 0.866573i \(-0.666318\pi\)
0.499051 0.866573i \(-0.333682\pi\)
\(14\) 0 0
\(15\) −41.7974 −0.719470
\(16\) 0 0
\(17\) −136.899 −1.95311 −0.976553 0.215277i \(-0.930934\pi\)
−0.976553 + 0.215277i \(0.930934\pi\)
\(18\) 0 0
\(19\) 114.510i 1.38265i 0.722542 + 0.691327i \(0.242974\pi\)
−0.722542 + 0.691327i \(0.757026\pi\)
\(20\) 0 0
\(21\) − 21.0000i − 0.218218i
\(22\) 0 0
\(23\) 67.0337 0.607717 0.303859 0.952717i \(-0.401725\pi\)
0.303859 + 0.952717i \(0.401725\pi\)
\(24\) 0 0
\(25\) −69.1139 −0.552912
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 222.366i 1.42388i 0.702243 + 0.711938i \(0.252182\pi\)
−0.702243 + 0.711938i \(0.747818\pi\)
\(30\) 0 0
\(31\) 135.161 0.783085 0.391542 0.920160i \(-0.371942\pi\)
0.391542 + 0.920160i \(0.371942\pi\)
\(32\) 0 0
\(33\) 133.760 0.705594
\(34\) 0 0
\(35\) − 97.5273i − 0.471004i
\(36\) 0 0
\(37\) 298.444i 1.32605i 0.748597 + 0.663025i \(0.230728\pi\)
−0.748597 + 0.663025i \(0.769272\pi\)
\(38\) 0 0
\(39\) −243.709 −1.00063
\(40\) 0 0
\(41\) 88.8987 0.338625 0.169313 0.985562i \(-0.445845\pi\)
0.169313 + 0.985562i \(0.445845\pi\)
\(42\) 0 0
\(43\) 241.435i 0.856245i 0.903721 + 0.428123i \(0.140825\pi\)
−0.903721 + 0.428123i \(0.859175\pi\)
\(44\) 0 0
\(45\) 125.392i 0.415386i
\(46\) 0 0
\(47\) −377.065 −1.17022 −0.585112 0.810953i \(-0.698949\pi\)
−0.585112 + 0.810953i \(0.698949\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 410.696i 1.12763i
\(52\) 0 0
\(53\) 174.621i 0.452567i 0.974062 + 0.226283i \(0.0726576\pi\)
−0.974062 + 0.226283i \(0.927342\pi\)
\(54\) 0 0
\(55\) 621.202 1.52296
\(56\) 0 0
\(57\) 343.531 0.798276
\(58\) 0 0
\(59\) − 26.8799i − 0.0593129i −0.999560 0.0296564i \(-0.990559\pi\)
0.999560 0.0296564i \(-0.00944132\pi\)
\(60\) 0 0
\(61\) − 51.2106i − 0.107489i −0.998555 0.0537446i \(-0.982884\pi\)
0.998555 0.0537446i \(-0.0171157\pi\)
\(62\) 0 0
\(63\) −63.0000 −0.125988
\(64\) 0 0
\(65\) −1131.82 −2.15977
\(66\) 0 0
\(67\) − 1011.62i − 1.84461i −0.386460 0.922306i \(-0.626302\pi\)
0.386460 0.922306i \(-0.373698\pi\)
\(68\) 0 0
\(69\) − 201.101i − 0.350866i
\(70\) 0 0
\(71\) −502.872 −0.840562 −0.420281 0.907394i \(-0.638069\pi\)
−0.420281 + 0.907394i \(0.638069\pi\)
\(72\) 0 0
\(73\) −709.769 −1.13798 −0.568988 0.822346i \(-0.692665\pi\)
−0.568988 + 0.822346i \(0.692665\pi\)
\(74\) 0 0
\(75\) 207.342i 0.319224i
\(76\) 0 0
\(77\) 312.106i 0.461919i
\(78\) 0 0
\(79\) −716.903 −1.02099 −0.510493 0.859882i \(-0.670537\pi\)
−0.510493 + 0.859882i \(0.670537\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 882.618i 1.16723i 0.812031 + 0.583614i \(0.198362\pi\)
−0.812031 + 0.583614i \(0.801638\pi\)
\(84\) 0 0
\(85\) 1907.34i 2.43388i
\(86\) 0 0
\(87\) 667.099 0.822075
\(88\) 0 0
\(89\) 1294.29 1.54151 0.770756 0.637131i \(-0.219879\pi\)
0.770756 + 0.637131i \(0.219879\pi\)
\(90\) 0 0
\(91\) − 568.654i − 0.655067i
\(92\) 0 0
\(93\) − 405.483i − 0.452114i
\(94\) 0 0
\(95\) 1595.41 1.72301
\(96\) 0 0
\(97\) −386.914 −0.405002 −0.202501 0.979282i \(-0.564907\pi\)
−0.202501 + 0.979282i \(0.564907\pi\)
\(98\) 0 0
\(99\) − 401.279i − 0.407375i
\(100\) 0 0
\(101\) 277.366i 0.273257i 0.990622 + 0.136629i \(0.0436267\pi\)
−0.990622 + 0.136629i \(0.956373\pi\)
\(102\) 0 0
\(103\) 626.953 0.599762 0.299881 0.953977i \(-0.403053\pi\)
0.299881 + 0.953977i \(0.403053\pi\)
\(104\) 0 0
\(105\) −292.582 −0.271934
\(106\) 0 0
\(107\) − 14.3217i − 0.0129396i −0.999979 0.00646979i \(-0.997941\pi\)
0.999979 0.00646979i \(-0.00205941\pi\)
\(108\) 0 0
\(109\) 467.212i 0.410558i 0.978704 + 0.205279i \(0.0658101\pi\)
−0.978704 + 0.205279i \(0.934190\pi\)
\(110\) 0 0
\(111\) 895.331 0.765595
\(112\) 0 0
\(113\) 1098.35 0.914375 0.457188 0.889370i \(-0.348857\pi\)
0.457188 + 0.889370i \(0.348857\pi\)
\(114\) 0 0
\(115\) − 933.946i − 0.757312i
\(116\) 0 0
\(117\) 731.126i 0.577715i
\(118\) 0 0
\(119\) −958.291 −0.738205
\(120\) 0 0
\(121\) −656.964 −0.493587
\(122\) 0 0
\(123\) − 266.696i − 0.195505i
\(124\) 0 0
\(125\) − 778.631i − 0.557143i
\(126\) 0 0
