Properties

Label 354.4.a.e.1.1
Level $354$
Weight $4$
Character 354.1
Self dual yes
Analytic conductor $20.887$
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] = [354,4,Mod(1,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, 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("354.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 354.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: \(20.8866761420\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.45581.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 37x + 90 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.82712\) of defining polynomial
Character \(\chi\) \(=\) 354.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -13.1803 q^{5} +6.00000 q^{6} -27.1877 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -13.1803 q^{5} +6.00000 q^{6} -27.1877 q^{7} -8.00000 q^{8} +9.00000 q^{9} +26.3605 q^{10} -25.5408 q^{11} -12.0000 q^{12} -70.8493 q^{13} +54.3753 q^{14} +39.5408 q^{15} +16.0000 q^{16} -16.3679 q^{17} -18.0000 q^{18} -93.5839 q^{19} -52.7211 q^{20} +81.5630 q^{21} +51.0816 q^{22} +42.9817 q^{23} +24.0000 q^{24} +48.7195 q^{25} +141.699 q^{26} -27.0000 q^{27} -108.751 q^{28} -46.2217 q^{29} -79.0816 q^{30} +47.2024 q^{31} -32.0000 q^{32} +76.6224 q^{33} +32.7358 q^{34} +358.341 q^{35} +36.0000 q^{36} -272.669 q^{37} +187.168 q^{38} +212.548 q^{39} +105.442 q^{40} -8.26282 q^{41} -163.126 q^{42} -408.690 q^{43} -102.163 q^{44} -118.622 q^{45} -85.9633 q^{46} +550.627 q^{47} -48.0000 q^{48} +396.169 q^{49} -97.4389 q^{50} +49.1038 q^{51} -283.397 q^{52} -45.1847 q^{53} +54.0000 q^{54} +336.635 q^{55} +217.501 q^{56} +280.752 q^{57} +92.4434 q^{58} -59.0000 q^{59} +158.163 q^{60} +507.849 q^{61} -94.4049 q^{62} -244.689 q^{63} +64.0000 q^{64} +933.813 q^{65} -153.245 q^{66} -9.08450 q^{67} -65.4717 q^{68} -128.945 q^{69} -716.681 q^{70} +57.0435 q^{71} -72.0000 q^{72} -31.6317 q^{73} +545.337 q^{74} -146.158 q^{75} -374.335 q^{76} +694.395 q^{77} -425.096 q^{78} +1066.85 q^{79} -210.884 q^{80} +81.0000 q^{81} +16.5256 q^{82} +1017.26 q^{83} +326.252 q^{84} +215.734 q^{85} +817.380 q^{86} +138.665 q^{87} +204.326 q^{88} -41.5553 q^{89} +237.245 q^{90} +1926.23 q^{91} +171.927 q^{92} -141.607 q^{93} -1101.25 q^{94} +1233.46 q^{95} +96.0000 q^{96} -1009.54 q^{97} -792.337 q^{98} -229.867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} + 4 q^{5} + 18 q^{6} + 13 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} + 4 q^{5} + 18 q^{6} + 13 q^{7} - 24 q^{8} + 27 q^{9} - 8 q^{10} + 54 q^{11} - 36 q^{12} - 52 q^{13} - 26 q^{14} - 12 q^{15} + 48 q^{16} + 89 q^{17} - 54 q^{18} - 163 q^{19} + 16 q^{20} - 39 q^{21} - 108 q^{22} - 31 q^{23} + 72 q^{24} - 9 q^{25} + 104 q^{26} - 81 q^{27} + 52 q^{28} - 127 q^{29} + 24 q^{30} - 55 q^{31} - 96 q^{32} - 162 q^{33} - 178 q^{34} + 847 q^{35} + 108 q^{36} - 401 q^{37} + 326 q^{38} + 156 q^{39} - 32 q^{40} + 255 q^{41} + 78 q^{42} - 400 q^{43} + 216 q^{44} + 36 q^{45} + 62 q^{46} + 1247 q^{47} - 144 q^{48} + 978 q^{49} + 18 q^{50} - 267 q^{51} - 208 q^{52} + 496 q^{53} + 162 q^{54} + 1154 q^{55} - 104 q^{56} + 489 q^{57} + 254 q^{58} - 177 q^{59} - 48 q^{60} + 1063 q^{61} + 110 q^{62} + 117 q^{63} + 192 q^{64} + 1446 q^{65} + 324 q^{66} - 72 q^{67} + 356 q^{68} + 93 q^{69} - 1694 q^{70} + 2234 q^{71} - 216 q^{72} - 577 q^{73} + 802 q^{74} + 27 q^{75} - 652 q^{76} + 2723 q^{77} - 312 q^{78} + 1164 q^{79} + 64 q^{80} + 243 q^{81} - 510 q^{82} + 85 q^{83} - 156 q^{84} + 1309 q^{85} + 800 q^{86} + 381 q^{87} - 432 q^{88} - 467 q^{89} - 72 q^{90} + 3429 q^{91} - 124 q^{92} + 165 q^{93} - 2494 q^{94} - 401 q^{95} + 288 q^{96} - 3498 q^{97} - 1956 q^{98} + 486 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −13.1803 −1.17888 −0.589439 0.807813i \(-0.700651\pi\)
−0.589439 + 0.807813i \(0.700651\pi\)
\(6\) 6.00000 0.408248
\(7\) −27.1877 −1.46800 −0.733998 0.679152i \(-0.762348\pi\)
−0.733998 + 0.679152i \(0.762348\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 26.3605 0.833593
\(11\) −25.5408 −0.700077 −0.350038 0.936735i \(-0.613831\pi\)
−0.350038 + 0.936735i \(0.613831\pi\)
\(12\) −12.0000 −0.288675
\(13\) −70.8493 −1.51154 −0.755771 0.654836i \(-0.772738\pi\)
−0.755771 + 0.654836i \(0.772738\pi\)
\(14\) 54.3753 1.03803
\(15\) 39.5408 0.680626
\(16\) 16.0000 0.250000
\(17\) −16.3679 −0.233518 −0.116759 0.993160i \(-0.537250\pi\)
−0.116759 + 0.993160i \(0.537250\pi\)
\(18\) −18.0000 −0.235702
\(19\) −93.5839 −1.12998 −0.564990 0.825098i \(-0.691120\pi\)
−0.564990 + 0.825098i \(0.691120\pi\)
\(20\) −52.7211 −0.589439
\(21\) 81.5630 0.847548
\(22\) 51.0816 0.495029
\(23\) 42.9817 0.389665 0.194833 0.980837i \(-0.437584\pi\)
0.194833 + 0.980837i \(0.437584\pi\)
\(24\) 24.0000 0.204124
\(25\) 48.7195 0.389756
\(26\) 141.699 1.06882
\(27\) −27.0000 −0.192450
\(28\) −108.751 −0.733998
\(29\) −46.2217 −0.295971 −0.147985 0.988990i \(-0.547279\pi\)
−0.147985 + 0.988990i \(0.547279\pi\)
\(30\) −79.0816 −0.481275
\(31\) 47.2024 0.273478 0.136739 0.990607i \(-0.456338\pi\)
0.136739 + 0.990607i \(0.456338\pi\)
\(32\) −32.0000 −0.176777
\(33\) 76.6224 0.404189
\(34\) 32.7358 0.165122
