Properties

Label 1323.4.a.bb.1.2
Level $1323$
Weight $4$
Character 1323.1
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,4,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1323.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.346909504.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 12x^{4} + 2x^{3} + 39x^{2} + 25x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.69129\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.78484 q^{2} -0.244684 q^{4} -20.8311 q^{5} +22.9601 q^{8} +O(q^{10})\) \(q-2.78484 q^{2} -0.244684 q^{4} -20.8311 q^{5} +22.9601 q^{8} +58.0112 q^{10} -65.7898 q^{11} -50.9140 q^{13} -61.9827 q^{16} +46.1303 q^{17} +62.1235 q^{19} +5.09704 q^{20} +183.214 q^{22} -125.329 q^{23} +308.935 q^{25} +141.787 q^{26} +168.840 q^{29} +187.101 q^{31} -11.0692 q^{32} -128.465 q^{34} -70.0464 q^{37} -173.004 q^{38} -478.284 q^{40} -188.366 q^{41} +239.590 q^{43} +16.0977 q^{44} +349.020 q^{46} -591.775 q^{47} -860.332 q^{50} +12.4579 q^{52} +464.090 q^{53} +1370.47 q^{55} -470.191 q^{58} +325.056 q^{59} -261.058 q^{61} -521.045 q^{62} +526.687 q^{64} +1060.59 q^{65} +340.985 q^{67} -11.2874 q^{68} +752.254 q^{71} +245.233 q^{73} +195.068 q^{74} -15.2006 q^{76} +546.822 q^{79} +1291.17 q^{80} +524.568 q^{82} +144.397 q^{83} -960.945 q^{85} -667.219 q^{86} -1510.54 q^{88} +603.758 q^{89} +30.6659 q^{92} +1648.00 q^{94} -1294.10 q^{95} +1355.27 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{4} - 24 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{4} - 24 q^{5} - 30 q^{16} - 42 q^{17} - 12 q^{20} + 132 q^{22} + 222 q^{25} - 366 q^{26} - 312 q^{37} - 336 q^{38} - 360 q^{41} + 654 q^{43} + 774 q^{46} - 1812 q^{47} - 378 q^{58} + 6 q^{59} - 2058 q^{62} + 66 q^{64} + 42 q^{67} - 2910 q^{68} + 1956 q^{79} + 2868 q^{80} - 2892 q^{83} - 1944 q^{85} - 2532 q^{88} - 1518 q^{89}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.78484 −0.984588 −0.492294 0.870429i \(-0.663842\pi\)
−0.492294 + 0.870429i \(0.663842\pi\)
\(3\) 0 0
\(4\) −0.244684 −0.0305855
\(5\) −20.8311 −1.86319 −0.931595 0.363498i \(-0.881582\pi\)
−0.931595 + 0.363498i \(0.881582\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 22.9601 1.01470
\(9\) 0 0
\(10\) 58.0112 1.83448
\(11\) −65.7898 −1.80331 −0.901654 0.432459i \(-0.857646\pi\)
−0.901654 + 0.432459i \(0.857646\pi\)
\(12\) 0 0
\(13\) −50.9140 −1.08623 −0.543115 0.839658i \(-0.682755\pi\)
−0.543115 + 0.839658i \(0.682755\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −61.9827 −0.968479
\(17\) 46.1303 0.658132 0.329066 0.944307i \(-0.393266\pi\)
0.329066 + 0.944307i \(0.393266\pi\)
\(18\) 0 0
\(19\) 62.1235 0.750110 0.375055 0.927002i \(-0.377624\pi\)
0.375055 + 0.927002i \(0.377624\pi\)
\(20\) 5.09704 0.0569867
\(21\) 0 0
\(22\) 183.214 1.77552
\(23\) −125.329 −1.13621 −0.568105 0.822956i \(-0.692323\pi\)
−0.568105 + 0.822956i \(0.692323\pi\)
\(24\) 0 0
\(25\) 308.935 2.47148
\(26\) 141.787 1.06949
\(27\) 0 0
\(28\) 0 0
\(29\) 168.840 1.08113 0.540565 0.841303i \(-0.318211\pi\)
0.540565 + 0.841303i \(0.318211\pi\)
\(30\) 0 0
\(31\) 187.101 1.08401 0.542005 0.840376i \(-0.317666\pi\)
0.542005 + 0.840376i \(0.317666\pi\)
\(32\) −11.0692 −0.0611494
\(33\) 0 0
\(34\) −128.465 −0.647989
\(35\) 0 0
\(36\) 0 0
\(37\) −70.0464 −0.311231 −0.155616 0.987818i \(-0.549736\pi\)
−0.155616 + 0.987818i \(0.549736\pi\)
\(38\) −173.004 −0.738550
\(39\) 0 0
\(40\) −478.284 −1.89058
\(41\) −188.366 −0.717507 −0.358753 0.933432i \(-0.616798\pi\)
−0.358753 + 0.933432i \(0.616798\pi\)
\(42\) 0 0
\(43\) 239.590 0.849701 0.424851 0.905264i \(-0.360327\pi\)
0.424851 + 0.905264i \(0.360327\pi\)
\(44\) 16.0977 0.0551551
\(45\) 0 0
\(46\) 349.020 1.11870
\(47\) −591.775 −1.83658 −0.918289 0.395910i \(-0.870429\pi\)
−0.918289 + 0.395910i \(0.870429\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −860.332 −2.43339
\(51\) 0 0
\(52\) 12.4579 0.0332230
\(53\) 464.090 1.20279 0.601393 0.798953i \(-0.294612\pi\)
0.601393 + 0.798953i \(0.294612\pi\)
\(54\) 0 0
\(55\) 1370.47 3.35990
\(56\) 0 0
\(57\) 0 0
\(58\) −470.191 −1.06447
\(59\) 325.056 0.717265 0.358633 0.933479i \(-0.383243\pi\)
0.358633 + 0.933479i \(0.383243\pi\)
\(60\) 0 0
\(61\) −261.058 −0.547951 −0.273976 0.961737i \(-0.588339\pi\)
−0.273976 + 0.961737i \(0.588339\pi\)
\(62\) −521.045 −1.06730
\(63\) 0 0
\(64\) 526.687 1.02869
\(65\) 1060.59 2.02385
\(66\) 0 0
