Properties

Label 1232.4.a.z.1.5
Level $1232$
Weight $4$
Character 1232.1
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
Dimension $5$
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] = [1232,4,Mod(1,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1232.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.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: \(72.6903531271\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 105x^{3} + 92x^{2} + 1858x - 2576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 616)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-8.22418\) of defining polynomial
Character \(\chi\) \(=\) 1232.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.22418 q^{3} -17.2362 q^{5} +7.00000 q^{7} +40.6371 q^{9} +O(q^{10})\) \(q+8.22418 q^{3} -17.2362 q^{5} +7.00000 q^{7} +40.6371 q^{9} +11.0000 q^{11} +62.9306 q^{13} -141.754 q^{15} -80.9209 q^{17} -44.9110 q^{19} +57.5693 q^{21} -55.0637 q^{23} +172.088 q^{25} +112.154 q^{27} +310.218 q^{29} +7.80993 q^{31} +90.4660 q^{33} -120.654 q^{35} +400.886 q^{37} +517.553 q^{39} -470.422 q^{41} +551.684 q^{43} -700.431 q^{45} -36.8998 q^{47} +49.0000 q^{49} -665.508 q^{51} +71.3509 q^{53} -189.599 q^{55} -369.356 q^{57} +623.175 q^{59} +326.021 q^{61} +284.460 q^{63} -1084.69 q^{65} -617.038 q^{67} -452.854 q^{69} +719.391 q^{71} -244.700 q^{73} +1415.28 q^{75} +77.0000 q^{77} +232.011 q^{79} -174.827 q^{81} -1045.98 q^{83} +1394.77 q^{85} +2551.29 q^{87} +927.468 q^{89} +440.514 q^{91} +64.2303 q^{93} +774.096 q^{95} +1153.90 q^{97} +447.008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{3} - 14 q^{5} + 35 q^{7} + 79 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{3} - 14 q^{5} + 35 q^{7} + 79 q^{9} + 55 q^{11} + 194 q^{13} + 118 q^{15} - 248 q^{17} + 190 q^{19} - 14 q^{21} + 34 q^{23} + 529 q^{25} - 254 q^{27} - 334 q^{29} + 172 q^{31} - 22 q^{33} - 98 q^{35} + 360 q^{37} + 360 q^{39} - 852 q^{41} + 1016 q^{43} - 2164 q^{45} + 194 q^{47} + 245 q^{49} + 836 q^{51} - 934 q^{53} - 154 q^{55} - 980 q^{57} + 542 q^{59} + 838 q^{61} + 553 q^{63} - 1780 q^{65} + 1830 q^{67} + 50 q^{69} - 422 q^{71} - 68 q^{73} - 320 q^{75} + 385 q^{77} + 8 q^{79} + 1889 q^{81} + 366 q^{83} + 2056 q^{85} + 4384 q^{87} - 524 q^{89} + 1358 q^{91} - 2242 q^{93} + 2884 q^{95} - 1336 q^{97} + 869 q^{99}+O(q^{100}) \) 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
\(3\) 8.22418 1.58274 0.791372 0.611335i \(-0.209367\pi\)
0.791372 + 0.611335i \(0.209367\pi\)
\(4\) 0 0
\(5\) −17.2362 −1.54166 −0.770828 0.637044i \(-0.780157\pi\)
−0.770828 + 0.637044i \(0.780157\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 40.6371 1.50508
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 62.9306 1.34260 0.671301 0.741185i \(-0.265736\pi\)
0.671301 + 0.741185i \(0.265736\pi\)
\(14\) 0 0
\(15\) −141.754 −2.44005
\(16\) 0 0
\(17\) −80.9209 −1.15448 −0.577241 0.816574i \(-0.695871\pi\)
−0.577241 + 0.816574i \(0.695871\pi\)
\(18\) 0 0
\(19\) −44.9110 −0.542278 −0.271139 0.962540i \(-0.587400\pi\)
−0.271139 + 0.962540i \(0.587400\pi\)
\(20\) 0 0
\(21\) 57.5693 0.598221
\(22\) 0 0
\(23\) −55.0637 −0.499199 −0.249599 0.968349i \(-0.580299\pi\)
−0.249599 + 0.968349i \(0.580299\pi\)
\(24\) 0 0
\(25\) 172.088 1.37670
\(26\) 0 0
\(27\) 112.154 0.799410
\(28\) 0 0
\(29\) 310.218 1.98642 0.993208 0.116349i \(-0.0371192\pi\)
0.993208 + 0.116349i \(0.0371192\pi\)
\(30\) 0 0
\(31\) 7.80993 0.0452486 0.0226243 0.999744i \(-0.492798\pi\)
0.0226243 + 0.999744i \(0.492798\pi\)
\(32\) 0 0
\(33\) 90.4660 0.477215
\(34\) 0 0
\(35\) −120.654 −0.582691
\(36\) 0 0
\(37\) 400.886 1.78122 0.890612 0.454764i \(-0.150276\pi\)
0.890612 + 0.454764i \(0.150276\pi\)
\(38\) 0 0
\(39\) 517.553 2.12499
\(40\) 0 0
\(41\) −470.422 −1.79189 −0.895946 0.444164i \(-0.853501\pi\)
−0.895946 + 0.444164i \(0.853501\pi\)
\(42\) 0 0
\(43\) 551.684 1.95653 0.978267 0.207348i \(-0.0664831\pi\)
0.978267 + 0.207348i \(0.0664831\pi\)
\(44\) 0 0
\(45\) −700.431 −2.32031
\(46\) 0 0
\(47\) −36.8998 −0.114519 −0.0572594 0.998359i \(-0.518236\pi\)
−0.0572594 + 0.998359i \(0.518236\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −665.508 −1.82725
\(52\) 0 0
\(53\) 71.3509 0.184921 0.0924604 0.995716i \(-0.470527\pi\)