\(35\) 358.341 1.73059
\(36\) 36.0000 0.166667
\(37\) −272.669 −1.21153 −0.605763 0.795645i \(-0.707132\pi\)
−0.605763 + 0.795645i \(0.707132\pi\)
\(38\) 187.168 0.799016
\(39\) 212.548 0.872690
\(40\) 105.442 0.416797
\(41\) −8.26282 −0.0314741 −0.0157370 0.999876i \(-0.505009\pi\)
−0.0157370 + 0.999876i \(0.505009\pi\)
\(42\) −163.126 −0.599307
\(43\) −408.690 −1.44941 −0.724705 0.689059i \(-0.758024\pi\)
−0.724705 + 0.689059i \(0.758024\pi\)
\(44\) −102.163 −0.350038
\(45\) −118.622 −0.392960
\(46\) −85.9633 −0.275535
\(47\) 550.627 1.70888 0.854438 0.519554i \(-0.173902\pi\)
0.854438 + 0.519554i \(0.173902\pi\)
\(48\) −48.0000 −0.144338
\(49\) 396.169 1.15501
\(50\) −97.4389 −0.275599
\(51\) 49.1038 0.134822
\(52\) −283.397 −0.755771
\(53\) −45.1847 −0.117106 −0.0585528 0.998284i \(-0.518649\pi\)
−0.0585528 + 0.998284i \(0.518649\pi\)
\(54\) 54.0000 0.136083
\(55\) 336.635 0.825306
\(56\) 217.501 0.519015
\(57\) 280.752 0.652394
\(58\) 92.4434 0.209283
\(59\) −59.0000 −0.130189
\(60\) 158.163 0.340313
\(61\) 507.849 1.06596 0.532979 0.846129i \(-0.321073\pi\)
0.532979 + 0.846129i \(0.321073\pi\)
\(62\) −94.4049 −0.193378
\(63\) −244.689 −0.489332
\(64\) 64.0000 0.125000
\(65\) 933.813 1.78193
\(66\) −153.245 −0.285805
\(67\) −9.08450 −0.0165649 −0.00828245 0.999966i \(-0.502636\pi\)
−0.00828245 + 0.999966i \(0.502636\pi\)
\(68\) −65.4717 −0.116759
\(69\) −128.945 −0.224973
\(70\) −716.681 −1.22371
\(71\) 57.0435 0.0953495 0.0476748 0.998863i \(-0.484819\pi\)
0.0476748 + 0.998863i \(0.484819\pi\)
\(72\) −72.0000 −0.117851
\(73\) −31.6317 −0.0507153 −0.0253576 0.999678i \(-0.508072\pi\)
−0.0253576 + 0.999678i \(0.508072\pi\)
\(74\) 545.337 0.856678
\(75\) −146.158 −0.225026
\(76\) −374.335 −0.564990
\(77\) 694.395 1.02771
\(78\) −425.096 −0.617085
\(79\) 1066.85 1.51937 0.759686 0.650290i \(-0.225353\pi\)
0.759686 + 0.650290i \(0.225353\pi\)
\(80\) −210.884 −0.294720
\(81\) 81.0000 0.111111
\(82\) 16.5256 0.0222555
\(83\) 1017.26 1.34528 0.672640 0.739969i \(-0.265160\pi\)
0.672640 + 0.739969i \(0.265160\pi\)
\(84\) 326.252 0.423774
\(85\) 215.734 0.275289
\(86\) 817.380 1.02489
\(87\) 138.665 0.170879
\(88\) 204.326 0.247514
\(89\) −41.5553 −0.0494927 −0.0247464 0.999694i \(-0.507878\pi\)
−0.0247464 + 0.999694i \(0.507878\pi\)
\(90\) 237.245 0.277864
\(91\) 1926.23 2.21894
\(92\) 171.927 0.194833
\(93\) −141.607 −0.157892
\(94\) −1101.25 −1.20836
\(95\) 1233.46 1.33211
\(96\) 96.0000 0.102062
\(97\) −1009.54 −1.05674 −0.528368 0.849015i \(-0.677196\pi\)
−0.528368 + 0.849015i \(0.677196\pi\)
\(98\) −792.337 −0.816716
\(99\) −229.867 −0.233359
\(100\) 194.878 0.194878
\(101\) −863.519 −0.850726 −0.425363 0.905023i \(-0.639854\pi\)
−0.425363 + 0.905023i \(0.639854\pi\)
\(102\) −98.2075 −0.0953333
\(103\) −1870.17 −1.78906 −0.894532 0.447003i \(-0.852491\pi\)
−0.894532 + 0.447003i \(0.852491\pi\)
\(104\) 566.794 0.534411
\(105\) −1075.02 −0.999156
\(106\) 90.3694 0.0828062
\(107\) −1147.67 −1.03691 −0.518457 0.855104i \(-0.673493\pi\)
−0.518457 + 0.855104i \(0.673493\pi\)
\(108\) −108.000 −0.0962250
\(109\) −1210.19 −1.06345 −0.531723 0.846918i \(-0.678455\pi\)
−0.531723 + 0.846918i \(0.678455\pi\)
\(110\) −673.269 −0.583579
\(111\) 818.006 0.699475
\(112\) −435.003 −0.366999
\(113\) 620.165 0.516285 0.258142 0.966107i \(-0.416890\pi\)
0.258142 + 0.966107i \(0.416890\pi\)
\(114\) −561.503 −0.461312
\(115\) −566.510 −0.459368
\(116\) −184.887 −0.147985
\(117\) −637.644 −0.503848
\(118\) 118.000 0.0920575
\(119\) 445.006 0.342803
\(120\) −316.326 −0.240638
\(121\) −678.667 −0.509893
\(122\) −1015.70 −0.753746
\(123\) 24.7885 0.0181716
\(124\) 188.810 0.136739
\(125\) 1005.40 0.719404
\(126\) 489.378 0.346010
\(127\) −1267.72 −0.885763 −0.442882 0.896580i \(-0.646044\pi\)
−0.442882 + 0.896580i \(0.646044\pi\)
\(128\) −128.000 −0.0883883
\(129\) 1226.07 0.836817
\(130\) −1867.63 −1.26001
\(131\) −1615.99 −1.07778 −0.538892 0.842375i \(-0.681157\pi\)
−0.538892 + 0.842375i \(0.681157\pi\)
\(132\) 306.490 0.202095
\(133\) 2544.33 1.65880
\(134\) 18.1690 0.0117132
\(135\) 355.867 0.226875
\(136\) 130.943 0.0825611
\(137\) −2033.70 −1.26826 −0.634128 0.773228i \(-0.718641\pi\)
−0.634128 + 0.773228i \(0.718641\pi\)
\(138\) 257.890 0.159080
\(139\) 2082.43 1.27072 0.635358 0.772218i \(-0.280853\pi\)
0.635358 + 0.772218i \(0.280853\pi\)
\(140\) 1433.36 0.865295
\(141\) −1651.88 −0.986620
\(142\) −114.087 −0.0674223
\(143\) 1809.55 1.05820
\(144\) 144.000 0.0833333
\(145\) 609.215 0.348914
\(146\) 63.2635 0.0358611
\(147\) −1188.51 −0.666846
\(148\) −1090.67 −0.605763
\(149\) −611.676 −0.336312 −0.168156 0.985760i \(-0.553781\pi\)
−0.168156 + 0.985760i \(0.553781\pi\)
\(150\) 292.317 0.159117
\(151\) −42.0765 −0.0226764 −0.0113382 0.999936i \(-0.503609\pi\)
−0.0113382 + 0.999936i \(0.503609\pi\)
\(152\) 748.671 0.399508
\(153\) −147.311 −0.0778393
\(154\) −1388.79 −0.726700
\(155\) −622.141 −0.322397
\(156\) 850.191 0.436345
\(157\) 251.374 0.127782 0.0638912 0.997957i \(-0.479649\pi\)
0.0638912 + 0.997957i \(0.479649\pi\)
\(158\) −2133.71 −1.07436
\(159\) 135.554 0.0676110
\(160\) 421.769 0.208398
\(161\) −1168.57 −0.572027