\(67\) 340.985 0.621760 0.310880 0.950449i \(-0.399376\pi\)
0.310880 + 0.950449i \(0.399376\pi\)
\(68\) −11.2874 −0.0201293
\(69\) 0 0
\(70\) 0 0
\(71\) 752.254 1.25741 0.628705 0.777644i \(-0.283585\pi\)
0.628705 + 0.777644i \(0.283585\pi\)
\(72\) 0 0
\(73\) 245.233 0.393183 0.196591 0.980485i \(-0.437013\pi\)
0.196591 + 0.980485i \(0.437013\pi\)
\(74\) 195.068 0.306435
\(75\) 0 0
\(76\) −15.2006 −0.0229425
\(77\) 0 0
\(78\) 0 0
\(79\) 546.822 0.778763 0.389382 0.921076i \(-0.372689\pi\)
0.389382 + 0.921076i \(0.372689\pi\)
\(80\) 1291.17 1.80446
\(81\) 0 0
\(82\) 524.568 0.706449
\(83\) 144.397 0.190959 0.0954794 0.995431i \(-0.469562\pi\)
0.0954794 + 0.995431i \(0.469562\pi\)
\(84\) 0 0
\(85\) −960.945 −1.22622
\(86\) −667.219 −0.836606
\(87\) 0 0
\(88\) −1510.54 −1.82982
\(89\) 603.758 0.719081 0.359540 0.933129i \(-0.382933\pi\)
0.359540 + 0.933129i \(0.382933\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 30.6659 0.0347516
\(93\) 0 0
\(94\) 1648.00 1.80827
\(95\) −1294.10 −1.39760
\(96\) 0 0
\(97\) 1355.27 1.41863 0.709315 0.704891i \(-0.249004\pi\)
0.709315 + 0.704891i \(0.249004\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −75.5914 −0.0755914
\(101\) −1690.26 −1.66522 −0.832608 0.553863i \(-0.813153\pi\)
−0.832608 + 0.553863i \(0.813153\pi\)
\(102\) 0 0
\(103\) 189.737 0.181508 0.0907542 0.995873i \(-0.471072\pi\)
0.0907542 + 0.995873i \(0.471072\pi\)
\(104\) −1168.99 −1.10220
\(105\) 0 0
\(106\) −1292.42 −1.18425
\(107\) −741.325 −0.669781 −0.334891 0.942257i \(-0.608699\pi\)
−0.334891 + 0.942257i \(0.608699\pi\)
\(108\) 0 0
\(109\) −607.599 −0.533921 −0.266961 0.963707i \(-0.586019\pi\)
−0.266961 + 0.963707i \(0.586019\pi\)
\(110\) −3816.55 −3.30812
\(111\) 0 0
\(112\) 0 0
\(113\) 1475.76 1.22856 0.614281 0.789088i \(-0.289446\pi\)
0.614281 + 0.789088i \(0.289446\pi\)
\(114\) 0 0
\(115\) 2610.73 2.11697
\(116\) −41.3124 −0.0330669
\(117\) 0 0
\(118\) −905.227 −0.706211
\(119\) 0 0
\(120\) 0 0
\(121\) 2997.30 2.25192
\(122\) 727.004 0.539507
\(123\) 0 0
\(124\) −45.7806 −0.0331550
\(125\) −3831.56 −2.74164
\(126\) 0 0
\(127\) −1560.29 −1.09018 −0.545090 0.838377i \(-0.683505\pi\)
−0.545090 + 0.838377i \(0.683505\pi\)
\(128\) −1378.18 −0.951683
\(129\) 0 0
\(130\) −2953.58 −1.99266
\(131\) −736.880 −0.491462 −0.245731 0.969338i \(-0.579028\pi\)
−0.245731 + 0.969338i \(0.579028\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −949.587 −0.612178
\(135\) 0 0
\(136\) 1059.16 0.667808
\(137\) 373.279 0.232784 0.116392 0.993203i \(-0.462867\pi\)
0.116392 + 0.993203i \(0.462867\pi\)
\(138\) 0 0
\(139\) 1292.10 0.788450 0.394225 0.919014i \(-0.371013\pi\)
0.394225 + 0.919014i \(0.371013\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2094.90 −1.23803
\(143\) 3349.62 1.95881
\(144\) 0 0
\(145\) −3517.12 −2.01435
\(146\) −682.934 −0.387123
\(147\) 0 0
\(148\) 17.1393 0.00951918
\(149\) −593.676 −0.326415 −0.163207 0.986592i \(-0.552184\pi\)
−0.163207 + 0.986592i \(0.552184\pi\)
\(150\) 0 0
\(151\) 1172.62 0.631965 0.315982 0.948765i \(-0.397666\pi\)
0.315982 + 0.948765i \(0.397666\pi\)
\(152\) 1426.36 0.761139
\(153\) 0 0
\(154\) 0 0
\(155\) −3897.51 −2.01971
\(156\) 0 0
\(157\) −2308.19 −1.17333 −0.586667 0.809828i \(-0.699560\pi\)
−0.586667 + 0.809828i \(0.699560\pi\)
\(158\) −1522.81 −0.766762
\(159\) 0 0
\(160\) 230.584 0.113933
\(161\) 0 0
\(162\) 0 0
\(163\) 550.709 0.264631 0.132316 0.991208i \(-0.457759\pi\)
0.132316 + 0.991208i \(0.457759\pi\)
\(164\) 46.0901 0.0219453
\(165\) 0 0
\(166\) −402.121 −0.188016
\(167\) −1875.76 −0.869167 −0.434584 0.900631i \(-0.643104\pi\)
−0.434584 + 0.900631i \(0.643104\pi\)
\(168\) 0 0
\(169\) 395.234 0.179897
\(170\) 2676.07 1.20733
\(171\) 0 0
\(172\) −58.6239 −0.0259886
\(173\) −3576.61 −1.57182 −0.785910 0.618341i \(-0.787805\pi\)
−0.785910 + 0.618341i \(0.787805\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4077.83 1.74646
\(177\) 0 0
\(178\) −1681.37 −0.707999
\(179\) 1847.79 0.771567 0.385783 0.922589i \(-0.373931\pi\)
0.385783 + 0.922589i \(0.373931\pi\)
\(180\) 0 0
\(181\) −3789.93 −1.55637 −0.778187 0.628033i \(-0.783860\pi\)
−0.778187 + 0.628033i \(0.783860\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2877.56 −1.15291
\(185\) 1459.14 0.579883
\(186\) 0 0
\(187\) −3034.90 −1.18681
\(188\) 144.798 0.0561728
\(189\) 0 0
\(190\) 3603.86 1.37606
\(191\) 3446.71 1.30573 0.652867 0.757472i \(-0.273566\pi\)