\(127\) 1213.62 0.847961 0.423980 0.905671i \(-0.360633\pi\)
0.423980 + 0.905671i \(0.360633\pi\)
\(128\) 0 0
\(129\) 724.306 0.494353
\(130\) 0 0
\(131\) 1391.85i 0.928295i 0.885758 + 0.464147i \(0.153639\pi\)
−0.885758 + 0.464147i \(0.846361\pi\)
\(132\) 0 0
\(133\) 801.571i 0.522594i
\(134\) 0 0
\(135\) 376.177 0.239823
\(136\) 0 0
\(137\) 1455.00 0.907363 0.453681 0.891164i \(-0.350110\pi\)
0.453681 + 0.891164i \(0.350110\pi\)
\(138\) 0 0
\(139\) − 2512.86i − 1.53337i −0.642025 0.766684i \(-0.721905\pi\)
0.642025 0.766684i \(-0.278095\pi\)
\(140\) 0 0
\(141\) 1131.19i 0.675629i
\(142\) 0 0
\(143\) 3622.05 2.11812
\(144\) 0 0
\(145\) 3098.11 1.77437
\(146\) 0 0
\(147\) − 147.000i − 0.0824786i
\(148\) 0 0
\(149\) 2790.64i 1.53435i 0.641438 + 0.767174i \(0.278338\pi\)
−0.641438 + 0.767174i \(0.721662\pi\)
\(150\) 0 0
\(151\) −15.1006 −0.00813819 −0.00406910 0.999992i \(-0.501295\pi\)
−0.00406910 + 0.999992i \(0.501295\pi\)
\(152\) 0 0
\(153\) 1232.09 0.651035
\(154\) 0 0
\(155\) − 1883.13i − 0.975848i
\(156\) 0 0
\(157\) − 3608.78i − 1.83447i −0.398345 0.917236i \(-0.630415\pi\)
0.398345 0.917236i \(-0.369585\pi\)
\(158\) 0 0
\(159\) 523.863 0.261290
\(160\) 0 0
\(161\) 469.236 0.229695
\(162\) 0 0
\(163\) 2469.78i 1.18680i 0.804908 + 0.593400i \(0.202215\pi\)
−0.804908 + 0.593400i \(0.797785\pi\)
\(164\) 0 0
\(165\) − 1863.61i − 0.879282i
\(166\) 0 0
\(167\) −1617.23 −0.749372 −0.374686 0.927152i \(-0.622250\pi\)
−0.374686 + 0.927152i \(0.622250\pi\)
\(168\) 0 0
\(169\) −4402.33 −2.00379
\(170\) 0 0
\(171\) − 1030.59i − 0.460885i
\(172\) 0 0
\(173\) 3969.48i 1.74448i 0.489082 + 0.872238i \(0.337332\pi\)
−0.489082 + 0.872238i \(0.662668\pi\)
\(174\) 0 0
\(175\) −483.798 −0.208981
\(176\) 0 0
\(177\) −80.6396 −0.0342443
\(178\) 0 0
\(179\) 2232.89i 0.932368i 0.884688 + 0.466184i \(0.154371\pi\)
−0.884688 + 0.466184i \(0.845629\pi\)
\(180\) 0 0
\(181\) 2565.36i 1.05349i 0.850024 + 0.526745i \(0.176588\pi\)
−0.850024 + 0.526745i \(0.823412\pi\)
\(182\) 0 0
\(183\) −153.632 −0.0620590
\(184\) 0 0
\(185\) 4158.06 1.65247
\(186\) 0 0
\(187\) − 6103.84i − 2.38694i
\(188\) 0 0
\(189\) 189.000i 0.0727393i
\(190\) 0 0
\(191\) 2884.75 1.09285 0.546423 0.837510i \(-0.315989\pi\)
0.546423 + 0.837510i \(0.315989\pi\)
\(192\) 0 0
\(193\) −2841.50 −1.05977 −0.529886 0.848069i \(-0.677765\pi\)
−0.529886 + 0.848069i \(0.677765\pi\)
\(194\) 0 0
\(195\) 3395.47i 1.24695i
\(196\) 0 0
\(197\) 512.567i 0.185375i 0.995695 + 0.0926876i \(0.0295458\pi\)
−0.995695 + 0.0926876i \(0.970454\pi\)
\(198\) 0 0
\(199\) 2817.72 1.00373 0.501866 0.864946i \(-0.332647\pi\)
0.501866 + 0.864946i \(0.332647\pi\)
\(200\) 0 0
\(201\) −3034.86 −1.06499
\(202\) 0 0
\(203\) 1556.56i 0.538174i
\(204\) 0 0
\(205\) − 1238.58i − 0.421981i
\(206\) 0 0
\(207\) −603.303 −0.202572
\(208\) 0 0
\(209\) −5105.62 −1.68978
\(210\) 0 0
\(211\) 4903.85i 1.59998i 0.600016 + 0.799988i \(0.295161\pi\)
−0.600016 + 0.799988i \(0.704839\pi\)
\(212\) 0 0
\(213\) 1508.62i 0.485299i
\(214\) 0 0
\(215\) 3363.79 1.06702
\(216\) 0 0
\(217\) 946.127 0.295978
\(218\) 0 0
\(219\) 2129.31i 0.657011i
\(220\) 0 0
\(221\) 11121.1i 3.38502i
\(222\) 0 0
\(223\) 5283.25 1.58651 0.793257 0.608887i \(-0.208384\pi\)
0.793257 + 0.608887i \(0.208384\pi\)
\(224\) 0 0
\(225\) 622.025 0.184304
\(226\) 0 0
\(227\) − 348.589i − 0.101924i −0.998701 0.0509619i \(-0.983771\pi\)
0.998701 0.0509619i \(-0.0162287\pi\)
\(228\) 0 0
\(229\) − 4394.50i − 1.26811i −0.773289 0.634054i \(-0.781390\pi\)
0.773289 0.634054i \(-0.218610\pi\)
\(230\) 0 0
\(231\) 936.318 0.266689
\(232\) 0 0
\(233\) −3230.16 −0.908219 −0.454109 0.890946i \(-0.650043\pi\)
−0.454109 + 0.890946i \(0.650043\pi\)
\(234\) 0 0
\(235\) 5253.44i 1.45828i
\(236\) 0 0
\(237\) 2150.71i 0.589466i
\(238\) 0 0
\(239\) −3610.19 −0.977087 −0.488543 0.872540i \(-0.662472\pi\)
−0.488543 + 0.872540i \(0.662472\pi\)
\(240\) 0 0
\(241\) −4528.15 −1.21031 −0.605153 0.796109i \(-0.706888\pi\)
