Properties

Label 1216.4.a.w.1.2
Level $1216$
Weight $4$
Character 1216.1
Self dual yes
Analytic conductor $71.746$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,4,Mod(1,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1216.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(71.7463225670\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7057.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 22x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.36181\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.33180 q^{3} -18.4427 q^{5} -10.3872 q^{7} -25.2263 q^{9} +O(q^{10})\) \(q+1.33180 q^{3} -18.4427 q^{5} -10.3872 q^{7} -25.2263 q^{9} +50.4519 q^{11} +61.8352 q^{13} -24.5619 q^{15} +68.1070 q^{17} -19.0000 q^{19} -13.8337 q^{21} -145.632 q^{23} +215.132 q^{25} -69.5550 q^{27} -42.6097 q^{29} -91.6582 q^{31} +67.1918 q^{33} +191.568 q^{35} +400.965 q^{37} +82.3522 q^{39} -123.355 q^{41} +449.802 q^{43} +465.240 q^{45} +453.075 q^{47} -235.105 q^{49} +90.7050 q^{51} -437.142 q^{53} -930.466 q^{55} -25.3042 q^{57} -159.352 q^{59} +476.816 q^{61} +262.031 q^{63} -1140.41 q^{65} -629.682 q^{67} -193.953 q^{69} -471.459 q^{71} -725.055 q^{73} +286.512 q^{75} -524.055 q^{77} +1057.66 q^{79} +588.477 q^{81} -726.957 q^{83} -1256.07 q^{85} -56.7476 q^{87} -468.065 q^{89} -642.297 q^{91} -122.071 q^{93} +350.410 q^{95} -891.891 q^{97} -1272.71 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{3} - 7 q^{5} - 7 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{3} - 7 q^{5} - 7 q^{7} + 21 q^{9} + 103 q^{11} - 32 q^{13} - 122 q^{15} + 11 q^{17} - 57 q^{19} - 114 q^{21} - 316 q^{23} + 162 q^{25} + 178 q^{27} + 138 q^{29} - 420 q^{31} - 330 q^{33} + 333 q^{35} - 102 q^{37} - 164 q^{39} - 370 q^{41} + 431 q^{43} + 429 q^{45} + 199 q^{47} - 802 q^{49} + 272 q^{51} + 308 q^{53} - 85 q^{55} - 76 q^{57} - 188 q^{59} + 609 q^{61} + 61 q^{63} - 1536 q^{65} - 246 q^{67} + 892 q^{69} + 954 q^{71} - 629 q^{73} - 1074 q^{75} + 71 q^{77} - 452 q^{79} - 361 q^{81} - 780 q^{83} - 1883 q^{85} + 96 q^{87} - 1356 q^{89} - 670 q^{91} - 1092 q^{93} + 133 q^{95} - 548 q^{97} - 1305 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.33180 0.256305 0.128153 0.991754i \(-0.459095\pi\)
0.128153 + 0.991754i \(0.459095\pi\)
\(4\) 0 0
\(5\) −18.4427 −1.64956 −0.824781 0.565453i \(-0.808701\pi\)
−0.824781 + 0.565453i \(0.808701\pi\)
\(6\) 0 0
\(7\) −10.3872 −0.560858 −0.280429 0.959875i \(-0.590477\pi\)
−0.280429 + 0.959875i \(0.590477\pi\)
\(8\) 0 0
\(9\) −25.2263 −0.934308
\(10\) 0 0
\(11\) 50.4519 1.38289 0.691446 0.722428i \(-0.256974\pi\)
0.691446 + 0.722428i \(0.256974\pi\)
\(12\) 0 0
\(13\) 61.8352 1.31923 0.659616 0.751603i \(-0.270719\pi\)
0.659616 + 0.751603i \(0.270719\pi\)
\(14\) 0 0
\(15\) −24.5619 −0.422791
\(16\) 0 0
\(17\) 68.1070 0.971669 0.485835 0.874051i \(-0.338516\pi\)
0.485835 + 0.874051i \(0.338516\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) −13.8337 −0.143751
\(22\) 0 0
\(23\) −145.632 −1.32028 −0.660138 0.751144i \(-0.729502\pi\)
−0.660138 + 0.751144i \(0.729502\pi\)
\(24\) 0 0
\(25\) 215.132 1.72105
\(26\) 0 0
\(27\) −69.5550 −0.495773
\(28\) 0 0
\(29\) −42.6097 −0.272842 −0.136421 0.990651i \(-0.543560\pi\)
−0.136421 + 0.990651i \(0.543560\pi\)
\(30\) 0 0
\(31\) −91.6582 −0.531042 −0.265521 0.964105i \(-0.585544\pi\)
−0.265521 + 0.964105i \(0.585544\pi\)
\(32\) 0 0
\(33\) 67.1918 0.354442
\(34\) 0 0
\(35\) 191.568 0.925169
\(36\) 0 0
\(37\) 400.965 1.78158 0.890788 0.454419i \(-0.150153\pi\)
0.890788 + 0.454419i \(0.150153\pi\)
\(38\) 0 0
\(39\) 82.3522 0.338126
\(40\) 0 0
\(41\) −123.355 −0.469873 −0.234936 0.972011i \(-0.575488\pi\)
−0.234936 + 0.972011i \(0.575488\pi\)
\(42\) 0 0
\(43\) 449.802 1.59521 0.797607 0.603178i \(-0.206099\pi\)
0.797607 + 0.603178i \(0.206099\pi\)
\(44\) 0 0
\(45\) 465.240 1.54120
\(46\) 0 0
\(47\) 453.075 1.40612 0.703062 0.711129i \(-0.251816\pi\)
0.703062 + 0.711129i \(0.251816\pi\)
\(48\) 0 0
\(49\) −235.105 −0.685439
\(50\) 0 0
\(51\) 90.7050 0.249044
\(52\) 0 0
\(53\) −437.142 −1.13294 −0.566472 0.824081i \(-0.691692\pi\)
−0.566472 + 0.824081i \(0.691692\pi\)
\(54\) 0 0
\(55\) −930.466 −2.28116
\(56\) 0 0
\(57\) −25.3042 −0.0588004
\(58\) 0 0
