Properties

Label 1088.4.a.w.1.3
Level $1088$
Weight $4$
Character 1088.1
Self dual yes
Analytic conductor $64.194$
Analytic rank $0$
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] = [1088,4,Mod(1,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, 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("1088.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1088.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: \(64.1940780862\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 68)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.27307\) of defining polynomial
Character \(\chi\) \(=\) 1088.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.87987 q^{3} -19.0923 q^{5} +34.1567 q^{7} +20.3326 q^{9} +O(q^{10})\) \(q+6.87987 q^{3} -19.0923 q^{5} +34.1567 q^{7} +20.3326 q^{9} +35.3047 q^{11} +16.3715 q^{13} -131.352 q^{15} +17.0000 q^{17} -91.4591 q^{19} +234.994 q^{21} -69.1353 q^{23} +239.516 q^{25} -45.8711 q^{27} +72.8010 q^{29} +314.737 q^{31} +242.892 q^{33} -652.130 q^{35} +103.729 q^{37} +112.634 q^{39} -34.8358 q^{41} +316.936 q^{43} -388.195 q^{45} +190.937 q^{47} +823.683 q^{49} +116.958 q^{51} -476.856 q^{53} -674.048 q^{55} -629.227 q^{57} -351.708 q^{59} +34.1756 q^{61} +694.494 q^{63} -312.569 q^{65} -114.361 q^{67} -475.642 q^{69} -611.520 q^{71} -11.4137 q^{73} +1647.84 q^{75} +1205.89 q^{77} +266.137 q^{79} -864.566 q^{81} +386.656 q^{83} -324.569 q^{85} +500.861 q^{87} +1654.30 q^{89} +559.197 q^{91} +2165.35 q^{93} +1746.16 q^{95} +1406.50 q^{97} +717.835 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{3} - 26 q^{5} - 8 q^{7} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{3} - 26 q^{5} - 8 q^{7} + 63 q^{9} + 60 q^{11} - 82 q^{13} - 40 q^{15} + 51 q^{17} - 124 q^{19} + 144 q^{21} + 120 q^{23} + 85 q^{25} - 296 q^{27} + 46 q^{29} + 272 q^{31} - 48 q^{33} - 640 q^{35} - 186 q^{37} + 728 q^{39} - 82 q^{41} + 300 q^{43} - 770 q^{45} + 224 q^{47} + 1275 q^{49} + 68 q^{51} - 202 q^{53} - 984 q^{55} + 624 q^{57} - 612 q^{59} - 786 q^{61} + 232 q^{63} + 444 q^{65} + 916 q^{67} + 240 q^{69} + 1272 q^{71} - 306 q^{73} + 1308 q^{75} + 1104 q^{77} - 1568 q^{79} - 1797 q^{81} + 948 q^{83} - 442 q^{85} + 2264 q^{87} + 478 q^{89} + 1856 q^{91} + 4688 q^{93} + 2920 q^{95} + 1318 q^{97} + 1980 q^{99}+O(q^{100}) \) 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
\(3\) 6.87987 1.32403 0.662015 0.749490i \(-0.269701\pi\)
0.662015 + 0.749490i \(0.269701\pi\)
\(4\) 0 0
\(5\) −19.0923 −1.70767 −0.853833 0.520547i \(-0.825728\pi\)
−0.853833 + 0.520547i \(0.825728\pi\)
\(6\) 0 0
\(7\) 34.1567 1.84429 0.922145 0.386844i \(-0.126435\pi\)
0.922145 + 0.386844i \(0.126435\pi\)
\(8\) 0 0
\(9\) 20.3326 0.753058
\(10\) 0 0
\(11\) 35.3047 0.967707 0.483853 0.875149i \(-0.339237\pi\)
0.483853 + 0.875149i \(0.339237\pi\)
\(12\) 0 0
\(13\) 16.3715 0.349280 0.174640 0.984632i \(-0.444124\pi\)
0.174640 + 0.984632i \(0.444124\pi\)
\(14\) 0 0
\(15\) −131.352 −2.26100
\(16\) 0 0
\(17\) 17.0000 0.242536
\(18\) 0 0
\(19\) −91.4591 −1.10432 −0.552162 0.833737i \(-0.686197\pi\)
−0.552162 + 0.833737i \(0.686197\pi\)
\(20\) 0 0
\(21\) 234.994 2.44190
\(22\) 0 0
\(23\) −69.1353 −0.626770 −0.313385 0.949626i \(-0.601463\pi\)
−0.313385 + 0.949626i \(0.601463\pi\)
\(24\) 0 0
\(25\) 239.516 1.91612
\(26\) 0 0
\(27\) −45.8711 −0.326959
\(28\) 0 0
\(29\) 72.8010 0.466166 0.233083 0.972457i \(-0.425119\pi\)
0.233083 + 0.972457i \(0.425119\pi\)
\(30\) 0 0
\(31\) 314.737 1.82350 0.911750 0.410746i \(-0.134732\pi\)
0.911750 + 0.410746i \(0.134732\pi\)
\(32\) 0 0
\(33\) 242.892 1.28127
\(34\) 0 0
\(35\) −652.130 −3.14943
\(36\) 0 0
\(37\) 103.729 0.460888 0.230444 0.973086i \(-0.425982\pi\)
0.230444 + 0.973086i \(0.425982\pi\)
\(38\) 0 0
\(39\) 112.634 0.462457
\(40\) 0 0
\(41\) −34.8358 −0.132693 −0.0663467 0.997797i \(-0.521134\pi\)
−0.0663467 + 0.997797i \(0.521134\pi\)
\(42\) 0 0
\(43\) 316.936 1.12401 0.562003 0.827135i \(-0.310031\pi\)
0.562003 + 0.827135i \(0.310031\pi\)
\(44\) 0 0
\(45\) −388.195 −1.28597
\(46\) 0 0
\(47\) 190.937 0.592575 0.296287 0.955099i \(-0.404251\pi\)
0.296287 + 0.955099i \(0.404251\pi\)
\(48\) 0 0
\(49\) 823.683 2.40141
\(50\) 0 0
\(51\) 116.958 0.321125
\(52\) 0 0
\(53\) −476.856 −1.23587 −0.617936 0.786229i \(-0.712031\pi\)
−0.617936 + 0.786229i \(0.712031\pi\)
