Properties

Label 2100.4.k.g.1849.1
Level $2100$
Weight $4$
Character 2100.1849
Analytic conductor $123.904$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,4,Mod(1849,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1849");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2100.k (of order \(2\), degree \(1\), not 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: \(123.904011012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1849
Dual form 2100.4.k.g.1849.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} +4.00000 q^{11} -54.0000i q^{13} -14.0000i q^{17} -92.0000 q^{19} -21.0000 q^{21} +152.000i q^{23} +27.0000i q^{27} +106.000 q^{29} -144.000 q^{31} -12.0000i q^{33} +158.000i q^{37} -162.000 q^{39} -390.000 q^{41} +508.000i q^{43} -528.000i q^{47} -49.0000 q^{49} -42.0000 q^{51} -606.000i q^{53} +276.000i q^{57} +364.000 q^{59} +678.000 q^{61} +63.0000i q^{63} +844.000i q^{67} +456.000 q^{69} -8.00000 q^{71} +422.000i q^{73} -28.0000i q^{77} -384.000 q^{79} +81.0000 q^{81} +548.000i q^{83} -318.000i q^{87} -1194.00 q^{89} -378.000 q^{91} +432.000i q^{93} -1502.00i q^{97} -36.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} + 8 q^{11} - 184 q^{19} - 42 q^{21} + 212 q^{29} - 288 q^{31} - 324 q^{39} - 780 q^{41} - 98 q^{49} - 84 q^{51} + 728 q^{59} + 1356 q^{61} + 912 q^{69} - 16 q^{71} - 768 q^{79} + 162 q^{81} - 2388 q^{89} - 756 q^{91} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) − 7.00000i − 0.377964i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 0.109640 0.0548202 0.998496i \(-0.482541\pi\)
0.0548202 + 0.998496i \(0.482541\pi\)
\(12\) 0 0
\(13\) − 54.0000i − 1.15207i −0.817425 0.576035i \(-0.804599\pi\)
0.817425 0.576035i \(-0.195401\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 14.0000i − 0.199735i −0.995001 0.0998676i \(-0.968158\pi\)
0.995001 0.0998676i \(-0.0318419\pi\)
\(18\) 0 0
\(19\) −92.0000 −1.11086 −0.555428 0.831565i \(-0.687445\pi\)
−0.555428 + 0.831565i \(0.687445\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 0 0
\(23\) 152.000i 1.37801i 0.724757 + 0.689004i \(0.241952\pi\)
−0.724757 + 0.689004i \(0.758048\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 106.000 0.678748 0.339374 0.940651i \(-0.389785\pi\)
0.339374 + 0.940651i \(0.389785\pi\)
\(30\) 0 0
\(31\) −144.000 −0.834296 −0.417148 0.908839i \(-0.636970\pi\)
−0.417148 + 0.908839i \(0.636970\pi\)
\(32\) 0 0
\(33\) − 12.0000i − 0.0633010i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 158.000i 0.702028i 0.936370 + 0.351014i \(0.114163\pi\)
−0.936370 + 0.351014i \(0.885837\pi\)
\(38\) 0 0
\(39\) −162.000 −0.665148
\(40\) 0 0
\(41\) −390.000 −1.48556 −0.742778 0.669538i \(-0.766492\pi\)
−0.742778 + 0.669538i \(0.766492\pi\)
\(42\) 0 0
\(43\) 508.000i 1.80161i 0.434223 + 0.900806i \(0.357023\pi\)
−0.434223 + 0.900806i \(0.642977\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 528.000i − 1.63865i −0.573327 0.819327i \(-0.694347\pi\)
0.573327 0.819327i \(-0.305653\pi\)
\(48\) 0 0
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −42.0000 −0.115317
\(52\) 0 0
\(53\) − 606.000i − 1.57058i −0.619131 0.785288i \(-0.712515\pi\)
0.619131 0.785288i \(-0.287485\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 276.000i 0.641353i
\(58\) 0 0
\(59\) 364.000 0.803199 0.401600 0.915815i \(-0.368454\pi\)
0.401600 + 0.915815i \(0.368454\pi\)
\(60\) 0 0
\(61\) 678.000 1.42310 0.711549 0.702636i \(-0.247994\pi\)
0.711549 + 0.702636i \(0.247994\pi\)
\(62\) 0 0
\(63\) 63.0000i 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 844.000i 1.53897i 0.638665 + 0.769485i \(0.279487\pi\)
−0.638665 + 0.769485i \(0.720513\pi\)
\(68\) 0 0
\(69\) 456.000 0.795593
\(70\) 0 0
\(71\) −8.00000 −0.0133722 −0.00668609 0.999978i \(-0.502128\pi\)
−0.00668609 + 0.999978i \(0.502128\pi\)
\(72\) 0 0
\(73\) 422.000i 0.676594i 0.941039 + 0.338297i \(0.109851\pi\)
−0.941039 + 0.338297i \(0.890149\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 28.0000i − 0.0414402i
\(78\) 0 0
\(79\) −384.000 −0.546878 −0.273439 0.961889i \(-0.588161\pi\)
−0.273439 + 0.961889i \(0.588161\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 548.000i 0.724709i 0.932040 + 0.362354i \(0.118027\pi\)