\(162\) −162.000 −0.0785674
\(163\) −3291.77 −1.58179 −0.790893 0.611954i \(-0.790384\pi\)
−0.790893 + 0.611954i \(0.790384\pi\)
\(164\) −33.0513 −0.0157370
\(165\) −1009.90 −0.476490
\(166\) −2034.51 −0.951257
\(167\) −1372.32 −0.635887 −0.317944 0.948110i \(-0.602992\pi\)
−0.317944 + 0.948110i \(0.602992\pi\)
\(168\) −652.504 −0.299653
\(169\) 2822.62 1.28476
\(170\) −431.467 −0.194659
\(171\) −842.255 −0.376660
\(172\) −1634.76 −0.724705
\(173\) −1188.59 −0.522354 −0.261177 0.965291i \(-0.584111\pi\)
−0.261177 + 0.965291i \(0.584111\pi\)
\(174\) −277.330 −0.120830
\(175\) −1324.57 −0.572160
\(176\) −408.653 −0.175019
\(177\) 177.000 0.0751646
\(178\) 83.1106 0.0349966
\(179\) −2.50060 −0.00104416 −0.000522078 1.00000i \(-0.500166\pi\)
−0.000522078 1.00000i \(0.500166\pi\)
\(180\) −474.490 −0.196480
\(181\) 1504.51 0.617841 0.308920 0.951088i \(-0.400032\pi\)
0.308920 + 0.951088i \(0.400032\pi\)
\(182\) −3852.45 −1.56903
\(183\) −1523.55 −0.615431
\(184\) −343.853 −0.137767
\(185\) 3593.85 1.42824
\(186\) 283.215 0.111647
\(187\) 418.050 0.163480
\(188\) 2202.51 0.854438
\(189\) 734.067 0.282516
\(190\) −2466.92 −0.941943
\(191\) 4474.89 1.69525 0.847623 0.530599i \(-0.178033\pi\)
0.847623 + 0.530599i \(0.178033\pi\)
\(192\) −192.000 −0.0721688
\(193\) −4906.36 −1.82988 −0.914942 0.403586i \(-0.867764\pi\)
−0.914942 + 0.403586i \(0.867764\pi\)
\(194\) 2019.08 0.747225
\(195\) −2801.44 −1.02880
\(196\) 1584.67 0.577505
\(197\) −2489.12 −0.900216 −0.450108 0.892974i \(-0.648615\pi\)
−0.450108 + 0.892974i \(0.648615\pi\)
\(198\) 459.734 0.165010
\(199\) 185.142 0.0659517 0.0329758 0.999456i \(-0.489502\pi\)
0.0329758 + 0.999456i \(0.489502\pi\)
\(200\) −389.756 −0.137799
\(201\) 27.2535 0.00956375
\(202\) 1727.04 0.601554
\(203\) 1256.66 0.434484
\(204\) 196.415 0.0674108
\(205\) 108.906 0.0371041
\(206\) 3740.35 1.26506
\(207\) 386.835 0.129888
\(208\) −1133.59 −0.377886
\(209\) 2390.21 0.791072
\(210\) 2150.04 0.706510
\(211\) −5736.47 −1.87164 −0.935818 0.352483i \(-0.885337\pi\)
−0.935818 + 0.352483i \(0.885337\pi\)
\(212\) −180.739 −0.0585528
\(213\) −171.130 −0.0550501
\(214\) 2295.35 0.733209
\(215\) 5386.64 1.70868
\(216\) 216.000 0.0680414
\(217\) −1283.32 −0.401464
\(218\) 2420.39 0.751970
\(219\) 94.8952 0.0292805
\(220\) 1346.54 0.412653
\(221\) 1159.66 0.352972
\(222\) −1636.01 −0.494603
\(223\) 4228.14 1.26967 0.634837 0.772646i \(-0.281067\pi\)
0.634837 + 0.772646i \(0.281067\pi\)
\(224\) 870.005 0.259507
\(225\) 438.475 0.129919
\(226\) −1240.33 −0.365069
\(227\) 5589.59 1.63434 0.817168 0.576399i \(-0.195543\pi\)
0.817168 + 0.576399i \(0.195543\pi\)
\(228\) 1123.01 0.326197
\(229\) 2306.37 0.665543 0.332771 0.943008i \(-0.392016\pi\)
0.332771 + 0.943008i \(0.392016\pi\)
\(230\) 1133.02 0.324822
\(231\) −2083.18 −0.593348
\(232\) 369.774 0.104642
\(233\) −1393.27 −0.391744 −0.195872 0.980629i \(-0.562754\pi\)
−0.195872 + 0.980629i \(0.562754\pi\)
\(234\) 1275.29 0.356274
\(235\) −7257.41 −2.01456
\(236\) −236.000 −0.0650945
\(237\) −3200.56 −0.877210
\(238\) −890.011 −0.242399
\(239\) −2293.76 −0.620800 −0.310400 0.950606i \(-0.600463\pi\)
−0.310400 + 0.950606i \(0.600463\pi\)
\(240\) 632.653 0.170157
\(241\) −4418.38 −1.18097 −0.590483 0.807050i \(-0.701063\pi\)
−0.590483 + 0.807050i \(0.701063\pi\)
\(242\) 1357.33 0.360549
\(243\) −243.000 −0.0641500
\(244\) 2031.40 0.532979
\(245\) −5221.61 −1.36162
\(246\) −49.5769 −0.0128492
\(247\) 6630.35 1.70801
\(248\) −377.619 −0.0966890
\(249\) −3051.77 −0.776698
\(250\) −2010.80 −0.508696
\(251\) 2942.59 0.739978 0.369989 0.929036i \(-0.379361\pi\)
0.369989 + 0.929036i \(0.379361\pi\)
\(252\) −978.756 −0.244666
\(253\) −1097.79 −0.272795
\(254\) 2535.44 0.626329
\(255\) −647.201 −0.158938
\(256\) 256.000 0.0625000
\(257\) −5519.95 −1.33979 −0.669894 0.742457i \(-0.733660\pi\)
−0.669894 + 0.742457i \(0.733660\pi\)
\(258\) −2452.14 −0.591719
\(259\) 7413.22 1.77851
\(260\) 3735.25 0.890963
\(261\) −415.995 −0.0986570
\(262\) 3231.98 0.762109
\(263\) 5184.34 1.21551 0.607757 0.794123i \(-0.292069\pi\)
0.607757 + 0.794123i \(0.292069\pi\)
\(264\) −612.979 −0.142903
\(265\) 595.547 0.138053
\(266\) −5088.65 −1.17295
\(267\) 124.666 0.0285746
\(268\) −36.3380 −0.00828245
\(269\) 5819.75 1.31910 0.659548 0.751663i \(-0.270748\pi\)
0.659548 + 0.751663i \(0.270748\pi\)
\(270\) −711.734 −0.160425
\(271\) 5740.44 1.28674 0.643371 0.765554i \(-0.277535\pi\)
0.643371 + 0.765554i \(0.277535\pi\)
\(272\) −261.887 −0.0583795
\(273\) −5778.68 −1.28110
\(274\) 4067.41 0.896793
\(275\) −1244.33 −0.272859
\(276\) −515.780 −0.112487
\(277\) 3439.33 0.746027 0.373013 0.927826i \(-0.378325\pi\)
0.373013 + 0.927826i \(0.378325\pi\)
\(278\) −4164.86 −0.898531
\(279\) 424.822 0.0911592
\(280\) −2866.72 −0.611856
\(281\) −2630.81 −0.558508 −0.279254 0.960217i \(-0.590087\pi\)
−0.279254 + 0.960217i \(0.590087\pi\)
\(282\) 3303.76 0.697645
\(283\) −2835.99 −0.595696 −0.297848 0.954613i \(-0.596269\pi\)
−0.297848 + 0.954613i \(0.596269\pi\)
\(284\) 228.174 0.0476748
\(285\) −3700.38 −0.769094
\(286\) −3619.10 −0.748257
\(287\) 224.647 0.0462038
\(288\) −288.000 −0.0589256