\(54\) 0 0
\(55\) −674.048 −1.65252
\(56\) 0 0
\(57\) −629.227 −1.46216
\(58\) 0 0
\(59\) −351.708 −0.776075 −0.388038 0.921644i \(-0.626847\pi\)
−0.388038 + 0.921644i \(0.626847\pi\)
\(60\) 0 0
\(61\) 34.1756 0.0717333 0.0358667 0.999357i \(-0.488581\pi\)
0.0358667 + 0.999357i \(0.488581\pi\)
\(62\) 0 0
\(63\) 694.494 1.38886
\(64\) 0 0
\(65\) −312.569 −0.596453
\(66\) 0 0
\(67\) −114.361 −0.208529 −0.104265 0.994550i \(-0.533249\pi\)
−0.104265 + 0.994550i \(0.533249\pi\)
\(68\) 0 0
\(69\) −475.642 −0.829863
\(70\) 0 0
\(71\) −611.520 −1.02217 −0.511085 0.859530i \(-0.670756\pi\)
−0.511085 + 0.859530i \(0.670756\pi\)
\(72\) 0 0
\(73\) −11.4137 −0.0182997 −0.00914984 0.999958i \(-0.502913\pi\)
−0.00914984 + 0.999958i \(0.502913\pi\)
\(74\) 0 0
\(75\) 1647.84 2.53701
\(76\) 0 0
\(77\) 1205.89 1.78473
\(78\) 0 0
\(79\) 266.137 0.379022 0.189511 0.981879i \(-0.439310\pi\)
0.189511 + 0.981879i \(0.439310\pi\)
\(80\) 0 0
\(81\) −864.566 −1.18596
\(82\) 0 0
\(83\) 386.656 0.511337 0.255669 0.966764i \(-0.417704\pi\)
0.255669 + 0.966764i \(0.417704\pi\)
\(84\) 0 0
\(85\) −324.569 −0.414170
\(86\) 0 0
\(87\) 500.861 0.617218
\(88\) 0 0
\(89\) 1654.30 1.97028 0.985140 0.171753i \(-0.0549430\pi\)
0.985140 + 0.171753i \(0.0549430\pi\)
\(90\) 0 0
\(91\) 559.197 0.644173
\(92\) 0 0
\(93\) 2165.35 2.41437
\(94\) 0 0
\(95\) 1746.16 1.88582
\(96\) 0 0
\(97\) 1406.50 1.47225 0.736126 0.676844i \(-0.236653\pi\)
0.736126 + 0.676844i \(0.236653\pi\)
\(98\) 0 0
\(99\) 717.835 0.728739
\(100\) 0 0
\(101\) 347.407 0.342261 0.171130 0.985248i \(-0.445258\pi\)
0.171130 + 0.985248i \(0.445258\pi\)
\(102\) 0 0
\(103\) 957.419 0.915896 0.457948 0.888979i \(-0.348585\pi\)
0.457948 + 0.888979i \(0.348585\pi\)
\(104\) 0 0
\(105\) −4486.57 −4.16995
\(106\) 0 0
\(107\) 1035.16 0.935262 0.467631 0.883924i \(-0.345108\pi\)
0.467631 + 0.883924i \(0.345108\pi\)
\(108\) 0 0
\(109\) 89.0964 0.0782925 0.0391463 0.999233i \(-0.487536\pi\)
0.0391463 + 0.999233i \(0.487536\pi\)
\(110\) 0 0
\(111\) 713.639 0.610230
\(112\) 0 0
\(113\) 981.016 0.816692 0.408346 0.912827i \(-0.366106\pi\)
0.408346 + 0.912827i \(0.366106\pi\)
\(114\) 0 0
\(115\) 1319.95 1.07031
\(116\) 0 0
\(117\) 332.874 0.263028
\(118\) 0 0
\(119\) 580.665 0.447306
\(120\) 0 0
\(121\) −84.5770 −0.0635439
\(122\) 0 0
\(123\) −239.665 −0.175690
\(124\) 0 0
\(125\) −2186.37 −1.56444
\(126\) 0 0
\(127\) −102.803 −0.0718289 −0.0359144 0.999355i \(-0.511434\pi\)
−0.0359144 + 0.999355i \(0.511434\pi\)
\(128\) 0 0
\(129\) 2180.48 1.48822
\(130\) 0 0
\(131\) −284.690 −0.189874 −0.0949368 0.995483i \(-0.530265\pi\)
−0.0949368 + 0.995483i \(0.530265\pi\)
\(132\) 0 0
\(133\) −3123.95 −2.03670
\(134\) 0 0
\(135\) 875.784 0.558337
\(136\) 0 0
\(137\) 646.494 0.403166 0.201583 0.979471i \(-0.435391\pi\)
0.201583 + 0.979471i \(0.435391\pi\)
\(138\) 0 0
\(139\) 902.289 0.550584 0.275292 0.961361i \(-0.411226\pi\)
0.275292 + 0.961361i \(0.411226\pi\)
\(140\) 0 0
\(141\) 1313.62 0.784587
\(142\) 0 0
\(143\) 577.991 0.338000
\(144\) 0 0
\(145\) −1389.94 −0.796055
\(146\) 0 0
\(147\) 5666.83 3.17954
\(148\) 0 0
\(149\) 2592.83 1.42559 0.712794 0.701373i \(-0.247429\pi\)
0.712794 + 0.701373i \(0.247429\pi\)
\(150\) 0 0
\(151\) −3182.05 −1.71491 −0.857455 0.514560i \(-0.827955\pi\)
−0.857455 + 0.514560i \(0.827955\pi\)
\(152\) 0 0
\(153\) 345.654 0.182643
\(154\) 0 0
\(155\) −6009.06 −3.11393
\(156\) 0 0
\(157\) 1465.86 0.745148 0.372574 0.928002i \(-0.378475\pi\)
0.372574 + 0.928002i \(0.378475\pi\)
\(158\) 0 0
\(159\) −3280.70 −1.63633
\(160\) 0 0
\(161\) −2361.44 −1.15595
\(162\) 0 0
\(163\) 1541.32 0.740647 0.370323 0.928903i \(-0.379247\pi\)
0.370323 + 0.928903i \(0.379247\pi\)
\(164\) 0 0
\(165\) −4637.36 −2.18799
\(166\) 0 0
\(167\) −2870.97 −1.33031 −0.665157 0.746703i \(-0.731636\pi\)
−0.665157 + 0.746703i \(0.731636\pi\)
\(168\) 0 0
\(169\) −1928.97 −0.878004
\(170\) 0 0
\(171\) −1859.60 −0.831620
\(172\) 0 0
\(173\) −778.716 −0.342224 −0.171112 0.985252i \(-0.554736\pi\)
−0.171112 + 0.985252i \(0.554736\pi\)
\(174\) 0 0
\(175\) 8181.07 3.53389
\(176\) 0 0
\(177\) −2419.70 −1.02755
\(178\) 0 0
\(179\) 3179.68 1.32771 0.663856 0.747860i \(-0.268919\pi\)
0.663856 + 0.747860i \(0.268919\pi\)
\(180\) 0 0
\(181\) 147.672 0.0606429 0.0303214 0.999540i \(-0.490347\pi\)