0.652867 + 0.757472i \(0.273566\pi\)
\(192\) 0 0
\(193\) 1264.46 0.471594 0.235797 0.971802i \(-0.424230\pi\)
0.235797 + 0.971802i \(0.424230\pi\)
\(194\) −3774.22 −1.39677
\(195\) 0 0
\(196\) 0 0
\(197\) −2591.45 −0.937226 −0.468613 0.883404i \(-0.655246\pi\)
−0.468613 + 0.883404i \(0.655246\pi\)
\(198\) 0 0
\(199\) −86.8060 −0.0309222 −0.0154611 0.999880i \(-0.504922\pi\)
−0.0154611 + 0.999880i \(0.504922\pi\)
\(200\) 7093.17 2.50781
\(201\) 0 0
\(202\) 4707.09 1.63955
\(203\) 0 0
\(204\) 0 0
\(205\) 3923.86 1.33685
\(206\) −528.388 −0.178711
\(207\) 0 0
\(208\) 3155.78 1.05199
\(209\) −4087.09 −1.35268
\(210\) 0 0
\(211\) −1586.52 −0.517633 −0.258817 0.965927i \(-0.583333\pi\)
−0.258817 + 0.965927i \(0.583333\pi\)
\(212\) −113.556 −0.0367879
\(213\) 0 0
\(214\) 2064.47 0.659459
\(215\) −4990.92 −1.58315
\(216\) 0 0
\(217\) 0 0
\(218\) 1692.06 0.525693
\(219\) 0 0
\(220\) −335.334 −0.102764
\(221\) −2348.68 −0.714883
\(222\) 0 0
\(223\) 519.207 0.155913 0.0779567 0.996957i \(-0.475160\pi\)
0.0779567 + 0.996957i \(0.475160\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −4109.74 −1.20963
\(227\) 931.422 0.272338 0.136169 0.990686i \(-0.456521\pi\)
0.136169 + 0.990686i \(0.456521\pi\)
\(228\) 0 0
\(229\) 5040.17 1.45443 0.727213 0.686412i \(-0.240815\pi\)
0.727213 + 0.686412i \(0.240815\pi\)
\(230\) −7270.46 −2.08435
\(231\) 0 0
\(232\) 3876.58 1.09702
\(233\) 191.642 0.0538835 0.0269418 0.999637i \(-0.491423\pi\)
0.0269418 + 0.999637i \(0.491423\pi\)
\(234\) 0 0
\(235\) 12327.3 3.42190
\(236\) −79.5361 −0.0219380
\(237\) 0 0
\(238\) 0 0
\(239\) 2806.82 0.759658 0.379829 0.925057i \(-0.375983\pi\)
0.379829 + 0.925057i \(0.375983\pi\)
\(240\) 0 0
\(241\) −3827.25 −1.02297 −0.511484 0.859293i \(-0.670904\pi\)
−0.511484 + 0.859293i \(0.670904\pi\)
\(242\) −8346.99 −2.21721
\(243\) 0 0
\(244\) 63.8768 0.0167594
\(245\) 0 0
\(246\) 0 0
\(247\) −3162.95 −0.814793
\(248\) 4295.85 1.09995
\(249\) 0 0
\(250\) 10670.3 2.69939
\(251\) −5242.63 −1.31837 −0.659187 0.751979i \(-0.729100\pi\)
−0.659187 + 0.751979i \(0.729100\pi\)
\(252\) 0 0
\(253\) 8245.34 2.04893
\(254\) 4345.14 1.07338
\(255\) 0 0
\(256\) −375.480 −0.0916699
\(257\) 1722.90 0.418176 0.209088 0.977897i \(-0.432950\pi\)
0.209088 + 0.977897i \(0.432950\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −259.511 −0.0619007
\(261\) 0 0
\(262\) 2052.09 0.483888
\(263\) −1478.24 −0.346586 −0.173293 0.984870i \(-0.555441\pi\)
−0.173293 + 0.984870i \(0.555441\pi\)
\(264\) 0 0
\(265\) −9667.51 −2.24102
\(266\) 0 0
\(267\) 0 0
\(268\) −83.4337 −0.0190169
\(269\) −2540.01 −0.575714 −0.287857 0.957673i \(-0.592943\pi\)
−0.287857 + 0.957673i \(0.592943\pi\)
\(270\) 0 0
\(271\) −2439.03 −0.546718 −0.273359 0.961912i \(-0.588135\pi\)
−0.273359 + 0.961912i \(0.588135\pi\)
\(272\) −2859.28 −0.637387
\(273\) 0 0
\(274\) −1039.52 −0.229196
\(275\) −20324.7 −4.45683
\(276\) 0 0
\(277\) −4322.03 −0.937493 −0.468747 0.883333i \(-0.655294\pi\)
−0.468747 + 0.883333i \(0.655294\pi\)
\(278\) −3598.29 −0.776299
\(279\) 0 0
\(280\) 0 0
\(281\) −2793.78 −0.593107 −0.296553 0.955016i \(-0.595837\pi\)
−0.296553 + 0.955016i \(0.595837\pi\)
\(282\) 0 0
\(283\) −4897.86 −1.02879 −0.514395 0.857553i \(-0.671984\pi\)
−0.514395 + 0.857553i \(0.671984\pi\)
\(284\) −184.065 −0.0384586
\(285\) 0 0
\(286\) −9328.15 −1.92862
\(287\) 0 0
\(288\) 0 0
\(289\) −2785.00 −0.566863
\(290\) 9794.59 1.98330
\(291\) 0 0
\(292\) −60.0047 −0.0120257
\(293\) 6039.14 1.20413 0.602065 0.798447i \(-0.294345\pi\)
0.602065 + 0.798447i \(0.294345\pi\)
\(294\) 0 0
\(295\) −6771.27 −1.33640
\(296\) −1608.27 −0.315807
\(297\) 0 0
\(298\) 1653.29 0.321384
\(299\) 6380.98 1.23419
\(300\) 0 0
\(301\) 0 0
\(302\) −3265.56 −0.622225
\(303\) 0 0
\(304\) −3850.58 −0.726466
\(305\) 5438.12 1.02094
\(306\) 0 0
\(307\) 7116.12 1.32293 0.661463 0.749978i \(-0.269936\pi\)
0.661463 + 0.749978i \(0.269936\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 10853.9 1.98859
\(311\) −10565.3 −1.92638 −0.963191 0.268819i \(-0.913366\pi\)
−0.963191 + 0.268819i \(0.913366\pi\)
\(312\) 0 0
\(313\) −3419.87 −0.617580 −0.308790 0.951130i \(-0.599924\pi\)
−0.308790 + 0.951130i \(0.599924\pi\)
\(314\) 6427.93 1.15525
\(315\) 0 0
\(316\) −133.799 −0.0238189
\(317\) 10098.2 1.78918 0.894589 0.446889i \(-0.147468\pi\)
0.894589 + 0.446889i \(0.147468\pi\)
\(318\) 0 0
\(319\) −11107.9 −1.94961