\(289\) −4645.09 −0.945469
\(290\) −1218.43 −0.246719
\(291\) 3028.62 0.610107
\(292\) −126.527 −0.0253576
\(293\) −7754.35 −1.54612 −0.773062 0.634331i \(-0.781276\pi\)
−0.773062 + 0.634331i \(0.781276\pi\)
\(294\) 2377.01 0.471531
\(295\) 777.636 0.153477
\(296\) 2181.35 0.428339
\(297\) 689.602 0.134730
\(298\) 1223.35 0.237808
\(299\) −3045.22 −0.588995
\(300\) −584.634 −0.112513
\(301\) 11111.3 2.12773
\(302\) 84.1530 0.0160346
\(303\) 2590.56 0.491167
\(304\) −1497.34 −0.282495
\(305\) −6693.59 −1.25663
\(306\) 294.623 0.0550407
\(307\) 4407.11 0.819307 0.409653 0.912241i \(-0.365650\pi\)
0.409653 + 0.912241i \(0.365650\pi\)
\(308\) 2777.58 0.513855
\(309\) 5610.52 1.03292
\(310\) 1244.28 0.227969
\(311\) 1940.72 0.353853 0.176926 0.984224i \(-0.443385\pi\)
0.176926 + 0.984224i \(0.443385\pi\)
\(312\) −1700.38 −0.308542
\(313\) 6120.82 1.10533 0.552667 0.833402i \(-0.313610\pi\)
0.552667 + 0.833402i \(0.313610\pi\)
\(314\) −502.749 −0.0903559
\(315\) 3225.07 0.576863
\(316\) 4267.41 0.759686
\(317\) −3220.49 −0.570602 −0.285301 0.958438i \(-0.592093\pi\)
−0.285301 + 0.958438i \(0.592093\pi\)
\(318\) −271.108 −0.0478082
\(319\) 1180.54 0.207202
\(320\) −843.537 −0.147360
\(321\) 3443.02 0.598663
\(322\) 2337.14 0.404484
\(323\) 1531.77 0.263870
\(324\) 324.000 0.0555556
\(325\) −3451.74 −0.589132
\(326\) 6583.54 1.11849
\(327\) 3630.58 0.613981
\(328\) 66.1026 0.0111278
\(329\) −14970.2 −2.50862
\(330\) 2019.81 0.336930
\(331\) 21.1960 0.00351976 0.00175988 0.999998i \(-0.499440\pi\)
0.00175988 + 0.999998i \(0.499440\pi\)
\(332\) 4069.02 0.672640
\(333\) −2454.02 −0.403842
\(334\) 2744.64 0.449640
\(335\) 119.736 0.0195280
\(336\) 1305.01 0.211887
\(337\) 7864.17 1.27118 0.635591 0.772026i \(-0.280756\pi\)
0.635591 + 0.772026i \(0.280756\pi\)
\(338\) −5645.24 −0.908464
\(339\) −1860.49 −0.298077
\(340\) 862.935 0.137645
\(341\) −1205.59 −0.191455
\(342\) 1684.51 0.266339
\(343\) −1445.53 −0.227555
\(344\) 3269.52 0.512444
\(345\) 1699.53 0.265216
\(346\) 2377.19 0.369360
\(347\) 11432.1 1.76862 0.884308 0.466904i \(-0.154631\pi\)
0.884308 + 0.466904i \(0.154631\pi\)
\(348\) 554.661 0.0854394
\(349\) 7424.04 1.13868 0.569341 0.822102i \(-0.307199\pi\)
0.569341 + 0.822102i \(0.307199\pi\)
\(350\) 2649.14 0.404578
\(351\) 1912.93 0.290897
\(352\) 817.306 0.123757
\(353\) −9914.93 −1.49495 −0.747476 0.664288i \(-0.768735\pi\)
−0.747476 + 0.664288i \(0.768735\pi\)
\(354\) −354.000 −0.0531494
\(355\) −751.848 −0.112406
\(356\) −166.221 −0.0247464
\(357\) −1335.02 −0.197918
\(358\) 5.00121 0.000738330 0
\(359\) −5706.08 −0.838874 −0.419437 0.907785i \(-0.637772\pi\)
−0.419437 + 0.907785i \(0.637772\pi\)
\(360\) 948.979 0.138932
\(361\) 1898.94 0.276854
\(362\) −3009.02 −0.436879
\(363\) 2036.00 0.294387
\(364\) 7704.90 1.10947
\(365\) 416.915 0.0597872
\(366\) 3047.09 0.435175
\(367\) 10571.3 1.50359 0.751796 0.659395i \(-0.229188\pi\)
0.751796 + 0.659395i \(0.229188\pi\)
\(368\) 687.707 0.0974163
\(369\) −74.3654 −0.0104914
\(370\) −7187.69 −1.00992
\(371\) 1228.47 0.171911
\(372\) −566.429 −0.0789462
\(373\) 1164.85 0.161699 0.0808495 0.996726i \(-0.474237\pi\)
0.0808495 + 0.996726i \(0.474237\pi\)
\(374\) −836.100 −0.115598
\(375\) −3016.19 −0.415348
\(376\) −4405.01 −0.604179
\(377\) 3274.78 0.447373
\(378\) −1468.13 −0.199769
\(379\) −5178.94 −0.701911 −0.350956 0.936392i \(-0.614143\pi\)
−0.350956 + 0.936392i \(0.614143\pi\)
\(380\) 4933.84 0.666055
\(381\) 3803.16 0.511396
\(382\) −8949.79 −1.19872
\(383\) 10634.4 1.41878 0.709392 0.704814i \(-0.248970\pi\)
0.709392 + 0.704814i \(0.248970\pi\)
\(384\) 384.000 0.0510310
\(385\) −9152.31 −1.21154
\(386\) 9812.72 1.29392
\(387\) −3678.21 −0.483137
\(388\) −4038.17 −0.528368
\(389\) −1177.81 −0.153514 −0.0767572 0.997050i \(-0.524457\pi\)
−0.0767572 + 0.997050i \(0.524457\pi\)
\(390\) 5602.88 0.727468
\(391\) −703.521 −0.0909938
\(392\) −3169.35 −0.408358
\(393\) 4847.97 0.622259
\(394\) 4978.24 0.636549
\(395\) −14061.4 −1.79116
\(396\) −919.469 −0.116679
\(397\) −2986.65 −0.377570 −0.188785 0.982018i \(-0.560455\pi\)
−0.188785 + 0.982018i \(0.560455\pi\)
\(398\) −370.284 −0.0466349
\(399\) −7632.98 −0.957711
\(400\) 779.511 0.0974389
\(401\) −4191.02 −0.521919 −0.260960 0.965350i \(-0.584039\pi\)
−0.260960 + 0.965350i \(0.584039\pi\)
\(402\) −54.5070 −0.00676259
\(403\) −3344.26 −0.413373
\(404\) −3454.08 −0.425363
\(405\) −1067.60 −0.130987
\(406\) −2513.32 −0.307227
\(407\) 6964.18 0.848161
\(408\) −392.830 −0.0476666
\(409\) 582.890 0.0704696 0.0352348 0.999379i \(-0.488782\pi\)
0.0352348 + 0.999379i \(0.488782\pi\)
\(410\) −217.812 −0.0262366
\(411\) 6101.11 0.732228
\(412\) −7480.69 −0.894532
\(413\) 1604.07 0.191117
\(414\) −773.670 −0.0918449
\(415\) −13407.7 −1.58592
\(416\) 2267.18 0.267206
\(417\) −6247.29 −0.733648
\(418\) −4780.41 −0.559372
\(419\) 15862.1 1.84944 0.924718 0.380652i \(-0.124300\pi\)
0.924718 + 0.380652i \(0.124300\pi\)
\(420\) −4300.09 −0.499578
\(421\) −1998.86 −0.231398 −0.115699 0.993284i \(-0.536911\pi\)
−0.115699 + 0.993284i \(0.536911\pi\)
\(422\) 11472.9 1.32345
\(423\) 4955.64 0.569625
\(424\) 361.478 0.0414031
\(425\) −797.436 −0.0910149