−0.605153 + 0.796109i \(0.706888\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) − 682.691i − 0.178023i
\(246\) 0 0
\(247\) 9302.38 2.39634
\(248\) 0 0
\(249\) 2647.85 0.673899
\(250\) 0 0
\(251\) 804.637i 0.202344i 0.994869 + 0.101172i \(0.0322592\pi\)
−0.994869 + 0.101172i \(0.967741\pi\)
\(252\) 0 0
\(253\) 2988.80i 0.742706i
\(254\) 0 0
\(255\) 5722.01 1.40520
\(256\) 0 0
\(257\) −1150.13 −0.279156 −0.139578 0.990211i \(-0.544575\pi\)
−0.139578 + 0.990211i \(0.544575\pi\)
\(258\) 0 0
\(259\) 2089.11i 0.501200i
\(260\) 0 0
\(261\) − 2001.30i − 0.474625i
\(262\) 0 0
\(263\) 548.868 0.128687 0.0643434 0.997928i \(-0.479505\pi\)
0.0643434 + 0.997928i \(0.479505\pi\)
\(264\) 0 0
\(265\) 2432.90 0.563970
\(266\) 0 0
\(267\) − 3882.87i − 0.889992i
\(268\) 0 0
\(269\) − 8040.48i − 1.82244i −0.411919 0.911221i \(-0.635141\pi\)
0.411919 0.911221i \(-0.364859\pi\)
\(270\) 0 0
\(271\) 3517.37 0.788431 0.394215 0.919018i \(-0.371016\pi\)
0.394215 + 0.919018i \(0.371016\pi\)
\(272\) 0 0
\(273\) −1705.96 −0.378203
\(274\) 0 0
\(275\) − 3081.56i − 0.675726i
\(276\) 0 0
\(277\) 1995.99i 0.432952i 0.976288 + 0.216476i \(0.0694563\pi\)
−0.976288 + 0.216476i \(0.930544\pi\)
\(278\) 0 0
\(279\) −1216.45 −0.261028
\(280\) 0 0
\(281\) −4672.94 −0.992043 −0.496022 0.868310i \(-0.665206\pi\)
−0.496022 + 0.868310i \(0.665206\pi\)
\(282\) 0 0
\(283\) 2886.12i 0.606227i 0.952954 + 0.303114i \(0.0980261\pi\)
−0.952954 + 0.303114i \(0.901974\pi\)
\(284\) 0 0
\(285\) − 4786.23i − 0.994779i
\(286\) 0 0
\(287\) 622.291 0.127988
\(288\) 0 0
\(289\) 13828.2 2.81462
\(290\) 0 0
\(291\) 1160.74i 0.233828i
\(292\) 0 0
\(293\) 6019.24i 1.20016i 0.799939 + 0.600082i \(0.204865\pi\)
−0.799939 + 0.600082i \(0.795135\pi\)
\(294\) 0 0
\(295\) −374.503 −0.0739133
\(296\) 0 0
\(297\) −1203.84 −0.235198
\(298\) 0 0
\(299\) − 5445.57i − 1.05326i
\(300\) 0 0
\(301\) 1690.05i 0.323630i
\(302\) 0 0
\(303\) 832.098 0.157765
\(304\) 0 0
\(305\) −713.491 −0.133949
\(306\) 0 0
\(307\) − 5862.19i − 1.08981i −0.838497 0.544907i \(-0.816565\pi\)
0.838497 0.544907i \(-0.183435\pi\)
\(308\) 0 0
\(309\) − 1880.86i − 0.346273i
\(310\) 0 0
\(311\) −5811.82 −1.05967 −0.529836 0.848100i \(-0.677747\pi\)
−0.529836 + 0.848100i \(0.677747\pi\)
\(312\) 0 0
\(313\) 2373.97 0.428705 0.214352 0.976756i \(-0.431236\pi\)
0.214352 + 0.976756i \(0.431236\pi\)
\(314\) 0 0
\(315\) 877.746i 0.157001i
\(316\) 0 0
\(317\) − 7220.24i − 1.27927i −0.768678 0.639636i \(-0.779085\pi\)
0.768678 0.639636i \(-0.220915\pi\)
\(318\) 0 0
\(319\) −9914.55 −1.74015
\(320\) 0 0
\(321\) −42.9652 −0.00747067
\(322\) 0 0
\(323\) − 15676.3i − 2.70047i
\(324\) 0 0
\(325\) 5614.56i 0.958276i
\(326\) 0 0
\(327\) 1401.64 0.237035
\(328\) 0 0
\(329\) −2639.45 −0.442303
\(330\) 0 0
\(331\) 4717.39i 0.783357i 0.920102 + 0.391678i \(0.128105\pi\)
−0.920102 + 0.391678i \(0.871895\pi\)
\(332\) 0 0
\(333\) − 2685.99i − 0.442017i
\(334\) 0 0
\(335\) −14094.4 −2.29868
\(336\) 0 0
\(337\) 4639.32 0.749911 0.374955 0.927043i \(-0.377658\pi\)
0.374955 + 0.927043i \(0.377658\pi\)
\(338\) 0 0
\(339\) − 3295.06i − 0.527915i
\(340\) 0 0
\(341\) 6026.37i 0.957027i
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) −2801.84 −0.437234
\(346\) 0 0
\(347\) 3262.45i 0.504718i 0.967634 + 0.252359i \(0.0812065\pi\)
−0.967634 + 0.252359i \(0.918794\pi\)
\(348\) 0 0
\(349\) 12732.2i 1.95283i 0.215892 + 0.976417i \(0.430734\pi\)
−0.215892 + 0.976417i \(0.569266\pi\)
\(350\) 0 0
\(351\) 2193.38 0.333544
\(352\) 0 0
\(353\) −2903.14 −0.437729 −0.218865 0.975755i \(-0.570235\pi\)
−0.218865 + 0.975755i \(0.570235\pi\)
\(354\) 0 0
\(355\) 7006.25i 1.04747i
\(356\) 0 0
\(357\) 2874.87i 0.426203i
\(358\) 0 0
\(359\) −3001.72 −0.441295 −0.220648 0.975354i \(-0.570817\pi\)
−0.220648 + 0.975354i \(0.570817\pi\)
\(360\) 0 0
\(361\) −6253.58 −0.911734
\(362\) 0 0
\(363\) 1970.89i 0.284973i
\(364\) 0 0
\(365\) 9888.85i 1.41810i
\(366\) 0 0
\(367\) 9286.04 1.32078 0.660391 0.750922i \(-0.270390\pi\)
0.660391 + 0.750922i \(0.270390\pi\)
\(368\) 0 0
\(369\) −800.088 −0.112875