\(320\) −10971.5 −1.91664
\(321\) 0 0
\(322\) 0 0
\(323\) 2865.77 0.493672
\(324\) 0 0
\(325\) −15729.1 −2.68459
\(326\) −1533.64 −0.260553
\(327\) 0 0
\(328\) −4324.90 −0.728056
\(329\) 0 0
\(330\) 0 0
\(331\) −6978.18 −1.15878 −0.579389 0.815051i \(-0.696709\pi\)
−0.579389 + 0.815051i \(0.696709\pi\)
\(332\) −35.3316 −0.00584058
\(333\) 0 0
\(334\) 5223.69 0.855772
\(335\) −7103.09 −1.15846
\(336\) 0 0
\(337\) −3376.48 −0.545782 −0.272891 0.962045i \(-0.587980\pi\)
−0.272891 + 0.962045i \(0.587980\pi\)
\(338\) −1100.66 −0.177125
\(339\) 0 0
\(340\) 235.128 0.0375047
\(341\) −12309.3 −1.95480
\(342\) 0 0
\(343\) 0 0
\(344\) 5501.01 0.862194
\(345\) 0 0
\(346\) 9960.28 1.54760
\(347\) −3781.51 −0.585020 −0.292510 0.956262i \(-0.594490\pi\)
−0.292510 + 0.956262i \(0.594490\pi\)
\(348\) 0 0
\(349\) −985.586 −0.151167 −0.0755834 0.997139i \(-0.524082\pi\)
−0.0755834 + 0.997139i \(0.524082\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 728.242 0.110271
\(353\) 5674.48 0.855587 0.427793 0.903877i \(-0.359291\pi\)
0.427793 + 0.903877i \(0.359291\pi\)
\(354\) 0 0
\(355\) −15670.3 −2.34279
\(356\) −147.730 −0.0219935
\(357\) 0 0
\(358\) −5145.80 −0.759676
\(359\) 8038.50 1.18177 0.590886 0.806755i \(-0.298778\pi\)
0.590886 + 0.806755i \(0.298778\pi\)
\(360\) 0 0
\(361\) −2999.68 −0.437334
\(362\) 10554.3 1.53239
\(363\) 0 0
\(364\) 0 0
\(365\) −5108.47 −0.732574
\(366\) 0 0
\(367\) −3466.45 −0.493044 −0.246522 0.969137i \(-0.579288\pi\)
−0.246522 + 0.969137i \(0.579288\pi\)
\(368\) 7768.20 1.10039
\(369\) 0 0
\(370\) −4063.48 −0.570946
\(371\) 0 0
\(372\) 0 0
\(373\) 10339.9 1.43534 0.717670 0.696384i \(-0.245209\pi\)
0.717670 + 0.696384i \(0.245209\pi\)
\(374\) 8451.71 1.16852
\(375\) 0 0
\(376\) −13587.2 −1.86358
\(377\) −8596.30 −1.17436
\(378\) 0 0
\(379\) −1299.74 −0.176156 −0.0880780 0.996114i \(-0.528072\pi\)
−0.0880780 + 0.996114i \(0.528072\pi\)
\(380\) 316.646 0.0427463
\(381\) 0 0
\(382\) −9598.53 −1.28561
\(383\) 10646.4 1.42038 0.710190 0.704010i \(-0.248609\pi\)
0.710190 + 0.704010i \(0.248609\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3521.31 −0.464326
\(387\) 0 0
\(388\) −331.614 −0.0433896
\(389\) 11895.3 1.55042 0.775212 0.631701i \(-0.217643\pi\)
0.775212 + 0.631701i \(0.217643\pi\)
\(390\) 0 0
\(391\) −5781.44 −0.747775
\(392\) 0 0
\(393\) 0 0
\(394\) 7216.78 0.922781
\(395\) −11390.9 −1.45098
\(396\) 0 0
\(397\) 12111.6 1.53114 0.765570 0.643353i \(-0.222457\pi\)
0.765570 + 0.643353i \(0.222457\pi\)
\(398\) 241.741 0.0304456
\(399\) 0 0
\(400\) −19148.6 −2.39357
\(401\) 7394.86 0.920902 0.460451 0.887685i \(-0.347688\pi\)
0.460451 + 0.887685i \(0.347688\pi\)
\(402\) 0 0
\(403\) −9526.04 −1.17748
\(404\) 413.579 0.0509315
\(405\) 0 0
\(406\) 0 0
\(407\) 4608.34 0.561246
\(408\) 0 0
\(409\) −15003.4 −1.81386 −0.906931 0.421280i \(-0.861581\pi\)
−0.906931 + 0.421280i \(0.861581\pi\)
\(410\) −10927.3 −1.31625
\(411\) 0 0
\(412\) −46.4258 −0.00555154
\(413\) 0 0
\(414\) 0 0
\(415\) −3007.94 −0.355793
\(416\) 563.578 0.0664224
\(417\) 0 0
\(418\) 11381.9 1.33183
\(419\) −1102.65 −0.128563 −0.0642815 0.997932i \(-0.520476\pi\)
−0.0642815 + 0.997932i \(0.520476\pi\)
\(420\) 0 0
\(421\) 7596.04 0.879355 0.439678 0.898156i \(-0.355093\pi\)
0.439678 + 0.898156i \(0.355093\pi\)
\(422\) 4418.20 0.509656
\(423\) 0 0
\(424\) 10655.6 1.22047
\(425\) 14251.2 1.62656
\(426\) 0 0
\(427\) 0 0
\(428\) 181.391 0.0204856
\(429\) 0 0
\(430\) 13898.9 1.55876
\(431\) 2127.52 0.237770 0.118885 0.992908i \(-0.462068\pi\)
0.118885 + 0.992908i \(0.462068\pi\)
\(432\) 0 0
\(433\) 6315.20 0.700899 0.350449 0.936582i \(-0.386029\pi\)
0.350449 + 0.936582i \(0.386029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 148.670 0.0163303
\(437\) −7785.84 −0.852282
\(438\) 0 0
\(439\) 10983.8 1.19414 0.597068 0.802190i \(-0.296332\pi\)
0.597068 + 0.802190i \(0.296332\pi\)
\(440\) 31466.2 3.40930
\(441\) 0 0
\(442\) 6540.68 0.703865
\(443\) 11095.3 1.18997 0.594983 0.803738i \(-0.297159\pi\)
0.594983 + 0.803738i \(0.297159\pi\)
\(444\) 0 0
\(445\) −12576.9 −1.33978
\(446\) −1445.91 −0.153510
\(447\) 0 0
\(448\) 0 0
\(449\) 9190.22 0.965954 0.482977 0.875633i \(-0.339555\pi\)
0.482977 + 0.875633i \(0.339555\pi\)
\(450\) 0 0
\(451\) 12392.5 1.29389
\(452\) −361.094 −0.0375762
\(453\) 0 0
\(454\) −2593.86 −0.268140
\(455\) 0 0
\(456\) 0 0