\(426\) 342.261 0.0389263
\(427\) −13807.2 −1.56482
\(428\) −4590.70 −0.518457
\(429\) −5428.64 −0.610950
\(430\) −10773.3 −1.20822
\(431\) −8287.44 −0.926199 −0.463100 0.886306i \(-0.653263\pi\)
−0.463100 + 0.886306i \(0.653263\pi\)
\(432\) −432.000 −0.0481125
\(433\) −1060.75 −0.117728 −0.0588639 0.998266i \(-0.518748\pi\)
−0.0588639 + 0.998266i \(0.518748\pi\)
\(434\) 2566.65 0.283878
\(435\) −1827.64 −0.201446
\(436\) −4840.78 −0.531723
\(437\) −4022.39 −0.440314
\(438\) −189.790 −0.0207044
\(439\) 1340.69 0.145757 0.0728787 0.997341i \(-0.476781\pi\)
0.0728787 + 0.997341i \(0.476781\pi\)
\(440\) −2693.08 −0.291790
\(441\) 3565.52 0.385004
\(442\) −2319.31 −0.249589
\(443\) −16182.5 −1.73556 −0.867782 0.496945i \(-0.834455\pi\)
−0.867782 + 0.496945i \(0.834455\pi\)
\(444\) 3272.02 0.349737
\(445\) 547.710 0.0583459
\(446\) −8456.28 −0.897795
\(447\) 1835.03 0.194170
\(448\) −1740.01 −0.183499
\(449\) 993.687 0.104443 0.0522216 0.998636i \(-0.483370\pi\)
0.0522216 + 0.998636i \(0.483370\pi\)
\(450\) −876.950 −0.0918663
\(451\) 211.039 0.0220342
\(452\) 2480.66 0.258142
\(453\) 126.230 0.0130922
\(454\) −11179.2 −1.15565
\(455\) −25388.2 −2.61586
\(456\) −2246.01 −0.230656
\(457\) 833.216 0.0852871 0.0426436 0.999090i \(-0.486422\pi\)
0.0426436 + 0.999090i \(0.486422\pi\)
\(458\) −4612.74 −0.470610
\(459\) 441.934 0.0449405
\(460\) −2266.04 −0.229684
\(461\) −17336.5 −1.75150 −0.875748 0.482769i \(-0.839631\pi\)
−0.875748 + 0.482769i \(0.839631\pi\)
\(462\) 4166.37 0.419561
\(463\) 15278.3 1.53357 0.766785 0.641904i \(-0.221855\pi\)
0.766785 + 0.641904i \(0.221855\pi\)
\(464\) −739.547 −0.0739927
\(465\) 1866.42 0.186136
\(466\) 2786.55 0.277005
\(467\) −15151.7 −1.50136 −0.750681 0.660665i \(-0.770274\pi\)
−0.750681 + 0.660665i \(0.770274\pi\)
\(468\) −2550.57 −0.251924
\(469\) 246.986 0.0243172
\(470\) 14514.8 1.42451
\(471\) −754.123 −0.0737753
\(472\) 472.000 0.0460287
\(473\) 10438.3 1.01470
\(474\) 6401.12 0.620281
\(475\) −4559.36 −0.440416
\(476\) 1780.02 0.171402
\(477\) −406.662 −0.0390352
\(478\) 4587.53 0.438972
\(479\) −18559.6 −1.77038 −0.885189 0.465231i \(-0.845971\pi\)
−0.885189 + 0.465231i \(0.845971\pi\)
\(480\) −1265.31 −0.120319
\(481\) 19318.4 1.83127
\(482\) 8836.75 0.835069
\(483\) 3505.71 0.330260
\(484\) −2714.67 −0.254946
\(485\) 13306.0 1.24576
\(486\) 486.000 0.0453609
\(487\) −12782.0 −1.18934 −0.594671 0.803969i \(-0.702718\pi\)
−0.594671 + 0.803969i \(0.702718\pi\)
\(488\) −4062.79 −0.376873
\(489\) 9875.30 0.913245
\(490\) 10443.2 0.962809
\(491\) 7846.62 0.721207 0.360604 0.932719i \(-0.382571\pi\)
0.360604 + 0.932719i \(0.382571\pi\)
\(492\) 99.1539 0.00908578
\(493\) 756.553 0.0691145
\(494\) −13260.7 −1.20775
\(495\) 3029.71 0.275102
\(496\) 755.239 0.0683694
\(497\) −1550.88 −0.139973
\(498\) 6103.53 0.549209
\(499\) 20002.4 1.79445 0.897226 0.441571i \(-0.145579\pi\)
0.897226 + 0.441571i \(0.145579\pi\)
\(500\) 4021.59 0.359702
\(501\) 4116.96 0.367130
\(502\) −5885.18 −0.523244
\(503\) 1592.53 0.141168 0.0705839 0.997506i \(-0.477514\pi\)
0.0705839 + 0.997506i \(0.477514\pi\)
\(504\) 1957.51 0.173005
\(505\) 11381.4 1.00290
\(506\) 2195.57 0.192895
\(507\) −8467.86 −0.741757
\(508\) −5070.88 −0.442882
\(509\) −6909.08 −0.601649 −0.300825 0.953679i \(-0.597262\pi\)
−0.300825 + 0.953679i \(0.597262\pi\)
\(510\) 1294.40 0.112386
\(511\) 859.993 0.0744498
\(512\) −512.000 −0.0441942
\(513\) 2526.76 0.217465
\(514\) 11039.9 0.947373
\(515\) 24649.4 2.10909
\(516\) 4904.28 0.418409
\(517\) −14063.4 −1.19634
\(518\) −14826.4 −1.25760
\(519\) 3565.78 0.301581
\(520\) −7470.50 −0.630006
\(521\) 5022.19 0.422315 0.211158 0.977452i \(-0.432277\pi\)
0.211158 + 0.977452i \(0.432277\pi\)
\(522\) 831.991 0.0697610
\(523\) 1346.37 0.112567 0.0562835 0.998415i \(-0.482075\pi\)
0.0562835 + 0.998415i \(0.482075\pi\)
\(524\) −6463.97 −0.538892
\(525\) 3973.70 0.330336
\(526\) −10368.7 −0.859498
\(527\) −772.606 −0.0638619
\(528\) 1225.96 0.101047
\(529\) −10319.6 −0.848161
\(530\) −1191.09 −0.0976185
\(531\) −531.000 −0.0433963
\(532\) 10177.3 0.829402
\(533\) 585.415 0.0475744
\(534\) −249.332 −0.0202053
\(535\) 15126.7 1.22240
\(536\) 72.6760 0.00585658
\(537\) 7.50181 0.000602844 0
\(538\) −11639.5 −0.932741
\(539\) −10118.5 −0.808596
\(540\) 1423.47 0.113438
\(541\) −15356.6 −1.22039 −0.610196 0.792251i \(-0.708909\pi\)
−0.610196 + 0.792251i \(0.708909\pi\)
\(542\) −11480.9 −0.909864
\(543\) −4513.52 −0.356711
\(544\) 523.774 0.0412805
\(545\) 15950.7 1.25367
\(546\) 11557.4 0.905878
\(547\) 13305.2 1.04002 0.520009 0.854160i \(-0.325928\pi\)
0.520009 + 0.854160i \(0.325928\pi\)
\(548\) −8134.82 −0.634128
\(549\) 4570.64 0.355319
\(550\) 2488.67 0.192940
\(551\) 4325.61 0.334441
\(552\) 1031.56 0.0795401
\(553\) −29005.3 −2.23043
\(554\) −6878.67 −0.527521
\(555\) −10781.5 −0.824596
\(556\) 8329.72 0.635358
\(557\) 5729.87 0.435875 0.217937 0.975963i \(-0.430067\pi\)
0.217937 + 0.975963i \(0.430067\pi\)
\(558\) −849.644 −0.0644593
\(559\) 28955.4 2.19084
\(560\) 5733.45 0.432647
\(561\) −1254.15 −0.0943855
\(562\) 5261.61 0.394925
\(563\) 6028.34 0.451269 0.225634 0.974212i \(-0.427554\pi\)