\(370\) 0 0
\(371\) 1222.35i 0.171054i
\(372\) 0 0
\(373\) 2865.96i 0.397838i 0.980016 + 0.198919i \(0.0637431\pi\)
−0.980016 + 0.198919i \(0.936257\pi\)
\(374\) 0 0
\(375\) −2335.89 −0.321667
\(376\) 0 0
\(377\) 18064.2 2.46778
\(378\) 0 0
\(379\) 3785.48i 0.513054i 0.966537 + 0.256527i \(0.0825782\pi\)
−0.966537 + 0.256527i \(0.917422\pi\)
\(380\) 0 0
\(381\) − 3640.85i − 0.489570i
\(382\) 0 0
\(383\) 11670.1 1.55696 0.778478 0.627671i \(-0.215992\pi\)
0.778478 + 0.627671i \(0.215992\pi\)
\(384\) 0 0
\(385\) 4348.41 0.575625
\(386\) 0 0
\(387\) − 2172.92i − 0.285415i
\(388\) 0 0
\(389\) 9918.63i 1.29279i 0.763004 + 0.646394i \(0.223724\pi\)
−0.763004 + 0.646394i \(0.776276\pi\)
\(390\) 0 0
\(391\) −9176.82 −1.18694
\(392\) 0 0
\(393\) 4175.55 0.535951
\(394\) 0 0
\(395\) 9988.23i 1.27231i
\(396\) 0 0
\(397\) − 5101.56i − 0.644937i −0.946580 0.322468i \(-0.895487\pi\)
0.946580 0.322468i \(-0.104513\pi\)
\(398\) 0 0
\(399\) 2404.71 0.301720
\(400\) 0 0
\(401\) −1288.07 −0.160407 −0.0802035 0.996779i \(-0.525557\pi\)
−0.0802035 + 0.996779i \(0.525557\pi\)
\(402\) 0 0
\(403\) − 10980.0i − 1.35720i
\(404\) 0 0
\(405\) − 1128.53i − 0.138462i
\(406\) 0 0
\(407\) −13306.6 −1.62060
\(408\) 0 0
\(409\) −7250.96 −0.876619 −0.438309 0.898824i \(-0.644423\pi\)
−0.438309 + 0.898824i \(0.644423\pi\)
\(410\) 0 0
\(411\) − 4364.99i − 0.523866i
\(412\) 0 0
\(413\) − 188.159i − 0.0224182i
\(414\) 0 0
\(415\) 12297.1 1.45455
\(416\) 0 0
\(417\) −7538.59 −0.885291
\(418\) 0 0
\(419\) 3254.76i 0.379488i 0.981834 + 0.189744i \(0.0607659\pi\)
−0.981834 + 0.189744i \(0.939234\pi\)
\(420\) 0 0
\(421\) − 9719.85i − 1.12522i −0.826723 0.562609i \(-0.809798\pi\)
0.826723 0.562609i \(-0.190202\pi\)
\(422\) 0 0
\(423\) 3393.58 0.390075
\(424\) 0 0
\(425\) 9461.61 1.07989
\(426\) 0 0
\(427\) − 358.474i − 0.0406271i
\(428\) 0 0
\(429\) − 10866.1i − 1.22290i
\(430\) 0 0
\(431\) −940.465 −0.105106 −0.0525529 0.998618i \(-0.516736\pi\)
−0.0525529 + 0.998618i \(0.516736\pi\)
\(432\) 0 0
\(433\) 1346.44 0.149436 0.0747179 0.997205i \(-0.476194\pi\)
0.0747179 + 0.997205i \(0.476194\pi\)
\(434\) 0 0
\(435\) − 9294.34i − 1.02444i
\(436\) 0 0
\(437\) 7676.04i 0.840263i
\(438\) 0 0
\(439\) −13724.4 −1.49209 −0.746047 0.665894i \(-0.768050\pi\)
−0.746047 + 0.665894i \(0.768050\pi\)
\(440\) 0 0
\(441\) −441.000 −0.0476190
\(442\) 0 0
\(443\) − 4032.04i − 0.432433i −0.976345 0.216217i \(-0.930628\pi\)
0.976345 0.216217i \(-0.0693718\pi\)
\(444\) 0 0
\(445\) − 18032.7i − 1.92097i
\(446\) 0 0
\(447\) 8371.91 0.885857
\(448\) 0 0
\(449\) 4730.47 0.497204 0.248602 0.968606i \(-0.420029\pi\)
0.248602 + 0.968606i \(0.420029\pi\)
\(450\) 0 0
\(451\) 3963.69i 0.413842i
\(452\) 0 0
\(453\) 45.3017i 0.00469859i
\(454\) 0 0
\(455\) −7922.76 −0.816318
\(456\) 0 0
\(457\) −3220.47 −0.329644 −0.164822 0.986323i \(-0.552705\pi\)
−0.164822 + 0.986323i \(0.552705\pi\)
\(458\) 0 0
\(459\) − 3696.26i − 0.375875i
\(460\) 0 0
\(461\) 967.010i 0.0976967i 0.998806 + 0.0488483i \(0.0155551\pi\)
−0.998806 + 0.0488483i \(0.984445\pi\)
\(462\) 0 0
\(463\) 9771.32 0.980803 0.490401 0.871497i \(-0.336850\pi\)
0.490401 + 0.871497i \(0.336850\pi\)
\(464\) 0 0
\(465\) −5649.38 −0.563406
\(466\) 0 0
\(467\) 12153.9i 1.20432i 0.798376 + 0.602160i \(0.205693\pi\)
−0.798376 + 0.602160i \(0.794307\pi\)
\(468\) 0 0
\(469\) − 7081.34i − 0.697198i
\(470\) 0 0
\(471\) −10826.3 −1.05913
\(472\) 0 0
\(473\) −10764.8 −1.04644
\(474\) 0 0
\(475\) − 7914.25i − 0.764486i
\(476\) 0 0
\(477\) − 1571.59i − 0.150856i
\(478\) 0 0
\(479\) −6972.65 −0.665112 −0.332556 0.943084i \(-0.607911\pi\)
−0.332556 + 0.943084i \(0.607911\pi\)
\(480\) 0 0
\(481\) 24244.5 2.29824
\(482\) 0 0
\(483\) − 1407.71i − 0.132615i
\(484\) 0 0
\(485\) 5390.67i 0.504697i
\(486\) 0 0
\(487\) −7725.74 −0.718864 −0.359432 0.933171i \(-0.617030\pi\)
−0.359432 + 0.933171i \(0.617030\pi\)
\(488\) 0 0
\(489\) 7409.35 0.685200
\(490\) 0 0
\(491\) 7923.75i 0.728297i 0.931341 + 0.364149i \(0.118640\pi\)