−0.932040 + 0.362354i \(0.881973\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 318.000i − 0.391876i
\(88\) 0 0
\(89\) −1194.00 −1.42206 −0.711032 0.703159i \(-0.751772\pi\)
−0.711032 + 0.703159i \(0.751772\pi\)
\(90\) 0 0
\(91\) −378.000 −0.435441
\(92\) 0 0
\(93\) 432.000i 0.481681i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1502.00i − 1.57222i −0.618089 0.786108i \(-0.712093\pi\)
0.618089 0.786108i \(-0.287907\pi\)
\(98\) 0 0
\(99\) −36.0000 −0.0365468
\(100\) 0 0
\(101\) 398.000 0.392104 0.196052 0.980594i \(-0.437188\pi\)
0.196052 + 0.980594i \(0.437188\pi\)
\(102\) 0 0
\(103\) − 1160.00i − 1.10969i −0.831953 0.554846i \(-0.812777\pi\)
0.831953 0.554846i \(-0.187223\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 324.000i 0.292731i 0.989231 + 0.146366i \(0.0467576\pi\)
−0.989231 + 0.146366i \(0.953242\pi\)
\(108\) 0 0
\(109\) 938.000 0.824258 0.412129 0.911126i \(-0.364785\pi\)
0.412129 + 0.911126i \(0.364785\pi\)
\(110\) 0 0
\(111\) 474.000 0.405316
\(112\) 0 0
\(113\) 622.000i 0.517813i 0.965902 + 0.258906i \(0.0833621\pi\)
−0.965902 + 0.258906i \(0.916638\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 486.000i 0.384023i
\(118\) 0 0
\(119\) −98.0000 −0.0754928
\(120\) 0 0
\(121\) −1315.00 −0.987979
\(122\) 0 0
\(123\) 1170.00i 0.857686i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1200.00i 0.838447i 0.907883 + 0.419224i \(0.137698\pi\)
−0.907883 + 0.419224i \(0.862302\pi\)
\(128\) 0 0
\(129\) 1524.00 1.04016
\(130\) 0 0
\(131\) −1396.00 −0.931062 −0.465531 0.885032i \(-0.654137\pi\)
−0.465531 + 0.885032i \(0.654137\pi\)
\(132\) 0 0
\(133\) 644.000i 0.419864i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2810.00i 1.75237i 0.481976 + 0.876184i \(0.339919\pi\)
−0.481976 + 0.876184i \(0.660081\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.00244083 −0.00122042 0.999999i \(-0.500388\pi\)
−0.00122042 + 0.999999i \(0.500388\pi\)
\(140\) 0 0
\(141\) −1584.00 −0.946077
\(142\) 0 0
\(143\) − 216.000i − 0.126313i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 147.000i 0.0824786i
\(148\) 0 0
\(149\) −1374.00 −0.755453 −0.377726 0.925917i \(-0.623294\pi\)
−0.377726 + 0.925917i \(0.623294\pi\)
\(150\) 0 0
\(151\) 2104.00 1.13391 0.566957 0.823747i \(-0.308120\pi\)
0.566957 + 0.823747i \(0.308120\pi\)
\(152\) 0 0
\(153\) 126.000i 0.0665784i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3206.00i 1.62972i 0.579655 + 0.814862i \(0.303187\pi\)
−0.579655 + 0.814862i \(0.696813\pi\)
\(158\) 0 0
\(159\) −1818.00 −0.906772
\(160\) 0 0
\(161\) 1064.00 0.520838
\(162\) 0 0
\(163\) − 332.000i − 0.159535i −0.996813 0.0797676i \(-0.974582\pi\)
0.996813 0.0797676i \(-0.0254178\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1496.00i 0.693197i 0.938014 + 0.346599i \(0.112663\pi\)
−0.938014 + 0.346599i \(0.887337\pi\)
\(168\) 0 0
\(169\) −719.000 −0.327264
\(170\) 0 0
\(171\) 828.000 0.370285
\(172\) 0 0
\(173\) 3322.00i 1.45992i 0.683487 + 0.729962i \(0.260462\pi\)
−0.683487 + 0.729962i \(0.739538\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1092.00i − 0.463727i
\(178\) 0 0
\(179\) 900.000 0.375805 0.187903 0.982188i \(-0.439831\pi\)
0.187903 + 0.982188i \(0.439831\pi\)
\(180\) 0 0
\(181\) 1902.00 0.781075 0.390537 0.920587i \(-0.372289\pi\)
0.390537 + 0.920587i \(0.372289\pi\)
\(182\) 0 0
\(183\) − 2034.00i − 0.821626i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 56.0000i − 0.0218991i
\(188\) 0 0
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 4128.00 1.56383 0.781915 0.623385i \(-0.214243\pi\)
0.781915 + 0.623385i \(0.214243\pi\)
\(192\) 0 0
\(193\) 1342.00i 0.500514i 0.968179 + 0.250257i \(0.0805152\pi\)
−0.968179 + 0.250257i \(0.919485\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3506.00i − 1.26798i −0.773341 0.633990i \(-0.781416\pi\)
0.773341 0.633990i \(-0.218584\pi\)
\(198\) 0 0
\(199\) −680.000 −0.242231 −0.121115 0.992638i \(-0.538647\pi\)
−0.121115 + 0.992638i \(0.538647\pi\)
\(200\) 0 0
\(201\) 2532.00 0.888525
\(202\) 0 0
\(203\) − 742.000i − 0.256543i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1368.00i − 0.459336i
\(208\) 0 0
\(209\) −368.000 −0.121795
\(210\) 0 0
\(211\) 5372.00 1.75272 0.876360 0.481657i \(-0.159965\pi\)