0.0924604 + 0.995716i \(0.470527\pi\)
\(54\) 0 0
\(55\) −189.599 −0.464827
\(56\) 0 0
\(57\) −369.356 −0.858287
\(58\) 0 0
\(59\) 623.175 1.37509 0.687546 0.726141i \(-0.258688\pi\)
0.687546 + 0.726141i \(0.258688\pi\)
\(60\) 0 0
\(61\) 326.021 0.684307 0.342153 0.939644i \(-0.388844\pi\)
0.342153 + 0.939644i \(0.388844\pi\)
\(62\) 0 0
\(63\) 284.460 0.568866
\(64\) 0 0
\(65\) −1084.69 −2.06983
\(66\) 0 0
\(67\) −617.038 −1.12512 −0.562561 0.826756i \(-0.690184\pi\)
−0.562561 + 0.826756i \(0.690184\pi\)
\(68\) 0 0
\(69\) −452.854 −0.790104
\(70\) 0 0
\(71\) 719.391 1.20248 0.601240 0.799069i \(-0.294674\pi\)
0.601240 + 0.799069i \(0.294674\pi\)
\(72\) 0 0
\(73\) −244.700 −0.392329 −0.196164 0.980571i \(-0.562849\pi\)
−0.196164 + 0.980571i \(0.562849\pi\)
\(74\) 0 0
\(75\) 1415.28 2.17897
\(76\) 0 0
\(77\) 77.0000 0.113961
\(78\) 0 0
\(79\) 232.011 0.330422 0.165211 0.986258i \(-0.447170\pi\)
0.165211 + 0.986258i \(0.447170\pi\)
\(80\) 0 0
\(81\) −174.827 −0.239818
\(82\) 0 0
\(83\) −1045.98 −1.38327 −0.691633 0.722249i \(-0.743108\pi\)
−0.691633 + 0.722249i \(0.743108\pi\)
\(84\) 0 0
\(85\) 1394.77 1.77981
\(86\) 0 0
\(87\) 2551.29 3.14399
\(88\) 0 0
\(89\) 927.468 1.10462 0.552311 0.833638i \(-0.313746\pi\)
0.552311 + 0.833638i \(0.313746\pi\)
\(90\) 0 0
\(91\) 440.514 0.507456
\(92\) 0 0
\(93\) 64.2303 0.0716169
\(94\) 0 0
\(95\) 774.096 0.836006
\(96\) 0 0
\(97\) 1153.90 1.20785 0.603924 0.797042i \(-0.293603\pi\)
0.603924 + 0.797042i \(0.293603\pi\)
\(98\) 0 0
\(99\) 447.008 0.453798
\(100\) 0 0
\(101\) −1770.78 −1.74455 −0.872273 0.489019i \(-0.837355\pi\)
−0.872273 + 0.489019i \(0.837355\pi\)
\(102\) 0 0
\(103\) 673.475 0.644266 0.322133 0.946694i \(-0.395600\pi\)
0.322133 + 0.946694i \(0.395600\pi\)
\(104\) 0 0
\(105\) −992.277 −0.922251
\(106\) 0 0
\(107\) 740.840 0.669343 0.334672 0.942335i \(-0.391375\pi\)
0.334672 + 0.942335i \(0.391375\pi\)
\(108\) 0 0
\(109\) 973.956 0.855854 0.427927 0.903813i \(-0.359244\pi\)
0.427927 + 0.903813i \(0.359244\pi\)
\(110\) 0 0
\(111\) 3296.96 2.81922
\(112\) 0 0
\(113\) 1246.55 1.03775 0.518874 0.854851i \(-0.326351\pi\)
0.518874 + 0.854851i \(0.326351\pi\)
\(114\) 0 0
\(115\) 949.091 0.769593
\(116\) 0 0
\(117\) 2557.32 2.02072
\(118\) 0 0
\(119\) −566.446 −0.436353
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −3868.83 −2.83610
\(124\) 0 0
\(125\) −811.613 −0.580743
\(126\) 0 0
\(127\) 2438.63 1.70388 0.851942 0.523636i \(-0.175425\pi\)
0.851942 + 0.523636i \(0.175425\pi\)
\(128\) 0 0
\(129\) 4537.15 3.09669
\(130\) 0 0
\(131\) 1946.49 1.29821 0.649106 0.760698i \(-0.275143\pi\)
0.649106 + 0.760698i \(0.275143\pi\)
\(132\) 0 0
\(133\) −314.377 −0.204962
\(134\) 0 0
\(135\) −1933.11 −1.23241
\(136\) 0 0
\(137\) −124.996 −0.0779500 −0.0389750 0.999240i \(-0.512409\pi\)
−0.0389750 + 0.999240i \(0.512409\pi\)
\(138\) 0 0
\(139\) −567.992 −0.346593 −0.173297 0.984870i \(-0.555442\pi\)
−0.173297 + 0.984870i \(0.555442\pi\)
\(140\) 0 0
\(141\) −303.470 −0.181254
\(142\) 0 0
\(143\) 692.237 0.404809
\(144\) 0 0
\(145\) −5346.99 −3.06237
\(146\) 0 0
\(147\) 402.985 0.226106
\(148\) 0 0
\(149\) 3.14245 0.00172778 0.000863890 1.00000i \(-0.499725\pi\)
0.000863890 1.00000i \(0.499725\pi\)
\(150\) 0 0
\(151\) 506.799 0.273130 0.136565 0.990631i \(-0.456394\pi\)
0.136565 + 0.990631i \(0.456394\pi\)
\(152\) 0 0
\(153\) −3288.39 −1.73759
\(154\) 0 0
\(155\) −134.614 −0.0697577
\(156\) 0 0
\(157\) −1390.34 −0.706759 −0.353379 0.935480i \(-0.614967\pi\)
−0.353379 + 0.935480i \(0.614967\pi\)
\(158\) 0 0
\(159\) 586.803 0.292682
\(160\) 0 0
\(161\) −385.446 −0.188679
\(162\) 0 0
\(163\) −3101.28 −1.49025 −0.745126 0.666924i \(-0.767610\pi\)
−0.745126 + 0.666924i \(0.767610\pi\)
\(164\) 0 0
\(165\) −1559.29 −0.735701
\(166\) 0 0
\(167\) −947.307 −0.438951 −0.219475 0.975618i \(-0.570435\pi\)
−0.219475 + 0.975618i \(0.570435\pi\)
\(168\) 0 0
\(169\) 1763.26 0.802578
\(170\) 0 0
\(171\) −1825.05 −0.816171
\(172\) 0 0
\(173\) 1506.99 0.662278 0.331139 0.943582i \(-0.392567\pi\)
0.331139 + 0.943582i \(0.392567\pi\)
\(174\) 0 0
\(175\) 1204.61 0.520344
\(176\) 0 0
\(177\) 5125.10 2.17642
\(178\) 0 0
\(179\) 1007.93 0.420871 0.210436 0.977608i \(-0.432512\pi\)