−0.931341 + 0.364149i \(0.881360\pi\)
\(492\) 0 0
\(493\) − 30441.6i − 2.78098i
\(494\) 0 0
\(495\) −5590.82 −0.507653
\(496\) 0 0
\(497\) −3520.10 −0.317703
\(498\) 0 0
\(499\) 370.120i 0.0332041i 0.999862 + 0.0166021i \(0.00528484\pi\)
−0.999862 + 0.0166021i \(0.994715\pi\)
\(500\) 0 0
\(501\) 4851.70i 0.432650i
\(502\) 0 0
\(503\) −8617.94 −0.763927 −0.381963 0.924177i \(-0.624752\pi\)
−0.381963 + 0.924177i \(0.624752\pi\)
\(504\) 0 0
\(505\) 3864.40 0.340522
\(506\) 0 0
\(507\) 13207.0i 1.15689i
\(508\) 0 0
\(509\) 6721.34i 0.585301i 0.956219 + 0.292650i \(0.0945372\pi\)
−0.956219 + 0.292650i \(0.905463\pi\)
\(510\) 0 0
\(511\) −4968.39 −0.430114
\(512\) 0 0
\(513\) −3091.78 −0.266092
\(514\) 0 0
\(515\) − 8735.01i − 0.747399i
\(516\) 0 0
\(517\) − 16812.0i − 1.43016i
\(518\) 0 0
\(519\) 11908.5 1.00717
\(520\) 0 0
\(521\) −19093.7 −1.60558 −0.802792 0.596260i \(-0.796653\pi\)
−0.802792 + 0.596260i \(0.796653\pi\)
\(522\) 0 0
\(523\) 12741.7i 1.06530i 0.846334 + 0.532652i \(0.178805\pi\)
−0.846334 + 0.532652i \(0.821195\pi\)
\(524\) 0 0
\(525\) 1451.39i 0.120655i
\(526\) 0 0
\(527\) −18503.4 −1.52945
\(528\) 0 0
\(529\) −7673.48 −0.630680
\(530\) 0 0
\(531\) 241.919i 0.0197710i
\(532\) 0 0
\(533\) − 7221.80i − 0.586887i
\(534\) 0 0
\(535\) −199.537 −0.0161248
\(536\) 0 0
\(537\) 6698.66 0.538303
\(538\) 0 0
\(539\) 2184.74i 0.174589i
\(540\) 0 0
\(541\) 7575.99i 0.602065i 0.953614 + 0.301033i \(0.0973313\pi\)
−0.953614 + 0.301033i \(0.902669\pi\)
\(542\) 0 0
\(543\) 7696.07 0.608232
\(544\) 0 0
\(545\) 6509.42 0.511620
\(546\) 0 0
\(547\) − 22515.9i − 1.75998i −0.474989 0.879992i \(-0.657548\pi\)
0.474989 0.879992i \(-0.342452\pi\)
\(548\) 0 0
\(549\) 460.895i 0.0358298i
\(550\) 0 0
\(551\) −25463.2 −1.96873
\(552\) 0 0
\(553\) −5018.32 −0.385896
\(554\) 0 0
\(555\) − 12474.2i − 0.954053i
\(556\) 0 0
\(557\) 11313.2i 0.860605i 0.902685 + 0.430302i \(0.141593\pi\)
−0.902685 + 0.430302i \(0.858407\pi\)
\(558\) 0 0
\(559\) 19613.3 1.48400
\(560\) 0 0
\(561\) −18311.5 −1.37810
\(562\) 0 0
\(563\) − 23180.4i − 1.73523i −0.497234 0.867616i \(-0.665651\pi\)
0.497234 0.867616i \(-0.334349\pi\)
\(564\) 0 0
\(565\) − 15302.8i − 1.13946i
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) −4268.03 −0.314455 −0.157228 0.987562i \(-0.550256\pi\)
−0.157228 + 0.987562i \(0.550256\pi\)
\(570\) 0 0
\(571\) 7080.94i 0.518963i 0.965748 + 0.259482i \(0.0835517\pi\)
−0.965748 + 0.259482i \(0.916448\pi\)
\(572\) 0 0
\(573\) − 8654.26i − 0.630954i
\(574\) 0 0
\(575\) −4632.96 −0.336014
\(576\) 0 0
\(577\) −10667.7 −0.769676 −0.384838 0.922984i \(-0.625743\pi\)
−0.384838 + 0.922984i \(0.625743\pi\)
\(578\) 0 0
\(579\) 8524.51i 0.611859i
\(580\) 0 0
\(581\) 6178.33i 0.441171i
\(582\) 0 0
\(583\) −7785.76 −0.553093
\(584\) 0 0
\(585\) 10186.4 0.719925
\(586\) 0 0
\(587\) 13401.8i 0.942333i 0.882044 + 0.471166i \(0.156167\pi\)
−0.882044 + 0.471166i \(0.843833\pi\)
\(588\) 0 0
\(589\) 15477.3i 1.08274i
\(590\) 0 0
\(591\) 1537.70 0.107026
\(592\) 0 0
\(593\) −207.518 −0.0143705 −0.00718526 0.999974i \(-0.502287\pi\)
−0.00718526 + 0.999974i \(0.502287\pi\)
\(594\) 0 0
\(595\) 13351.4i 0.919920i
\(596\) 0 0
\(597\) − 8453.15i − 0.579505i
\(598\) 0 0
\(599\) 624.815 0.0426198 0.0213099 0.999773i \(-0.493216\pi\)
0.0213099 + 0.999773i \(0.493216\pi\)
\(600\) 0 0
\(601\) −15573.3 −1.05699 −0.528494 0.848937i \(-0.677243\pi\)
−0.528494 + 0.848937i \(0.677243\pi\)
\(602\) 0 0
\(603\) 9104.58i 0.614871i
\(604\) 0 0
\(605\) 9153.14i 0.615088i
\(606\) 0 0
\(607\) −7372.78 −0.493002 −0.246501 0.969143i \(-0.579281\pi\)
−0.246501 + 0.969143i \(0.579281\pi\)
\(608\) 0 0
\(609\) 4669.69 0.310715
\(610\) 0 0
\(611\) 30631.3i 2.02817i
\(612\) 0 0
\(613\) − 20963.2i − 1.38123i −0.723223 0.690615i \(-0.757340\pi\)
0.723223 0.690615i \(-0.242660\pi\)
\(614\) 0 0
\(615\) −3715.74 −0.243631
\(616\) 0 0
\(617\) −19585.5 −1.27793 −0.638966 0.769235i \(-0.720638\pi\)
−0.638966 + 0.769235i \(0.720638\pi\)
\(618\) 0 0
\(619\) − 11978.4i − 0.777792i −0.921282 0.388896i \(-0.872856\pi\)