0.876360 + 0.481657i \(0.159965\pi\)
\(212\) 0 0
\(213\) 24.0000i 0.00772044i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1008.00i 0.315334i
\(218\) 0 0
\(219\) 1266.00 0.390632
\(220\) 0 0
\(221\) −756.000 −0.230109
\(222\) 0 0
\(223\) 1072.00i 0.321912i 0.986962 + 0.160956i \(0.0514578\pi\)
−0.986962 + 0.160956i \(0.948542\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2868.00i − 0.838572i −0.907854 0.419286i \(-0.862280\pi\)
0.907854 0.419286i \(-0.137720\pi\)
\(228\) 0 0
\(229\) −4798.00 −1.38454 −0.692272 0.721636i \(-0.743390\pi\)
−0.692272 + 0.721636i \(0.743390\pi\)
\(230\) 0 0
\(231\) −84.0000 −0.0239255
\(232\) 0 0
\(233\) 5126.00i 1.44127i 0.693316 + 0.720634i \(0.256149\pi\)
−0.693316 + 0.720634i \(0.743851\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1152.00i 0.315740i
\(238\) 0 0
\(239\) 528.000 0.142902 0.0714508 0.997444i \(-0.477237\pi\)
0.0714508 + 0.997444i \(0.477237\pi\)
\(240\) 0 0
\(241\) −814.000 −0.217570 −0.108785 0.994065i \(-0.534696\pi\)
−0.108785 + 0.994065i \(0.534696\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4968.00i 1.27978i
\(248\) 0 0
\(249\) 1644.00 0.418411
\(250\) 0 0
\(251\) −1932.00 −0.485844 −0.242922 0.970046i \(-0.578106\pi\)
−0.242922 + 0.970046i \(0.578106\pi\)
\(252\) 0 0
\(253\) 608.000i 0.151086i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 3294.00i − 0.799510i −0.916622 0.399755i \(-0.869095\pi\)
0.916622 0.399755i \(-0.130905\pi\)
\(258\) 0 0
\(259\) 1106.00 0.265342
\(260\) 0 0
\(261\) −954.000 −0.226249
\(262\) 0 0
\(263\) 7080.00i 1.65997i 0.557787 + 0.829984i \(0.311650\pi\)
−0.557787 + 0.829984i \(0.688350\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3582.00i 0.821029i
\(268\) 0 0
\(269\) −7814.00 −1.77111 −0.885554 0.464537i \(-0.846221\pi\)
−0.885554 + 0.464537i \(0.846221\pi\)
\(270\) 0 0
\(271\) 3168.00 0.710119 0.355060 0.934844i \(-0.384461\pi\)
0.355060 + 0.934844i \(0.384461\pi\)
\(272\) 0 0
\(273\) 1134.00i 0.251402i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 7858.00i − 1.70448i −0.523150 0.852241i \(-0.675243\pi\)
0.523150 0.852241i \(-0.324757\pi\)
\(278\) 0 0
\(279\) 1296.00 0.278099
\(280\) 0 0
\(281\) 6730.00 1.42875 0.714374 0.699764i \(-0.246712\pi\)
0.714374 + 0.699764i \(0.246712\pi\)
\(282\) 0 0
\(283\) 3020.00i 0.634348i 0.948367 + 0.317174i \(0.102734\pi\)
−0.948367 + 0.317174i \(0.897266\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2730.00i 0.561487i
\(288\) 0 0
\(289\) 4717.00 0.960106
\(290\) 0 0
\(291\) −4506.00 −0.907720
\(292\) 0 0
\(293\) 6834.00i 1.36262i 0.731997 + 0.681308i \(0.238589\pi\)
−0.731997 + 0.681308i \(0.761411\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 108.000i 0.0211003i
\(298\) 0 0
\(299\) 8208.00 1.58756
\(300\) 0 0
\(301\) 3556.00 0.680945
\(302\) 0 0
\(303\) − 1194.00i − 0.226381i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2332.00i 0.433532i 0.976224 + 0.216766i \(0.0695508\pi\)
−0.976224 + 0.216766i \(0.930449\pi\)
\(308\) 0 0
\(309\) −3480.00 −0.640681
\(310\) 0 0
\(311\) 8840.00 1.61180 0.805901 0.592050i \(-0.201681\pi\)
0.805901 + 0.592050i \(0.201681\pi\)
\(312\) 0 0
\(313\) 1046.00i 0.188893i 0.995530 + 0.0944464i \(0.0301081\pi\)
−0.995530 + 0.0944464i \(0.969892\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7542.00i 1.33628i 0.744035 + 0.668140i \(0.232909\pi\)
−0.744035 + 0.668140i \(0.767091\pi\)
\(318\) 0 0
\(319\) 424.000 0.0744183
\(320\) 0 0
\(321\) 972.000 0.169009
\(322\) 0 0
\(323\) 1288.00i 0.221877i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2814.00i − 0.475885i
\(328\) 0 0
\(329\) −3696.00 −0.619353
\(330\) 0 0
\(331\) 2756.00 0.457654 0.228827 0.973467i \(-0.426511\pi\)
0.228827 + 0.973467i \(0.426511\pi\)
\(332\) 0 0
\(333\) − 1422.00i − 0.234009i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3954.00i 0.639134i 0.947564 + 0.319567i \(0.103537\pi\)
−0.947564 + 0.319567i \(0.896463\pi\)
\(338\) 0 0
\(339\) 1866.00 0.298959
\(340\) 0 0
\(341\) −576.000 −0.0914726
\(342\) 0 0
\(343\) 343.000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6900.00i 1.06747i 0.845652 + 0.533734i \(0.179212\pi\)
−0.845652 + 0.533734i \(0.820788\pi\)
\(348\) 0 0