\(59\) −159.352 −0.351624 −0.175812 0.984424i \(-0.556255\pi\)
−0.175812 + 0.984424i \(0.556255\pi\)
\(60\) 0 0
\(61\) 476.816 1.00082 0.500410 0.865789i \(-0.333183\pi\)
0.500410 + 0.865789i \(0.333183\pi\)
\(62\) 0 0
\(63\) 262.031 0.524014
\(64\) 0 0
\(65\) −1140.41 −2.17615
\(66\) 0 0
\(67\) −629.682 −1.14818 −0.574089 0.818793i \(-0.694643\pi\)
−0.574089 + 0.818793i \(0.694643\pi\)
\(68\) 0 0
\(69\) −193.953 −0.338394
\(70\) 0 0
\(71\) −471.459 −0.788056 −0.394028 0.919099i \(-0.628919\pi\)
−0.394028 + 0.919099i \(0.628919\pi\)
\(72\) 0 0
\(73\) −725.055 −1.16248 −0.581241 0.813731i \(-0.697433\pi\)
−0.581241 + 0.813731i \(0.697433\pi\)
\(74\) 0 0
\(75\) 286.512 0.441115
\(76\) 0 0
\(77\) −524.055 −0.775605
\(78\) 0 0
\(79\) 1057.66 1.50628 0.753139 0.657861i \(-0.228539\pi\)
0.753139 + 0.657861i \(0.228539\pi\)
\(80\) 0 0
\(81\) 588.477 0.807238
\(82\) 0 0
\(83\) −726.957 −0.961373 −0.480686 0.876893i \(-0.659612\pi\)
−0.480686 + 0.876893i \(0.659612\pi\)
\(84\) 0 0
\(85\) −1256.07 −1.60283
\(86\) 0 0
\(87\) −56.7476 −0.0699309
\(88\) 0 0
\(89\) −468.065 −0.557469 −0.278735 0.960368i \(-0.589915\pi\)
−0.278735 + 0.960368i \(0.589915\pi\)
\(90\) 0 0
\(91\) −642.297 −0.739901
\(92\) 0 0
\(93\) −122.071 −0.136109
\(94\) 0 0
\(95\) 350.410 0.378435
\(96\) 0 0
\(97\) −891.891 −0.933585 −0.466793 0.884367i \(-0.654591\pi\)
−0.466793 + 0.884367i \(0.654591\pi\)
\(98\) 0 0
\(99\) −1272.71 −1.29205
\(100\) 0 0
\(101\) 16.6636 0.0164167 0.00820835 0.999966i \(-0.497387\pi\)
0.00820835 + 0.999966i \(0.497387\pi\)
\(102\) 0 0
\(103\) −1292.68 −1.23662 −0.618309 0.785935i \(-0.712182\pi\)
−0.618309 + 0.785935i \(0.712182\pi\)
\(104\) 0 0
\(105\) 255.130 0.237126
\(106\) 0 0
\(107\) −176.185 −0.159182 −0.0795911 0.996828i \(-0.525361\pi\)
−0.0795911 + 0.996828i \(0.525361\pi\)
\(108\) 0 0
\(109\) −1378.74 −1.21155 −0.605776 0.795635i \(-0.707137\pi\)
−0.605776 + 0.795635i \(0.707137\pi\)
\(110\) 0 0
\(111\) 534.006 0.456627
\(112\) 0 0
\(113\) −601.795 −0.500992 −0.250496 0.968118i \(-0.580594\pi\)
−0.250496 + 0.968118i \(0.580594\pi\)
\(114\) 0 0
\(115\) 2685.84 2.17788
\(116\) 0 0
\(117\) −1559.87 −1.23257
\(118\) 0 0
\(119\) −707.443 −0.544968
\(120\) 0 0
\(121\) 1214.39 0.912389
\(122\) 0 0
\(123\) −164.284 −0.120431
\(124\) 0 0
\(125\) −1662.26 −1.18942
\(126\) 0 0
\(127\) −1890.38 −1.32082 −0.660409 0.750906i \(-0.729617\pi\)
−0.660409 + 0.750906i \(0.729617\pi\)
\(128\) 0 0
\(129\) 599.047 0.408861
\(130\) 0 0
\(131\) 331.964 0.221403 0.110701 0.993854i \(-0.464690\pi\)
0.110701 + 0.993854i \(0.464690\pi\)
\(132\) 0 0
\(133\) 197.357 0.128670
\(134\) 0 0
\(135\) 1282.78 0.817808
\(136\) 0 0
\(137\) −1956.20 −1.21992 −0.609962 0.792430i \(-0.708815\pi\)
−0.609962 + 0.792430i \(0.708815\pi\)
\(138\) 0 0
\(139\) 547.821 0.334285 0.167142 0.985933i \(-0.446546\pi\)
0.167142 + 0.985933i \(0.446546\pi\)
\(140\) 0 0
\(141\) 603.406 0.360397
\(142\) 0 0
\(143\) 3119.70 1.82435
\(144\) 0 0
\(145\) 785.836 0.450070
\(146\) 0 0
\(147\) −313.114 −0.175682
\(148\) 0 0
\(149\) 992.480 0.545686 0.272843 0.962059i \(-0.412036\pi\)
0.272843 + 0.962059i \(0.412036\pi\)
\(150\) 0 0
\(151\) 148.167 0.0798520 0.0399260 0.999203i \(-0.487288\pi\)
0.0399260 + 0.999203i \(0.487288\pi\)
\(152\) 0 0
\(153\) −1718.09 −0.907838
\(154\) 0 0
\(155\) 1690.42 0.875987
\(156\) 0 0
\(157\) −2356.36 −1.19782 −0.598910 0.800817i \(-0.704399\pi\)
−0.598910 + 0.800817i \(0.704399\pi\)
\(158\) 0 0
\(159\) −582.186 −0.290380
\(160\) 0 0
\(161\) 1512.71 0.740487
\(162\) 0 0
\(163\) −2974.86 −1.42950 −0.714751 0.699379i \(-0.753460\pi\)
−0.714751 + 0.699379i \(0.753460\pi\)
\(164\) 0 0
\(165\) −1239.20 −0.584674
\(166\) 0 0
\(167\) 3133.09 1.45177 0.725885 0.687816i \(-0.241430\pi\)
0.725885 + 0.687816i \(0.241430\pi\)
\(168\) 0 0
\(169\) 1626.60 0.740372
\(170\) 0 0
\(171\) 479.300 0.214345
\(172\) 0 0
\(173\) −3415.40 −1.50097 −0.750485 0.660887i \(-0.770180\pi\)
−0.750485 + 0.660887i \(0.770180\pi\)
\(174\) 0 0
\(175\) −2234.62 −0.965265
\(176\) 0 0
\(177\) −212.225 −0.0901231
\(178\) 0 0
\(179\) 80.0505 0.0334260 0.0167130 0.999860i \(-0.494680\pi\)
0.0167130 + 0.999860i \(0.494680\pi\)
\(180\) 0 0
\(181\) −3379.91 −1.38799 −0.693996 0.719978i \(-0.744152\pi\)
−0.693996 + 0.719978i \(0.744152\pi\)