0.210436 + 0.977608i \(0.432512\pi\)
\(180\) 0 0
\(181\) −542.440 −0.222758 −0.111379 0.993778i \(-0.535527\pi\)
−0.111379 + 0.993778i \(0.535527\pi\)
\(182\) 0 0
\(183\) 2681.25 1.08308
\(184\) 0 0
\(185\) −6909.77 −2.74603
\(186\) 0 0
\(187\) −890.130 −0.348089
\(188\) 0 0
\(189\) 785.078 0.302148
\(190\) 0 0
\(191\) −648.159 −0.245545 −0.122773 0.992435i \(-0.539179\pi\)
−0.122773 + 0.992435i \(0.539179\pi\)
\(192\) 0 0
\(193\) 3158.55 1.17802 0.589009 0.808126i \(-0.299518\pi\)
0.589009 + 0.808126i \(0.299518\pi\)
\(194\) 0 0
\(195\) −8920.66 −3.27601
\(196\) 0 0
\(197\) 2286.70 0.827008 0.413504 0.910502i \(-0.364305\pi\)
0.413504 + 0.910502i \(0.364305\pi\)
\(198\) 0 0
\(199\) 2106.32 0.750316 0.375158 0.926961i \(-0.377588\pi\)
0.375158 + 0.926961i \(0.377588\pi\)
\(200\) 0 0
\(201\) −5074.63 −1.78078
\(202\) 0 0
\(203\) 2171.53 0.750795
\(204\) 0 0
\(205\) 8108.30 2.76248
\(206\) 0 0
\(207\) −2237.63 −0.751333
\(208\) 0 0
\(209\) −494.021 −0.163503
\(210\) 0 0
\(211\) −1304.93 −0.425759 −0.212880 0.977078i \(-0.568284\pi\)
−0.212880 + 0.977078i \(0.568284\pi\)
\(212\) 0 0
\(213\) 5916.40 1.90322
\(214\) 0 0
\(215\) −9508.95 −3.01630
\(216\) 0 0
\(217\) 54.6695 0.0171024
\(218\) 0 0
\(219\) −2012.46 −0.620956
\(220\) 0 0
\(221\) −5092.40 −1.55001
\(222\) 0 0
\(223\) 4779.08 1.43511 0.717557 0.696499i \(-0.245260\pi\)
0.717557 + 0.696499i \(0.245260\pi\)
\(224\) 0 0
\(225\) 6993.15 2.07204
\(226\) 0 0
\(227\) −2523.58 −0.737868 −0.368934 0.929456i \(-0.620277\pi\)
−0.368934 + 0.929456i \(0.620277\pi\)
\(228\) 0 0
\(229\) −3831.98 −1.10578 −0.552892 0.833253i \(-0.686476\pi\)
−0.552892 + 0.833253i \(0.686476\pi\)
\(230\) 0 0
\(231\) 633.262 0.180370
\(232\) 0 0
\(233\) −1317.12 −0.370333 −0.185167 0.982707i \(-0.559282\pi\)
−0.185167 + 0.982707i \(0.559282\pi\)
\(234\) 0 0
\(235\) 636.013 0.176549
\(236\) 0 0
\(237\) 1908.10 0.522973
\(238\) 0 0
\(239\) −3152.14 −0.853117 −0.426559 0.904460i \(-0.640274\pi\)
−0.426559 + 0.904460i \(0.640274\pi\)
\(240\) 0 0
\(241\) 3979.97 1.06379 0.531893 0.846812i \(-0.321481\pi\)
0.531893 + 0.846812i \(0.321481\pi\)
\(242\) 0 0
\(243\) −4465.97 −1.17898
\(244\) 0 0
\(245\) −844.575 −0.220236
\(246\) 0 0
\(247\) −2826.27 −0.728063
\(248\) 0 0
\(249\) −8602.32 −2.18936
\(250\) 0 0
\(251\) 4434.79 1.11522 0.557612 0.830101i \(-0.311718\pi\)
0.557612 + 0.830101i \(0.311718\pi\)
\(252\) 0 0
\(253\) −605.701 −0.150514
\(254\) 0 0
\(255\) 11470.8 2.81699
\(256\) 0 0
\(257\) −4581.08 −1.11191 −0.555953 0.831214i \(-0.687646\pi\)
−0.555953 + 0.831214i \(0.687646\pi\)
\(258\) 0 0
\(259\) 2806.20 0.673239
\(260\) 0 0
\(261\) 12606.4 2.98971
\(262\) 0 0
\(263\) 2208.39 0.517776 0.258888 0.965907i \(-0.416644\pi\)
0.258888 + 0.965907i \(0.416644\pi\)
\(264\) 0 0
\(265\) −1229.82 −0.285084
\(266\) 0 0
\(267\) 7627.66 1.74833
\(268\) 0 0
\(269\) 772.816 0.175165 0.0875826 0.996157i \(-0.472086\pi\)
0.0875826 + 0.996157i \(0.472086\pi\)
\(270\) 0 0
\(271\) −6671.13 −1.49536 −0.747679 0.664060i \(-0.768832\pi\)
−0.747679 + 0.664060i \(0.768832\pi\)
\(272\) 0 0
\(273\) 3622.87 0.803172
\(274\) 0 0
\(275\) 1892.96 0.415091
\(276\) 0 0
\(277\) −4620.41 −1.00222 −0.501108 0.865385i \(-0.667074\pi\)
−0.501108 + 0.865385i \(0.667074\pi\)
\(278\) 0 0
\(279\) 317.373 0.0681026
\(280\) 0 0
\(281\) −3501.89 −0.743436 −0.371718 0.928346i \(-0.621231\pi\)
−0.371718 + 0.928346i \(0.621231\pi\)
\(282\) 0 0
\(283\) 4388.45 0.921789 0.460895 0.887455i \(-0.347529\pi\)
0.460895 + 0.887455i \(0.347529\pi\)
\(284\) 0 0
\(285\) 6366.30 1.32318
\(286\) 0 0
\(287\) −3292.95 −0.677271
\(288\) 0 0
\(289\) 1635.19 0.332829
\(290\) 0 0
\(291\) 9489.91 1.91171
\(292\) 0 0
\(293\) 647.804 0.129164 0.0645821 0.997912i \(-0.479429\pi\)
0.0645821 + 0.997912i \(0.479429\pi\)
\(294\) 0 0
\(295\) −10741.2 −2.11992
\(296\) 0 0
\(297\) 1233.69 0.241031
\(298\) 0 0
\(299\) −3465.19 −0.670225
\(300\) 0 0
\(301\) 3861.79 0.739501
\(302\) 0 0
\(303\) −14563.2 −2.76117
\(304\) 0 0
\(305\) −5619.37 −1.05497
\(306\) 0 0
\(307\) 4964.98 0.923017 0.461508 0.887136i \(-0.347308\pi\)
0.461508 + 0.887136i \(0.347308\pi\)
\(308\) 0 0
\(309\) 5538.78 1.01971
\(310\) 0 0
\(311\) 2882.26 0.525524 0.262762 0.964861i \(-0.415367\pi\)