\(349\) 2426.00 0.372094 0.186047 0.982541i \(-0.440432\pi\)
0.186047 + 0.982541i \(0.440432\pi\)
\(350\) 0 0
\(351\) 1458.00 0.221716
\(352\) 0 0
\(353\) 1470.00i 0.221644i 0.993840 + 0.110822i \(0.0353483\pi\)
−0.993840 + 0.110822i \(0.964652\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 294.000i 0.0435858i
\(358\) 0 0
\(359\) −6872.00 −1.01028 −0.505140 0.863038i \(-0.668559\pi\)
−0.505140 + 0.863038i \(0.668559\pi\)
\(360\) 0 0
\(361\) 1605.00 0.233999
\(362\) 0 0
\(363\) 3945.00i 0.570410i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 7072.00i − 1.00587i −0.864323 0.502937i \(-0.832253\pi\)
0.864323 0.502937i \(-0.167747\pi\)
\(368\) 0 0
\(369\) 3510.00 0.495185
\(370\) 0 0
\(371\) −4242.00 −0.593622
\(372\) 0 0
\(373\) 818.000i 0.113551i 0.998387 + 0.0567754i \(0.0180819\pi\)
−0.998387 + 0.0567754i \(0.981918\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 5724.00i − 0.781966i
\(378\) 0 0
\(379\) 5132.00 0.695549 0.347775 0.937578i \(-0.386937\pi\)
0.347775 + 0.937578i \(0.386937\pi\)
\(380\) 0 0
\(381\) 3600.00 0.484078
\(382\) 0 0
\(383\) 8576.00i 1.14416i 0.820198 + 0.572080i \(0.193863\pi\)
−0.820198 + 0.572080i \(0.806137\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4572.00i − 0.600537i
\(388\) 0 0
\(389\) 3730.00 0.486166 0.243083 0.970006i \(-0.421841\pi\)
0.243083 + 0.970006i \(0.421841\pi\)
\(390\) 0 0
\(391\) 2128.00 0.275237
\(392\) 0 0
\(393\) 4188.00i 0.537549i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6678.00i 0.844230i 0.906542 + 0.422115i \(0.138712\pi\)
−0.906542 + 0.422115i \(0.861288\pi\)
\(398\) 0 0
\(399\) 1932.00 0.242408
\(400\) 0 0
\(401\) −3054.00 −0.380323 −0.190161 0.981753i \(-0.560901\pi\)
−0.190161 + 0.981753i \(0.560901\pi\)
\(402\) 0 0
\(403\) 7776.00i 0.961167i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 632.000i 0.0769707i
\(408\) 0 0
\(409\) −266.000 −0.0321586 −0.0160793 0.999871i \(-0.505118\pi\)
−0.0160793 + 0.999871i \(0.505118\pi\)
\(410\) 0 0
\(411\) 8430.00 1.01173
\(412\) 0 0
\(413\) − 2548.00i − 0.303581i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000i 0.00140921i
\(418\) 0 0
\(419\) −8844.00 −1.03116 −0.515582 0.856840i \(-0.672424\pi\)
−0.515582 + 0.856840i \(0.672424\pi\)
\(420\) 0 0
\(421\) −4482.00 −0.518858 −0.259429 0.965762i \(-0.583534\pi\)
−0.259429 + 0.965762i \(0.583534\pi\)
\(422\) 0 0
\(423\) 4752.00i 0.546218i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 4746.00i − 0.537881i
\(428\) 0 0
\(429\) −648.000 −0.0729271
\(430\) 0 0
\(431\) 9936.00 1.11044 0.555221 0.831703i \(-0.312634\pi\)
0.555221 + 0.831703i \(0.312634\pi\)
\(432\) 0 0
\(433\) 11758.0i 1.30497i 0.757800 + 0.652487i \(0.226274\pi\)
−0.757800 + 0.652487i \(0.773726\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 13984.0i − 1.53077i
\(438\) 0 0
\(439\) 4104.00 0.446180 0.223090 0.974798i \(-0.428386\pi\)
0.223090 + 0.974798i \(0.428386\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) − 9748.00i − 1.04547i −0.852496 0.522733i \(-0.824912\pi\)
0.852496 0.522733i \(-0.175088\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4122.00i 0.436161i
\(448\) 0 0
\(449\) 478.000 0.0502410 0.0251205 0.999684i \(-0.492003\pi\)
0.0251205 + 0.999684i \(0.492003\pi\)
\(450\) 0 0
\(451\) −1560.00 −0.162877
\(452\) 0 0
\(453\) − 6312.00i − 0.654666i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 11174.0i − 1.14376i −0.820338 0.571879i \(-0.806215\pi\)
0.820338 0.571879i \(-0.193785\pi\)
\(458\) 0 0
\(459\) 378.000 0.0384391
\(460\) 0 0
\(461\) −11674.0 −1.17942 −0.589710 0.807615i \(-0.700758\pi\)
−0.589710 + 0.807615i \(0.700758\pi\)
\(462\) 0 0
\(463\) − 10528.0i − 1.05676i −0.849009 0.528378i \(-0.822801\pi\)
0.849009 0.528378i \(-0.177199\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16604.0i 1.64527i 0.568569 + 0.822635i \(0.307497\pi\)
−0.568569 + 0.822635i \(0.692503\pi\)
\(468\) 0 0
\(469\) 5908.00 0.581676
\(470\) 0 0
\(471\) 9618.00 0.940922
\(472\) 0 0
\(473\) 2032.00i 0.197530i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 5454.00i 0.523525i
\(478\) 0 0
\(479\) 8576.00 0.818053 0.409027 0.912522i \(-0.365868\pi\)
0.409027 + 0.912522i \(0.365868\pi\)
\(480\) 0 0
\(481\) 8532.00 0.808785
