Properties

Label 152.4.a.c.1.2
Level $152$
Weight $4$
Character 152.1
Self dual yes
Analytic conductor $8.968$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [152,4,Mod(1,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, 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("152.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 152.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: \(8.96829032087\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.36181\) of defining polynomial
Character \(\chi\) \(=\) 152.1

$q$-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 \(-0.191299\pi\)
0.824781 + 0.565453i \(0.191299\pi\)
\(6\) 0 0
\(7\) 10.3872 0.560858 0.280429 0.959875i \(-0.409523\pi\)
0.280429 + 0.959875i \(0.409523\pi\)
\(8\) 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 \(-0.729281\pi\)
−0.659616 + 0.751603i \(0.729281\pi\)
\(14\) 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 \(-0.270498\pi\)
0.660138 + 0.751144i \(0.270498\pi\)
\(24\) 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 \(-0.456440\pi\)
0.136421 + 0.990651i \(0.456440\pi\)
\(30\) 0 0
\(31\) 91.6582 0.531042 0.265521 0.964105i \(-0.414456\pi\)
0.265521 + 0.964105i \(0.414456\pi\)
\(32\) 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 \(-0.849847\pi\)
−0.890788 + 0.454419i \(0.849847\pi\)
\(38\) 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 \(-0.748184\pi\)
−0.703062 + 0.711129i \(0.748184\pi\)
\(48\) 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 \(-0.308308\pi\)
0.566472 + 0.824081i \(0.308308\pi\)
\(54\) 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 \(-0.666817\pi\)
−0.500410 + 0.865789i \(0.666817\pi\)
\(62\) 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 \(-0.371081\pi\)
0.394028 + 0.919099i \(0.371081\pi\)
\(72\) 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 \(-0.771461\pi\)
−0.753139 + 0.657861i \(0.771461\pi\)
\(80\) 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 \(-0.502613\pi\)
−0.00820835 + 0.999966i \(0.502613\pi\)
\(102\) 0 0
\(103\) 1292.68 1.23662 0.618309 0.785935i \(-0.287818\pi\)
0.618309 + 0.785935i \(0.287818\pi\)
\(104\) 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 \(-0.292863\pi\)
0.605776 + 0.795635i \(0.292863\pi\)
\(110\) 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 \(-0.270383\pi\)
0.660409 + 0.750906i \(0.270383\pi\)
\(128\) 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 \(-0.587964\pi\)
−0.272843 + 0.962059i \(0.587964\pi\)
\(150\) 0 0
\(151\) −148.167 −0.0798520 −0.0399260 0.999203i \(-0.512712\pi\)
−0.0399260 + 0.999203i \(0.512712\pi\)
\(152\) 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 \(-0.295601\pi\)
0.598910 + 0.800817i \(0.295601\pi\)
\(158\) 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 \(-0.758570\pi\)
−0.725885 + 0.687816i \(0.758570\pi\)
\(168\) 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 \(-0.229820\pi\)
0.750485 + 0.660887i \(0.229820\pi\)
\(174\) 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 \(-0.255848\pi\)
0.693996 + 0.719978i \(0.255848\pi\)
\(182\) 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 \(-0.714067\pi\)
−0.622952 + 0.782260i \(0.714067\pi\)
\(192\) 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 \(-0.500989\pi\)
−0.00310808 + 0.999995i \(0.500989\pi\)
\(198\) 0 0
\(199\) −613.599 −0.218577 −0.109289 0.994010i \(-0.534857\pi\)
−0.109289 + 0.994010i \(0.534857\pi\)
\(200\) 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 \(-0.550449\pi\)
−0.157827 + 0.987467i \(0.550449\pi\)
\(224\) 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 \(-0.824849\pi\)
−0.852392 + 0.522904i \(0.824849\pi\)
\(230\) 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 \(-0.371710\pi\)
0.392213 + 0.919875i \(0.371710\pi\)
\(240\) 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 \(-0.370246\pi\)
0.396439 + 0.918061i \(0.370246\pi\)
\(264\) 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 \(-0.596269\pi\)
−0.297849 + 0.954613i \(0.596269\pi\)
\(270\) 0 0
\(271\) 1104.06 0.247480 0.123740 0.992315i \(-0.460511\pi\)
0.123740 + 0.992315i \(0.460511\pi\)
\(272\) 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 \(-0.518988\pi\)
−0.0596177 + 0.998221i \(0.518988\pi\)
\(278\) 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 \(-0.289506\pi\)
0.614131 + 0.789204i \(0.289506\pi\)
\(294\) 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 \(-0.227698\pi\)
0.754873 + 0.655870i \(0.227698\pi\)
\(312\) 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 \(-0.176983\pi\)
0.849369 + 0.527800i \(0.176983\pi\)
\(318\) 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 \(-0.557883\pi\)
−0.180843 + 0.983512i \(0.557883\pi\)
\(350\) 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 \(-0.455566\pi\)
0.139141 + 0.990273i \(0.455566\pi\)
\(360\) 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 \(-0.416567\pi\)
0.259122 + 0.965844i \(0.416567\pi\)
\(368\) 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 \(-0.391400\pi\)
0.334598 + 0.942361i \(0.391400\pi\)
\(374\) 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 \(-0.480773\pi\)
0.0603679 + 0.998176i \(0.480773\pi\)
\(384\) 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 \(-0.303424\pi\)
0.579050 + 0.815292i \(0.303424\pi\)
\(390\) 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 \(-0.519768\pi\)
−0.0620619 + 0.998072i \(0.519768\pi\)
\(398\) 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 \(-0.476369\pi\)