0.225634 + 0.974212i \(0.427554\pi\)
\(564\) −6607.52 −0.493310
\(565\) −8173.94 −0.608637
\(566\) 5671.97 0.421220
\(567\) −2202.20 −0.163111
\(568\) −456.348 −0.0337112
\(569\) 5645.33 0.415931 0.207965 0.978136i \(-0.433316\pi\)
0.207965 + 0.978136i \(0.433316\pi\)
\(570\) 7400.76 0.543831
\(571\) −8432.15 −0.617994 −0.308997 0.951063i \(-0.599993\pi\)
−0.308997 + 0.951063i \(0.599993\pi\)
\(572\) 7238.19 0.529098
\(573\) −13424.7 −0.978751
\(574\) −449.294 −0.0326710
\(575\) 2094.04 0.151874
\(576\) 576.000 0.0416667
\(577\) 7885.56 0.568943 0.284472 0.958684i \(-0.408182\pi\)
0.284472 + 0.958684i \(0.408182\pi\)
\(578\) 9290.18 0.668548
\(579\) 14719.1 1.05648
\(580\) 2436.86 0.174457
\(581\) −27656.8 −1.97487
\(582\) −6057.25 −0.431411
\(583\) 1154.05 0.0819829
\(584\) 253.054 0.0179306
\(585\) 8404.31 0.593975
\(586\) 15508.7 1.09327
\(587\) 10360.5 0.728490 0.364245 0.931303i \(-0.381327\pi\)
0.364245 + 0.931303i \(0.381327\pi\)
\(588\) −4754.02 −0.333423
\(589\) −4417.39 −0.309024
\(590\) −1555.27 −0.108525
\(591\) 7467.37 0.519740
\(592\) −4362.70 −0.302881
\(593\) −11986.1 −0.830032 −0.415016 0.909814i \(-0.636224\pi\)
−0.415016 + 0.909814i \(0.636224\pi\)
\(594\) −1379.20 −0.0952684
\(595\) −5865.29 −0.404124
\(596\) −2446.71 −0.168156
\(597\) −555.427 −0.0380772
\(598\) 6090.44 0.416483
\(599\) −6426.26 −0.438347 −0.219173 0.975686i \(-0.570336\pi\)
−0.219173 + 0.975686i \(0.570336\pi\)
\(600\) 1169.27 0.0795585
\(601\) 23916.6 1.62326 0.811630 0.584172i \(-0.198580\pi\)
0.811630 + 0.584172i \(0.198580\pi\)
\(602\) −22222.6 −1.50453
\(603\) −81.7605 −0.00552163
\(604\) −168.306 −0.0113382
\(605\) 8945.02 0.601102
\(606\) −5181.11 −0.347308
\(607\) −23005.0 −1.53829 −0.769145 0.639074i \(-0.779318\pi\)
−0.769145 + 0.639074i \(0.779318\pi\)
\(608\) 2994.68 0.199754
\(609\) −3769.98 −0.250849
\(610\) 13387.2 0.888575
\(611\) −39011.5 −2.58304
\(612\) −589.245 −0.0389197
\(613\) 336.519 0.0221727 0.0110863 0.999939i \(-0.496471\pi\)
0.0110863 + 0.999939i \(0.496471\pi\)
\(614\) −8814.22 −0.579337
\(615\) −326.719 −0.0214221
\(616\) −5555.16 −0.363350
\(617\) 10401.1 0.678659 0.339329 0.940668i \(-0.389800\pi\)
0.339329 + 0.940668i \(0.389800\pi\)
\(618\) −11221.0 −0.730383
\(619\) −13538.7 −0.879106 −0.439553 0.898217i \(-0.644863\pi\)
−0.439553 + 0.898217i \(0.644863\pi\)
\(620\) −2488.56 −0.161199
\(621\) −1160.51 −0.0749911
\(622\) −3881.44 −0.250212
\(623\) 1129.79 0.0726551
\(624\) 3400.77 0.218172
\(625\) −19341.3 −1.23785
\(626\) −12241.6 −0.781589
\(627\) −7170.62 −0.456726
\(628\) 1005.50 0.0638912
\(629\) 4463.02 0.282913
\(630\) −6450.13 −0.407904
\(631\) 25233.4 1.59196 0.795978 0.605325i \(-0.206957\pi\)
0.795978 + 0.605325i \(0.206957\pi\)
\(632\) −8534.83 −0.537179
\(633\) 17209.4 1.08059
\(634\) 6440.98 0.403476
\(635\) 16708.9 1.04421
\(636\) 542.217 0.0338055
\(637\) −28068.3 −1.74585
\(638\) −2361.08 −0.146514
\(639\) 513.391 0.0317832
\(640\) 1687.07 0.104199
\(641\) 26721.1 1.64652 0.823260 0.567664i \(-0.192153\pi\)
0.823260 + 0.567664i \(0.192153\pi\)
\(642\) −6886.05 −0.423319
\(643\) −10266.9 −0.629683 −0.314842 0.949144i \(-0.601951\pi\)
−0.314842 + 0.949144i \(0.601951\pi\)
\(644\) −4674.28 −0.286013
\(645\) −16159.9 −0.986506
\(646\) −3063.55 −0.186585
\(647\) 6973.79 0.423753 0.211876 0.977297i \(-0.432043\pi\)
0.211876 + 0.977297i \(0.432043\pi\)
\(648\) −648.000 −0.0392837
\(649\) 1506.91 0.0911422
\(650\) 6903.48 0.416580
\(651\) 3849.97 0.231785
\(652\) −13167.1 −0.790893
\(653\) −5925.24 −0.355088 −0.177544 0.984113i \(-0.556815\pi\)
−0.177544 + 0.984113i \(0.556815\pi\)
\(654\) −7261.17 −0.434150
\(655\) 21299.2 1.27058
\(656\) −132.205 −0.00786851
\(657\) −284.686 −0.0169051
\(658\) 29940.5 1.77386
\(659\) 2302.35 0.136095 0.0680476 0.997682i \(-0.478323\pi\)
0.0680476 + 0.997682i \(0.478323\pi\)
\(660\) −4039.62 −0.238245
\(661\) −8569.20 −0.504241 −0.252121 0.967696i \(-0.581128\pi\)
−0.252121 + 0.967696i \(0.581128\pi\)
\(662\) −42.3921 −0.00248884
\(663\) −3478.97 −0.203789
\(664\) −8138.05 −0.475629
\(665\) −33534.9 −1.95553
\(666\) 4908.04 0.285559
\(667\) −1986.69 −0.115330
\(668\) −5489.27 −0.317944
\(669\) −12684.4 −0.733047
\(670\) −239.472 −0.0138084
\(671\) −12970.9 −0.746252
\(672\) −2610.02 −0.149827
\(673\) −4227.05 −0.242111 −0.121056 0.992646i \(-0.538628\pi\)
−0.121056 + 0.992646i \(0.538628\pi\)
\(674\) −15728.3 −0.898862
\(675\) −1315.43 −0.0750085
\(676\) 11290.5 0.642381
\(677\) 20850.6 1.18368 0.591841 0.806055i \(-0.298401\pi\)
0.591841 + 0.806055i \(0.298401\pi\)
\(678\) 3720.99 0.210772
\(679\) 27447.1 1.55128
\(680\) −1725.87 −0.0973295
\(681\) −16768.8 −0.943584
\(682\) 2411.18 0.135379
\(683\) 7979.20 0.447022 0.223511 0.974701i \(-0.428248\pi\)
0.223511 + 0.974701i \(0.428248\pi\)
\(684\) −3369.02 −0.188330
\(685\) 26804.8 1.49512
\(686\) 2891.06 0.160906
\(687\) −6919.11 −0.384251
\(688\) −6539.04 −0.362352
\(689\) 3201.30 0.177010
\(690\) −3399.06 −0.187536
\(691\) 19.1250 0.00105289 0.000526447 1.00000i \(-0.499832\pi\)
0.000526447 1.00000i \(0.499832\pi\)
\(692\) −4754.38 −0.261177
\(693\) 6249.55 0.342570