\(482\) 0 0
\(483\) − 3192.00i − 0.300706i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 9704.00i 0.902937i 0.892287 + 0.451468i \(0.149100\pi\)
−0.892287 + 0.451468i \(0.850900\pi\)
\(488\) 0 0
\(489\) −996.000 −0.0921077
\(490\) 0 0
\(491\) −4092.00 −0.376109 −0.188054 0.982159i \(-0.560218\pi\)
−0.188054 + 0.982159i \(0.560218\pi\)
\(492\) 0 0
\(493\) − 1484.00i − 0.135570i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 56.0000i 0.00505421i
\(498\) 0 0
\(499\) −17884.0 −1.60440 −0.802202 0.597052i \(-0.796338\pi\)
−0.802202 + 0.597052i \(0.796338\pi\)
\(500\) 0 0
\(501\) 4488.00 0.400218
\(502\) 0 0
\(503\) 7704.00i 0.682911i 0.939898 + 0.341456i \(0.110920\pi\)
−0.939898 + 0.341456i \(0.889080\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2157.00i 0.188946i
\(508\) 0 0
\(509\) −14358.0 −1.25031 −0.625154 0.780501i \(-0.714964\pi\)
−0.625154 + 0.780501i \(0.714964\pi\)
\(510\) 0 0
\(511\) 2954.00 0.255729
\(512\) 0 0
\(513\) − 2484.00i − 0.213784i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 2112.00i − 0.179663i
\(518\) 0 0
\(519\) 9966.00 0.842888
\(520\) 0 0
\(521\) 5082.00 0.427344 0.213672 0.976905i \(-0.431458\pi\)
0.213672 + 0.976905i \(0.431458\pi\)
\(522\) 0 0
\(523\) 1756.00i 0.146816i 0.997302 + 0.0734078i \(0.0233875\pi\)
−0.997302 + 0.0734078i \(0.976613\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2016.00i 0.166638i
\(528\) 0 0
\(529\) −10937.0 −0.898907
\(530\) 0 0
\(531\) −3276.00 −0.267733
\(532\) 0 0
\(533\) 21060.0i 1.71146i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 2700.00i − 0.216971i
\(538\) 0 0
\(539\) −196.000 −0.0156629
\(540\) 0 0
\(541\) 16230.0 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) − 5706.00i − 0.450954i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 17676.0i 1.38167i 0.723014 + 0.690833i \(0.242756\pi\)
−0.723014 + 0.690833i \(0.757244\pi\)
\(548\) 0 0
\(549\) −6102.00 −0.474366
\(550\) 0 0
\(551\) −9752.00 −0.753991
\(552\) 0 0
\(553\) 2688.00i 0.206701i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 12250.0i − 0.931866i −0.884820 0.465933i \(-0.845719\pi\)
0.884820 0.465933i \(-0.154281\pi\)
\(558\) 0 0
\(559\) 27432.0 2.07558
\(560\) 0 0
\(561\) −168.000 −0.0126434
\(562\) 0 0
\(563\) 10052.0i 0.752471i 0.926524 + 0.376236i \(0.122782\pi\)
−0.926524 + 0.376236i \(0.877218\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 567.000i − 0.0419961i
\(568\) 0 0
\(569\) −25674.0 −1.89158 −0.945791 0.324776i \(-0.894711\pi\)
−0.945791 + 0.324776i \(0.894711\pi\)
\(570\) 0 0
\(571\) 3732.00 0.273519 0.136759 0.990604i \(-0.456331\pi\)
0.136759 + 0.990604i \(0.456331\pi\)
\(572\) 0 0
\(573\) − 12384.0i − 0.902878i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 1214.00i − 0.0875901i −0.999041 0.0437950i \(-0.986055\pi\)
0.999041 0.0437950i \(-0.0139449\pi\)
\(578\) 0 0
\(579\) 4026.00 0.288972
\(580\) 0 0
\(581\) 3836.00 0.273914
\(582\) 0 0
\(583\) − 2424.00i − 0.172199i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7108.00i 0.499793i 0.968273 + 0.249897i \(0.0803966\pi\)
−0.968273 + 0.249897i \(0.919603\pi\)
\(588\) 0 0
\(589\) 13248.0 0.926782
\(590\) 0 0
\(591\) −10518.0 −0.732069
\(592\) 0 0
\(593\) − 6162.00i − 0.426717i −0.976974 0.213358i \(-0.931560\pi\)
0.976974 0.213358i \(-0.0684402\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2040.00i 0.139852i
\(598\) 0 0
\(599\) −2472.00 −0.168620 −0.0843098 0.996440i \(-0.526869\pi\)
−0.0843098 + 0.996440i \(0.526869\pi\)
\(600\) 0 0
\(601\) −13750.0 −0.933235 −0.466617 0.884459i \(-0.654528\pi\)
−0.466617 + 0.884459i \(0.654528\pi\)
\(602\) 0 0
\(603\) − 7596.00i − 0.512990i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 11376.0i − 0.760688i −0.924845 0.380344i \(-0.875806\pi\)
0.924845 0.380344i \(-0.124194\pi\)
\(608\) 0 0
\(609\) −2226.00 −0.148115
\(610\) 0 0
\(611\) −28512.0 −1.88784
\(612\) 0 0
\(613\) − 20382.0i − 1.34294i −0.741032 0.671469i \(-0.765664\pi\)
0.741032 0.671469i \(-0.234336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21178.0i 1.38184i 0.722932 + 0.690919i \(0.242794\pi\)
−0.722932 + 0.690919i \(0.757206\pi\)
\(618\) 0 0
\(619\) 4700.00 0.305184 0.152592 0.988289i \(-0.451238\pi\)