0.921282 0.388896i \(-0.127144\pi\)
\(620\) 0 0
\(621\) 1809.91i 0.116955i
\(622\) 0 0
\(623\) 9060.03 0.582637
\(624\) 0 0
\(625\) −19487.5 −1.24720
\(626\) 0 0
\(627\) 15316.9i 0.975592i
\(628\) 0 0
\(629\) − 40856.6i − 2.58992i
\(630\) 0 0
\(631\) −25115.4 −1.58452 −0.792258 0.610187i \(-0.791094\pi\)
−0.792258 + 0.610187i \(0.791094\pi\)
\(632\) 0 0
\(633\) 14711.5 0.923747
\(634\) 0 0
\(635\) − 16908.7i − 1.05669i
\(636\) 0 0
\(637\) − 3980.58i − 0.247592i
\(638\) 0 0
\(639\) 4525.85 0.280187
\(640\) 0 0
\(641\) −10400.7 −0.640877 −0.320438 0.947269i \(-0.603830\pi\)
−0.320438 + 0.947269i \(0.603830\pi\)
\(642\) 0 0
\(643\) 27866.0i 1.70906i 0.519398 + 0.854532i \(0.326156\pi\)
−0.519398 + 0.854532i \(0.673844\pi\)
\(644\) 0 0
\(645\) − 10091.4i − 0.616043i
\(646\) 0 0
\(647\) −20115.3 −1.22228 −0.611139 0.791523i \(-0.709288\pi\)
−0.611139 + 0.791523i \(0.709288\pi\)
\(648\) 0 0
\(649\) 1198.48 0.0724877
\(650\) 0 0
\(651\) − 2838.38i − 0.170883i
\(652\) 0 0
\(653\) − 3473.50i − 0.208160i −0.994569 0.104080i \(-0.966810\pi\)
0.994569 0.104080i \(-0.0331898\pi\)
\(654\) 0 0
\(655\) 19391.9 1.15680
\(656\) 0 0
\(657\) 6387.93 0.379325
\(658\) 0 0
\(659\) − 1216.52i − 0.0719102i −0.999353 0.0359551i \(-0.988553\pi\)
0.999353 0.0359551i \(-0.0114473\pi\)
\(660\) 0 0
\(661\) 94.5174i 0.00556173i 0.999996 + 0.00278086i \(0.000885177\pi\)
−0.999996 + 0.00278086i \(0.999115\pi\)
\(662\) 0 0
\(663\) 33363.4 1.95434
\(664\) 0 0
\(665\) 11167.9 0.651236
\(666\) 0 0
\(667\) 14906.0i 0.865313i
\(668\) 0 0
\(669\) − 15849.8i − 0.915974i
\(670\) 0 0
\(671\) 2283.31 0.131365
\(672\) 0 0
\(673\) −3226.20 −0.184786 −0.0923928 0.995723i \(-0.529452\pi\)
−0.0923928 + 0.995723i \(0.529452\pi\)
\(674\) 0 0
\(675\) − 1866.08i − 0.106408i
\(676\) 0 0
\(677\) − 17261.8i − 0.979949i −0.871737 0.489975i \(-0.837006\pi\)
0.871737 0.489975i \(-0.162994\pi\)
\(678\) 0 0
\(679\) −2708.40 −0.153076
\(680\) 0 0
\(681\) −1045.77 −0.0588457
\(682\) 0 0
\(683\) 8923.41i 0.499919i 0.968256 + 0.249960i \(0.0804173\pi\)
−0.968256 + 0.249960i \(0.919583\pi\)
\(684\) 0 0
\(685\) − 20271.7i − 1.13072i
\(686\) 0 0
\(687\) −13183.5 −0.732142
\(688\) 0 0
\(689\) 14185.6 0.784364
\(690\) 0 0
\(691\) − 20790.4i − 1.14458i −0.820052 0.572289i \(-0.806056\pi\)
0.820052 0.572289i \(-0.193944\pi\)
\(692\) 0 0
\(693\) − 2808.96i − 0.153973i
\(694\) 0 0
\(695\) −35010.4 −1.91082
\(696\) 0 0
\(697\) −12170.1 −0.661371
\(698\) 0 0
\(699\) 9690.49i 0.524360i
\(700\) 0 0
\(701\) 5947.55i 0.320450i 0.987080 + 0.160225i \(0.0512221\pi\)
−0.987080 + 0.160225i \(0.948778\pi\)
\(702\) 0 0
\(703\) −34174.9 −1.83347
\(704\) 0 0
\(705\) 15760.3 0.841941
\(706\) 0 0
\(707\) 1941.56i 0.103281i
\(708\) 0 0
\(709\) 25595.4i 1.35579i 0.735157 + 0.677896i \(0.237108\pi\)
−0.735157 + 0.677896i \(0.762892\pi\)
\(710\) 0 0
\(711\) 6452.12 0.340328
\(712\) 0 0
\(713\) 9060.34 0.475894
\(714\) 0 0
\(715\) − 50464.1i − 2.63951i
\(716\) 0 0
\(717\) 10830.6i 0.564121i
\(718\) 0 0
\(719\) −19000.8 −0.985552 −0.492776 0.870156i \(-0.664018\pi\)
−0.492776 + 0.870156i \(0.664018\pi\)
\(720\) 0 0
\(721\) 4388.67 0.226689
\(722\) 0 0
\(723\) 13584.5i 0.698771i
\(724\) 0 0
\(725\) − 15368.6i − 0.787277i
\(726\) 0 0
\(727\) 17873.0 0.911790 0.455895 0.890033i \(-0.349319\pi\)
0.455895 + 0.890033i \(0.349319\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) − 33052.2i − 1.67234i
\(732\) 0 0
\(733\) 1862.48i 0.0938503i 0.998898 + 0.0469252i \(0.0149422\pi\)
−0.998898 + 0.0469252i \(0.985058\pi\)
\(734\) 0 0
\(735\) −2048.07 −0.102781
\(736\) 0 0
\(737\) 45104.7 2.25435
\(738\) 0 0
\(739\) − 21671.9i − 1.07877i −0.842058 0.539386i \(-0.818656\pi\)
0.842058 0.539386i \(-0.181344\pi\)
\(740\) 0 0
\(741\) − 27907.1i − 1.38353i
\(742\) 0 0
\(743\) −12546.0 −0.619474 −0.309737 0.950822i \(-0.600241\pi\)
−0.309737 + 0.950822i \(0.600241\pi\)
\(744\) 0 0
\(745\) 38880.5 1.91204
\(746\) 0 0
\(747\) − 7943.56i − 0.389076i
\(748\) 0 0
\(749\) − 100.252i − 0.00489070i