0.0303214 + 0.999540i \(0.490347\pi\)
\(182\) 0 0
\(183\) 235.123 0.0949771
\(184\) 0 0
\(185\) −1980.42 −0.787044
\(186\) 0 0
\(187\) 600.180 0.234703
\(188\) 0 0
\(189\) −1566.81 −0.603008
\(190\) 0 0
\(191\) 23.3680 0.00885260 0.00442630 0.999990i \(-0.498591\pi\)
0.00442630 + 0.999990i \(0.498591\pi\)
\(192\) 0 0
\(193\) 394.629 0.147181 0.0735907 0.997289i \(-0.476554\pi\)
0.0735907 + 0.997289i \(0.476554\pi\)
\(194\) 0 0
\(195\) −2150.43 −0.789722
\(196\) 0 0
\(197\) 3656.93 1.32257 0.661283 0.750136i \(-0.270012\pi\)
0.661283 + 0.750136i \(0.270012\pi\)
\(198\) 0 0
\(199\) 2612.26 0.930545 0.465272 0.885168i \(-0.345956\pi\)
0.465272 + 0.885168i \(0.345956\pi\)
\(200\) 0 0
\(201\) −786.791 −0.276099
\(202\) 0 0
\(203\) 2486.64 0.859745
\(204\) 0 0
\(205\) 665.095 0.226596
\(206\) 0 0
\(207\) −1405.70 −0.471994
\(208\) 0 0
\(209\) −3228.94 −1.06866
\(210\) 0 0
\(211\) −4008.42 −1.30782 −0.653912 0.756570i \(-0.726873\pi\)
−0.653912 + 0.756570i \(0.726873\pi\)
\(212\) 0 0
\(213\) −4207.18 −1.35339
\(214\) 0 0
\(215\) −6051.03 −1.91943
\(216\) 0 0
\(217\) 10750.4 3.36306
\(218\) 0 0
\(219\) −78.5249 −0.0242293
\(220\) 0 0
\(221\) 278.315 0.0847127
\(222\) 0 0
\(223\) 3531.76 1.06056 0.530278 0.847824i \(-0.322088\pi\)
0.530278 + 0.847824i \(0.322088\pi\)
\(224\) 0 0
\(225\) 4869.96 1.44295
\(226\) 0 0
\(227\) 336.895 0.0985046 0.0492523 0.998786i \(-0.484316\pi\)
0.0492523 + 0.998786i \(0.484316\pi\)
\(228\) 0 0
\(229\) 5253.56 1.51600 0.758002 0.652252i \(-0.226176\pi\)
0.758002 + 0.652252i \(0.226176\pi\)
\(230\) 0 0
\(231\) 8296.39 2.36304
\(232\) 0 0
\(233\) −3486.05 −0.980167 −0.490084 0.871675i \(-0.663034\pi\)
−0.490084 + 0.871675i \(0.663034\pi\)
\(234\) 0 0
\(235\) −3645.42 −1.01192
\(236\) 0 0
\(237\) 1830.98 0.501836
\(238\) 0 0
\(239\) 2435.01 0.659028 0.329514 0.944151i \(-0.393115\pi\)
0.329514 + 0.944151i \(0.393115\pi\)
\(240\) 0 0
\(241\) −6151.11 −1.64410 −0.822049 0.569416i \(-0.807169\pi\)
−0.822049 + 0.569416i \(0.807169\pi\)
\(242\) 0 0
\(243\) −4709.58 −1.24329
\(244\) 0 0
\(245\) −15726.0 −4.10080
\(246\) 0 0
\(247\) −1497.32 −0.385718
\(248\) 0 0
\(249\) 2660.14 0.677026
\(250\) 0 0
\(251\) −1609.83 −0.404827 −0.202413 0.979300i \(-0.564878\pi\)
−0.202413 + 0.979300i \(0.564878\pi\)
\(252\) 0 0
\(253\) −2440.80 −0.606530
\(254\) 0 0
\(255\) −2232.99 −0.548374
\(256\) 0 0
\(257\) −2328.06 −0.565060 −0.282530 0.959259i \(-0.591174\pi\)
−0.282530 + 0.959259i \(0.591174\pi\)
\(258\) 0 0
\(259\) 3543.03 0.850012
\(260\) 0 0
\(261\) 1480.23 0.351050
\(262\) 0 0
\(263\) −3138.99 −0.735964 −0.367982 0.929833i \(-0.619951\pi\)
−0.367982 + 0.929833i \(0.619951\pi\)
\(264\) 0 0
\(265\) 9104.27 2.11046
\(266\) 0 0
\(267\) 11381.3 2.60871
\(268\) 0 0
\(269\) −4887.94 −1.10789 −0.553946 0.832552i \(-0.686879\pi\)
−0.553946 + 0.832552i \(0.686879\pi\)
\(270\) 0 0
\(271\) 2746.40 0.615617 0.307808 0.951448i \(-0.400404\pi\)
0.307808 + 0.951448i \(0.400404\pi\)
\(272\) 0 0
\(273\) 3847.20 0.852905
\(274\) 0 0
\(275\) 8456.03 1.85425
\(276\) 0 0
\(277\) −8743.46 −1.89655 −0.948274 0.317454i \(-0.897172\pi\)
−0.948274 + 0.317454i \(0.897172\pi\)
\(278\) 0 0
\(279\) 6399.41 1.37320
\(280\) 0 0
\(281\) −6214.16 −1.31924 −0.659618 0.751601i \(-0.729282\pi\)
−0.659618 + 0.751601i \(0.729282\pi\)
\(282\) 0 0
\(283\) −6138.63 −1.28941 −0.644706 0.764430i \(-0.723020\pi\)
−0.644706 + 0.764430i \(0.723020\pi\)
\(284\) 0 0
\(285\) 12013.4 2.49688
\(286\) 0 0
\(287\) −1189.88 −0.244725
\(288\) 0 0
\(289\) 289.000 0.0588235
\(290\) 0 0
\(291\) 9676.53 1.94931
\(292\) 0 0
\(293\) 2033.08 0.405371 0.202685 0.979244i \(-0.435033\pi\)
0.202685 + 0.979244i \(0.435033\pi\)
\(294\) 0 0
\(295\) 6714.90 1.32528
\(296\) 0 0
\(297\) −1619.47 −0.316401
\(298\) 0 0
\(299\) −1131.85 −0.218918
\(300\) 0 0
\(301\) 10825.5 2.07299
\(302\) 0 0
\(303\) 2390.12 0.453164
\(304\) 0 0
\(305\) −652.490 −0.122497
\(306\) 0 0
\(307\) −3139.90 −0.583725 −0.291862 0.956460i \(-0.594275\pi\)
−0.291862 + 0.956460i \(0.594275\pi\)
\(308\) 0 0
\(309\) 6586.91 1.21267
\(310\) 0 0
\(311\) −9907.83 −1.80650 −0.903250 0.429114i \(-0.858826\pi\)
−0.903250 + 0.429114i \(0.858826\pi\)
\(312\) 0 0
\(313\) 1445.19 0.260981 0.130491 0.991450i \(-0.458345\pi\)
0.130491 + 0.991450i \(0.458345\pi\)
\(314\) 0 0
\(315\) −13259.5 −2.37171
\(316\) 0 0