0.152592 + 0.988289i \(0.451238\pi\)
\(620\) 0 0
\(621\) −4104.00 −0.265198
\(622\) 0 0
\(623\) 8358.00i 0.537490i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1104.00i 0.0703182i
\(628\) 0 0
\(629\) 2212.00 0.140220
\(630\) 0 0
\(631\) −21736.0 −1.37131 −0.685655 0.727927i \(-0.740484\pi\)
−0.685655 + 0.727927i \(0.740484\pi\)
\(632\) 0 0
\(633\) − 16116.0i − 1.01193i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2646.00i 0.164581i
\(638\) 0 0
\(639\) 72.0000 0.00445740
\(640\) 0 0
\(641\) −13022.0 −0.802399 −0.401200 0.915991i \(-0.631407\pi\)
−0.401200 + 0.915991i \(0.631407\pi\)
\(642\) 0 0
\(643\) − 3308.00i − 0.202885i −0.994841 0.101442i \(-0.967654\pi\)
0.994841 0.101442i \(-0.0323457\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 13800.0i − 0.838538i −0.907862 0.419269i \(-0.862286\pi\)
0.907862 0.419269i \(-0.137714\pi\)
\(648\) 0 0
\(649\) 1456.00 0.0880632
\(650\) 0 0
\(651\) 3024.00 0.182058
\(652\) 0 0
\(653\) 2682.00i 0.160727i 0.996766 + 0.0803635i \(0.0256081\pi\)
−0.996766 + 0.0803635i \(0.974392\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 3798.00i − 0.225531i
\(658\) 0 0
\(659\) −23836.0 −1.40898 −0.704491 0.709713i \(-0.748825\pi\)
−0.704491 + 0.709713i \(0.748825\pi\)
\(660\) 0 0
\(661\) −11282.0 −0.663871 −0.331936 0.943302i \(-0.607702\pi\)
−0.331936 + 0.943302i \(0.607702\pi\)
\(662\) 0 0
\(663\) 2268.00i 0.132853i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16112.0i 0.935321i
\(668\) 0 0
\(669\) 3216.00 0.185856
\(670\) 0 0
\(671\) 2712.00 0.156029
\(672\) 0 0
\(673\) 13726.0i 0.786179i 0.919500 + 0.393089i \(0.128594\pi\)
−0.919500 + 0.393089i \(0.871406\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4974.00i 0.282373i 0.989983 + 0.141186i \(0.0450917\pi\)
−0.989983 + 0.141186i \(0.954908\pi\)
\(678\) 0 0
\(679\) −10514.0 −0.594242
\(680\) 0 0
\(681\) −8604.00 −0.484150
\(682\) 0 0
\(683\) 8988.00i 0.503538i 0.967787 + 0.251769i \(0.0810123\pi\)
−0.967787 + 0.251769i \(0.918988\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14394.0i 0.799367i
\(688\) 0 0
\(689\) −32724.0 −1.80941
\(690\) 0 0
\(691\) 10172.0 0.560002 0.280001 0.960000i \(-0.409665\pi\)
0.280001 + 0.960000i \(0.409665\pi\)
\(692\) 0 0
\(693\) 252.000i 0.0138134i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5460.00i 0.296718i
\(698\) 0 0
\(699\) 15378.0 0.832116
\(700\) 0 0
\(701\) 27446.0 1.47877 0.739387 0.673280i \(-0.235115\pi\)
0.739387 + 0.673280i \(0.235115\pi\)
\(702\) 0 0
\(703\) − 14536.0i − 0.779852i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 2786.00i − 0.148201i
\(708\) 0 0
\(709\) −3998.00 −0.211774 −0.105887 0.994378i \(-0.533768\pi\)
−0.105887 + 0.994378i \(0.533768\pi\)
\(710\) 0 0
\(711\) 3456.00 0.182293
\(712\) 0 0
\(713\) − 21888.0i − 1.14967i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 1584.00i − 0.0825043i
\(718\) 0 0
\(719\) −25872.0 −1.34195 −0.670976 0.741480i \(-0.734124\pi\)
−0.670976 + 0.741480i \(0.734124\pi\)
\(720\) 0 0
\(721\) −8120.00 −0.419424
\(722\) 0 0
\(723\) 2442.00i 0.125614i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 12088.0i 0.616670i 0.951278 + 0.308335i \(0.0997718\pi\)
−0.951278 + 0.308335i \(0.900228\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 7112.00 0.359845
\(732\) 0 0
\(733\) − 7974.00i − 0.401810i −0.979611 0.200905i \(-0.935612\pi\)
0.979611 0.200905i \(-0.0643882\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3376.00i 0.168733i
\(738\) 0 0
\(739\) 31764.0 1.58113 0.790567 0.612376i \(-0.209786\pi\)
0.790567 + 0.612376i \(0.209786\pi\)
\(740\) 0 0
\(741\) 14904.0 0.738883
\(742\) 0 0
\(743\) − 888.000i − 0.0438460i −0.999760 0.0219230i \(-0.993021\pi\)
0.999760 0.0219230i \(-0.00697886\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 4932.00i − 0.241570i
\(748\) 0 0
\(749\) 2268.00 0.110642
\(750\) 0 0
\(751\) −34656.0 −1.68391 −0.841954 0.539549i \(-0.818595\pi\)
−0.841954 + 0.539549i \(0.818595\pi\)
\(752\) 0 0
\(753\) 5796.00i 0.280502i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 22866.0i − 1.09786i −0.835869 0.548929i \(-0.815036\pi\)
0.835869 0.548929i \(-0.184964\pi\)
\(758\) 0 0
\(759\) 1824.00 0.0872293
\(760\) 0 0
\(761\) 22570.0 1.07511 0.537557 0.843227i \(-0.319347\pi\)