\(182\) 0 0
\(183\) 635.023 0.256515
\(184\) 0 0
\(185\) −7394.87 −2.93882
\(186\) 0 0
\(187\) 3436.13 1.34371
\(188\) 0 0
\(189\) 722.484 0.278058
\(190\) 0 0
\(191\) 3288.78 1.24590 0.622952 0.782260i \(-0.285933\pi\)
0.622952 + 0.782260i \(0.285933\pi\)
\(192\) 0 0
\(193\) 2906.89 1.08416 0.542079 0.840327i \(-0.317637\pi\)
0.542079 + 0.840327i \(0.317637\pi\)
\(194\) 0 0
\(195\) −1518.79 −0.557759
\(196\) 0 0
\(197\) 17.1879 0.00621617 0.00310808 0.999995i \(-0.499011\pi\)
0.00310808 + 0.999995i \(0.499011\pi\)
\(198\) 0 0
\(199\) 613.599 0.218577 0.109289 0.994010i \(-0.465143\pi\)
0.109289 + 0.994010i \(0.465143\pi\)
\(200\) 0 0
\(201\) −838.611 −0.294284
\(202\) 0 0
\(203\) 442.597 0.153026
\(204\) 0 0
\(205\) 2274.99 0.775084
\(206\) 0 0
\(207\) 3673.76 1.23354
\(208\) 0 0
\(209\) −958.585 −0.317257
\(210\) 0 0
\(211\) −3991.85 −1.30242 −0.651209 0.758898i \(-0.725738\pi\)
−0.651209 + 0.758898i \(0.725738\pi\)
\(212\) 0 0
\(213\) −627.890 −0.201983
\(214\) 0 0
\(215\) −8295.54 −2.63140
\(216\) 0 0
\(217\) 952.075 0.297839
\(218\) 0 0
\(219\) −965.628 −0.297950
\(220\) 0 0
\(221\) 4211.41 1.28186
\(222\) 0 0
\(223\) 1051.16 0.315653 0.157827 0.987467i \(-0.449551\pi\)
0.157827 + 0.987467i \(0.449551\pi\)
\(224\) 0 0
\(225\) −5426.97 −1.60799
\(226\) 0 0
\(227\) 3430.64 1.00308 0.501541 0.865134i \(-0.332767\pi\)
0.501541 + 0.865134i \(0.332767\pi\)
\(228\) 0 0
\(229\) 5907.75 1.70478 0.852392 0.522904i \(-0.175151\pi\)
0.852392 + 0.522904i \(0.175151\pi\)
\(230\) 0 0
\(231\) −697.937 −0.198792
\(232\) 0 0
\(233\) −2109.67 −0.593171 −0.296586 0.955006i \(-0.595848\pi\)
−0.296586 + 0.955006i \(0.595848\pi\)
\(234\) 0 0
\(235\) −8355.91 −2.31949
\(236\) 0 0
\(237\) 1408.59 0.386067
\(238\) 0 0
\(239\) −2898.33 −0.784426 −0.392213 0.919875i \(-0.628290\pi\)
−0.392213 + 0.919875i \(0.628290\pi\)
\(240\) 0 0
\(241\) −734.884 −0.196423 −0.0982117 0.995166i \(-0.531312\pi\)
−0.0982117 + 0.995166i \(0.531312\pi\)
\(242\) 0 0
\(243\) 2661.72 0.702672
\(244\) 0 0
\(245\) 4335.97 1.13067
\(246\) 0 0
\(247\) −1174.87 −0.302652
\(248\) 0 0
\(249\) −968.162 −0.246405
\(250\) 0 0
\(251\) 2758.13 0.693593 0.346796 0.937940i \(-0.387270\pi\)
0.346796 + 0.937940i \(0.387270\pi\)
\(252\) 0 0
\(253\) −7347.40 −1.82580
\(254\) 0 0
\(255\) −1672.84 −0.410813
\(256\) 0 0
\(257\) 486.309 0.118036 0.0590178 0.998257i \(-0.481203\pi\)
0.0590178 + 0.998257i \(0.481203\pi\)
\(258\) 0 0
\(259\) −4164.92 −0.999211
\(260\) 0 0
\(261\) 1074.89 0.254919
\(262\) 0 0
\(263\) −3381.74 −0.792878 −0.396439 0.918061i \(-0.629754\pi\)
−0.396439 + 0.918061i \(0.629754\pi\)
\(264\) 0 0
\(265\) 8062.06 1.86886
\(266\) 0 0
\(267\) −623.369 −0.142882
\(268\) 0 0
\(269\) 2628.18 0.595699 0.297849 0.954613i \(-0.403731\pi\)
0.297849 + 0.954613i \(0.403731\pi\)
\(270\) 0 0
\(271\) −1104.06 −0.247480 −0.123740 0.992315i \(-0.539489\pi\)
−0.123740 + 0.992315i \(0.539489\pi\)
\(272\) 0 0
\(273\) −855.411 −0.189640
\(274\) 0 0
\(275\) 10853.8 2.38003
\(276\) 0 0
\(277\) 549.699 0.119235 0.0596177 0.998221i \(-0.481012\pi\)
0.0596177 + 0.998221i \(0.481012\pi\)
\(278\) 0 0
\(279\) 2312.20 0.496157
\(280\) 0 0
\(281\) −182.930 −0.0388352 −0.0194176 0.999811i \(-0.506181\pi\)
−0.0194176 + 0.999811i \(0.506181\pi\)
\(282\) 0 0
\(283\) 5218.27 1.09609 0.548046 0.836448i \(-0.315372\pi\)
0.548046 + 0.836448i \(0.315372\pi\)
\(284\) 0 0
\(285\) 466.677 0.0969949
\(286\) 0 0
\(287\) 1281.31 0.263532
\(288\) 0 0
\(289\) −274.434 −0.0558587
\(290\) 0 0
\(291\) −1187.82 −0.239283
\(292\) 0 0
\(293\) −6160.17 −1.22826 −0.614131 0.789204i \(-0.710494\pi\)
−0.614131 + 0.789204i \(0.710494\pi\)
\(294\) 0 0
\(295\) 2938.87 0.580026
\(296\) 0 0
\(297\) −3509.18 −0.685600
\(298\) 0 0
\(299\) −9005.19 −1.74175
\(300\) 0 0
\(301\) −4672.20 −0.894687
\(302\) 0 0
\(303\) 22.1925 0.00420768
\(304\) 0 0
\(305\) −8793.75 −1.65091
\(306\) 0 0
\(307\) 6620.89 1.23086 0.615430 0.788192i \(-0.288982\pi\)
0.615430 + 0.788192i \(0.288982\pi\)
\(308\) 0 0
\(309\) −1721.59 −0.316952
\(310\) 0 0
\(311\) −8280.27 −1.50975 −0.754873 0.655870i \(-0.772302\pi\)
−0.754873 + 0.655870i \(0.772302\pi\)
\(312\) 0 0
\(313\) 3338.77 0.602934 0.301467 0.953477i \(-0.402524\pi\)