0.262762 + 0.964861i \(0.415367\pi\)
\(312\) 0 0
\(313\) −6569.36 −1.18633 −0.593167 0.805080i \(-0.702123\pi\)
−0.593167 + 0.805080i \(0.702123\pi\)
\(314\) 0 0
\(315\) −4903.01 −0.876996
\(316\) 0 0
\(317\) −10191.8 −1.80576 −0.902880 0.429892i \(-0.858552\pi\)
−0.902880 + 0.429892i \(0.858552\pi\)
\(318\) 0 0
\(319\) 3412.40 0.598927
\(320\) 0 0
\(321\) 6092.80 1.05940
\(322\) 0 0
\(323\) 3634.23 0.626050
\(324\) 0 0
\(325\) 10829.6 1.84836
\(326\) 0 0
\(327\) 8009.99 1.35460
\(328\) 0 0
\(329\) −258.298 −0.0432840
\(330\) 0 0
\(331\) −8701.15 −1.44489 −0.722445 0.691429i \(-0.756982\pi\)
−0.722445 + 0.691429i \(0.756982\pi\)
\(332\) 0 0
\(333\) 16290.9 2.68088
\(334\) 0 0
\(335\) 10635.4 1.73455
\(336\) 0 0
\(337\) 8820.95 1.42584 0.712919 0.701246i \(-0.247373\pi\)
0.712919 + 0.701246i \(0.247373\pi\)
\(338\) 0 0
\(339\) 10251.9 1.64249
\(340\) 0 0
\(341\) 85.9093 0.0136430
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 7805.49 1.21807
\(346\) 0 0
\(347\) 444.321 0.0687389 0.0343694 0.999409i \(-0.489058\pi\)
0.0343694 + 0.999409i \(0.489058\pi\)
\(348\) 0 0
\(349\) −3522.75 −0.540311 −0.270156 0.962817i \(-0.587075\pi\)
−0.270156 + 0.962817i \(0.587075\pi\)
\(350\) 0 0
\(351\) 7057.93 1.07329
\(352\) 0 0
\(353\) −8301.61 −1.25170 −0.625850 0.779944i \(-0.715248\pi\)
−0.625850 + 0.779944i \(0.715248\pi\)
\(354\) 0 0
\(355\) −12399.6 −1.85381
\(356\) 0 0
\(357\) −4658.55 −0.690635
\(358\) 0 0
\(359\) 988.600 0.145338 0.0726690 0.997356i \(-0.476848\pi\)
0.0726690 + 0.997356i \(0.476848\pi\)
\(360\) 0 0
\(361\) −4842.01 −0.705935
\(362\) 0 0
\(363\) 995.126 0.143886
\(364\) 0 0
\(365\) 4217.71 0.604835
\(366\) 0 0
\(367\) −539.911 −0.0767932 −0.0383966 0.999263i \(-0.512225\pi\)
−0.0383966 + 0.999263i \(0.512225\pi\)
\(368\) 0 0
\(369\) −19116.6 −2.69694
\(370\) 0 0
\(371\) 499.457 0.0698935
\(372\) 0 0
\(373\) −7443.64 −1.03329 −0.516645 0.856200i \(-0.672819\pi\)
−0.516645 + 0.856200i \(0.672819\pi\)
\(374\) 0 0
\(375\) −6674.85 −0.919168
\(376\) 0 0
\(377\) 19522.2 2.66697
\(378\) 0 0
\(379\) 10087.0 1.36711 0.683553 0.729901i \(-0.260434\pi\)
0.683553 + 0.729901i \(0.260434\pi\)
\(380\) 0 0
\(381\) 20055.7 2.69681
\(382\) 0 0
\(383\) 7400.26 0.987299 0.493650 0.869661i \(-0.335663\pi\)
0.493650 + 0.869661i \(0.335663\pi\)
\(384\) 0 0
\(385\) −1327.19 −0.175688
\(386\) 0 0
\(387\) 22418.8 2.94474
\(388\) 0 0
\(389\) 459.213 0.0598536 0.0299268 0.999552i \(-0.490473\pi\)
0.0299268 + 0.999552i \(0.490473\pi\)
\(390\) 0 0
\(391\) 4455.80 0.576316
\(392\) 0 0
\(393\) 16008.3 2.05474
\(394\) 0 0
\(395\) −3999.00 −0.509397
\(396\) 0 0
\(397\) 5539.55 0.700308 0.350154 0.936692i \(-0.386129\pi\)
0.350154 + 0.936692i \(0.386129\pi\)
\(398\) 0 0
\(399\) −2585.49 −0.324402
\(400\) 0 0
\(401\) −5070.83 −0.631485 −0.315742 0.948845i \(-0.602253\pi\)
−0.315742 + 0.948845i \(0.602253\pi\)
\(402\) 0 0
\(403\) 491.484 0.0607508
\(404\) 0 0
\(405\) 3013.36 0.369716
\(406\) 0 0
\(407\) 4409.75 0.537059
\(408\) 0 0
\(409\) −10086.7 −1.21944 −0.609722 0.792615i \(-0.708719\pi\)
−0.609722 + 0.792615i \(0.708719\pi\)
\(410\) 0 0
\(411\) −1027.99 −0.123375
\(412\) 0 0
\(413\) 4362.22 0.519736
\(414\) 0 0
\(415\) 18028.7 2.13252
\(416\) 0 0
\(417\) −4671.27 −0.548568
\(418\) 0 0
\(419\) −14633.3 −1.70616 −0.853081 0.521779i \(-0.825269\pi\)
−0.853081 + 0.521779i \(0.825269\pi\)
\(420\) 0 0
\(421\) −11472.5 −1.32811 −0.664054 0.747685i \(-0.731165\pi\)
−0.664054 + 0.747685i \(0.731165\pi\)
\(422\) 0 0
\(423\) −1499.50 −0.172360
\(424\) 0 0
\(425\) −13925.5 −1.58938
\(426\) 0 0
\(427\) 2282.15 0.258644
\(428\) 0 0
\(429\) 5693.08 0.640710
\(430\) 0 0
\(431\) 9873.29 1.10343 0.551717 0.834032i \(-0.313973\pi\)
0.551717 + 0.834032i \(0.313973\pi\)
\(432\) 0 0
\(433\) 9246.94 1.02628 0.513140 0.858305i \(-0.328482\pi\)
0.513140 + 0.858305i \(0.328482\pi\)
\(434\) 0 0
\(435\) −43974.6 −4.84695
\(436\) 0 0
\(437\) 2472.96 0.270705
\(438\) 0 0
\(439\) −2826.08 −0.307247 −0.153624 0.988129i \(-0.549094\pi\)
−0.153624 + 0.988129i \(0.549094\pi\)
\(440\) 0 0
\(441\) 1991.22 0.215011
\(442\) 0 0
\(443\) −18182.2 −1.95003 −0.975014 0.222142i \(-0.928695\pi\)
−0.975014 + 0.222142i \(0.928695\pi\)
\(444\) 0 0
\(445\) −15986.0 −1.70295