\(457\) −295.522 −0.0302493 −0.0151247 0.999886i \(-0.504815\pi\)
−0.0151247 + 0.999886i \(0.504815\pi\)
\(458\) −14036.0 −1.43201
\(459\) 0 0
\(460\) −638.805 −0.0647488
\(461\) 10319.9 1.04262 0.521309 0.853368i \(-0.325444\pi\)
0.521309 + 0.853368i \(0.325444\pi\)
\(462\) 0 0
\(463\) −10886.8 −1.09277 −0.546385 0.837534i \(-0.683996\pi\)
−0.546385 + 0.837534i \(0.683996\pi\)
\(464\) −10465.1 −1.04705
\(465\) 0 0
\(466\) −533.691 −0.0530531
\(467\) −5056.59 −0.501052 −0.250526 0.968110i \(-0.580604\pi\)
−0.250526 + 0.968110i \(0.580604\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −34329.6 −3.36916
\(471\) 0 0
\(472\) 7463.31 0.727811
\(473\) −15762.6 −1.53227
\(474\) 0 0
\(475\) 19192.1 1.85388
\(476\) 0 0
\(477\) 0 0
\(478\) −7816.54 −0.747950
\(479\) −389.667 −0.0371698 −0.0185849 0.999827i \(-0.505916\pi\)
−0.0185849 + 0.999827i \(0.505916\pi\)
\(480\) 0 0
\(481\) 3566.34 0.338069
\(482\) 10658.3 1.00720
\(483\) 0 0
\(484\) −733.392 −0.0688761
\(485\) −28231.8 −2.64318
\(486\) 0 0
\(487\) 15067.5 1.40200 0.700998 0.713163i \(-0.252738\pi\)
0.700998 + 0.713163i \(0.252738\pi\)
\(488\) −5993.91 −0.556008
\(489\) 0 0
\(490\) 0 0
\(491\) 17552.7 1.61333 0.806663 0.591012i \(-0.201271\pi\)
0.806663 + 0.591012i \(0.201271\pi\)
\(492\) 0 0
\(493\) 7788.63 0.711525
\(494\) 8808.31 0.802236
\(495\) 0 0
\(496\) −11597.0 −1.04984
\(497\) 0 0
\(498\) 0 0
\(499\) −19298.9 −1.73133 −0.865667 0.500620i \(-0.833105\pi\)
−0.865667 + 0.500620i \(0.833105\pi\)
\(500\) 937.522 0.0838545
\(501\) 0 0
\(502\) 14599.9 1.29806
\(503\) −10038.8 −0.889881 −0.444940 0.895560i \(-0.646775\pi\)
−0.444940 + 0.895560i \(0.646775\pi\)
\(504\) 0 0
\(505\) 35209.9 3.10261
\(506\) −22961.9 −2.01736
\(507\) 0 0
\(508\) 381.777 0.0333438
\(509\) 7594.91 0.661372 0.330686 0.943741i \(-0.392720\pi\)
0.330686 + 0.943741i \(0.392720\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 12071.1 1.04194
\(513\) 0 0
\(514\) −4797.98 −0.411732
\(515\) −3952.44 −0.338185
\(516\) 0 0
\(517\) 38932.7 3.31192
\(518\) 0 0
\(519\) 0 0
\(520\) 24351.3 2.05361
\(521\) 13207.5 1.11062 0.555309 0.831644i \(-0.312600\pi\)
0.555309 + 0.831644i \(0.312600\pi\)
\(522\) 0 0
\(523\) −21135.4 −1.76709 −0.883544 0.468349i \(-0.844849\pi\)
−0.883544 + 0.468349i \(0.844849\pi\)
\(524\) 180.303 0.0150316
\(525\) 0 0
\(526\) 4116.66 0.341245
\(527\) 8631.01 0.713421
\(528\) 0 0
\(529\) 3540.25 0.290971
\(530\) 26922.4 2.20648
\(531\) 0 0
\(532\) 0 0
\(533\) 9590.45 0.779378
\(534\) 0 0
\(535\) 15442.6 1.24793
\(536\) 7829.05 0.630902
\(537\) 0 0
\(538\) 7073.51 0.566841
\(539\) 0 0
\(540\) 0 0
\(541\) −16839.0 −1.33820 −0.669099 0.743173i \(-0.733320\pi\)
−0.669099 + 0.743173i \(0.733320\pi\)
\(542\) 6792.30 0.538292
\(543\) 0 0
\(544\) −510.627 −0.0402444
\(545\) 12656.9 0.994797
\(546\) 0 0
\(547\) −6413.01 −0.501281 −0.250640 0.968080i \(-0.580641\pi\)
−0.250640 + 0.968080i \(0.580641\pi\)
\(548\) −91.3355 −0.00711982
\(549\) 0 0
\(550\) 56601.1 4.38814
\(551\) 10488.9 0.810966
\(552\) 0 0
\(553\) 0 0
\(554\) 12036.2 0.923045
\(555\) 0 0
\(556\) −316.157 −0.0241152
\(557\) −18970.8 −1.44312 −0.721562 0.692350i \(-0.756575\pi\)
−0.721562 + 0.692350i \(0.756575\pi\)
\(558\) 0 0
\(559\) −12198.5 −0.922971
\(560\) 0 0
\(561\) 0 0
\(562\) 7780.23 0.583966
\(563\) −11466.8 −0.858378 −0.429189 0.903215i \(-0.641201\pi\)
−0.429189 + 0.903215i \(0.641201\pi\)
\(564\) 0 0
\(565\) −30741.6 −2.28904
\(566\) 13639.7 1.01294
\(567\) 0 0
\(568\) 17271.8 1.27590
\(569\) −1329.17 −0.0979293 −0.0489647 0.998801i \(-0.515592\pi\)
−0.0489647 + 0.998801i \(0.515592\pi\)
\(570\) 0 0
\(571\) −5171.50 −0.379020 −0.189510 0.981879i \(-0.560690\pi\)
−0.189510 + 0.981879i \(0.560690\pi\)
\(572\) −819.600 −0.0599112
\(573\) 0 0
\(574\) 0 0
\(575\) −38718.3 −2.80811
\(576\) 0 0
\(577\) 18629.2 1.34410 0.672048 0.740507i \(-0.265415\pi\)
0.672048 + 0.740507i \(0.265415\pi\)
\(578\) 7755.76 0.558126
\(579\) 0 0
\(580\) 860.583 0.0616099
\(581\) 0 0
\(582\) 0 0
\(583\) −30532.4 −2.16899
\(584\) 5630.57 0.398964
\(585\) 0 0
\(586\) −16818.0 −1.18557
\(587\) 21184.8 1.48959 0.744795 0.667293i \(-0.232547\pi\)
0.744795 + 0.667293i \(0.232547\pi\)
\(588\) 0 0
\(589\) 11623.3 0.813127
\(590\) 18856.9 1.31581
\(591\) 0 0
\(592\) 4341.66 0.301421
\(593\) 25230.1 1.74718 0.873588 0.486666i \(-0.161787\pi\)
0.873588 + 0.486666i \(0.161787\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 145.263 0.00998357