0.0741710 + 0.997246i \(0.476369\pi\)
\(422\) 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 \(-0.316752\pi\)
0.544414 + 0.838817i \(0.316752\pi\)
\(432\) 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 \(-0.803947\pi\)
−0.816243 + 0.577708i \(0.803947\pi\)
\(440\) 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 \(-0.594637\pi\)
−0.292949 + 0.956128i \(0.594637\pi\)
\(462\) 0 0
\(463\) 620.160 0.0622490 0.0311245 0.999516i \(-0.490091\pi\)
0.0311245 + 0.999516i \(0.490091\pi\)
\(464\) 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 \(-0.228016\pi\)
0.754218 + 0.656625i \(0.228016\pi\)
\(480\) 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 \(-0.602967\pi\)
−0.317868 + 0.948135i \(0.602967\pi\)
\(488\) 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 \(-0.106937\pi\)
0.944097 + 0.329668i \(0.106937\pi\)
\(504\) 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 \(-0.743359\pi\)
−0.692202 + 0.721704i \(0.743359\pi\)
\(510\) 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 \(-0.718231\pi\)
−0.633131 + 0.774045i \(0.718231\pi\)
\(542\) 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 \(-0.344507\pi\)
0.469299 + 0.883039i \(0.344507\pi\)
\(558\) 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 \(-0.475040\pi\)
0.0783340 + 0.996927i \(0.475040\pi\)
\(600\) 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 \(-0.629008\pi\)
−0.394285 + 0.918988i \(0.629008\pi\)
\(608\) 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 \(-0.257992\pi\)
0.689131 + 0.724637i \(0.257992\pi\)
\(614\) 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 \(-0.380267\pi\)
0.367344 + 0.930085i \(0.380267\pi\)
\(632\) 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 \(-0.283505\pi\)
0.628902 + 0.777485i \(0.283505\pi\)
\(648\) 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 \(-0.327675\pi\)
0.515315 + 0.857001i \(0.327675\pi\)
\(654\) 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 \(-0.588058\pi\)
−0.273128 + 0.961978i \(0.588058\pi\)
\(662\) 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 \(-0.419561\pi\)
0.250025 + 0.968239i \(0.419561\pi\)
\(678\) 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 \(-0.607385\pi\)
−0.330997 + 0.943632i \(0.607385\pi\)
\(702\) 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 \(-0.611158\pi\)
−0.342158 + 0.939643i \(0.611158\pi\)
\(710\) 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 \(-0.634569\pi\)
−0.410279 + 0.911960i \(0.634569\pi\)
\(720\) 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 \(-0.485430\pi\)
0.0457556 + 0.998953i \(0.485430\pi\)
\(728\) 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 \(-0.415858\pi\)
0.261271 + 0.965266i \(0.415858\pi\)
\(734\) 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 \(-0.620177\pi\)
−0.368643 + 0.929571i \(0.620177\pi\)
\(744\) 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 \(-0.321267\pi\)
0.532462 + 0.846454i \(0.321267\pi\)
\(752\) 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 \(-0.724199\pi\)
−0.647532 + 0.762038i \(0.724199\pi\)
\(758\) 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 \(-0.274544\pi\)
0.650537 + 0.759475i \(0.274544\pi\)
\(774\) 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 \(-0.646526\pi\)
−0.444240 + 0.895908i \(0.646526\pi\)
\(798\) 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 \(-0.284682\pi\)
0.626021 + 0.779806i \(0.284682\pi\)
\(822\) 0 0
\(823\) −43227.2 −1.83087 −0.915435 0.402465i \(-0.868153\pi\)
−0.915435 + 0.402465i \(0.868153\pi\)
\(824\) 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 \(-0.262447\pi\)
0.678923 + 0.734209i \(0.262447\pi\)
\(830\) 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 \(-0.541795\pi\)
−0.130925 + 0.991392i \(0.541795\pi\)
\(840\) 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 \(-0.348724\pi\)
0.457558 + 0.889180i \(0.348724\pi\)
\(854\) 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 \(-0.643294\pi\)
−0.435121 + 0.900372i \(0.643294\pi\)
\(864\) 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 \(-0.499894\pi\)
0.000332658 1.00000i \(0.499894\pi\)
\(878\) 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 \(-0.0823280\pi\)
0.966738 + 0.255767i \(0.0823280\pi\)
\(888\) 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 \(-0.399067\pi\)
0.311802 + 0.950147i \(0.399067\pi\)
\(912\) 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 \(-0.557517\pi\)
−0.179712 + 0.983719i \(0.557517\pi\)
\(920\) 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 \(-0.418163\pi\)
0.254275 + 0.967132i \(0.418163\pi\)
\(942\) 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 \(-0.471164\pi\)
0.0904661 + 0.995900i \(0.471164\pi\)
\(968\) 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 \(-0.571214\pi\)
−0.221862 + 0.975078i \(0.571214\pi\)
\(984\) 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 \(-0.379729\pi\)
0.368916 + 0.929463i \(0.379729\pi\)
\(992\) 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 \(-0.582776\pi\)
−0.257128 + 0.966377i \(0.582776\pi\)
\(998\) 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 152.4.a.c.1.2 3
3.2 odd 2 1368.4.a.d.1.1 3
4.3 odd 2 304.4.a.g.1.2 3
8.3 odd 2 1216.4.a.w.1.2 3
8.5 even 2 1216.4.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.4.a.c.1.2 3 1.1 even 1 trivial
304.4.a.g.1.2 3 4.3 odd 2
1216.4.a.r.1.2 3 8.5 even 2
1216.4.a.w.1.2 3 8.3 odd 2
1368.4.a.d.1.1 3 3.2 odd 2