0.301467 + 0.953477i \(0.402524\pi\)
\(314\) 0 0
\(315\) −4832.55 −0.864392
\(316\) 0 0
\(317\) −9587.72 −1.69874 −0.849369 0.527800i \(-0.823017\pi\)
−0.849369 + 0.527800i \(0.823017\pi\)
\(318\) 0 0
\(319\) −2149.74 −0.377311
\(320\) 0 0
\(321\) −234.644 −0.0407992
\(322\) 0 0
\(323\) −1294.03 −0.222916
\(324\) 0 0
\(325\) 13302.7 2.27047
\(326\) 0 0
\(327\) −1836.20 −0.310527
\(328\) 0 0
\(329\) −4706.19 −0.788635
\(330\) 0 0
\(331\) 11195.2 1.85905 0.929526 0.368756i \(-0.120216\pi\)
0.929526 + 0.368756i \(0.120216\pi\)
\(332\) 0 0
\(333\) −10114.9 −1.66454
\(334\) 0 0
\(335\) 11613.0 1.89399
\(336\) 0 0
\(337\) 5991.21 0.968432 0.484216 0.874948i \(-0.339105\pi\)
0.484216 + 0.874948i \(0.339105\pi\)
\(338\) 0 0
\(339\) −801.470 −0.128407
\(340\) 0 0
\(341\) −4624.33 −0.734374
\(342\) 0 0
\(343\) 6004.91 0.945291
\(344\) 0 0
\(345\) 3577.00 0.558201
\(346\) 0 0
\(347\) 6755.35 1.04509 0.522545 0.852612i \(-0.324983\pi\)
0.522545 + 0.852612i \(0.324983\pi\)
\(348\) 0 0
\(349\) 2358.14 0.361685 0.180843 0.983512i \(-0.442117\pi\)
0.180843 + 0.983512i \(0.442117\pi\)
\(350\) 0 0
\(351\) −4300.95 −0.654039
\(352\) 0 0
\(353\) −5950.00 −0.897129 −0.448565 0.893750i \(-0.648065\pi\)
−0.448565 + 0.893750i \(0.648065\pi\)
\(354\) 0 0
\(355\) 8694.96 1.29995
\(356\) 0 0
\(357\) −942.173 −0.139678
\(358\) 0 0
\(359\) −1892.90 −0.278283 −0.139141 0.990273i \(-0.544434\pi\)
−0.139141 + 0.990273i \(0.544434\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 1617.32 0.233850
\(364\) 0 0
\(365\) 13371.9 1.91759
\(366\) 0 0
\(367\) −3643.63 −0.518245 −0.259122 0.965844i \(-0.583433\pi\)
−0.259122 + 0.965844i \(0.583433\pi\)
\(368\) 0 0
\(369\) 3111.79 0.439006
\(370\) 0 0
\(371\) 4540.69 0.635421
\(372\) 0 0
\(373\) −4820.77 −0.669195 −0.334598 0.942361i \(-0.608600\pi\)
−0.334598 + 0.942361i \(0.608600\pi\)
\(374\) 0 0
\(375\) −2213.81 −0.304854
\(376\) 0 0
\(377\) −2634.78 −0.359942
\(378\) 0 0
\(379\) −8138.02 −1.10296 −0.551480 0.834188i \(-0.685937\pi\)
−0.551480 + 0.834188i \(0.685937\pi\)
\(380\) 0 0
\(381\) −2517.61 −0.338532
\(382\) 0 0
\(383\) −904.971 −0.120736 −0.0603679 0.998176i \(-0.519227\pi\)
−0.0603679 + 0.998176i \(0.519227\pi\)
\(384\) 0 0
\(385\) 9664.96 1.27941
\(386\) 0 0
\(387\) −11346.8 −1.49042
\(388\) 0 0
\(389\) −8885.27 −1.15810 −0.579050 0.815292i \(-0.696576\pi\)
−0.579050 + 0.815292i \(0.696576\pi\)
\(390\) 0 0
\(391\) −9918.56 −1.28287
\(392\) 0 0
\(393\) 442.109 0.0567467
\(394\) 0 0
\(395\) −19506.1 −2.48470
\(396\) 0 0
\(397\) 981.841 0.124124 0.0620619 0.998072i \(-0.480232\pi\)
0.0620619 + 0.998072i \(0.480232\pi\)
\(398\) 0 0
\(399\) 262.841 0.0329787
\(400\) 0 0
\(401\) −15512.7 −1.93184 −0.965919 0.258845i \(-0.916658\pi\)
−0.965919 + 0.258845i \(0.916658\pi\)
\(402\) 0 0
\(403\) −5667.71 −0.700568
\(404\) 0 0
\(405\) −10853.1 −1.33159
\(406\) 0 0
\(407\) 20229.5 2.46373
\(408\) 0 0
\(409\) −13347.9 −1.61371 −0.806857 0.590747i \(-0.798833\pi\)
−0.806857 + 0.590747i \(0.798833\pi\)
\(410\) 0 0
\(411\) −2605.27 −0.312673
\(412\) 0 0
\(413\) 1655.22 0.197211
\(414\) 0 0
\(415\) 13407.0 1.58584
\(416\) 0 0
\(417\) 729.588 0.0856789
\(418\) 0 0
\(419\) 1557.12 0.181552 0.0907758 0.995871i \(-0.471065\pi\)
0.0907758 + 0.995871i \(0.471065\pi\)
\(420\) 0 0
\(421\) −1281.41 −0.148342 −0.0741710 0.997246i \(-0.523631\pi\)
−0.0741710 + 0.997246i \(0.523631\pi\)
\(422\) 0 0
\(423\) −11429.4 −1.31375
\(424\) 0 0
\(425\) 14652.0 1.67229
\(426\) 0 0
\(427\) −4952.79 −0.561317
\(428\) 0 0
\(429\) 4154.82 0.467591
\(430\) 0 0
\(431\) −9742.61 −1.08883 −0.544414 0.838817i \(-0.683248\pi\)
−0.544414 + 0.838817i \(0.683248\pi\)
\(432\) 0 0
\(433\) −5262.21 −0.584031 −0.292016 0.956414i \(-0.594326\pi\)
−0.292016 + 0.956414i \(0.594326\pi\)
\(434\) 0 0
\(435\) 1046.58 0.115355
\(436\) 0 0
\(437\) 2767.01 0.302892
\(438\) 0 0
\(439\) 15015.7 1.63249 0.816243 0.577708i \(-0.196053\pi\)
0.816243 + 0.577708i \(0.196053\pi\)
\(440\) 0 0
\(441\) 5930.84 0.640411
\(442\) 0 0
\(443\) −4296.96 −0.460846 −0.230423 0.973091i \(-0.574011\pi\)
−0.230423 + 0.973091i \(0.574011\pi\)
\(444\) 0 0
\(445\) 8632.36 0.919580
\(446\) 0 0
\(447\) 1321.79 0.139862
\(448\) 0 0
\(449\) 6749.23 0.709389 0.354695 0.934982i \(-0.384585\pi\)