\(694\) −22864.3 −1.25060
\(695\) −27447.0 −1.49802
\(696\) −1109.32 −0.0604148
\(697\) 135.245 0.00734975
\(698\) −14848.1 −0.805169
\(699\) 4179.82 0.226174
\(700\) −5298.27 −0.286080
\(701\) 32156.1 1.73255 0.866277 0.499564i \(-0.166507\pi\)
0.866277 + 0.499564i \(0.166507\pi\)
\(702\) −3825.86 −0.205695
\(703\) 25517.4 1.36900
\(704\) −1634.61 −0.0875096
\(705\) 21772.2 1.16311
\(706\) 19829.9 1.05709
\(707\) 23477.1 1.24886
\(708\) 708.000 0.0375823
\(709\) 34125.4 1.80763 0.903814 0.427926i \(-0.140756\pi\)
0.903814 + 0.427926i \(0.140756\pi\)
\(710\) 1503.70 0.0794827
\(711\) 9601.68 0.506457
\(712\) 332.442 0.0174983
\(713\) 2028.84 0.106565
\(714\) 2670.03 0.139949
\(715\) −23850.3 −1.24748
\(716\) −10.0024 −0.000522078 0
\(717\) 6881.29 0.358419
\(718\) 11412.2 0.593173
\(719\) 22246.9 1.15392 0.576960 0.816772i \(-0.304239\pi\)
0.576960 + 0.816772i \(0.304239\pi\)
\(720\) −1897.96 −0.0982399
\(721\) 50845.6 2.62634
\(722\) −3797.88 −0.195765
\(723\) 13255.1 0.681831
\(724\) 6018.03 0.308920
\(725\) −2251.90 −0.115356
\(726\) −4072.00 −0.208163
\(727\) −2935.22 −0.149740 −0.0748702 0.997193i \(-0.523854\pi\)
−0.0748702 + 0.997193i \(0.523854\pi\)
\(728\) −15409.8 −0.784513
\(729\) 729.000 0.0370370
\(730\) −833.830 −0.0422759
\(731\) 6689.40 0.338463
\(732\) −6094.19 −0.307715
\(733\) −9687.75 −0.488166 −0.244083 0.969754i \(-0.578487\pi\)
−0.244083 + 0.969754i \(0.578487\pi\)
\(734\) −21142.6 −1.06320
\(735\) 15664.8 0.786130
\(736\) −1375.41 −0.0688837
\(737\) 232.025 0.0115967
\(738\) 148.731 0.00741850
\(739\) −26669.2 −1.32753 −0.663764 0.747942i \(-0.731042\pi\)
−0.663764 + 0.747942i \(0.731042\pi\)
\(740\) 14375.4 0.714121
\(741\) −19891.0 −0.986121
\(742\) −2456.93 −0.121559
\(743\) 29864.3 1.47458 0.737291 0.675575i \(-0.236105\pi\)
0.737291 + 0.675575i \(0.236105\pi\)
\(744\) 1132.86 0.0558234
\(745\) 8062.06 0.396471
\(746\) −2329.70 −0.114338
\(747\) 9155.30 0.448427
\(748\) 1672.20 0.0817402
\(749\) 31202.6 1.52219
\(750\) 6032.39 0.293696
\(751\) 7452.40 0.362106 0.181053 0.983473i \(-0.442049\pi\)
0.181053 + 0.983473i \(0.442049\pi\)
\(752\) 8810.03 0.427219
\(753\) −8827.77 −0.427227
\(754\) −6549.55 −0.316340
\(755\) 554.580 0.0267327
\(756\) 2936.27 0.141258
\(757\) 7664.45 0.367991 0.183996 0.982927i \(-0.441097\pi\)
0.183996 + 0.982927i \(0.441097\pi\)
\(758\) 10357.9 0.496326
\(759\) 3293.36 0.157499
\(760\) −9867.68 −0.470972
\(761\) 17753.5 0.845684 0.422842 0.906204i \(-0.361033\pi\)
0.422842 + 0.906204i \(0.361033\pi\)
\(762\) −7606.32 −0.361611
\(763\) 32902.3 1.56113
\(764\) 17899.6 0.847623
\(765\) 1941.60 0.0917631
\(766\) −21268.9 −1.00323
\(767\) 4180.11 0.196786
\(768\) −768.000 −0.0360844
\(769\) −38103.4 −1.78679 −0.893396 0.449269i \(-0.851684\pi\)
−0.893396 + 0.449269i \(0.851684\pi\)
\(770\) 18304.6 0.856692
\(771\) 16559.9 0.773527
\(772\) −19625.4 −0.914942
\(773\) −36223.3 −1.68546 −0.842730 0.538337i \(-0.819053\pi\)
−0.842730 + 0.538337i \(0.819053\pi\)
\(774\) 7356.42 0.341629
\(775\) 2299.68 0.106589
\(776\) 8076.33 0.373613
\(777\) −22239.7 −1.02683
\(778\) 2355.61 0.108551
\(779\) 773.267 0.0355650
\(780\) −11205.8 −0.514398
\(781\) −1456.94 −0.0667520
\(782\) 1407.04 0.0643423
\(783\) 1247.99 0.0569596
\(784\) 6338.70 0.288753
\(785\) −3313.18 −0.150640
\(786\) −9695.95 −0.440004
\(787\) −12859.4 −0.582450 −0.291225 0.956655i \(-0.594063\pi\)
−0.291225 + 0.956655i \(0.594063\pi\)
\(788\) −9956.49 −0.450108
\(789\) −15553.0 −0.701777
\(790\) 28122.8 1.26654
\(791\) −16860.8 −0.757904
\(792\) 1838.94 0.0825048
\(793\) −35980.7 −1.61124
\(794\) 5973.29 0.266983
\(795\) −1786.64 −0.0797051
\(796\) 740.569 0.0329758
\(797\) 25823.5 1.14770 0.573849 0.818961i \(-0.305450\pi\)
0.573849 + 0.818961i \(0.305450\pi\)
\(798\) 15266.0 0.677204
\(799\) −9012.61 −0.399053
\(800\) −1559.02 −0.0688997
\(801\) −373.998 −0.0164976
\(802\) 8382.04 0.369053
\(803\) 807.900 0.0355046
\(804\) 109.014 0.00478187
\(805\) 15402.1 0.674350
\(806\) 6688.52 0.292299
\(807\) −17459.3 −0.761580
\(808\) 6908.15 0.300777
\(809\) −19693.4 −0.855850 −0.427925 0.903814i \(-0.640755\pi\)
−0.427925 + 0.903814i \(0.640755\pi\)
\(810\) 2135.20 0.0926215
\(811\) −32605.0 −1.41173 −0.705866 0.708345i \(-0.749442\pi\)
−0.705866 + 0.708345i \(0.749442\pi\)
\(812\) 5026.64 0.217242
\(813\) −17221.3 −0.742901
\(814\) −13928.4 −0.599740
\(815\) 43386.4 1.86473
\(816\) 785.660 0.0337054
\(817\) 38246.8 1.63780
\(818\) −1165.78 −0.0498295
\(819\) 17336.0 0.739646
\(820\) 435.625 0.0185520
\(821\) −13591.1 −0.577751 −0.288876 0.957367i \(-0.593281\pi\)
−0.288876 + 0.957367i \(0.593281\pi\)
\(822\) −12202.2 −0.517763
\(823\) −16953.0 −0.718035 −0.359018 0.933331i \(-0.616888\pi\)
−0.359018 + 0.933331i \(0.616888\pi\)
\(824\) 14961.4 0.632530
\(825\) 3733.00 0.157535
\(826\) −3208.14 −0.135140
\(827\) −20334.6 −0.855023 −0.427512 0.904010i \(-0.640610\pi\)
−0.427512 + 0.904010i \(0.640610\pi\)
\(828\) 1547.34 0.0649442
\(829\) 2011.07 0.0842551 0.0421275 0.999112i \(-0.486586\pi\)
0.0421275 + 0.999112i \(0.486586\pi\)
\(830\) 26815.4 1.12142
\(831\) −10318.0 −0.430719
\(832\) −4534.35 −0.188943