\(597\) 0 0
\(598\) −17770.0 −1.21516
\(599\) 1180.40 0.0805173 0.0402587 0.999189i \(-0.487182\pi\)
0.0402587 + 0.999189i \(0.487182\pi\)
\(600\) 0 0
\(601\) −14156.0 −0.960793 −0.480396 0.877052i \(-0.659507\pi\)
−0.480396 + 0.877052i \(0.659507\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −286.922 −0.0193290
\(605\) −62437.1 −4.19575
\(606\) 0 0
\(607\) −5158.89 −0.344964 −0.172482 0.985013i \(-0.555179\pi\)
−0.172482 + 0.985013i \(0.555179\pi\)
\(608\) −687.658 −0.0458688
\(609\) 0 0
\(610\) −15144.3 −1.00520
\(611\) 30129.6 1.99495
\(612\) 0 0
\(613\) −5700.96 −0.375628 −0.187814 0.982205i \(-0.560140\pi\)
−0.187814 + 0.982205i \(0.560140\pi\)
\(614\) −19817.2 −1.30254
\(615\) 0 0
\(616\) 0 0
\(617\) −7424.49 −0.484439 −0.242219 0.970221i \(-0.577875\pi\)
−0.242219 + 0.970221i \(0.577875\pi\)
\(618\) 0 0
\(619\) −25672.5 −1.66699 −0.833494 0.552529i \(-0.813663\pi\)
−0.833494 + 0.552529i \(0.813663\pi\)
\(620\) 953.660 0.0617741
\(621\) 0 0
\(622\) 29422.7 1.89669
\(623\) 0 0
\(624\) 0 0
\(625\) 41198.7 2.63672
\(626\) 9523.78 0.608062
\(627\) 0 0
\(628\) 564.778 0.0358871
\(629\) −3231.26 −0.204831
\(630\) 0 0
\(631\) −7093.10 −0.447499 −0.223749 0.974647i \(-0.571830\pi\)
−0.223749 + 0.974647i \(0.571830\pi\)
\(632\) 12555.1 0.790213
\(633\) 0 0
\(634\) −28121.8 −1.76160
\(635\) 32502.5 2.03121
\(636\) 0 0
\(637\) 0 0
\(638\) 30933.8 1.91956
\(639\) 0 0
\(640\) 28709.1 1.77317
\(641\) 3465.01 0.213510 0.106755 0.994285i \(-0.465954\pi\)
0.106755 + 0.994285i \(0.465954\pi\)
\(642\) 0 0
\(643\) 5460.79 0.334919 0.167459 0.985879i \(-0.446444\pi\)
0.167459 + 0.985879i \(0.446444\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −7980.71 −0.486063
\(647\) −4535.16 −0.275573 −0.137786 0.990462i \(-0.543999\pi\)
−0.137786 + 0.990462i \(0.543999\pi\)
\(648\) 0 0
\(649\) −21385.4 −1.29345
\(650\) 43802.9 2.64322
\(651\) 0 0
\(652\) −134.750 −0.00809389
\(653\) 961.151 0.0575999 0.0288000 0.999585i \(-0.490831\pi\)
0.0288000 + 0.999585i \(0.490831\pi\)
\(654\) 0 0
\(655\) 15350.0 0.915687
\(656\) 11675.4 0.694890
\(657\) 0 0
\(658\) 0 0
\(659\) 17654.3 1.04357 0.521787 0.853076i \(-0.325266\pi\)
0.521787 + 0.853076i \(0.325266\pi\)
\(660\) 0 0
\(661\) 27374.3 1.61080 0.805400 0.592732i \(-0.201951\pi\)
0.805400 + 0.592732i \(0.201951\pi\)
\(662\) 19433.1 1.14092
\(663\) 0 0
\(664\) 3315.36 0.193766
\(665\) 0 0
\(666\) 0 0
\(667\) −21160.4 −1.22839
\(668\) 458.970 0.0265839
\(669\) 0 0
\(670\) 19780.9 1.14060
\(671\) 17174.9 0.988125
\(672\) 0 0
\(673\) 23933.1 1.37080 0.685402 0.728165i \(-0.259626\pi\)
0.685402 + 0.728165i \(0.259626\pi\)
\(674\) 9402.94 0.537370
\(675\) 0 0
\(676\) −96.7075 −0.00550225
\(677\) −17537.7 −0.995610 −0.497805 0.867289i \(-0.665861\pi\)
−0.497805 + 0.867289i \(0.665861\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −22063.4 −1.24425
\(681\) 0 0
\(682\) 34279.5 1.92467
\(683\) 15170.6 0.849908 0.424954 0.905215i \(-0.360290\pi\)
0.424954 + 0.905215i \(0.360290\pi\)
\(684\) 0 0
\(685\) −7775.81 −0.433720
\(686\) 0 0
\(687\) 0 0
\(688\) −14850.4 −0.822918
\(689\) −23628.7 −1.30650
\(690\) 0 0
\(691\) 8107.90 0.446366 0.223183 0.974777i \(-0.428355\pi\)
0.223183 + 0.974777i \(0.428355\pi\)
\(692\) 875.141 0.0480750
\(693\) 0 0
\(694\) 10530.9 0.576004
\(695\) −26915.9 −1.46903
\(696\) 0 0
\(697\) −8689.37 −0.472214
\(698\) 2744.70 0.148837
\(699\) 0 0
\(700\) 0 0
\(701\) −24388.3 −1.31403 −0.657015 0.753877i \(-0.728181\pi\)
−0.657015 + 0.753877i \(0.728181\pi\)
\(702\) 0 0
\(703\) −4351.53 −0.233458
\(704\) −34650.7 −1.85504
\(705\) 0 0
\(706\) −15802.5 −0.842401
\(707\) 0 0
\(708\) 0 0
\(709\) 16879.7 0.894119 0.447060 0.894504i \(-0.352471\pi\)
0.447060 + 0.894504i \(0.352471\pi\)
\(710\) 43639.1 2.30669
\(711\) 0 0
\(712\) 13862.3 0.729653
\(713\) −23449.1 −1.23166
\(714\) 0 0
\(715\) −69776.3 −3.64963
\(716\) −452.126 −0.0235988
\(717\) 0 0
\(718\) −22385.9 −1.16356
\(719\) 18111.5 0.939421 0.469710 0.882821i \(-0.344358\pi\)
0.469710 + 0.882821i \(0.344358\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 8353.61 0.430594
\(723\) 0 0
\(724\) 927.338 0.0476025
\(725\) 52160.4 2.67198
\(726\) 0 0
\(727\) −18001.3 −0.918339 −0.459170 0.888349i \(-0.651853\pi\)
−0.459170 + 0.888349i \(0.651853\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 14226.3 0.721284
\(731\) 11052.4 0.559215
\(732\) 0 0