0.354695 + 0.934982i \(0.384585\pi\)
\(450\) 0 0
\(451\) −6223.48 −0.649783
\(452\) 0 0
\(453\) 197.329 0.0204665
\(454\) 0 0
\(455\) 11845.7 1.22051
\(456\) 0 0
\(457\) 11399.9 1.16689 0.583443 0.812154i \(-0.301705\pi\)
0.583443 + 0.812154i \(0.301705\pi\)
\(458\) 0 0
\(459\) −4737.19 −0.481727
\(460\) 0 0
\(461\) 5799.27 0.585898 0.292949 0.956128i \(-0.405363\pi\)
0.292949 + 0.956128i \(0.405363\pi\)
\(462\) 0 0
\(463\) −620.160 −0.0622490 −0.0311245 0.999516i \(-0.509909\pi\)
−0.0311245 + 0.999516i \(0.509909\pi\)
\(464\) 0 0
\(465\) 2251.30 0.224520
\(466\) 0 0
\(467\) 9426.16 0.934027 0.467014 0.884250i \(-0.345330\pi\)
0.467014 + 0.884250i \(0.345330\pi\)
\(468\) 0 0
\(469\) 6540.65 0.643964
\(470\) 0 0
\(471\) −3138.20 −0.307007
\(472\) 0 0
\(473\) 22693.3 2.20601
\(474\) 0 0
\(475\) −4087.50 −0.394836
\(476\) 0 0
\(477\) 11027.5 1.05852
\(478\) 0 0
\(479\) −15813.6 −1.50844 −0.754218 0.656625i \(-0.771984\pi\)
−0.754218 + 0.656625i \(0.771984\pi\)
\(480\) 0 0
\(481\) 24793.8 2.35031
\(482\) 0 0
\(483\) 2014.63 0.189791
\(484\) 0 0
\(485\) 16448.8 1.54001
\(486\) 0 0
\(487\) 6832.36 0.635736 0.317868 0.948135i \(-0.397033\pi\)
0.317868 + 0.948135i \(0.397033\pi\)
\(488\) 0 0
\(489\) −3961.92 −0.366389
\(490\) 0 0
\(491\) −15084.5 −1.38646 −0.693232 0.720714i \(-0.743814\pi\)
−0.693232 + 0.720714i \(0.743814\pi\)
\(492\) 0 0
\(493\) −2902.02 −0.265112
\(494\) 0 0
\(495\) 23472.2 2.13131
\(496\) 0 0
\(497\) 4897.16 0.441987
\(498\) 0 0
\(499\) 21771.2 1.95313 0.976564 0.215226i \(-0.0690490\pi\)
0.976564 + 0.215226i \(0.0690490\pi\)
\(500\) 0 0
\(501\) 4172.65 0.372096
\(502\) 0 0
\(503\) −21300.9 −1.88819 −0.944097 0.329668i \(-0.893063\pi\)
−0.944097 + 0.329668i \(0.893063\pi\)
\(504\) 0 0
\(505\) −307.320 −0.0270803
\(506\) 0 0
\(507\) 2166.30 0.189761
\(508\) 0 0
\(509\) 15897.9 1.38440 0.692202 0.721704i \(-0.256641\pi\)
0.692202 + 0.721704i \(0.256641\pi\)
\(510\) 0 0
\(511\) 7531.31 0.651987
\(512\) 0 0
\(513\) 1321.55 0.113738
\(514\) 0 0
\(515\) 23840.5 2.03988
\(516\) 0 0
\(517\) 22858.5 1.94452
\(518\) 0 0
\(519\) −4548.63 −0.384707
\(520\) 0 0
\(521\) −13864.6 −1.16587 −0.582937 0.812517i \(-0.698097\pi\)
−0.582937 + 0.812517i \(0.698097\pi\)
\(522\) 0 0
\(523\) −20503.0 −1.71421 −0.857106 0.515141i \(-0.827740\pi\)
−0.857106 + 0.515141i \(0.827740\pi\)
\(524\) 0 0
\(525\) −2976.07 −0.247402
\(526\) 0 0
\(527\) −6242.57 −0.515997
\(528\) 0 0
\(529\) 9041.66 0.743130
\(530\) 0 0
\(531\) 4019.86 0.328525
\(532\) 0 0
\(533\) −7627.67 −0.619871
\(534\) 0 0
\(535\) 3249.33 0.262581
\(536\) 0 0
\(537\) 106.611 0.00856726
\(538\) 0 0
\(539\) −11861.5 −0.947887
\(540\) 0 0
\(541\) 15933.8 1.26626 0.633131 0.774045i \(-0.281769\pi\)
0.633131 + 0.774045i \(0.281769\pi\)
\(542\) 0 0
\(543\) −4501.37 −0.355750
\(544\) 0 0
\(545\) 25427.6 1.99853
\(546\) 0 0
\(547\) 20542.0 1.60569 0.802847 0.596185i \(-0.203318\pi\)
0.802847 + 0.596185i \(0.203318\pi\)
\(548\) 0 0
\(549\) −12028.3 −0.935073
\(550\) 0 0
\(551\) 809.584 0.0625943
\(552\) 0 0
\(553\) −10986.2 −0.844808
\(554\) 0 0
\(555\) −9848.49 −0.753235
\(556\) 0 0
\(557\) −12338.5 −0.938599 −0.469299 0.883039i \(-0.655493\pi\)
−0.469299 + 0.883039i \(0.655493\pi\)
\(558\) 0 0
\(559\) 27813.6 2.10446
\(560\) 0 0
\(561\) 4576.23 0.344401
\(562\) 0 0
\(563\) 7611.22 0.569759 0.284880 0.958563i \(-0.408046\pi\)
0.284880 + 0.958563i \(0.408046\pi\)
\(564\) 0 0
\(565\) 11098.7 0.826417
\(566\) 0 0
\(567\) −6112.64 −0.452746
\(568\) 0 0
\(569\) 9237.35 0.680579 0.340290 0.940321i \(-0.389475\pi\)
0.340290 + 0.940321i \(0.389475\pi\)
\(570\) 0 0
\(571\) 7061.48 0.517537 0.258769 0.965939i \(-0.416683\pi\)
0.258769 + 0.965939i \(0.416683\pi\)
\(572\) 0 0
\(573\) 4380.00 0.319332
\(574\) 0 0
\(575\) −31330.0 −2.27226
\(576\) 0 0
\(577\) −12167.1 −0.877856 −0.438928 0.898522i \(-0.644642\pi\)
−0.438928 + 0.898522i \(0.644642\pi\)
\(578\) 0 0
\(579\) 3871.40 0.277876
\(580\) 0 0
\(581\) 7551.07 0.539193
\(582\) 0 0
\(583\) −22054.6 −1.56674
\(584\) 0 0
\(585\) 28768.2 2.03320
\(586\) 0 0
\(587\) −11396.4 −0.801327 −0.400663 0.916225i \(-0.631220\pi\)
−0.400663 + 0.916225i \(0.631220\pi\)
\(588\) 0 0
\(589\) 1741.51 0.121829
\(590\) 0 0
\(591\) 22.8908 0.00159324
\(592\) 0 0