0.537557 + 0.843227i \(0.319347\pi\)
\(762\) 0 0
\(763\) − 6566.00i − 0.311540i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 19656.0i − 0.925342i
\(768\) 0 0
\(769\) 1790.00 0.0839389 0.0419695 0.999119i \(-0.486637\pi\)
0.0419695 + 0.999119i \(0.486637\pi\)
\(770\) 0 0
\(771\) −9882.00 −0.461597
\(772\) 0 0
\(773\) − 2990.00i − 0.139124i −0.997578 0.0695620i \(-0.977840\pi\)
0.997578 0.0695620i \(-0.0221602\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 3318.00i − 0.153195i
\(778\) 0 0
\(779\) 35880.0 1.65024
\(780\) 0 0
\(781\) −32.0000 −0.00146613
\(782\) 0 0
\(783\) 2862.00i 0.130625i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 30756.0i − 1.39305i −0.717531 0.696527i \(-0.754728\pi\)
0.717531 0.696527i \(-0.245272\pi\)
\(788\) 0 0
\(789\) 21240.0 0.958383
\(790\) 0 0
\(791\) 4354.00 0.195715
\(792\) 0 0
\(793\) − 36612.0i − 1.63951i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15126.0i 0.672259i 0.941816 + 0.336129i \(0.109118\pi\)
−0.941816 + 0.336129i \(0.890882\pi\)
\(798\) 0 0
\(799\) −7392.00 −0.327297
\(800\) 0 0
\(801\) 10746.0 0.474022
\(802\) 0 0
\(803\) 1688.00i 0.0741821i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 23442.0i 1.02255i
\(808\) 0 0
\(809\) 6502.00 0.282569 0.141284 0.989969i \(-0.454877\pi\)
0.141284 + 0.989969i \(0.454877\pi\)
\(810\) 0 0
\(811\) −8252.00 −0.357296 −0.178648 0.983913i \(-0.557172\pi\)
−0.178648 + 0.983913i \(0.557172\pi\)
\(812\) 0 0
\(813\) − 9504.00i − 0.409987i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 46736.0i − 2.00133i
\(818\) 0 0
\(819\) 3402.00 0.145147
\(820\) 0 0
\(821\) −21986.0 −0.934612 −0.467306 0.884096i \(-0.654775\pi\)
−0.467306 + 0.884096i \(0.654775\pi\)
\(822\) 0 0
\(823\) − 3736.00i − 0.158237i −0.996865 0.0791183i \(-0.974790\pi\)
0.996865 0.0791183i \(-0.0252105\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 23820.0i − 1.00158i −0.865570 0.500788i \(-0.833044\pi\)
0.865570 0.500788i \(-0.166956\pi\)
\(828\) 0 0
\(829\) −7942.00 −0.332735 −0.166367 0.986064i \(-0.553204\pi\)
−0.166367 + 0.986064i \(0.553204\pi\)
\(830\) 0 0
\(831\) −23574.0 −0.984083
\(832\) 0 0
\(833\) 686.000i 0.0285336i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 3888.00i − 0.160560i
\(838\) 0 0
\(839\) −21016.0 −0.864783 −0.432391 0.901686i \(-0.642330\pi\)
−0.432391 + 0.901686i \(0.642330\pi\)
\(840\) 0 0
\(841\) −13153.0 −0.539301
\(842\) 0 0
\(843\) − 20190.0i − 0.824888i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 9205.00i 0.373421i
\(848\) 0 0
\(849\) 9060.00 0.366241
\(850\) 0 0
\(851\) −24016.0 −0.967401
\(852\) 0 0
\(853\) − 24878.0i − 0.998601i −0.866429 0.499300i \(-0.833590\pi\)
0.866429 0.499300i \(-0.166410\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 6390.00i − 0.254700i −0.991858 0.127350i \(-0.959353\pi\)
0.991858 0.127350i \(-0.0406472\pi\)
\(858\) 0 0
\(859\) 46444.0 1.84476 0.922380 0.386284i \(-0.126241\pi\)
0.922380 + 0.386284i \(0.126241\pi\)
\(860\) 0 0
\(861\) 8190.00 0.324175
\(862\) 0 0
\(863\) 25408.0i 1.00220i 0.865389 + 0.501100i \(0.167071\pi\)
−0.865389 + 0.501100i \(0.832929\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 14151.0i − 0.554317i
\(868\) 0 0
\(869\) −1536.00 −0.0599600
\(870\) 0 0
\(871\) 45576.0 1.77300
\(872\) 0 0
\(873\) 13518.0i 0.524072i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1078.00i 0.0415068i 0.999785 + 0.0207534i \(0.00660649\pi\)
−0.999785 + 0.0207534i \(0.993394\pi\)
\(878\) 0 0
\(879\) 20502.0 0.786707
\(880\) 0 0
\(881\) −45006.0 −1.72110 −0.860551 0.509364i \(-0.829881\pi\)
−0.860551 + 0.509364i \(0.829881\pi\)
\(882\) 0 0
\(883\) − 4028.00i − 0.153514i −0.997050 0.0767571i \(-0.975543\pi\)
0.997050 0.0767571i \(-0.0244566\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 29304.0i − 1.10928i −0.832090 0.554640i \(-0.812856\pi\)
0.832090 0.554640i \(-0.187144\pi\)
\(888\) 0 0
\(889\) 8400.00 0.316903
\(890\) 0 0
\(891\) 324.000 0.0121823
\(892\) 0 0
\(893\) 48576.0i 1.82031i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 24624.0i − 0.916579i
\(898\) 0 0
\(899\) −15264.0 −0.566277
\(900\) 0 0
\(901\) −8484.00 −0.313699
\(902\) 0 0