\(317\) −1315.32 −0.233046 −0.116523 0.993188i \(-0.537175\pi\)
−0.116523 + 0.993188i \(0.537175\pi\)
\(318\) 0 0
\(319\) 2570.22 0.451112
\(320\) 0 0
\(321\) 7121.79 1.23832
\(322\) 0 0
\(323\) −1554.81 −0.267838
\(324\) 0 0
\(325\) 3921.23 0.669263
\(326\) 0 0
\(327\) 612.971 0.103662
\(328\) 0 0
\(329\) 6521.78 1.09288
\(330\) 0 0
\(331\) 1679.47 0.278888 0.139444 0.990230i \(-0.455468\pi\)
0.139444 + 0.990230i \(0.455468\pi\)
\(332\) 0 0
\(333\) 2109.07 0.347076
\(334\) 0 0
\(335\) 2183.42 0.356099
\(336\) 0 0
\(337\) 6427.98 1.03903 0.519517 0.854460i \(-0.326112\pi\)
0.519517 + 0.854460i \(0.326112\pi\)
\(338\) 0 0
\(339\) 6749.26 1.08133
\(340\) 0 0
\(341\) 11111.7 1.76461
\(342\) 0 0
\(343\) 16418.6 2.58460
\(344\) 0 0
\(345\) 9081.09 1.41713
\(346\) 0 0
\(347\) −355.706 −0.0550296 −0.0275148 0.999621i \(-0.508759\pi\)
−0.0275148 + 0.999621i \(0.508759\pi\)
\(348\) 0 0
\(349\) 4790.26 0.734719 0.367359 0.930079i \(-0.380262\pi\)
0.367359 + 0.930079i \(0.380262\pi\)
\(350\) 0 0
\(351\) −750.978 −0.114200
\(352\) 0 0
\(353\) −9135.45 −1.37742 −0.688712 0.725035i \(-0.741824\pi\)
−0.688712 + 0.725035i \(0.741824\pi\)
\(354\) 0 0
\(355\) 11675.3 1.74553
\(356\) 0 0
\(357\) 3994.89 0.592247
\(358\) 0 0
\(359\) −10649.0 −1.56555 −0.782773 0.622308i \(-0.786195\pi\)
−0.782773 + 0.622308i \(0.786195\pi\)
\(360\) 0 0
\(361\) 1505.77 0.219532
\(362\) 0 0
\(363\) −581.878 −0.0841341
\(364\) 0 0
\(365\) 217.914 0.0312497
\(366\) 0 0
\(367\) −6913.73 −0.983362 −0.491681 0.870775i \(-0.663617\pi\)
−0.491681 + 0.870775i \(0.663617\pi\)
\(368\) 0 0
\(369\) −708.300 −0.0999258
\(370\) 0 0
\(371\) −16287.8 −2.27931
\(372\) 0 0
\(373\) −6993.44 −0.970795 −0.485398 0.874294i \(-0.661325\pi\)
−0.485398 + 0.874294i \(0.661325\pi\)
\(374\) 0 0
\(375\) −15041.9 −2.07136
\(376\) 0 0
\(377\) 1191.86 0.162822
\(378\) 0 0
\(379\) −602.412 −0.0816460 −0.0408230 0.999166i \(-0.512998\pi\)
−0.0408230 + 0.999166i \(0.512998\pi\)
\(380\) 0 0
\(381\) −707.269 −0.0951037
\(382\) 0 0
\(383\) −8931.74 −1.19162 −0.595810 0.803125i \(-0.703169\pi\)
−0.595810 + 0.803125i \(0.703169\pi\)
\(384\) 0 0
\(385\) −23023.3 −3.04773
\(386\) 0 0
\(387\) 6444.12 0.846441
\(388\) 0 0
\(389\) 12111.7 1.57864 0.789318 0.613985i \(-0.210434\pi\)
0.789318 + 0.613985i \(0.210434\pi\)
\(390\) 0 0
\(391\) −1175.30 −0.152014
\(392\) 0 0
\(393\) −1958.63 −0.251399
\(394\) 0 0
\(395\) −5081.16 −0.647243
\(396\) 0 0
\(397\) −9016.37 −1.13985 −0.569923 0.821698i \(-0.693027\pi\)
−0.569923 + 0.821698i \(0.693027\pi\)
\(398\) 0 0
\(399\) −21492.3 −2.69665
\(400\) 0 0
\(401\) 4046.55 0.503928 0.251964 0.967737i \(-0.418924\pi\)
0.251964 + 0.967737i \(0.418924\pi\)
\(402\) 0 0
\(403\) 5152.72 0.636911
\(404\) 0 0
\(405\) 16506.5 2.02523
\(406\) 0 0
\(407\) 3662.11 0.446005
\(408\) 0 0
\(409\) −3490.95 −0.422045 −0.211023 0.977481i \(-0.567679\pi\)
−0.211023 + 0.977481i \(0.567679\pi\)
\(410\) 0 0
\(411\) 4447.79 0.533804
\(412\) 0 0
\(413\) −12013.2 −1.43131
\(414\) 0 0
\(415\) −7382.15 −0.873194
\(416\) 0 0
\(417\) 6207.62 0.728990
\(418\) 0 0
\(419\) −6940.21 −0.809192 −0.404596 0.914496i \(-0.632588\pi\)
−0.404596 + 0.914496i \(0.632588\pi\)
\(420\) 0 0
\(421\) 867.961 0.100479 0.0502397 0.998737i \(-0.484001\pi\)
0.0502397 + 0.998737i \(0.484001\pi\)
\(422\) 0 0
\(423\) 3882.23 0.446243
\(424\) 0 0
\(425\) 4071.77 0.464729
\(426\) 0 0
\(427\) 1167.33 0.132297
\(428\) 0 0
\(429\) 3976.50 0.447523
\(430\) 0 0
\(431\) 2713.40 0.303248 0.151624 0.988438i \(-0.451550\pi\)
0.151624 + 0.988438i \(0.451550\pi\)
\(432\) 0 0
\(433\) 499.370 0.0554231 0.0277115 0.999616i \(-0.491178\pi\)
0.0277115 + 0.999616i \(0.491178\pi\)
\(434\) 0 0
\(435\) −9562.58 −1.05400
\(436\) 0 0
\(437\) 6323.06 0.692158
\(438\) 0 0
\(439\) −13153.8 −1.43006 −0.715029 0.699095i \(-0.753586\pi\)
−0.715029 + 0.699095i \(0.753586\pi\)
\(440\) 0 0
\(441\) 16747.6 1.80840
\(442\) 0 0
\(443\) 3950.78 0.423718 0.211859 0.977300i \(-0.432048\pi\)
0.211859 + 0.977300i \(0.432048\pi\)
\(444\) 0 0
\(445\) −31584.3 −3.36458
\(446\) 0 0
\(447\) 17838.3 1.88752
\(448\) 0 0
\(449\) −9373.70 −0.985239 −0.492619 0.870245i \(-0.663961\pi\)
−0.492619 + 0.870245i \(0.663961\pi\)
\(450\) 0 0
\(451\) −1229.87 −0.128408
\(452\) 0 0
\(453\) −21892.1 −2.27059
\(454\) 0 0
\(455\) −10676.3 −1.10003