\(750\) 0 0
\(751\) 32680.9 1.58794 0.793970 0.607957i \(-0.208011\pi\)
0.793970 + 0.607957i \(0.208011\pi\)
\(752\) 0 0
\(753\) 2413.91 0.116823
\(754\) 0 0
\(755\) 210.388i 0.0101415i
\(756\) 0 0
\(757\) − 15404.1i − 0.739592i −0.929113 0.369796i \(-0.879428\pi\)
0.929113 0.369796i \(-0.120572\pi\)
\(758\) 0 0
\(759\) 8966.41 0.428801
\(760\) 0 0
\(761\) 33194.8 1.58122 0.790612 0.612317i \(-0.209762\pi\)
0.790612 + 0.612317i \(0.209762\pi\)
\(762\) 0 0
\(763\) 3270.48i 0.155176i
\(764\) 0 0
\(765\) − 17166.0i − 0.811293i
\(766\) 0 0
\(767\) −2183.62 −0.102798
\(768\) 0 0
\(769\) 6375.25 0.298956 0.149478 0.988765i \(-0.452241\pi\)
0.149478 + 0.988765i \(0.452241\pi\)
\(770\) 0 0
\(771\) 3450.39i 0.161171i
\(772\) 0 0
\(773\) 30988.7i 1.44190i 0.692988 + 0.720949i \(0.256294\pi\)
−0.692988 + 0.720949i \(0.743706\pi\)
\(774\) 0 0
\(775\) −9341.51 −0.432977
\(776\) 0 0
\(777\) 6267.32 0.289368
\(778\) 0 0
\(779\) 10179.8i 0.468202i
\(780\) 0 0
\(781\) − 22421.3i − 1.02727i
\(782\) 0 0
\(783\) −6003.89 −0.274025
\(784\) 0 0
\(785\) −50279.3 −2.28604
\(786\) 0 0
\(787\) − 21179.0i − 0.959274i −0.877467 0.479637i \(-0.840768\pi\)
0.877467 0.479637i \(-0.159232\pi\)
\(788\) 0 0
\(789\) − 1646.60i − 0.0742974i
\(790\) 0 0
\(791\) 7688.47 0.345601
\(792\) 0 0
\(793\) −4160.16 −0.186295
\(794\) 0 0
\(795\) − 7298.71i − 0.325608i
\(796\) 0 0
\(797\) 38721.4i 1.72093i 0.509509 + 0.860465i \(0.329827\pi\)
−0.509509 + 0.860465i \(0.670173\pi\)
\(798\) 0 0
\(799\) 51619.6 2.28557
\(800\) 0 0
\(801\) −11648.6 −0.513837
\(802\) 0 0
\(803\) − 31646.2i − 1.39075i
\(804\) 0 0
\(805\) − 6537.62i − 0.286237i
\(806\) 0 0
\(807\) −24121.4 −1.05219
\(808\) 0 0
\(809\) 8127.66 0.353218 0.176609 0.984281i \(-0.443487\pi\)
0.176609 + 0.984281i \(0.443487\pi\)
\(810\) 0 0
\(811\) − 4986.26i − 0.215896i −0.994157 0.107948i \(-0.965572\pi\)
0.994157 0.107948i \(-0.0344280\pi\)
\(812\) 0 0
\(813\) − 10552.1i − 0.455201i
\(814\) 0 0
\(815\) 34410.2 1.47894
\(816\) 0 0
\(817\) −27646.8 −1.18389
\(818\) 0 0
\(819\) 5117.88i 0.218356i
\(820\) 0 0
\(821\) 12495.4i 0.531174i 0.964087 + 0.265587i \(0.0855658\pi\)
−0.964087 + 0.265587i \(0.914434\pi\)
\(822\) 0 0
\(823\) 12736.1 0.539432 0.269716 0.962940i \(-0.413070\pi\)
0.269716 + 0.962940i \(0.413070\pi\)
\(824\) 0 0
\(825\) −9244.67 −0.390131
\(826\) 0 0
\(827\) − 6806.65i − 0.286204i −0.989708 0.143102i \(-0.954292\pi\)
0.989708 0.143102i \(-0.0457076\pi\)
\(828\) 0 0
\(829\) 3291.88i 0.137915i 0.997620 + 0.0689577i \(0.0219674\pi\)
−0.997620 + 0.0689577i \(0.978033\pi\)
\(830\) 0 0
\(831\) 5987.98 0.249965
\(832\) 0 0
\(833\) −6708.03 −0.279015
\(834\) 0 0
\(835\) 22532.0i 0.933837i
\(836\) 0 0
\(837\) 3649.35i 0.150705i
\(838\) 0 0
\(839\) 3389.45 0.139472 0.0697360 0.997565i \(-0.477784\pi\)
0.0697360 + 0.997565i \(0.477784\pi\)
\(840\) 0 0
\(841\) −25057.8 −1.02742
\(842\) 0 0
\(843\) 14018.8i 0.572756i
\(844\) 0 0
\(845\) 61335.4i 2.49704i
\(846\) 0 0
\(847\) −4598.75 −0.186558
\(848\) 0 0
\(849\) 8658.37 0.350005
\(850\) 0 0
\(851\) 20005.8i 0.805863i
\(852\) 0 0
\(853\) 42012.1i 1.68636i 0.537629 + 0.843181i \(0.319320\pi\)
−0.537629 + 0.843181i \(0.680680\pi\)
\(854\) 0 0
\(855\) −14358.7 −0.574336
\(856\) 0 0
\(857\) −9947.12 −0.396484 −0.198242 0.980153i \(-0.563523\pi\)
−0.198242 + 0.980153i \(0.563523\pi\)
\(858\) 0 0
\(859\) 36287.5i 1.44134i 0.693276 + 0.720672i \(0.256167\pi\)
−0.693276 + 0.720672i \(0.743833\pi\)
\(860\) 0 0
\(861\) − 1866.87i − 0.0738941i
\(862\) 0 0
\(863\) −34147.4 −1.34692 −0.673459 0.739225i \(-0.735192\pi\)
−0.673459 + 0.739225i \(0.735192\pi\)
\(864\) 0 0
\(865\) 55304.7 2.17389
\(866\) 0 0
\(867\) − 41484.7i − 1.62502i
\(868\) 0 0
\(869\) − 31964.2i − 1.24777i
\(870\) 0 0
\(871\) −82180.2 −3.19698
\(872\) 0 0
\(873\) 3482.23 0.135001
\(874\) 0 0
\(875\) − 5450.42i − 0.210580i
\(876\) 0 0
\(877\) − 10423.3i − 0.401332i −0.979660 0.200666i \(-0.935689\pi\)
0.979660 0.200666i \(-0.0643107\pi\)
\(878\) 0 0
\(879\) 18057.7 0.692915