\(446\) 0 0
\(447\) 25.8440 0.00273463
\(448\) 0 0
\(449\) −1091.57 −0.114731 −0.0573656 0.998353i \(-0.518270\pi\)
−0.0573656 + 0.998353i \(0.518270\pi\)
\(450\) 0 0
\(451\) −5174.64 −0.540275
\(452\) 0 0
\(453\) 4168.00 0.432295
\(454\) 0 0
\(455\) −7592.81 −0.782321
\(456\) 0 0
\(457\) −7689.22 −0.787060 −0.393530 0.919312i \(-0.628746\pi\)
−0.393530 + 0.919312i \(0.628746\pi\)
\(458\) 0 0
\(459\) −9075.60 −0.922904
\(460\) 0 0
\(461\) −8159.13 −0.824313 −0.412157 0.911113i \(-0.635224\pi\)
−0.412157 + 0.911113i \(0.635224\pi\)
\(462\) 0 0
\(463\) −8483.01 −0.851488 −0.425744 0.904844i \(-0.639988\pi\)
−0.425744 + 0.904844i \(0.639988\pi\)
\(464\) 0 0
\(465\) −1107.09 −0.110409
\(466\) 0 0
\(467\) −1373.10 −0.136058 −0.0680292 0.997683i \(-0.521671\pi\)
−0.0680292 + 0.997683i \(0.521671\pi\)
\(468\) 0 0
\(469\) −4319.27 −0.425256
\(470\) 0 0
\(471\) −11434.4 −1.11862
\(472\) 0 0
\(473\) 6068.52 0.589917
\(474\) 0 0
\(475\) −7728.62 −0.746555
\(476\) 0 0
\(477\) 2899.50 0.278320
\(478\) 0 0
\(479\) −4492.84 −0.428566 −0.214283 0.976772i \(-0.568741\pi\)
−0.214283 + 0.976772i \(0.568741\pi\)
\(480\) 0 0
\(481\) 25228.0 2.39147
\(482\) 0 0
\(483\) −3169.98 −0.298631
\(484\) 0 0
\(485\) −19889.0 −1.86208
\(486\) 0 0
\(487\) −5602.38 −0.521290 −0.260645 0.965435i \(-0.583935\pi\)
−0.260645 + 0.965435i \(0.583935\pi\)
\(488\) 0 0
\(489\) −25505.5 −2.35869
\(490\) 0 0
\(491\) 10278.7 0.944747 0.472374 0.881398i \(-0.343397\pi\)
0.472374 + 0.881398i \(0.343397\pi\)
\(492\) 0 0
\(493\) −25103.1 −2.29328
\(494\) 0 0
\(495\) −7704.74 −0.699600
\(496\) 0 0
\(497\) 5035.74 0.454495
\(498\) 0 0
\(499\) 16802.7 1.50740 0.753698 0.657221i \(-0.228268\pi\)
0.753698 + 0.657221i \(0.228268\pi\)
\(500\) 0 0
\(501\) −7790.82 −0.694747
\(502\) 0 0
\(503\) 2510.54 0.222543 0.111272 0.993790i \(-0.464508\pi\)
0.111272 + 0.993790i \(0.464508\pi\)
\(504\) 0 0
\(505\) 30521.6 2.68949
\(506\) 0 0
\(507\) 14501.4 1.27027
\(508\) 0 0
\(509\) −13608.8 −1.18507 −0.592533 0.805546i \(-0.701872\pi\)
−0.592533 + 0.805546i \(0.701872\pi\)
\(510\) 0 0
\(511\) −1712.90 −0.148286
\(512\) 0 0
\(513\) −5036.95 −0.433502
\(514\) 0 0
\(515\) −11608.2 −0.993237
\(516\) 0 0
\(517\) −405.897 −0.0345287
\(518\) 0 0
\(519\) 12393.7 1.04822
\(520\) 0 0
\(521\) 212.794 0.0178938 0.00894689 0.999960i \(-0.497152\pi\)
0.00894689 + 0.999960i \(0.497152\pi\)
\(522\) 0 0
\(523\) −180.086 −0.0150566 −0.00752831 0.999972i \(-0.502396\pi\)
−0.00752831 + 0.999972i \(0.502396\pi\)
\(524\) 0 0
\(525\) 9906.96 0.823571
\(526\) 0 0
\(527\) −631.987 −0.0522387
\(528\) 0 0
\(529\) −9134.99 −0.750800
\(530\) 0 0
\(531\) 25324.0 2.06962
\(532\) 0 0
\(533\) −29603.9 −2.40579
\(534\) 0 0
\(535\) −12769.3 −1.03190
\(536\) 0 0
\(537\) 8289.37 0.666131
\(538\) 0 0
\(539\) 539.000 0.0430730
\(540\) 0 0
\(541\) −913.428 −0.0725903 −0.0362951 0.999341i \(-0.511556\pi\)
−0.0362951 + 0.999341i \(0.511556\pi\)
\(542\) 0 0
\(543\) −4461.13 −0.352570
\(544\) 0 0
\(545\) −16787.3 −1.31943
\(546\) 0 0
\(547\) −4777.46 −0.373436 −0.186718 0.982414i \(-0.559785\pi\)
−0.186718 + 0.982414i \(0.559785\pi\)
\(548\) 0 0
\(549\) 13248.6 1.02994
\(550\) 0 0
\(551\) −13932.2 −1.07719
\(552\) 0 0
\(553\) 1624.08 0.124888
\(554\) 0 0
\(555\) −56827.2 −4.34627
\(556\) 0 0
\(557\) −22543.7 −1.71492 −0.857458 0.514553i \(-0.827958\pi\)
−0.857458 + 0.514553i \(0.827958\pi\)
\(558\) 0 0
\(559\) 34717.8 2.62685
\(560\) 0 0
\(561\) −7320.58 −0.550936
\(562\) 0 0
\(563\) −4815.28 −0.360462 −0.180231 0.983624i \(-0.557685\pi\)
−0.180231 + 0.983624i \(0.557685\pi\)
\(564\) 0 0
\(565\) −21485.8 −1.59985
\(566\) 0 0
\(567\) −1223.79 −0.0906425
\(568\) 0 0
\(569\) −11312.7 −0.833482 −0.416741 0.909025i \(-0.636828\pi\)
−0.416741 + 0.909025i \(0.636828\pi\)
\(570\) 0 0
\(571\) 709.967 0.0520336 0.0260168 0.999662i \(-0.491718\pi\)
0.0260168 + 0.999662i \(0.491718\pi\)
\(572\) 0 0
\(573\) −5330.58 −0.388635
\(574\) 0 0
\(575\) −9475.78 −0.687248
\(576\) 0 0
\(577\) 22452.0 1.61991 0.809956 0.586490i \(-0.199491\pi\)
0.809956 + 0.586490i \(0.199491\pi\)
\(578\) 0 0
\(579\) 25976.5 1.86450
\(580\) 0 0
\(581\) −7321.85 −0.522825
\(582\) 0 0
\(583\) 784.860 0.0557557
\(584\) 0 0
\(585\) −44078.5 −3.11525
\(586\) 0 0