\(733\) −27617.2 −1.39163 −0.695816 0.718220i \(-0.744957\pi\)
−0.695816 + 0.718220i \(0.744957\pi\)
\(734\) 9653.49 0.485445
\(735\) 0 0
\(736\) 1387.29 0.0694785
\(737\) −22433.3 −1.12122
\(738\) 0 0
\(739\) 5128.96 0.255307 0.127653 0.991819i \(-0.459255\pi\)
0.127653 + 0.991819i \(0.459255\pi\)
\(740\) −357.030 −0.0177360
\(741\) 0 0
\(742\) 0 0
\(743\) −30347.6 −1.49845 −0.749224 0.662317i \(-0.769573\pi\)
−0.749224 + 0.662317i \(0.769573\pi\)
\(744\) 0 0
\(745\) 12366.9 0.608173
\(746\) −28795.0 −1.41322
\(747\) 0 0
\(748\) 742.593 0.0362993
\(749\) 0 0
\(750\) 0 0
\(751\) 23545.1 1.14404 0.572020 0.820239i \(-0.306160\pi\)
0.572020 + 0.820239i \(0.306160\pi\)
\(752\) 36679.8 1.77869
\(753\) 0 0
\(754\) 23939.3 1.15626
\(755\) −24427.0 −1.17747
\(756\) 0 0
\(757\) 10693.8 0.513439 0.256719 0.966486i \(-0.417358\pi\)
0.256719 + 0.966486i \(0.417358\pi\)
\(758\) 3619.56 0.173441
\(759\) 0 0
\(760\) −29712.7 −1.41815
\(761\) 6456.89 0.307572 0.153786 0.988104i \(-0.450853\pi\)
0.153786 + 0.988104i \(0.450853\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −843.356 −0.0399366
\(765\) 0 0
\(766\) −29648.5 −1.39849
\(767\) −16549.9 −0.779116
\(768\) 0 0
\(769\) 2661.03 0.124784 0.0623922 0.998052i \(-0.480127\pi\)
0.0623922 + 0.998052i \(0.480127\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −309.393 −0.0144239
\(773\) −28404.4 −1.32165 −0.660825 0.750540i \(-0.729794\pi\)
−0.660825 + 0.750540i \(0.729794\pi\)
\(774\) 0 0
\(775\) 57801.9 2.67910
\(776\) 31117.2 1.43949
\(777\) 0 0
\(778\) −33126.4 −1.52653
\(779\) −11701.9 −0.538209
\(780\) 0 0
\(781\) −49490.6 −2.26750
\(782\) 16100.4 0.736251
\(783\) 0 0
\(784\) 0 0
\(785\) 48082.1 2.18615
\(786\) 0 0
\(787\) 708.483 0.0320898 0.0160449 0.999871i \(-0.494893\pi\)
0.0160449 + 0.999871i \(0.494893\pi\)
\(788\) 634.088 0.0286656
\(789\) 0 0
\(790\) 31721.8 1.42862
\(791\) 0 0
\(792\) 0 0
\(793\) 13291.5 0.595202
\(794\) −33728.8 −1.50754
\(795\) 0 0
\(796\) 21.2401 0.000945772 0
\(797\) −10538.1 −0.468355 −0.234178 0.972194i \(-0.575240\pi\)
−0.234178 + 0.972194i \(0.575240\pi\)
\(798\) 0 0
\(799\) −27298.7 −1.20871
\(800\) −3419.67 −0.151129
\(801\) 0 0
\(802\) −20593.5 −0.906710
\(803\) −16133.8 −0.709030
\(804\) 0 0
\(805\) 0 0
\(806\) 26528.5 1.15934
\(807\) 0 0
\(808\) −38808.4 −1.68970
\(809\) −38211.3 −1.66061 −0.830307 0.557306i \(-0.811835\pi\)
−0.830307 + 0.557306i \(0.811835\pi\)
\(810\) 0 0
\(811\) 12627.8 0.546758 0.273379 0.961906i \(-0.411859\pi\)
0.273379 + 0.961906i \(0.411859\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −12833.5 −0.552596
\(815\) −11471.9 −0.493058
\(816\) 0 0
\(817\) 14884.2 0.637370
\(818\) 41782.0 1.78591
\(819\) 0 0
\(820\) −960.108 −0.0408883
\(821\) −15408.0 −0.654985 −0.327492 0.944854i \(-0.606204\pi\)
−0.327492 + 0.944854i \(0.606204\pi\)
\(822\) 0 0
\(823\) 12225.4 0.517803 0.258901 0.965904i \(-0.416640\pi\)
0.258901 + 0.965904i \(0.416640\pi\)
\(824\) 4356.39 0.184177
\(825\) 0 0
\(826\) 0 0
\(827\) −2156.41 −0.0906721 −0.0453360 0.998972i \(-0.514436\pi\)
−0.0453360 + 0.998972i \(0.514436\pi\)
\(828\) 0 0
\(829\) −20682.7 −0.866515 −0.433257 0.901270i \(-0.642636\pi\)
−0.433257 + 0.901270i \(0.642636\pi\)
\(830\) 8376.62 0.350309
\(831\) 0 0
\(832\) −26815.7 −1.11739
\(833\) 0 0
\(834\) 0 0
\(835\) 39074.2 1.61942
\(836\) 1000.05 0.0413724
\(837\) 0 0
\(838\) 3070.70 0.126582
\(839\) 1930.29 0.0794291 0.0397145 0.999211i \(-0.487355\pi\)
0.0397145 + 0.999211i \(0.487355\pi\)
\(840\) 0 0
\(841\) 4117.84 0.168840
\(842\) −21153.7 −0.865803
\(843\) 0 0
\(844\) 388.197 0.0158321
\(845\) −8233.15 −0.335182
\(846\) 0 0
\(847\) 0 0
\(848\) −28765.5 −1.16487
\(849\) 0 0
\(850\) −39687.4 −1.60149
\(851\) 8778.82 0.353624
\(852\) 0 0
\(853\) −11991.2 −0.481327 −0.240663 0.970609i \(-0.577365\pi\)
−0.240663 + 0.970609i \(0.577365\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −17020.9 −0.679629
\(857\) −32354.2 −1.28961 −0.644807 0.764346i \(-0.723062\pi\)
−0.644807 + 0.764346i \(0.723062\pi\)
\(858\) 0 0
\(859\) −7053.85 −0.280180 −0.140090 0.990139i \(-0.544739\pi\)
−0.140090 + 0.990139i \(0.544739\pi\)
\(860\) 1221.20 0.0484216
\(861\) 0 0
\(862\) −5924.78 −0.234105
\(863\) −19806.4 −0.781247 −0.390623 0.920551i \(-0.627741\pi\)
−0.390623 + 0.920551i \(0.627741\pi\)
\(864\) 0 0
\(865\) 74504.8 2.92860
\(866\) −17586.8 −0.690097
\(867\) 0 0
\(868\) 0 0