\(456\) 0 0
\(457\) −14528.6 −1.48713 −0.743566 0.668663i \(-0.766867\pi\)
−0.743566 + 0.668663i \(0.766867\pi\)
\(458\) 0 0
\(459\) −779.809 −0.0792992
\(460\) 0 0
\(461\) −694.967 −0.0702122 −0.0351061 0.999384i \(-0.511177\pi\)
−0.0351061 + 0.999384i \(0.511177\pi\)
\(462\) 0 0
\(463\) −2953.87 −0.296497 −0.148248 0.988950i \(-0.547364\pi\)
−0.148248 + 0.988950i \(0.547364\pi\)
\(464\) 0 0
\(465\) −41341.5 −4.12294
\(466\) 0 0
\(467\) −17509.4 −1.73498 −0.867491 0.497453i \(-0.834269\pi\)
−0.867491 + 0.497453i \(0.834269\pi\)
\(468\) 0 0
\(469\) −3906.21 −0.384589
\(470\) 0 0
\(471\) 10084.9 0.986599
\(472\) 0 0
\(473\) 11189.3 1.08771
\(474\) 0 0
\(475\) −21905.9 −2.11602
\(476\) 0 0
\(477\) −9695.70 −0.930683
\(478\) 0 0
\(479\) −14221.5 −1.35657 −0.678284 0.734800i \(-0.737276\pi\)
−0.678284 + 0.734800i \(0.737276\pi\)
\(480\) 0 0
\(481\) 1698.19 0.160979
\(482\) 0 0
\(483\) −16246.4 −1.53051
\(484\) 0 0
\(485\) −26853.3 −2.51412
\(486\) 0 0
\(487\) 14872.8 1.38388 0.691940 0.721955i \(-0.256756\pi\)
0.691940 + 0.721955i \(0.256756\pi\)
\(488\) 0 0
\(489\) 10604.1 0.980639
\(490\) 0 0
\(491\) 8765.65 0.805679 0.402839 0.915271i \(-0.368023\pi\)
0.402839 + 0.915271i \(0.368023\pi\)
\(492\) 0 0
\(493\) 1237.62 0.113062
\(494\) 0 0
\(495\) −13705.1 −1.24444
\(496\) 0 0
\(497\) −20887.5 −1.88518
\(498\) 0 0
\(499\) −8061.92 −0.723249 −0.361624 0.932324i \(-0.617778\pi\)
−0.361624 + 0.932324i \(0.617778\pi\)
\(500\) 0 0
\(501\) −19751.9 −1.76138
\(502\) 0 0
\(503\) 15748.0 1.39596 0.697980 0.716118i \(-0.254083\pi\)
0.697980 + 0.716118i \(0.254083\pi\)
\(504\) 0 0
\(505\) −6632.80 −0.584467
\(506\) 0 0
\(507\) −13271.1 −1.16250
\(508\) 0 0
\(509\) 11294.7 0.983557 0.491779 0.870720i \(-0.336347\pi\)
0.491779 + 0.870720i \(0.336347\pi\)
\(510\) 0 0
\(511\) −389.856 −0.0337499
\(512\) 0 0
\(513\) 4195.33 0.361069
\(514\) 0 0
\(515\) −18279.3 −1.56404
\(516\) 0 0
\(517\) 6740.97 0.573438
\(518\) 0 0
\(519\) −5357.46 −0.453115
\(520\) 0 0
\(521\) −12847.7 −1.08036 −0.540180 0.841550i \(-0.681644\pi\)
−0.540180 + 0.841550i \(0.681644\pi\)
\(522\) 0 0
\(523\) 22081.9 1.84622 0.923112 0.384531i \(-0.125637\pi\)
0.923112 + 0.384531i \(0.125637\pi\)
\(524\) 0 0
\(525\) 56284.7 4.67898
\(526\) 0 0
\(527\) 5350.53 0.442264
\(528\) 0 0
\(529\) −7387.30 −0.607159
\(530\) 0 0
\(531\) −7151.12 −0.584429
\(532\) 0 0
\(533\) −570.313 −0.0463471
\(534\) 0 0
\(535\) −19763.7 −1.59712
\(536\) 0 0
\(537\) 21875.8 1.75793
\(538\) 0 0
\(539\) 29079.9 2.32386
\(540\) 0 0
\(541\) 17994.4 1.43002 0.715008 0.699116i \(-0.246423\pi\)
0.715008 + 0.699116i \(0.246423\pi\)
\(542\) 0 0
\(543\) 1015.96 0.0802930
\(544\) 0 0
\(545\) −1701.05 −0.133698
\(546\) 0 0
\(547\) 11528.6 0.901147 0.450573 0.892739i \(-0.351220\pi\)
0.450573 + 0.892739i \(0.351220\pi\)
\(548\) 0 0
\(549\) 694.877 0.0540193
\(550\) 0 0
\(551\) −6658.32 −0.514798
\(552\) 0 0
\(553\) 9090.36 0.699026
\(554\) 0 0
\(555\) −13625.0 −1.04207
\(556\) 0 0
\(557\) −9312.88 −0.708437 −0.354219 0.935163i \(-0.615253\pi\)
−0.354219 + 0.935163i \(0.615253\pi\)
\(558\) 0 0
\(559\) 5188.71 0.392592
\(560\) 0 0
\(561\) 4129.16 0.310754
\(562\) 0 0
\(563\) 552.235 0.0413391 0.0206695 0.999786i \(-0.493420\pi\)
0.0206695 + 0.999786i \(0.493420\pi\)
\(564\) 0 0
\(565\) −18729.8 −1.39464
\(566\) 0 0
\(567\) −29530.8 −2.18726
\(568\) 0 0
\(569\) 13836.8 1.01945 0.509727 0.860336i \(-0.329746\pi\)
0.509727 + 0.860336i \(0.329746\pi\)
\(570\) 0 0
\(571\) 8568.10 0.627958 0.313979 0.949430i \(-0.398338\pi\)
0.313979 + 0.949430i \(0.398338\pi\)
\(572\) 0 0
\(573\) 160.768 0.0117211
\(574\) 0 0
\(575\) −16559.0 −1.20097
\(576\) 0 0
\(577\) 15080.1 1.08803 0.544014 0.839076i \(-0.316904\pi\)
0.544014 + 0.839076i \(0.316904\pi\)
\(578\) 0 0
\(579\) 2714.99 0.194873
\(580\) 0 0
\(581\) 13206.9 0.943055
\(582\) 0 0
\(583\) −16835.3 −1.19596
\(584\) 0 0
\(585\) −6355.33 −0.449164
\(586\) 0 0
\(587\) −19594.7 −1.37779 −0.688893 0.724863i \(-0.741903\pi\)
−0.688893 + 0.724863i \(0.741903\pi\)
\(588\) 0 0
\(589\) −28785.6 −2.01374
\(590\) 0 0
\(591\) 25159.2 1.75112
\(592\) 0 0
\(593\) 12775.1 0.884674 0.442337 0.896849i \(-0.354149\pi\)
0.442337 + 0.896849i \(0.354149\pi\)
\(594\) 0 0
\(595\) −11086.2 −0.763850
\(596\) 0 0
\(597\) 17972.0 1.23207
\(598\) 0 0