\(587\) 6232.49 0.438233 0.219116 0.975699i \(-0.429683\pi\)
0.219116 + 0.975699i \(0.429683\pi\)
\(588\) 0 0
\(589\) −350.752 −0.0245373
\(590\) 0 0
\(591\) 18806.2 1.30894
\(592\) 0 0
\(593\) 2650.46 0.183544 0.0917718 0.995780i \(-0.470747\pi\)
0.0917718 + 0.995780i \(0.470747\pi\)
\(594\) 0 0
\(595\) 9763.40 0.672706
\(596\) 0 0
\(597\) 17322.7 1.18756
\(598\) 0 0
\(599\) 19416.6 1.32444 0.662220 0.749310i \(-0.269615\pi\)
0.662220 + 0.749310i \(0.269615\pi\)
\(600\) 0 0
\(601\) −28676.0 −1.94628 −0.973142 0.230204i \(-0.926061\pi\)
−0.973142 + 0.230204i \(0.926061\pi\)
\(602\) 0 0
\(603\) −25074.6 −1.69340
\(604\) 0 0
\(605\) −2085.58 −0.140150
\(606\) 0 0
\(607\) 5980.13 0.399878 0.199939 0.979808i \(-0.435926\pi\)
0.199939 + 0.979808i \(0.435926\pi\)
\(608\) 0 0
\(609\) 17859.0 1.18832
\(610\) 0 0
\(611\) −2322.13 −0.153753
\(612\) 0 0
\(613\) −24033.0 −1.58350 −0.791750 0.610845i \(-0.790830\pi\)
−0.791750 + 0.610845i \(0.790830\pi\)
\(614\) 0 0
\(615\) 66684.1 4.37230
\(616\) 0 0
\(617\) 2485.18 0.162155 0.0810776 0.996708i \(-0.474164\pi\)
0.0810776 + 0.996708i \(0.474164\pi\)
\(618\) 0 0
\(619\) 21047.6 1.36668 0.683339 0.730101i \(-0.260527\pi\)
0.683339 + 0.730101i \(0.260527\pi\)
\(620\) 0 0
\(621\) −6175.62 −0.399064
\(622\) 0 0
\(623\) 6492.27 0.417508
\(624\) 0 0
\(625\) −7521.80 −0.481395
\(626\) 0 0
\(627\) −4062.91 −0.258783
\(628\) 0 0
\(629\) −32440.1 −2.05639
\(630\) 0 0
\(631\) −3086.63 −0.194733 −0.0973667 0.995249i \(-0.531042\pi\)
−0.0973667 + 0.995249i \(0.531042\pi\)
\(632\) 0 0
\(633\) −10732.0 −0.673868
\(634\) 0 0
\(635\) −42032.8 −2.62680
\(636\) 0 0
\(637\) 3083.60 0.191800
\(638\) 0 0
\(639\) 29234.0 1.80983
\(640\) 0 0
\(641\) 1120.56 0.0690475 0.0345238 0.999404i \(-0.489009\pi\)
0.0345238 + 0.999404i \(0.489009\pi\)
\(642\) 0 0
\(643\) −5004.81 −0.306952 −0.153476 0.988152i \(-0.549047\pi\)
−0.153476 + 0.988152i \(0.549047\pi\)
\(644\) 0 0
\(645\) −78203.3 −4.77403
\(646\) 0 0
\(647\) −14422.2 −0.876348 −0.438174 0.898890i \(-0.644375\pi\)
−0.438174 + 0.898890i \(0.644375\pi\)
\(648\) 0 0
\(649\) 6854.92 0.414606
\(650\) 0 0
\(651\) 449.612 0.0270686
\(652\) 0 0
\(653\) −1456.18 −0.0872660 −0.0436330 0.999048i \(-0.513893\pi\)
−0.0436330 + 0.999048i \(0.513893\pi\)
\(654\) 0 0
\(655\) −33550.2 −2.00140
\(656\) 0 0
\(657\) −9943.91 −0.590485
\(658\) 0 0
\(659\) 30655.2 1.81207 0.906037 0.423198i \(-0.139092\pi\)
0.906037 + 0.423198i \(0.139092\pi\)
\(660\) 0 0
\(661\) 15079.2 0.887311 0.443656 0.896197i \(-0.353681\pi\)
0.443656 + 0.896197i \(0.353681\pi\)
\(662\) 0 0
\(663\) −41880.8 −2.45327
\(664\) 0 0
\(665\) 5418.67 0.315980
\(666\) 0 0
\(667\) −17081.8 −0.991617
\(668\) 0 0
\(669\) 39304.0 2.27142
\(670\) 0 0
\(671\) 3586.23 0.206326
\(672\) 0 0
\(673\) 1713.08 0.0981197 0.0490599 0.998796i \(-0.484377\pi\)
0.0490599 + 0.998796i \(0.484377\pi\)
\(674\) 0 0
\(675\) 19300.3 1.10055
\(676\) 0 0
\(677\) 26093.5 1.48132 0.740661 0.671879i \(-0.234513\pi\)
0.740661 + 0.671879i \(0.234513\pi\)
\(678\) 0 0
\(679\) 8077.33 0.456523
\(680\) 0 0
\(681\) −20754.4 −1.16786
\(682\) 0 0
\(683\) −5309.57 −0.297460 −0.148730 0.988878i \(-0.547518\pi\)
−0.148730 + 0.988878i \(0.547518\pi\)
\(684\) 0 0
\(685\) 2154.46 0.120172
\(686\) 0 0
\(687\) −31514.9 −1.75017
\(688\) 0 0
\(689\) 4490.16 0.248275
\(690\) 0 0
\(691\) −658.594 −0.0362577 −0.0181289 0.999836i \(-0.505771\pi\)
−0.0181289 + 0.999836i \(0.505771\pi\)
\(692\) 0 0
\(693\) 3129.06 0.171520
\(694\) 0 0
\(695\) 9790.04 0.534327
\(696\) 0 0
\(697\) 38066.9 2.06871
\(698\) 0 0
\(699\) −10832.3 −0.586142
\(700\) 0 0
\(701\) 29457.6 1.58716 0.793580 0.608466i \(-0.208215\pi\)
0.793580 + 0.608466i \(0.208215\pi\)
\(702\) 0 0
\(703\) −18004.2 −0.965919
\(704\) 0 0
\(705\) 5230.68 0.279431
\(706\) 0 0
\(707\) −12395.5 −0.659377
\(708\) 0 0
\(709\) 6831.93 0.361888 0.180944 0.983493i \(-0.442085\pi\)
0.180944 + 0.983493i \(0.442085\pi\)
\(710\) 0 0
\(711\) 9428.28 0.497311
\(712\) 0 0
\(713\) −430.044 −0.0225880
\(714\) 0 0
\(715\) −11931.6 −0.624077
\(716\) 0 0
\(717\) −25923.8 −1.35027
\(718\) 0 0
\(719\) −21721.3 −1.12666 −0.563330 0.826232i \(-0.690480\pi\)
−0.563330 + 0.826232i \(0.690480\pi\)
\(720\) 0 0
\(721\) 4714.32 0.243510