\(869\) −35975.3 −1.40435
\(870\) 0 0
\(871\) −17360.9 −0.675375
\(872\) −13950.5 −0.541771
\(873\) 0 0
\(874\) 21682.3 0.839147
\(875\) 0 0
\(876\) 0 0
\(877\) 7276.80 0.280183 0.140091 0.990139i \(-0.455260\pi\)
0.140091 + 0.990139i \(0.455260\pi\)
\(878\) −30588.0 −1.17573
\(879\) 0 0
\(880\) −84945.6 −3.25400
\(881\) −25903.2 −0.990580 −0.495290 0.868728i \(-0.664938\pi\)
−0.495290 + 0.868728i \(0.664938\pi\)
\(882\) 0 0
\(883\) 14502.1 0.552701 0.276351 0.961057i \(-0.410875\pi\)
0.276351 + 0.961057i \(0.410875\pi\)
\(884\) 574.684 0.0218651
\(885\) 0 0
\(886\) −30898.7 −1.17163
\(887\) −4327.73 −0.163823 −0.0819115 0.996640i \(-0.526102\pi\)
−0.0819115 + 0.996640i \(0.526102\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 35024.7 1.31914
\(891\) 0 0
\(892\) −127.042 −0.00476869
\(893\) −36763.1 −1.37764
\(894\) 0 0
\(895\) −38491.5 −1.43758
\(896\) 0 0
\(897\) 0 0
\(898\) −25593.3 −0.951067
\(899\) 31590.0 1.17195
\(900\) 0 0
\(901\) 21408.6 0.791592
\(902\) −34511.2 −1.27394
\(903\) 0 0
\(904\) 33883.5 1.24662
\(905\) 78948.5 2.89982
\(906\) 0 0
\(907\) −7456.79 −0.272986 −0.136493 0.990641i \(-0.543583\pi\)
−0.136493 + 0.990641i \(0.543583\pi\)
\(908\) −227.904 −0.00832959
\(909\) 0 0
\(910\) 0 0
\(911\) 27823.5 1.01189 0.505946 0.862565i \(-0.331143\pi\)
0.505946 + 0.862565i \(0.331143\pi\)
\(912\) 0 0
\(913\) −9499.82 −0.344357
\(914\) 822.981 0.0297831
\(915\) 0 0
\(916\) −1233.25 −0.0444844
\(917\) 0 0
\(918\) 0 0
\(919\) −27876.2 −1.00060 −0.500299 0.865853i \(-0.666777\pi\)
−0.500299 + 0.865853i \(0.666777\pi\)
\(920\) 59942.7 2.14810
\(921\) 0 0
\(922\) −28739.3 −1.02655
\(923\) −38300.2 −1.36584
\(924\) 0 0
\(925\) −21639.8 −0.769201
\(926\) 30317.9 1.07593
\(927\) 0 0
\(928\) −1868.92 −0.0661104
\(929\) 26442.1 0.933842 0.466921 0.884299i \(-0.345363\pi\)
0.466921 + 0.884299i \(0.345363\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −46.8917 −0.00164806
\(933\) 0 0
\(934\) 14081.8 0.493330
\(935\) 63220.4 2.21126
\(936\) 0 0
\(937\) 1548.38 0.0539843 0.0269922 0.999636i \(-0.491407\pi\)
0.0269922 + 0.999636i \(0.491407\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −3016.30 −0.104661
\(941\) 57386.7 1.98805 0.994024 0.109163i \(-0.0348171\pi\)
0.994024 + 0.109163i \(0.0348171\pi\)
\(942\) 0 0
\(943\) 23607.6 0.815238
\(944\) −20147.8 −0.694657
\(945\) 0 0
\(946\) 43896.2 1.50866
\(947\) 7852.31 0.269446 0.134723 0.990883i \(-0.456985\pi\)
0.134723 + 0.990883i \(0.456985\pi\)
\(948\) 0 0
\(949\) −12485.8 −0.427087
\(950\) −53446.8 −1.82531
\(951\) 0 0
\(952\) 0 0
\(953\) 17617.8 0.598842 0.299421 0.954121i \(-0.403206\pi\)
0.299421 + 0.954121i \(0.403206\pi\)
\(954\) 0 0
\(955\) −71798.8 −2.43283
\(956\) −686.785 −0.0232345
\(957\) 0 0
\(958\) 1085.16 0.0365970
\(959\) 0 0
\(960\) 0 0
\(961\) 5215.69 0.175076
\(962\) −9931.68 −0.332859
\(963\) 0 0
\(964\) 936.469 0.0312880
\(965\) −26340.0 −0.878669
\(966\) 0 0
\(967\) 40773.5 1.35593 0.677966 0.735093i \(-0.262862\pi\)
0.677966 + 0.735093i \(0.262862\pi\)
\(968\) 68818.3 2.28503
\(969\) 0 0
\(970\) 78621.0 2.60244
\(971\) 4951.31 0.163641 0.0818203 0.996647i \(-0.473927\pi\)
0.0818203 + 0.996647i \(0.473927\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −41960.4 −1.38039
\(975\) 0 0
\(976\) 16181.1 0.530679
\(977\) 54810.5 1.79483 0.897413 0.441192i \(-0.145444\pi\)
0.897413 + 0.441192i \(0.145444\pi\)
\(978\) 0 0
\(979\) −39721.1 −1.29672
\(980\) 0 0
\(981\) 0 0
\(982\) −48881.4 −1.58846
\(983\) 27663.4 0.897583 0.448792 0.893636i \(-0.351855\pi\)
0.448792 + 0.893636i \(0.351855\pi\)
\(984\) 0 0
\(985\) 53982.8 1.74623
\(986\) −21690.0 −0.700560
\(987\) 0 0
\(988\) 773.925 0.0249209
\(989\) −30027.5 −0.965438
\(990\) 0 0
\(991\) 45685.2 1.46442 0.732210 0.681079i \(-0.238489\pi\)
0.732210 + 0.681079i \(0.238489\pi\)
\(992\) −2071.06 −0.0662865
\(993\) 0 0
\(994\) 0 0
\(995\) 1808.26 0.0576139
\(996\) 0 0
\(997\) 17641.7 0.560399 0.280199 0.959942i \(-0.409599\pi\)
0.280199 + 0.959942i \(0.409599\pi\)
\(998\) 53744.2 1.70465
\(999\) 0 0
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 1323.4.a.bb.1.2 6
3.2 odd 2 1323.4.a.bc.1.5 yes 6
7.6 odd 2 1323.4.a.bc.1.2 yes 6
21.20 even 2 inner 1323.4.a.bb.1.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bb.1.2 6 1.1 even 1 trivial
1323.4.a.bb.1.5 yes 6 21.20 even 2 inner
1323.4.a.bc.1.2 yes 6 7.6 odd 2
1323.4.a.bc.1.5 yes 6 3.2 odd 2