\(599\) 17535.3 1.19612 0.598058 0.801452i \(-0.295939\pi\)
0.598058 + 0.801452i \(0.295939\pi\)
\(600\) 0 0
\(601\) 20352.9 1.38138 0.690691 0.723150i \(-0.257306\pi\)
0.690691 + 0.723150i \(0.257306\pi\)
\(602\) 0 0
\(603\) −2325.26 −0.157035
\(604\) 0 0
\(605\) 1614.77 0.108512
\(606\) 0 0
\(607\) −748.483 −0.0500494 −0.0250247 0.999687i \(-0.507966\pi\)
−0.0250247 + 0.999687i \(0.507966\pi\)
\(608\) 0 0
\(609\) 17107.8 1.13833
\(610\) 0 0
\(611\) 3125.92 0.206974
\(612\) 0 0
\(613\) −28364.1 −1.86887 −0.934434 0.356136i \(-0.884094\pi\)
−0.934434 + 0.356136i \(0.884094\pi\)
\(614\) 0 0
\(615\) 4575.76 0.300020
\(616\) 0 0
\(617\) −27240.5 −1.77741 −0.888703 0.458483i \(-0.848393\pi\)
−0.888703 + 0.458483i \(0.848393\pi\)
\(618\) 0 0
\(619\) −7222.04 −0.468947 −0.234474 0.972122i \(-0.575337\pi\)
−0.234474 + 0.972122i \(0.575337\pi\)
\(620\) 0 0
\(621\) 3171.31 0.204928
\(622\) 0 0
\(623\) 56505.3 3.63377
\(624\) 0 0
\(625\) 11803.3 0.755409
\(626\) 0 0
\(627\) −22214.7 −1.41494
\(628\) 0 0
\(629\) 1763.39 0.111782
\(630\) 0 0
\(631\) 1099.75 0.0693823 0.0346911 0.999398i \(-0.488955\pi\)
0.0346911 + 0.999398i \(0.488955\pi\)
\(632\) 0 0
\(633\) −27577.4 −1.73160
\(634\) 0 0
\(635\) 1962.74 0.122660
\(636\) 0 0
\(637\) 13484.9 0.838763
\(638\) 0 0
\(639\) −12433.8 −0.769754
\(640\) 0 0
\(641\) −1846.78 −0.113796 −0.0568982 0.998380i \(-0.518121\pi\)
−0.0568982 + 0.998380i \(0.518121\pi\)
\(642\) 0 0
\(643\) 1995.66 0.122397 0.0611984 0.998126i \(-0.480508\pi\)
0.0611984 + 0.998126i \(0.480508\pi\)
\(644\) 0 0
\(645\) −41630.3 −2.54138
\(646\) 0 0
\(647\) −21623.2 −1.31390 −0.656952 0.753933i \(-0.728154\pi\)
−0.656952 + 0.753933i \(0.728154\pi\)
\(648\) 0 0
\(649\) −12416.9 −0.751013
\(650\) 0 0
\(651\) 73961.3 4.45280
\(652\) 0 0
\(653\) 9359.42 0.560892 0.280446 0.959870i \(-0.409518\pi\)
0.280446 + 0.959870i \(0.409518\pi\)
\(654\) 0 0
\(655\) 5435.38 0.324241
\(656\) 0 0
\(657\) −232.070 −0.0137807
\(658\) 0 0
\(659\) −14648.4 −0.865888 −0.432944 0.901421i \(-0.642525\pi\)
−0.432944 + 0.901421i \(0.642525\pi\)
\(660\) 0 0
\(661\) −15894.8 −0.935306 −0.467653 0.883912i \(-0.654900\pi\)
−0.467653 + 0.883912i \(0.654900\pi\)
\(662\) 0 0
\(663\) 1914.77 0.112162
\(664\) 0 0
\(665\) 59643.3 3.47800
\(666\) 0 0
\(667\) −5033.12 −0.292179
\(668\) 0 0
\(669\) 24298.0 1.40421
\(670\) 0 0
\(671\) 1206.56 0.0694168
\(672\) 0 0
\(673\) 1312.42 0.0751712 0.0375856 0.999293i \(-0.488033\pi\)
0.0375856 + 0.999293i \(0.488033\pi\)
\(674\) 0 0
\(675\) −10986.8 −0.626495
\(676\) 0 0
\(677\) −18835.2 −1.06927 −0.534635 0.845083i \(-0.679551\pi\)
−0.534635 + 0.845083i \(0.679551\pi\)
\(678\) 0 0
\(679\) 48041.5 2.71526
\(680\) 0 0
\(681\) 2317.80 0.130423
\(682\) 0 0
\(683\) 9093.24 0.509434 0.254717 0.967016i \(-0.418018\pi\)
0.254717 + 0.967016i \(0.418018\pi\)
\(684\) 0 0
\(685\) −12343.1 −0.688473
\(686\) 0 0
\(687\) 36143.8 2.00724
\(688\) 0 0
\(689\) −7806.84 −0.431665
\(690\) 0 0
\(691\) 10920.5 0.601210 0.300605 0.953749i \(-0.402811\pi\)
0.300605 + 0.953749i \(0.402811\pi\)
\(692\) 0 0
\(693\) 24518.9 1.34401
\(694\) 0 0
\(695\) −17226.8 −0.940213
\(696\) 0 0
\(697\) −592.208 −0.0321829
\(698\) 0 0
\(699\) −23983.6 −1.29777
\(700\) 0 0
\(701\) 14249.4 0.767750 0.383875 0.923385i \(-0.374589\pi\)
0.383875 + 0.923385i \(0.374589\pi\)
\(702\) 0 0
\(703\) −9486.92 −0.508970
\(704\) 0 0
\(705\) −25080.0 −1.33981
\(706\) 0 0
\(707\) 11866.3 0.631228
\(708\) 0 0
\(709\) 19183.7 1.01616 0.508082 0.861309i \(-0.330355\pi\)
0.508082 + 0.861309i \(0.330355\pi\)
\(710\) 0 0
\(711\) 5411.24 0.285425
\(712\) 0 0
\(713\) −21759.5 −1.14292
\(714\) 0 0
\(715\) −11035.2 −0.577191
\(716\) 0 0
\(717\) 16752.6 0.872574
\(718\) 0 0
\(719\) 1872.24 0.0971112 0.0485556 0.998820i \(-0.484538\pi\)
0.0485556 + 0.998820i \(0.484538\pi\)
\(720\) 0 0
\(721\) 32702.3 1.68918
\(722\) 0 0
\(723\) −42318.8 −2.17684
\(724\) 0 0
\(725\) 17437.0 0.893232
\(726\) 0 0
\(727\) 33391.6 1.70347 0.851737 0.523969i \(-0.175549\pi\)
0.851737 + 0.523969i \(0.175549\pi\)
\(728\) 0 0
\(729\) −9057.99 −0.460194
\(730\) 0 0
\(731\) 5387.91 0.272611
\(732\) 0 0
\(733\) 30489.9 1.53639 0.768193 0.640218i \(-0.221156\pi\)
0.768193 + 0.640218i \(0.221156\pi\)
\(734\) 0 0
\(735\) −108193. −5.42959
\(736\) 0 0
\(737\) −4037.50 −0.201795
\(738\) 0 0