\(593\) −10401.1 −0.720276 −0.360138 0.932899i \(-0.617270\pi\)
−0.360138 + 0.932899i \(0.617270\pi\)
\(594\) 0 0
\(595\) 13047.1 0.898958
\(596\) 0 0
\(597\) 817.192 0.0560225
\(598\) 0 0
\(599\) −2296.79 −0.156668 −0.0783340 0.996927i \(-0.524960\pi\)
−0.0783340 + 0.996927i \(0.524960\pi\)
\(600\) 0 0
\(601\) −7376.50 −0.500655 −0.250328 0.968161i \(-0.580538\pi\)
−0.250328 + 0.968161i \(0.580538\pi\)
\(602\) 0 0
\(603\) 15884.5 1.07275
\(604\) 0 0
\(605\) −22396.6 −1.50504
\(606\) 0 0
\(607\) 11793.0 0.788570 0.394285 0.918988i \(-0.370992\pi\)
0.394285 + 0.918988i \(0.370992\pi\)
\(608\) 0 0
\(609\) 589.451 0.0392213
\(610\) 0 0
\(611\) 28016.0 1.85500
\(612\) 0 0
\(613\) −20918.1 −1.37826 −0.689131 0.724637i \(-0.742008\pi\)
−0.689131 + 0.724637i \(0.742008\pi\)
\(614\) 0 0
\(615\) 3029.83 0.198658
\(616\) 0 0
\(617\) −3951.13 −0.257806 −0.128903 0.991657i \(-0.541146\pi\)
−0.128903 + 0.991657i \(0.541146\pi\)
\(618\) 0 0
\(619\) −22159.8 −1.43890 −0.719450 0.694544i \(-0.755606\pi\)
−0.719450 + 0.694544i \(0.755606\pi\)
\(620\) 0 0
\(621\) 10129.4 0.654557
\(622\) 0 0
\(623\) 4861.90 0.312661
\(624\) 0 0
\(625\) 3765.14 0.240969
\(626\) 0 0
\(627\) −1276.64 −0.0813146
\(628\) 0 0
\(629\) 27308.6 1.73110
\(630\) 0 0
\(631\) −11645.2 −0.734687 −0.367344 0.930085i \(-0.619733\pi\)
−0.367344 + 0.930085i \(0.619733\pi\)
\(632\) 0 0
\(633\) −5316.35 −0.333817
\(634\) 0 0
\(635\) 34863.6 2.17877
\(636\) 0 0
\(637\) −14537.8 −0.904252
\(638\) 0 0
\(639\) 11893.2 0.736286
\(640\) 0 0
\(641\) −21970.1 −1.35377 −0.676885 0.736089i \(-0.736671\pi\)
−0.676885 + 0.736089i \(0.736671\pi\)
\(642\) 0 0
\(643\) −14047.2 −0.861537 −0.430769 0.902462i \(-0.641757\pi\)
−0.430769 + 0.902462i \(0.641757\pi\)
\(644\) 0 0
\(645\) −11048.0 −0.674442
\(646\) 0 0
\(647\) −20700.0 −1.25780 −0.628902 0.777485i \(-0.716495\pi\)
−0.628902 + 0.777485i \(0.716495\pi\)
\(648\) 0 0
\(649\) −8039.59 −0.486258
\(650\) 0 0
\(651\) 1267.97 0.0763377
\(652\) 0 0
\(653\) −17197.8 −1.03063 −0.515315 0.857001i \(-0.672325\pi\)
−0.515315 + 0.857001i \(0.672325\pi\)
\(654\) 0 0
\(655\) −6122.29 −0.365218
\(656\) 0 0
\(657\) 18290.5 1.08612
\(658\) 0 0
\(659\) −23084.0 −1.36453 −0.682264 0.731106i \(-0.739005\pi\)
−0.682264 + 0.731106i \(0.739005\pi\)
\(660\) 0 0
\(661\) 9283.22 0.546256 0.273128 0.961978i \(-0.411942\pi\)
0.273128 + 0.961978i \(0.411942\pi\)
\(662\) 0 0
\(663\) 5608.76 0.328547
\(664\) 0 0
\(665\) −3639.79 −0.212248
\(666\) 0 0
\(667\) 6205.33 0.360227
\(668\) 0 0
\(669\) 1399.93 0.0809036
\(670\) 0 0
\(671\) 24056.2 1.38402
\(672\) 0 0
\(673\) 11565.0 0.662403 0.331202 0.943560i \(-0.392546\pi\)
0.331202 + 0.943560i \(0.392546\pi\)
\(674\) 0 0
\(675\) −14963.5 −0.853251
\(676\) 0 0
\(677\) −8808.39 −0.500050 −0.250025 0.968239i \(-0.580439\pi\)
−0.250025 + 0.968239i \(0.580439\pi\)
\(678\) 0 0
\(679\) 9264.27 0.523608
\(680\) 0 0
\(681\) 4568.93 0.257095
\(682\) 0 0
\(683\) 16282.3 0.912186 0.456093 0.889932i \(-0.349248\pi\)
0.456093 + 0.889932i \(0.349248\pi\)
\(684\) 0 0
\(685\) 36077.6 2.01234
\(686\) 0 0
\(687\) 7867.95 0.436945
\(688\) 0 0
\(689\) −27030.8 −1.49462
\(690\) 0 0
\(691\) −21900.4 −1.20569 −0.602845 0.797859i \(-0.705966\pi\)
−0.602845 + 0.797859i \(0.705966\pi\)
\(692\) 0 0
\(693\) 13220.0 0.724654
\(694\) 0 0
\(695\) −10103.3 −0.551423
\(696\) 0 0
\(697\) −8401.33 −0.456561
\(698\) 0 0
\(699\) −2809.66 −0.152033
\(700\) 0 0
\(701\) 12286.6 0.661994 0.330997 0.943632i \(-0.392615\pi\)
0.330997 + 0.943632i \(0.392615\pi\)
\(702\) 0 0
\(703\) −7618.34 −0.408722
\(704\) 0 0
\(705\) −11128.4 −0.594496
\(706\) 0 0
\(707\) −173.088 −0.00920743
\(708\) 0 0
\(709\) 12918.9 0.684315 0.342158 0.939643i \(-0.388842\pi\)
0.342158 + 0.939643i \(0.388842\pi\)
\(710\) 0 0
\(711\) −26680.8 −1.40733
\(712\) 0 0
\(713\) 13348.4 0.701122
\(714\) 0 0
\(715\) −57535.6 −3.00938
\(716\) 0 0
\(717\) −3860.00 −0.201052
\(718\) 0 0
\(719\) 15819.9 0.820558 0.410279 0.911960i \(-0.365431\pi\)
0.410279 + 0.911960i \(0.365431\pi\)
\(720\) 0 0
\(721\) 13427.4 0.693567
\(722\) 0 0
\(723\) −978.719 −0.0503443
\(724\) 0 0
\(725\) −9166.69 −0.469576
\(726\) 0 0
\(727\) −1793.81 −0.0915112 −0.0457556 0.998953i \(-0.514570\pi\)
−0.0457556 + 0.998953i \(0.514570\pi\)
\(728\) 0 0