\(722\) 0 0
\(723\) 32732.0 1.68370
\(724\) 0 0
\(725\) 53384.7 2.73470
\(726\) 0 0
\(727\) −19491.9 −0.994382 −0.497191 0.867641i \(-0.665635\pi\)
−0.497191 + 0.867641i \(0.665635\pi\)
\(728\) 0 0
\(729\) −32008.6 −1.62620
\(730\) 0 0
\(731\) −44642.7 −2.25878
\(732\) 0 0
\(733\) 14205.8 0.715829 0.357915 0.933754i \(-0.383488\pi\)
0.357915 + 0.933754i \(0.383488\pi\)
\(734\) 0 0
\(735\) −6945.94 −0.348578
\(736\) 0 0
\(737\) −6787.42 −0.339237
\(738\) 0 0
\(739\) 12836.5 0.638970 0.319485 0.947591i \(-0.396490\pi\)
0.319485 + 0.947591i \(0.396490\pi\)
\(740\) 0 0
\(741\) −23243.8 −1.15234
\(742\) 0 0
\(743\) 19694.2 0.972421 0.486211 0.873842i \(-0.338379\pi\)
0.486211 + 0.873842i \(0.338379\pi\)
\(744\) 0 0
\(745\) −54.1639 −0.00266364
\(746\) 0 0
\(747\) −42505.6 −2.08192
\(748\) 0 0
\(749\) 5185.88 0.252988
\(750\) 0 0
\(751\) −11383.4 −0.553111 −0.276555 0.960998i \(-0.589193\pi\)
−0.276555 + 0.960998i \(0.589193\pi\)
\(752\) 0 0
\(753\) 36472.5 1.76512
\(754\) 0 0
\(755\) −8735.30 −0.421073
\(756\) 0 0
\(757\) −16665.7 −0.800165 −0.400083 0.916479i \(-0.631019\pi\)
−0.400083 + 0.916479i \(0.631019\pi\)
\(758\) 0 0
\(759\) −4981.39 −0.238225
\(760\) 0 0
\(761\) 3634.33 0.173120 0.0865600 0.996247i \(-0.472413\pi\)
0.0865600 + 0.996247i \(0.472413\pi\)
\(762\) 0 0
\(763\) 6817.69 0.323482
\(764\) 0 0
\(765\) 56679.5 2.67876
\(766\) 0 0
\(767\) 39216.8 1.84620
\(768\) 0 0
\(769\) 18467.7 0.866012 0.433006 0.901391i \(-0.357453\pi\)
0.433006 + 0.901391i \(0.357453\pi\)
\(770\) 0 0
\(771\) −37675.6 −1.75986
\(772\) 0 0
\(773\) −2076.66 −0.0966265 −0.0483133 0.998832i \(-0.515385\pi\)
−0.0483133 + 0.998832i \(0.515385\pi\)
\(774\) 0 0
\(775\) 1343.99 0.0622938
\(776\) 0 0
\(777\) 23078.7 1.06557
\(778\) 0 0
\(779\) 21127.1 0.971703
\(780\) 0 0
\(781\) 7913.31 0.362561
\(782\) 0 0
\(783\) 34792.2 1.58796
\(784\) 0 0
\(785\) 23964.2 1.08958
\(786\) 0 0
\(787\) 424.318 0.0192189 0.00960947 0.999954i \(-0.496941\pi\)
0.00960947 + 0.999954i \(0.496941\pi\)
\(788\) 0 0
\(789\) 18162.2 0.819506
\(790\) 0 0
\(791\) 8725.85 0.392232
\(792\) 0 0
\(793\) 20516.7 0.918751
\(794\) 0 0
\(795\) −10114.3 −0.451215
\(796\) 0 0
\(797\) −7198.00 −0.319907 −0.159954 0.987125i \(-0.551134\pi\)
−0.159954 + 0.987125i \(0.551134\pi\)
\(798\) 0 0
\(799\) 2985.96 0.132210
\(800\) 0 0
\(801\) 37689.6 1.66254
\(802\) 0 0
\(803\) −2691.70 −0.118292
\(804\) 0 0
\(805\) 6643.63 0.290879
\(806\) 0 0
\(807\) 6355.78 0.277242
\(808\) 0 0
\(809\) −39155.0 −1.70163 −0.850813 0.525469i \(-0.823890\pi\)
−0.850813 + 0.525469i \(0.823890\pi\)
\(810\) 0 0
\(811\) 2976.22 0.128865 0.0644323 0.997922i \(-0.479476\pi\)
0.0644323 + 0.997922i \(0.479476\pi\)
\(812\) 0 0
\(813\) −54864.6 −2.36677
\(814\) 0 0
\(815\) 53454.4 2.29745
\(816\) 0 0
\(817\) −24776.7 −1.06099
\(818\) 0 0
\(819\) 17901.2 0.763760
\(820\) 0 0
\(821\) 35168.0 1.49497 0.747485 0.664279i \(-0.231261\pi\)
0.747485 + 0.664279i \(0.231261\pi\)
\(822\) 0 0
\(823\) −24599.8 −1.04192 −0.520958 0.853583i \(-0.674425\pi\)
−0.520958 + 0.853583i \(0.674425\pi\)
\(824\) 0 0
\(825\) 15568.1 0.656983
\(826\) 0 0
\(827\) 41171.7 1.73117 0.865586 0.500760i \(-0.166946\pi\)
0.865586 + 0.500760i \(0.166946\pi\)
\(828\) 0 0
\(829\) 31895.5 1.33628 0.668141 0.744035i \(-0.267090\pi\)
0.668141 + 0.744035i \(0.267090\pi\)
\(830\) 0 0
\(831\) −37999.1 −1.58625
\(832\) 0 0
\(833\) −3965.12 −0.164926
\(834\) 0 0
\(835\) 16328.0 0.676711
\(836\) 0 0
\(837\) 875.916 0.0361721
\(838\) 0 0
\(839\) 1010.79 0.0415927 0.0207964 0.999784i \(-0.493380\pi\)
0.0207964 + 0.999784i \(0.493380\pi\)
\(840\) 0 0
\(841\) 71846.4 2.94585
\(842\) 0 0
\(843\) −28800.2 −1.17667
\(844\) 0 0
\(845\) −30392.0 −1.23730
\(846\) 0 0
\(847\) 847.000 0.0343604
\(848\) 0 0
\(849\) 36091.4 1.45896
\(850\) 0 0
\(851\) −22074.3 −0.889185
\(852\) 0 0
\(853\) 25953.4 1.04177 0.520883 0.853628i \(-0.325603\pi\)
0.520883 + 0.853628i \(0.325603\pi\)
\(854\) 0 0
\(855\) 31457.0 1.25825
\(856\) 0 0
\(857\) 30216.2 1.20439 0.602197 0.798348i \(-0.294292\pi\)
0.602197 + 0.798348i \(0.294292\pi\)
\(858\) 0 0
\(859\) 847.560 0.0336652 0.0168326 0.999858i \(-0.494642\pi\)
0.0168326 + 0.999858i \(0.494642\pi\)
\(860\) 0 0
\(861\) −27081.8 −1.07195
\(862\) 0 0