\(739\) 12342.0 0.614353 0.307177 0.951652i \(-0.400616\pi\)
0.307177 + 0.951652i \(0.400616\pi\)
\(740\) 0 0
\(741\) −10301.4 −0.510702
\(742\) 0 0
\(743\) −18490.4 −0.912985 −0.456492 0.889727i \(-0.650894\pi\)
−0.456492 + 0.889727i \(0.650894\pi\)
\(744\) 0 0
\(745\) −49503.0 −2.43443
\(746\) 0 0
\(747\) 7861.70 0.385067
\(748\) 0 0
\(749\) 35357.8 1.72490
\(750\) 0 0
\(751\) −9321.89 −0.452944 −0.226472 0.974018i \(-0.572719\pi\)
−0.226472 + 0.974018i \(0.572719\pi\)
\(752\) 0 0
\(753\) −11075.4 −0.536003
\(754\) 0 0
\(755\) 60752.6 2.92849
\(756\) 0 0
\(757\) −28417.6 −1.36441 −0.682203 0.731163i \(-0.738978\pi\)
−0.682203 + 0.731163i \(0.738978\pi\)
\(758\) 0 0
\(759\) −16792.4 −0.803064
\(760\) 0 0
\(761\) 19383.8 0.923341 0.461670 0.887052i \(-0.347250\pi\)
0.461670 + 0.887052i \(0.347250\pi\)
\(762\) 0 0
\(763\) 3043.24 0.144394
\(764\) 0 0
\(765\) −6599.32 −0.311894
\(766\) 0 0
\(767\) −5757.98 −0.271067
\(768\) 0 0
\(769\) −29856.5 −1.40007 −0.700035 0.714108i \(-0.746832\pi\)
−0.700035 + 0.714108i \(0.746832\pi\)
\(770\) 0 0
\(771\) −16016.7 −0.748157
\(772\) 0 0
\(773\) −20965.5 −0.975518 −0.487759 0.872978i \(-0.662186\pi\)
−0.487759 + 0.872978i \(0.662186\pi\)
\(774\) 0 0
\(775\) 75384.5 3.49405
\(776\) 0 0
\(777\) 24375.6 1.12544
\(778\) 0 0
\(779\) 3186.05 0.146537
\(780\) 0 0
\(781\) −21589.6 −0.989161
\(782\) 0 0
\(783\) −3339.46 −0.152417
\(784\) 0 0
\(785\) −27986.6 −1.27246
\(786\) 0 0
\(787\) −13031.4 −0.590240 −0.295120 0.955460i \(-0.595360\pi\)
−0.295120 + 0.955460i \(0.595360\pi\)
\(788\) 0 0
\(789\) −21595.8 −0.974439
\(790\) 0 0
\(791\) 33508.3 1.50622
\(792\) 0 0
\(793\) 559.505 0.0250550
\(794\) 0 0
\(795\) 62636.2 2.79431
\(796\) 0 0
\(797\) 4156.24 0.184719 0.0923597 0.995726i \(-0.470559\pi\)
0.0923597 + 0.995726i \(0.470559\pi\)
\(798\) 0 0
\(799\) 3245.93 0.143720
\(800\) 0 0
\(801\) 33636.1 1.48373
\(802\) 0 0
\(803\) −402.959 −0.0177087
\(804\) 0 0
\(805\) 45085.3 1.97397
\(806\) 0 0
\(807\) −33628.4 −1.46688
\(808\) 0 0
\(809\) 38507.6 1.67349 0.836745 0.547592i \(-0.184456\pi\)
0.836745 + 0.547592i \(0.184456\pi\)
\(810\) 0 0
\(811\) −19824.6 −0.858367 −0.429183 0.903217i \(-0.641199\pi\)
−0.429183 + 0.903217i \(0.641199\pi\)
\(812\) 0 0
\(813\) 18894.9 0.815096
\(814\) 0 0
\(815\) −29427.3 −1.26478
\(816\) 0 0
\(817\) −28986.7 −1.24127
\(818\) 0 0
\(819\) 11369.9 0.485099
\(820\) 0 0
\(821\) −4055.06 −0.172378 −0.0861892 0.996279i \(-0.527469\pi\)
−0.0861892 + 0.996279i \(0.527469\pi\)
\(822\) 0 0
\(823\) −14743.6 −0.624460 −0.312230 0.950007i \(-0.601076\pi\)
−0.312230 + 0.950007i \(0.601076\pi\)
\(824\) 0 0
\(825\) 58176.4 2.45508
\(826\) 0 0
\(827\) 24850.8 1.04492 0.522459 0.852664i \(-0.325015\pi\)
0.522459 + 0.852664i \(0.325015\pi\)
\(828\) 0 0
\(829\) 12731.1 0.533377 0.266688 0.963783i \(-0.414071\pi\)
0.266688 + 0.963783i \(0.414071\pi\)
\(830\) 0 0
\(831\) −60153.8 −2.51109
\(832\) 0 0
\(833\) 14002.6 0.582427
\(834\) 0 0
\(835\) 54813.5 2.27173
\(836\) 0 0
\(837\) −14437.3 −0.596210
\(838\) 0 0
\(839\) 11989.1 0.493337 0.246669 0.969100i \(-0.420664\pi\)
0.246669 + 0.969100i \(0.420664\pi\)
\(840\) 0 0
\(841\) −19089.0 −0.782690
\(842\) 0 0
\(843\) −42752.6 −1.74671
\(844\) 0 0
\(845\) 36828.5 1.49934
\(846\) 0 0
\(847\) −2888.87 −0.117193
\(848\) 0 0
\(849\) −42233.0 −1.70722
\(850\) 0 0
\(851\) −7171.31 −0.288871
\(852\) 0 0
\(853\) −24744.0 −0.993224 −0.496612 0.867973i \(-0.665423\pi\)
−0.496612 + 0.867973i \(0.665423\pi\)
\(854\) 0 0
\(855\) 35504.0 1.42013
\(856\) 0 0
\(857\) −6396.71 −0.254968 −0.127484 0.991841i \(-0.540690\pi\)
−0.127484 + 0.991841i \(0.540690\pi\)
\(858\) 0 0
\(859\) 5535.27 0.219862 0.109931 0.993939i \(-0.464937\pi\)
0.109931 + 0.993939i \(0.464937\pi\)
\(860\) 0 0
\(861\) −8186.19 −0.324024
\(862\) 0 0
\(863\) 2478.07 0.0977458 0.0488729 0.998805i \(-0.484437\pi\)
0.0488729 + 0.998805i \(0.484437\pi\)
\(864\) 0 0
\(865\) 14867.5 0.584404
\(866\) 0 0
\(867\) 1988.28 0.0778842
\(868\) 0 0
\(869\) 9395.88 0.366782
\(870\) 0 0
\(871\) −1872.27 −0.0728351
\(872\) 0 0
\(873\) 28597.7 1.10869
\(874\) 0 0
\(875\) −74679.1 −2.88527
\(876\) 0 0
\(877\) 12732.5 0.490246 0.245123 0.969492i \(-0.421172\pi\)
0.245123 + 0.969492i \(0.421172\pi\)
\(878\) 0 0
\(879\) 13987.3 0.536723
\(880\) 0 0
\(881\) −4514.92 −0.172658 −0.0863289 0.996267i \(-0.527514\pi\)