\(729\) −12344.0 −0.627140
\(730\) 0 0
\(731\) 30634.7 1.55002
\(732\) 0 0
\(733\) −10370.0 −0.522541 −0.261271 0.965266i \(-0.584142\pi\)
−0.261271 + 0.965266i \(0.584142\pi\)
\(734\) 0 0
\(735\) 5774.65 0.289797
\(736\) 0 0
\(737\) −31768.6 −1.58780
\(738\) 0 0
\(739\) 29414.1 1.46416 0.732081 0.681218i \(-0.238549\pi\)
0.732081 + 0.681218i \(0.238549\pi\)
\(740\) 0 0
\(741\) −1564.69 −0.0775714
\(742\) 0 0
\(743\) 14932.0 0.737286 0.368643 0.929571i \(-0.379823\pi\)
0.368643 + 0.929571i \(0.379823\pi\)
\(744\) 0 0
\(745\) −18304.0 −0.900142
\(746\) 0 0
\(747\) 18338.4 0.898218
\(748\) 0 0
\(749\) 1830.08 0.0892785
\(750\) 0 0
\(751\) −21916.9 −1.06492 −0.532462 0.846454i \(-0.678733\pi\)
−0.532462 + 0.846454i \(0.678733\pi\)
\(752\) 0 0
\(753\) 3673.28 0.177771
\(754\) 0 0
\(755\) −2732.59 −0.131721
\(756\) 0 0
\(757\) 26973.4 1.29506 0.647532 0.762038i \(-0.275801\pi\)
0.647532 + 0.762038i \(0.275801\pi\)
\(758\) 0 0
\(759\) −9785.27 −0.467962
\(760\) 0 0
\(761\) −16145.3 −0.769076 −0.384538 0.923109i \(-0.625639\pi\)
−0.384538 + 0.923109i \(0.625639\pi\)
\(762\) 0 0
\(763\) 14321.3 0.679508
\(764\) 0 0
\(765\) 31686.1 1.49753
\(766\) 0 0
\(767\) −9853.55 −0.463874
\(768\) 0 0
\(769\) −23777.1 −1.11499 −0.557494 0.830181i \(-0.688237\pi\)
−0.557494 + 0.830181i \(0.688237\pi\)
\(770\) 0 0
\(771\) 647.667 0.0302531
\(772\) 0 0
\(773\) −27962.2 −1.30107 −0.650537 0.759475i \(-0.725456\pi\)
−0.650537 + 0.759475i \(0.725456\pi\)
\(774\) 0 0
\(775\) −19718.6 −0.913951
\(776\) 0 0
\(777\) −5546.84 −0.256103
\(778\) 0 0
\(779\) 2343.74 0.107796
\(780\) 0 0
\(781\) −23786.0 −1.08980
\(782\) 0 0
\(783\) 2963.72 0.135268
\(784\) 0 0
\(785\) 43457.5 1.97588
\(786\) 0 0
\(787\) 19000.4 0.860600 0.430300 0.902686i \(-0.358408\pi\)
0.430300 + 0.902686i \(0.358408\pi\)
\(788\) 0 0
\(789\) −4503.80 −0.203219
\(790\) 0 0
\(791\) 6250.98 0.280985
\(792\) 0 0
\(793\) 29484.0 1.32031
\(794\) 0 0
\(795\) 10737.1 0.478999
\(796\) 0 0
\(797\) 19991.0 0.888480 0.444240 0.895908i \(-0.353474\pi\)
0.444240 + 0.895908i \(0.353474\pi\)
\(798\) 0 0
\(799\) 30857.6 1.36629
\(800\) 0 0
\(801\) 11807.5 0.520848
\(802\) 0 0
\(803\) −36580.3 −1.60759
\(804\) 0 0
\(805\) −27898.4 −1.22148
\(806\) 0 0
\(807\) 3500.21 0.152681
\(808\) 0 0
\(809\) −40779.8 −1.77224 −0.886120 0.463455i \(-0.846609\pi\)
−0.886120 + 0.463455i \(0.846609\pi\)
\(810\) 0 0
\(811\) −23160.4 −1.00280 −0.501400 0.865216i \(-0.667181\pi\)
−0.501400 + 0.865216i \(0.667181\pi\)
\(812\) 0 0
\(813\) −1470.39 −0.0634304
\(814\) 0 0
\(815\) 54864.3 2.35805
\(816\) 0 0
\(817\) −8546.24 −0.365967
\(818\) 0 0
\(819\) 16202.8 0.691295
\(820\) 0 0
\(821\) −29453.3 −1.25204 −0.626021 0.779806i \(-0.715318\pi\)
−0.626021 + 0.779806i \(0.715318\pi\)
\(822\) 0 0
\(823\) 43227.2 1.83087 0.915435 0.402465i \(-0.131847\pi\)
0.915435 + 0.402465i \(0.131847\pi\)
\(824\) 0 0
\(825\) 14455.1 0.610014
\(826\) 0 0
\(827\) −34232.7 −1.43940 −0.719702 0.694283i \(-0.755722\pi\)
−0.719702 + 0.694283i \(0.755722\pi\)
\(828\) 0 0
\(829\) −32410.2 −1.35785 −0.678923 0.734209i \(-0.737553\pi\)
−0.678923 + 0.734209i \(0.737553\pi\)
\(830\) 0 0
\(831\) 732.089 0.0305606
\(832\) 0 0
\(833\) −16012.3 −0.666020
\(834\) 0 0
\(835\) −57782.5 −2.39478
\(836\) 0 0
\(837\) 6375.29 0.263276
\(838\) 0 0
\(839\) 6363.50 0.261851 0.130925 0.991392i \(-0.458205\pi\)
0.130925 + 0.991392i \(0.458205\pi\)
\(840\) 0 0
\(841\) −22573.4 −0.925557
\(842\) 0 0
\(843\) −243.626 −0.00995366
\(844\) 0 0
\(845\) −29998.8 −1.22129
\(846\) 0 0
\(847\) −12614.1 −0.511720
\(848\) 0 0
\(849\) 6949.69 0.280934
\(850\) 0 0
\(851\) −58393.4 −2.35217
\(852\) 0 0
\(853\) −22798.2 −0.915117 −0.457558 0.889180i \(-0.651276\pi\)
−0.457558 + 0.889180i \(0.651276\pi\)
\(854\) 0 0
\(855\) −8839.56 −0.353575
\(856\) 0 0
\(857\) 26426.8 1.05335 0.526675 0.850067i \(-0.323438\pi\)
0.526675 + 0.850067i \(0.323438\pi\)
\(858\) 0 0
\(859\) 22484.5 0.893087 0.446544 0.894762i \(-0.352655\pi\)
0.446544 + 0.894762i \(0.352655\pi\)
\(860\) 0 0
\(861\) 1706.46 0.0675445
\(862\) 0 0
\(863\) 22062.6 0.870242 0.435121 0.900372i \(-0.356706\pi\)
0.435121 + 0.900372i \(0.356706\pi\)
\(864\) 0 0
\(865\) 62989.0 2.47594
\(866\) 0 0
\(867\) −365.491 −0.0143169
\(868\) 0 0