\(863\) 48862.9 1.92736 0.963680 0.267060i \(-0.0860522\pi\)
0.963680 + 0.267060i \(0.0860522\pi\)
\(864\) 0 0
\(865\) −25974.8 −1.02100
\(866\) 0 0
\(867\) 13448.1 0.526783
\(868\) 0 0
\(869\) 2552.13 0.0996259
\(870\) 0 0
\(871\) −38830.6 −1.51059
\(872\) 0 0
\(873\) 46891.3 1.81790
\(874\) 0 0
\(875\) −5681.29 −0.219500
\(876\) 0 0
\(877\) 14942.0 0.575321 0.287660 0.957733i \(-0.407123\pi\)
0.287660 + 0.957733i \(0.407123\pi\)
\(878\) 0 0
\(879\) 5327.66 0.204434
\(880\) 0 0
\(881\) 40955.4 1.56620 0.783101 0.621895i \(-0.213637\pi\)
0.783101 + 0.621895i \(0.213637\pi\)
\(882\) 0 0
\(883\) −31473.5 −1.19951 −0.599755 0.800184i \(-0.704735\pi\)
−0.599755 + 0.800184i \(0.704735\pi\)
\(884\) 0 0
\(885\) −88337.4 −3.35529
\(886\) 0 0
\(887\) 29023.6 1.09867 0.549333 0.835604i \(-0.314882\pi\)
0.549333 + 0.835604i \(0.314882\pi\)
\(888\) 0 0
\(889\) 17070.4 0.644008
\(890\) 0 0
\(891\) −1923.10 −0.0723077
\(892\) 0 0
\(893\) 1657.20 0.0621010
\(894\) 0 0
\(895\) −17372.9 −0.648838
\(896\) 0 0
\(897\) −28498.4 −1.06079
\(898\) 0 0
\(899\) 2422.78 0.0898825
\(900\) 0 0
\(901\) −5773.78 −0.213488
\(902\) 0 0
\(903\) 31760.0 1.17044
\(904\) 0 0
\(905\) 9349.63 0.343417
\(906\) 0 0
\(907\) −33337.4 −1.22045 −0.610226 0.792227i \(-0.708921\pi\)
−0.610226 + 0.792227i \(0.708921\pi\)
\(908\) 0 0
\(909\) −71959.4 −2.62568
\(910\) 0 0
\(911\) 5676.54 0.206446 0.103223 0.994658i \(-0.467085\pi\)
0.103223 + 0.994658i \(0.467085\pi\)
\(912\) 0 0
\(913\) −11505.8 −0.417070
\(914\) 0 0
\(915\) −46214.7 −1.66974
\(916\) 0 0
\(917\) 13625.5 0.490678
\(918\) 0 0
\(919\) 28155.7 1.01063 0.505316 0.862934i \(-0.331376\pi\)
0.505316 + 0.862934i \(0.331376\pi\)
\(920\) 0 0
\(921\) 40832.8 1.46090
\(922\) 0 0
\(923\) 45271.7 1.61445
\(924\) 0 0
\(925\) 68987.6 2.45221
\(926\) 0 0
\(927\) 27368.1 0.969671
\(928\) 0 0
\(929\) −30763.7 −1.08646 −0.543232 0.839583i \(-0.682800\pi\)
−0.543232 + 0.839583i \(0.682800\pi\)
\(930\) 0 0
\(931\) −2200.64 −0.0774683
\(932\) 0 0
\(933\) 23704.2 0.831771
\(934\) 0 0
\(935\) 15342.5 0.536634
\(936\) 0 0
\(937\) 16888.6 0.588824 0.294412 0.955679i \(-0.404876\pi\)
0.294412 + 0.955679i \(0.404876\pi\)
\(938\) 0 0
\(939\) −54027.6 −1.87766
\(940\) 0 0
\(941\) −42479.9 −1.47163 −0.735816 0.677181i \(-0.763201\pi\)
−0.735816 + 0.677181i \(0.763201\pi\)
\(942\) 0 0
\(943\) 25903.2 0.894510
\(944\) 0 0
\(945\) −13531.8 −0.465809
\(946\) 0 0
\(947\) −47593.1 −1.63312 −0.816561 0.577259i \(-0.804122\pi\)
−0.816561 + 0.577259i \(0.804122\pi\)
\(948\) 0 0
\(949\) −15399.1 −0.526741
\(950\) 0 0
\(951\) −83818.8 −2.85806
\(952\) 0 0
\(953\) 31801.0 1.08094 0.540469 0.841364i \(-0.318247\pi\)
0.540469 + 0.841364i \(0.318247\pi\)
\(954\) 0 0
\(955\) 11171.8 0.378546
\(956\) 0 0
\(957\) 28064.2 0.947948
\(958\) 0 0
\(959\) −874.974 −0.0294623
\(960\) 0 0
\(961\) −29730.0 −0.997953
\(962\) 0 0
\(963\) 30105.6 1.00741
\(964\) 0 0
\(965\) −54441.5 −1.81610
\(966\) 0 0
\(967\) −45625.3 −1.51728 −0.758641 0.651509i \(-0.774136\pi\)
−0.758641 + 0.651509i \(0.774136\pi\)
\(968\) 0 0
\(969\) 29888.6 0.990877
\(970\) 0 0
\(971\) 32698.0 1.08067 0.540333 0.841451i \(-0.318298\pi\)
0.540333 + 0.841451i \(0.318298\pi\)
\(972\) 0 0
\(973\) −3975.94 −0.131000
\(974\) 0 0
\(975\) 89064.4 2.92548
\(976\) 0 0
\(977\) −43812.9 −1.43470 −0.717349 0.696714i \(-0.754645\pi\)
−0.717349 + 0.696714i \(0.754645\pi\)
\(978\) 0 0
\(979\) 10202.1 0.333056
\(980\) 0 0
\(981\) 39578.8 1.28813
\(982\) 0 0
\(983\) −37190.0 −1.20669 −0.603345 0.797480i \(-0.706166\pi\)
−0.603345 + 0.797480i \(0.706166\pi\)
\(984\) 0 0
\(985\) −39414.1 −1.27496
\(986\) 0 0
\(987\) −2124.29 −0.0685076
\(988\) 0 0
\(989\) −30377.7 −0.976700
\(990\) 0 0
\(991\) 40750.3 1.30623 0.653115 0.757258i \(-0.273462\pi\)
0.653115 + 0.757258i \(0.273462\pi\)
\(992\) 0 0
\(993\) −71559.8 −2.28689
\(994\) 0 0
\(995\) −36305.0 −1.15673
\(996\) 0 0
\(997\) 22567.1 0.716859 0.358429 0.933557i \(-0.383312\pi\)
0.358429 + 0.933557i \(0.383312\pi\)
\(998\) 0 0
\(999\) 44961.0 1.42393
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 1232.4.a.z.1.5 5
4.3 odd 2 616.4.a.g.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.4.a.g.1.1 5 4.3 odd 2
1232.4.a.z.1.5 5 1.1 even 1 trivial