−0.0863289 + 0.996267i \(0.527514\pi\)
\(882\) 0 0
\(883\) 29843.9 1.13740 0.568702 0.822544i \(-0.307446\pi\)
0.568702 + 0.822544i \(0.307446\pi\)
\(884\) 0 0
\(885\) 46197.6 1.75471
\(886\) 0 0
\(887\) −9244.69 −0.349951 −0.174975 0.984573i \(-0.555985\pi\)
−0.174975 + 0.984573i \(0.555985\pi\)
\(888\) 0 0
\(889\) −3511.41 −0.132473
\(890\) 0 0
\(891\) −30523.3 −1.14766
\(892\) 0 0
\(893\) −17462.9 −0.654395
\(894\) 0 0
\(895\) −60707.4 −2.26729
\(896\) 0 0
\(897\) −7786.97 −0.289854
\(898\) 0 0
\(899\) 22913.2 0.850053
\(900\) 0 0
\(901\) −8106.55 −0.299743
\(902\) 0 0
\(903\) 74477.9 2.74471
\(904\) 0 0
\(905\) −2819.39 −0.103558
\(906\) 0 0
\(907\) 17024.8 0.623263 0.311632 0.950203i \(-0.399125\pi\)
0.311632 + 0.950203i \(0.399125\pi\)
\(908\) 0 0
\(909\) 7063.68 0.257742
\(910\) 0 0
\(911\) 21104.9 0.767547 0.383773 0.923427i \(-0.374624\pi\)
0.383773 + 0.923427i \(0.374624\pi\)
\(912\) 0 0
\(913\) 13650.8 0.494825
\(914\) 0 0
\(915\) −4489.04 −0.162189
\(916\) 0 0
\(917\) −9724.07 −0.350182
\(918\) 0 0
\(919\) 16052.0 0.576176 0.288088 0.957604i \(-0.406980\pi\)
0.288088 + 0.957604i \(0.406980\pi\)
\(920\) 0 0
\(921\) −21602.1 −0.772870
\(922\) 0 0
\(923\) −10011.5 −0.357023
\(924\) 0 0
\(925\) 24844.6 0.883120
\(926\) 0 0
\(927\) 19466.8 0.689723
\(928\) 0 0
\(929\) −8279.43 −0.292400 −0.146200 0.989255i \(-0.546704\pi\)
−0.146200 + 0.989255i \(0.546704\pi\)
\(930\) 0 0
\(931\) −75333.3 −2.65193
\(932\) 0 0
\(933\) −68164.6 −2.39186
\(934\) 0 0
\(935\) −11458.8 −0.400795
\(936\) 0 0
\(937\) −47460.0 −1.65470 −0.827348 0.561689i \(-0.810152\pi\)
−0.827348 + 0.561689i \(0.810152\pi\)
\(938\) 0 0
\(939\) 9942.73 0.345547
\(940\) 0 0
\(941\) −47914.5 −1.65990 −0.829950 0.557837i \(-0.811631\pi\)
−0.829950 + 0.557837i \(0.811631\pi\)
\(942\) 0 0
\(943\) 2408.38 0.0831683
\(944\) 0 0
\(945\) 29913.9 1.02974
\(946\) 0 0
\(947\) −1854.15 −0.0636238 −0.0318119 0.999494i \(-0.510128\pi\)
−0.0318119 + 0.999494i \(0.510128\pi\)
\(948\) 0 0
\(949\) −186.860 −0.00639170
\(950\) 0 0
\(951\) −9049.21 −0.308560
\(952\) 0 0
\(953\) 18073.1 0.614319 0.307159 0.951658i \(-0.400622\pi\)
0.307159 + 0.951658i \(0.400622\pi\)
\(954\) 0 0
\(955\) −446.148 −0.0151173
\(956\) 0 0
\(957\) 17682.8 0.597286
\(958\) 0 0
\(959\) 22082.1 0.743555
\(960\) 0 0
\(961\) 69268.6 2.32515
\(962\) 0 0
\(963\) 21047.5 0.704307
\(964\) 0 0
\(965\) −7534.37 −0.251337
\(966\) 0 0
\(967\) 38564.7 1.28248 0.641239 0.767341i \(-0.278421\pi\)
0.641239 + 0.767341i \(0.278421\pi\)
\(968\) 0 0
\(969\) −10696.9 −0.354626
\(970\) 0 0
\(971\) −30157.3 −0.996698 −0.498349 0.866976i \(-0.666060\pi\)
−0.498349 + 0.866976i \(0.666060\pi\)
\(972\) 0 0
\(973\) 30819.2 1.01544
\(974\) 0 0
\(975\) 26977.5 0.886125
\(976\) 0 0
\(977\) 4765.98 0.156067 0.0780334 0.996951i \(-0.475136\pi\)
0.0780334 + 0.996951i \(0.475136\pi\)
\(978\) 0 0
\(979\) 58404.4 1.90665
\(980\) 0 0
\(981\) 1811.56 0.0589588
\(982\) 0 0
\(983\) −8485.97 −0.275341 −0.137671 0.990478i \(-0.543962\pi\)
−0.137671 + 0.990478i \(0.543962\pi\)
\(984\) 0 0
\(985\) −69819.2 −2.25850
\(986\) 0 0
\(987\) 44869.0 1.44701
\(988\) 0 0
\(989\) −21911.5 −0.704493
\(990\) 0 0
\(991\) 24385.9 0.781680 0.390840 0.920459i \(-0.372185\pi\)
0.390840 + 0.920459i \(0.372185\pi\)
\(992\) 0 0
\(993\) 11554.5 0.369257
\(994\) 0 0
\(995\) −49874.1 −1.58906
\(996\) 0 0
\(997\) −33548.8 −1.06570 −0.532848 0.846211i \(-0.678878\pi\)
−0.532848 + 0.846211i \(0.678878\pi\)
\(998\) 0 0
\(999\) −4758.14 −0.150692
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 1088.4.a.w.1.3 3
4.3 odd 2 1088.4.a.t.1.1 3
8.3 odd 2 68.4.a.b.1.3 3
8.5 even 2 272.4.a.i.1.1 3
24.5 odd 2 2448.4.a.ba.1.1 3
24.11 even 2 612.4.a.g.1.1 3
40.3 even 4 1700.4.e.d.749.5 6
40.19 odd 2 1700.4.a.d.1.1 3
40.27 even 4 1700.4.e.d.749.2 6
136.67 odd 2 1156.4.a.g.1.1 3
136.115 odd 4 1156.4.b.e.577.5 6
136.123 odd 4 1156.4.b.e.577.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.4.a.b.1.3 3 8.3 odd 2
272.4.a.i.1.1 3 8.5 even 2
612.4.a.g.1.1 3 24.11 even 2
1088.4.a.t.1.1 3 4.3 odd 2
1088.4.a.w.1.3 3 1.1 even 1 trivial
1156.4.a.g.1.1 3 136.67 odd 2
1156.4.b.e.577.2 6 136.123 odd 4
1156.4.b.e.577.5 6 136.115 odd 4
1700.4.a.d.1.1 3 40.19 odd 2
1700.4.e.d.749.2 6 40.27 even 4
1700.4.e.d.749.5 6 40.3 even 4
2448.4.a.ba.1.1 3 24.5 odd 2