\(869\) 53360.9 2.08302
\(870\) 0 0
\(871\) −38936.5 −1.51471
\(872\) 0 0
\(873\) 22499.1 0.872256
\(874\) 0 0
\(875\) 17266.3 0.667095
\(876\) 0 0
\(877\) −17.2794 −0.000665316 0 −0.000332658 1.00000i \(-0.500106\pi\)
−0.000332658 1.00000i \(0.500106\pi\)
\(878\) 0 0
\(879\) −8204.12 −0.314810
\(880\) 0 0
\(881\) 19338.2 0.739524 0.369762 0.929126i \(-0.379439\pi\)
0.369762 + 0.929126i \(0.379439\pi\)
\(882\) 0 0
\(883\) 35899.9 1.36821 0.684105 0.729384i \(-0.260193\pi\)
0.684105 + 0.729384i \(0.260193\pi\)
\(884\) 0 0
\(885\) 3913.99 0.148664
\(886\) 0 0
\(887\) −51076.9 −1.93348 −0.966738 0.255767i \(-0.917672\pi\)
−0.966738 + 0.255767i \(0.917672\pi\)
\(888\) 0 0
\(889\) 19635.8 0.740791
\(890\) 0 0
\(891\) 29689.7 1.11632
\(892\) 0 0
\(893\) −8608.43 −0.322587
\(894\) 0 0
\(895\) −1476.34 −0.0551383
\(896\) 0 0
\(897\) −11993.1 −0.446420
\(898\) 0 0
\(899\) 3905.53 0.144891
\(900\) 0 0
\(901\) −29772.4 −1.10085
\(902\) 0 0
\(903\) −6222.43 −0.229313
\(904\) 0 0
\(905\) 62334.5 2.28958
\(906\) 0 0
\(907\) 18999.5 0.695555 0.347778 0.937577i \(-0.386936\pi\)
0.347778 + 0.937577i \(0.386936\pi\)
\(908\) 0 0
\(909\) −420.360 −0.0153382
\(910\) 0 0
\(911\) −17147.0 −0.623605 −0.311802 0.950147i \(-0.600933\pi\)
−0.311802 + 0.950147i \(0.600933\pi\)
\(912\) 0 0
\(913\) −36676.3 −1.32947
\(914\) 0 0
\(915\) −11711.5 −0.423137
\(916\) 0 0
\(917\) −3448.18 −0.124176
\(918\) 0 0
\(919\) 10013.4 0.359425 0.179712 0.983719i \(-0.442483\pi\)
0.179712 + 0.983719i \(0.442483\pi\)
\(920\) 0 0
\(921\) 8817.70 0.315476
\(922\) 0 0
\(923\) −29152.8 −1.03963
\(924\) 0 0
\(925\) 86260.3 3.06619
\(926\) 0 0
\(927\) 32609.6 1.15538
\(928\) 0 0
\(929\) −22893.7 −0.808524 −0.404262 0.914643i \(-0.632472\pi\)
−0.404262 + 0.914643i \(0.632472\pi\)
\(930\) 0 0
\(931\) 4467.00 0.157250
\(932\) 0 0
\(933\) −11027.7 −0.386956
\(934\) 0 0
\(935\) −63371.3 −2.21654
\(936\) 0 0
\(937\) 1003.65 0.0349922 0.0174961 0.999847i \(-0.494431\pi\)
0.0174961 + 0.999847i \(0.494431\pi\)
\(938\) 0 0
\(939\) 4446.58 0.154535
\(940\) 0 0
\(941\) −14679.7 −0.508550 −0.254275 0.967132i \(-0.581837\pi\)
−0.254275 + 0.967132i \(0.581837\pi\)
\(942\) 0 0
\(943\) 17964.4 0.620362
\(944\) 0 0
\(945\) −13324.5 −0.458674
\(946\) 0 0
\(947\) 38570.8 1.32353 0.661765 0.749711i \(-0.269808\pi\)
0.661765 + 0.749711i \(0.269808\pi\)
\(948\) 0 0
\(949\) −44833.9 −1.53358
\(950\) 0 0
\(951\) −12768.9 −0.435395
\(952\) 0 0
\(953\) 2208.75 0.0750770 0.0375385 0.999295i \(-0.488048\pi\)
0.0375385 + 0.999295i \(0.488048\pi\)
\(954\) 0 0
\(955\) −60653.8 −2.05520
\(956\) 0 0
\(957\) −2863.02 −0.0967068
\(958\) 0 0
\(959\) 20319.5 0.684204
\(960\) 0 0
\(961\) −21389.8 −0.717994
\(962\) 0 0
\(963\) 4444.51 0.148725
\(964\) 0 0
\(965\) −53610.8 −1.78839
\(966\) 0 0
\(967\) −5440.71 −0.180932 −0.0904661 0.995900i \(-0.528836\pi\)
−0.0904661 + 0.995900i \(0.528836\pi\)
\(968\) 0 0
\(969\) −1723.39 −0.0571346
\(970\) 0 0
\(971\) 25018.4 0.826857 0.413428 0.910537i \(-0.364331\pi\)
0.413428 + 0.910537i \(0.364331\pi\)
\(972\) 0 0
\(973\) −5690.34 −0.187486
\(974\) 0 0
\(975\) 17716.6 0.581932
\(976\) 0 0
\(977\) −45055.8 −1.47540 −0.737698 0.675131i \(-0.764087\pi\)
−0.737698 + 0.675131i \(0.764087\pi\)
\(978\) 0 0
\(979\) −23614.7 −0.770920
\(980\) 0 0
\(981\) 34780.5 1.13196
\(982\) 0 0
\(983\) 13675.5 0.443725 0.221862 0.975078i \(-0.428786\pi\)
0.221862 + 0.975078i \(0.428786\pi\)
\(984\) 0 0
\(985\) −316.990 −0.0102539
\(986\) 0 0
\(987\) −6267.71 −0.202131
\(988\) 0 0
\(989\) −65505.5 −2.10612
\(990\) 0 0
\(991\) −23018.0 −0.737832 −0.368916 0.929463i \(-0.620271\pi\)
−0.368916 + 0.929463i \(0.620271\pi\)
\(992\) 0 0
\(993\) 14909.8 0.476485
\(994\) 0 0
\(995\) −11316.4 −0.360557
\(996\) 0 0
\(997\) 16189.1 0.514256 0.257128 0.966377i \(-0.417224\pi\)
0.257128 + 0.966377i \(0.417224\pi\)
\(998\) 0 0
\(999\) −27889.2 −0.883258
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 1216.4.a.w.1.2 3
4.3 odd 2 1216.4.a.r.1.2 3
8.3 odd 2 152.4.a.c.1.2 3
8.5 even 2 304.4.a.g.1.2 3
24.11 even 2 1368.4.a.d.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.4.a.c.1.2 3 8.3 odd 2
304.4.a.g.1.2 3 8.5 even 2
1216.4.a.r.1.2 3 4.3 odd 2
1216.4.a.w.1.2 3 1.1 even 1 trivial
1368.4.a.d.1.1 3 24.11 even 2