\(880\) 0 0
\(881\) 16973.7 0.649101 0.324551 0.945868i \(-0.394787\pi\)
0.324551 + 0.945868i \(0.394787\pi\)
\(882\) 0 0
\(883\) − 40684.6i − 1.55056i −0.631617 0.775281i \(-0.717608\pi\)
0.631617 0.775281i \(-0.282392\pi\)
\(884\) 0 0
\(885\) 1123.51i 0.0426738i
\(886\) 0 0
\(887\) −22220.1 −0.841125 −0.420563 0.907264i \(-0.638167\pi\)
−0.420563 + 0.907264i \(0.638167\pi\)
\(888\) 0 0
\(889\) 8495.31 0.320499
\(890\) 0 0
\(891\) 3611.51i 0.135792i
\(892\) 0 0
\(893\) − 43177.7i − 1.61802i
\(894\) 0 0
\(895\) 31109.6 1.16188
\(896\) 0 0
\(897\) −16336.7 −0.608101
\(898\) 0 0
\(899\) 30055.2i 1.11501i
\(900\) 0 0
\(901\) − 23905.4i − 0.883911i
\(902\) 0 0
\(903\) 5070.14 0.186848
\(904\) 0 0
\(905\) 35741.8 1.31281
\(906\) 0 0
\(907\) 8719.36i 0.319208i 0.987181 + 0.159604i \(0.0510217\pi\)
−0.987181 + 0.159604i \(0.948978\pi\)
\(908\) 0 0
\(909\) − 2496.30i − 0.0910857i
\(910\) 0 0
\(911\) 17835.7 0.648652 0.324326 0.945945i \(-0.394863\pi\)
0.324326 + 0.945945i \(0.394863\pi\)
\(912\) 0 0
\(913\) −39352.9 −1.42650
\(914\) 0 0
\(915\) 2140.47i 0.0773353i
\(916\) 0 0
\(917\) 9742.96i 0.350862i
\(918\) 0 0
\(919\) 17565.7 0.630509 0.315254 0.949007i \(-0.397910\pi\)
0.315254 + 0.949007i \(0.397910\pi\)
\(920\) 0 0
\(921\) −17586.6 −0.629204
\(922\) 0 0
\(923\) 40851.4i 1.45682i
\(924\) 0 0
\(925\) − 20626.6i − 0.733188i
\(926\) 0 0
\(927\) −5642.58 −0.199921
\(928\) 0 0
\(929\) 21421.2 0.756521 0.378260 0.925699i \(-0.376522\pi\)
0.378260 + 0.925699i \(0.376522\pi\)
\(930\) 0 0
\(931\) 5611.00i 0.197522i
\(932\) 0 0
\(933\) 17435.5i 0.611802i
\(934\) 0 0
\(935\) −85041.7 −2.97450
\(936\) 0 0
\(937\) −20707.3 −0.721962 −0.360981 0.932573i \(-0.617558\pi\)
−0.360981 + 0.932573i \(0.617558\pi\)
\(938\) 0 0
\(939\) − 7121.90i − 0.247513i
\(940\) 0 0
\(941\) 56436.2i 1.95512i 0.210654 + 0.977561i \(0.432441\pi\)
−0.210654 + 0.977561i \(0.567559\pi\)
\(942\) 0 0
\(943\) 5959.21 0.205788
\(944\) 0 0
\(945\) 2633.24 0.0906447
\(946\) 0 0
\(947\) 29599.4i 1.01568i 0.861451 + 0.507841i \(0.169556\pi\)
−0.861451 + 0.507841i \(0.830444\pi\)
\(948\) 0 0
\(949\) 57659.0i 1.97228i
\(950\) 0 0
\(951\) −21660.7 −0.738588
\(952\) 0 0
\(953\) −25206.5 −0.856786 −0.428393 0.903592i \(-0.640920\pi\)
−0.428393 + 0.903592i \(0.640920\pi\)
\(954\) 0 0
\(955\) − 40191.8i − 1.36186i
\(956\) 0 0
\(957\) 29743.7i 1.00468i
\(958\) 0 0
\(959\) 10185.0 0.342951
\(960\) 0 0
\(961\) −11522.5 −0.386778
\(962\) 0 0
\(963\) 128.896i 0.00431319i
\(964\) 0 0
\(965\) 39589.2i 1.32064i
\(966\) 0 0
\(967\) 31256.7 1.03945 0.519724 0.854334i \(-0.326035\pi\)
0.519724 + 0.854334i \(0.326035\pi\)
\(968\) 0 0
\(969\) −47028.9 −1.55912
\(970\) 0 0
\(971\) 31271.1i 1.03351i 0.856134 + 0.516754i \(0.172860\pi\)
−0.856134 + 0.516754i \(0.827140\pi\)
\(972\) 0 0
\(973\) − 17590.0i − 0.579559i
\(974\) 0 0
\(975\) 16843.7 0.553261
\(976\) 0 0
\(977\) −17413.6 −0.570224 −0.285112 0.958494i \(-0.592031\pi\)
−0.285112 + 0.958494i \(0.592031\pi\)
\(978\) 0 0
\(979\) 57708.0i 1.88392i
\(980\) 0 0
\(981\) − 4204.91i − 0.136853i
\(982\) 0 0
\(983\) −18772.3 −0.609098 −0.304549 0.952497i \(-0.598506\pi\)
−0.304549 + 0.952497i \(0.598506\pi\)
\(984\) 0 0
\(985\) 7141.33 0.231007
\(986\) 0 0
\(987\) 7918.36i 0.255364i
\(988\) 0 0
\(989\) 16184.3i 0.520355i
\(990\) 0 0
\(991\) −5324.36 −0.170670 −0.0853350 0.996352i \(-0.527196\pi\)
−0.0853350 + 0.996352i \(0.527196\pi\)
\(992\) 0 0
\(993\) 14152.2 0.452271
\(994\) 0 0
\(995\) − 39257.8i − 1.25081i
\(996\) 0 0
\(997\) − 48823.0i − 1.55089i −0.631413 0.775447i \(-0.717525\pi\)
0.631413 0.775447i \(-0.282475\pi\)
\(998\) 0 0
\(999\) −8057.98 −0.255198
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.c.673.1 yes 6
4.3 odd 2 1344.4.c.b.673.4 yes 6
8.3 odd 2 1344.4.c.b.673.3 6
8.5 even 2 inner 1344.4.c.c.673.6 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1344.4.c.b.673.3 6 8.3 odd 2
1344.4.c.b.673.4 yes 6 4.3 odd 2
1344.4.c.c.673.1 yes 6 1.1 even 1 trivial
1344.4.c.c.673.6 yes 6 8.5 even 2 inner