\(903\) − 10668.0i − 0.393144i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 50916.0i 1.86399i 0.362472 + 0.931995i \(0.381933\pi\)
−0.362472 + 0.931995i \(0.618067\pi\)
\(908\) 0 0
\(909\) −3582.00 −0.130701
\(910\) 0 0
\(911\) −24432.0 −0.888549 −0.444275 0.895891i \(-0.646539\pi\)
−0.444275 + 0.895891i \(0.646539\pi\)
\(912\) 0 0
\(913\) 2192.00i 0.0794574i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 9772.00i 0.351908i
\(918\) 0 0
\(919\) 20360.0 0.730810 0.365405 0.930849i \(-0.380930\pi\)
0.365405 + 0.930849i \(0.380930\pi\)
\(920\) 0 0
\(921\) 6996.00 0.250300
\(922\) 0 0
\(923\) 432.000i 0.0154057i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 10440.0i 0.369897i
\(928\) 0 0
\(929\) −23202.0 −0.819411 −0.409706 0.912218i \(-0.634369\pi\)
−0.409706 + 0.912218i \(0.634369\pi\)
\(930\) 0 0
\(931\) 4508.00 0.158694
\(932\) 0 0
\(933\) − 26520.0i − 0.930574i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 1990.00i − 0.0693815i −0.999398 0.0346908i \(-0.988955\pi\)
0.999398 0.0346908i \(-0.0110446\pi\)
\(938\) 0 0
\(939\) 3138.00 0.109057
\(940\) 0 0
\(941\) −51130.0 −1.77130 −0.885648 0.464356i \(-0.846286\pi\)
−0.885648 + 0.464356i \(0.846286\pi\)
\(942\) 0 0
\(943\) − 59280.0i − 2.04711i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 47044.0i − 1.61428i −0.590359 0.807141i \(-0.701014\pi\)
0.590359 0.807141i \(-0.298986\pi\)
\(948\) 0 0
\(949\) 22788.0 0.779483
\(950\) 0 0
\(951\) 22626.0 0.771502
\(952\) 0 0
\(953\) − 46858.0i − 1.59274i −0.604811 0.796369i \(-0.706751\pi\)
0.604811 0.796369i \(-0.293249\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 1272.00i − 0.0429654i
\(958\) 0 0
\(959\) 19670.0 0.662333
\(960\) 0 0
\(961\) −9055.00 −0.303951
\(962\) 0 0
\(963\) − 2916.00i − 0.0975771i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 30632.0i 1.01867i 0.860567 + 0.509337i \(0.170110\pi\)
−0.860567 + 0.509337i \(0.829890\pi\)
\(968\) 0 0
\(969\) 3864.00 0.128101
\(970\) 0 0
\(971\) −3804.00 −0.125722 −0.0628611 0.998022i \(-0.520022\pi\)
−0.0628611 + 0.998022i \(0.520022\pi\)
\(972\) 0 0
\(973\) 28.0000i 0 0.000922548i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 49326.0i − 1.61523i −0.589711 0.807614i \(-0.700758\pi\)
0.589711 0.807614i \(-0.299242\pi\)
\(978\) 0 0
\(979\) −4776.00 −0.155916
\(980\) 0 0
\(981\) −8442.00 −0.274753
\(982\) 0 0
\(983\) − 11112.0i − 0.360547i −0.983617 0.180274i \(-0.942302\pi\)
0.983617 0.180274i \(-0.0576983\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 11088.0i 0.357584i
\(988\) 0 0
\(989\) −77216.0 −2.48263
\(990\) 0 0
\(991\) 13616.0 0.436455 0.218227 0.975898i \(-0.429973\pi\)
0.218227 + 0.975898i \(0.429973\pi\)
\(992\) 0 0
\(993\) − 8268.00i − 0.264227i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 56674.0i − 1.80028i −0.435596 0.900142i \(-0.643462\pi\)
0.435596 0.900142i \(-0.356538\pi\)
\(998\) 0 0
\(999\) −4266.00 −0.135105
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 2100.4.k.g.1849.1 2
5.2 odd 4 2100.4.a.g.1.1 1
5.3 odd 4 84.4.a.b.1.1 1
5.4 even 2 inner 2100.4.k.g.1849.2 2
15.8 even 4 252.4.a.a.1.1 1
20.3 even 4 336.4.a.e.1.1 1
35.3 even 12 588.4.i.h.373.1 2
35.13 even 4 588.4.a.a.1.1 1
35.18 odd 12 588.4.i.a.373.1 2
35.23 odd 12 588.4.i.a.361.1 2
35.33 even 12 588.4.i.h.361.1 2
40.3 even 4 1344.4.a.p.1.1 1
40.13 odd 4 1344.4.a.b.1.1 1
60.23 odd 4 1008.4.a.d.1.1 1
105.23 even 12 1764.4.k.n.361.1 2
105.38 odd 12 1764.4.k.c.1549.1 2
105.53 even 12 1764.4.k.n.1549.1 2
105.68 odd 12 1764.4.k.c.361.1 2
105.83 odd 4 1764.4.a.l.1.1 1
140.83 odd 4 2352.4.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.a.b.1.1 1 5.3 odd 4
252.4.a.a.1.1 1 15.8 even 4
336.4.a.e.1.1 1 20.3 even 4
588.4.a.a.1.1 1 35.13 even 4
588.4.i.a.361.1 2 35.23 odd 12
588.4.i.a.373.1 2 35.18 odd 12
588.4.i.h.361.1 2 35.33 even 12
588.4.i.h.373.1 2 35.3 even 12
1008.4.a.d.1.1 1 60.23 odd 4
1344.4.a.b.1.1 1 40.13 odd 4
1344.4.a.p.1.1 1 40.3 even 4
1764.4.a.l.1.1 1 105.83 odd 4
1764.4.k.c.361.1 2 105.68 odd 12
1764.4.k.c.1549.1 2 105.38 odd 12
1764.4.k.n.361.1 2 105.23 even 12
1764.4.k.n.1549.1 2 105.53 even 12
2100.4.a.g.1.1 1 5.2 odd 4
2100.4.k.g.1849.1 2 1.1 even 1 trivial
2100.4.k.g.1849.2 2 5.4 even 2 inner
2352.4.a.v.1.1 1 140.83 odd 4