\(833\) −6484.46 −0.269716
\(834\) 12494.6 0.518767
\(835\) 18087.5 0.749634
\(836\) 9560.83 0.395536
\(837\) −1274.47 −0.0526308
\(838\) −31724.2 −1.30775
\(839\) 39355.1 1.61941 0.809707 0.586834i \(-0.199626\pi\)
0.809707 + 0.586834i \(0.199626\pi\)
\(840\) 8600.17 0.353255
\(841\) −22252.6 −0.912401
\(842\) 3997.71 0.163623
\(843\) 7892.42 0.322455
\(844\) −22945.9 −0.935818
\(845\) −37202.9 −1.51458
\(846\) −9911.28 −0.402786
\(847\) 18451.4 0.748520
\(848\) −722.955 −0.0292764
\(849\) 8507.96 0.343925
\(850\) 1594.87 0.0643573
\(851\) −11719.8 −0.472089
\(852\) −684.522 −0.0275250
\(853\) −39238.9 −1.57505 −0.787523 0.616285i \(-0.788637\pi\)
−0.787523 + 0.616285i \(0.788637\pi\)
\(854\) 27614.4 1.10650
\(855\) 11101.1 0.444036
\(856\) 9181.39 0.366605
\(857\) 31231.8 1.24487 0.622437 0.782670i \(-0.286143\pi\)
0.622437 + 0.782670i \(0.286143\pi\)
\(858\) 10857.3 0.432007
\(859\) −30254.4 −1.20171 −0.600853 0.799359i \(-0.705172\pi\)
−0.600853 + 0.799359i \(0.705172\pi\)
\(860\) 21546.6 0.854339
\(861\) −673.940 −0.0266758
\(862\) 16574.9 0.654922
\(863\) −15484.2 −0.610764 −0.305382 0.952230i \(-0.598784\pi\)
−0.305382 + 0.952230i \(0.598784\pi\)
\(864\) 864.000 0.0340207
\(865\) 15666.0 0.615792
\(866\) 2121.49 0.0832462
\(867\) 13935.3 0.545867
\(868\) −5133.29 −0.200732
\(869\) −27248.3 −1.06368
\(870\) 3655.29 0.142443
\(871\) 643.630 0.0250385
\(872\) 9681.56 0.375985
\(873\) −9085.87 −0.352245
\(874\) 8044.78 0.311349
\(875\) −27334.4 −1.05608
\(876\) 379.581 0.0146402
\(877\) −19408.1 −0.747282 −0.373641 0.927573i \(-0.621891\pi\)
−0.373641 + 0.927573i \(0.621891\pi\)
\(878\) −2681.38 −0.103066
\(879\) 23263.1 0.892655
\(880\) 5386.15 0.206326
\(881\) 50319.8 1.92431 0.962155 0.272504i \(-0.0878517\pi\)
0.962155 + 0.272504i \(0.0878517\pi\)
\(882\) −7131.04 −0.272239
\(883\) −2696.79 −0.102780 −0.0513898 0.998679i \(-0.516365\pi\)
−0.0513898 + 0.998679i \(0.516365\pi\)
\(884\) 4638.62 0.176486
\(885\) −2332.91 −0.0886100
\(886\) 32365.0 1.22723
\(887\) 6398.55 0.242212 0.121106 0.992640i \(-0.461356\pi\)
0.121106 + 0.992640i \(0.461356\pi\)
\(888\) −6544.05 −0.247302
\(889\) 34466.3 1.30030
\(890\) −1095.42 −0.0412568
\(891\) −2068.81 −0.0777863
\(892\) 16912.6 0.634837
\(893\) −51529.8 −1.93099
\(894\) −3670.06 −0.137299
\(895\) 32.9586 0.00123093
\(896\) 3480.02 0.129754
\(897\) 9135.66 0.340057
\(898\) −1987.37 −0.0738525
\(899\) −2181.78 −0.0809414
\(900\) 1753.90 0.0649593
\(901\) 739.580 0.0273463
\(902\) −422.078 −0.0155806
\(903\) −33334.0 −1.22844
\(904\) −4961.32 −0.182534
\(905\) −19829.8 −0.728360
\(906\) −252.459 −0.00925760
\(907\) 51527.9 1.88639 0.943195 0.332240i \(-0.107804\pi\)
0.943195 + 0.332240i \(0.107804\pi\)
\(908\) 22358.4 0.817168
\(909\) −7771.67 −0.283575
\(910\) 50776.4 1.84969
\(911\) 42103.4 1.53123 0.765614 0.643300i \(-0.222435\pi\)
0.765614 + 0.643300i \(0.222435\pi\)
\(912\) 4492.03 0.163098
\(913\) −25981.5 −0.941800
\(914\) −1666.43 −0.0603071
\(915\) 20080.8 0.725518
\(916\) 9225.48 0.332771
\(917\) 43935.0 1.58218
\(918\) −883.868 −0.0317778
\(919\) −25405.8 −0.911928 −0.455964 0.889998i \(-0.650705\pi\)
−0.455964 + 0.889998i \(0.650705\pi\)
\(920\) 4532.08 0.162411
\(921\) −13221.3 −0.473027
\(922\) 34672.9 1.23849
\(923\) −4041.49 −0.144125
\(924\) −8332.74 −0.296674
\(925\) −13284.3 −0.472199
\(926\) −30556.6 −1.08440
\(927\) −16831.6 −0.596355
\(928\) 1479.09 0.0523208
\(929\) −27234.2 −0.961813 −0.480907 0.876772i \(-0.659692\pi\)
−0.480907 + 0.876772i \(0.659692\pi\)
\(930\) −3732.84 −0.131618
\(931\) −37075.0 −1.30514
\(932\) −5573.10 −0.195872
\(933\) −5822.16 −0.204297
\(934\) 30303.4 1.06162
\(935\) −5510.01 −0.192724
\(936\) 5101.15 0.178137
\(937\) 52145.3 1.81805 0.909025 0.416742i \(-0.136828\pi\)
0.909025 + 0.416742i \(0.136828\pi\)
\(938\) −493.972 −0.0171949
\(939\) −18362.5 −0.638165
\(940\) −29029.6 −1.00728
\(941\) −35478.1 −1.22907 −0.614534 0.788890i \(-0.710656\pi\)
−0.614534 + 0.788890i \(0.710656\pi\)
\(942\) 1508.25 0.0521670
\(943\) −355.150 −0.0122643
\(944\) −944.000 −0.0325472
\(945\) −9675.20 −0.333052
\(946\) −20876.5 −0.717500
\(947\) 15928.3 0.546568 0.273284 0.961933i \(-0.411890\pi\)
0.273284 + 0.961933i \(0.411890\pi\)
\(948\) −12802.2 −0.438605
\(949\) 2241.09 0.0766583
\(950\) 9118.71 0.311421
\(951\) 9661.47 0.329437
\(952\) −3560.04 −0.121199
\(953\) −9122.33 −0.310075 −0.155037 0.987909i \(-0.549550\pi\)
−0.155037 + 0.987909i \(0.549550\pi\)
\(954\) 813.325 0.0276021
\(955\) −58980.3 −1.99849
\(956\) −9175.05 −0.310400
\(957\) −3541.62 −0.119628
\(958\) 37119.3 1.25185
\(959\) 55291.7 1.86179
\(960\) 2530.61 0.0850783
\(961\) −27562.9 −0.925210
\(962\) −38636.8 −1.29491
\(963\) −10329.1 −0.345638
\(964\) −17673.5 −0.590483
\(965\) 64667.1 2.15721
\(966\) −7011.43 −0.233529
\(967\) −39017.8 −1.29755 −0.648773 0.760982i \(-0.724718\pi\)
−0.648773 + 0.760982i \(0.724718\pi\)
\(968\) 5429.34 0.180274
\(969\) −4595.32 −0.152346
\(970\) −26612.1 −0.880888
\(971\) −4022.40 −0.132940 −0.0664700 0.997788i \(-0.521174\pi\)
−0.0664700 + 0.997788i \(0.521174\pi\)
\(972\) −972.000 −0.0320750
\(973\) −56616.4 −1.86540
\(974\) 25564.1 0.840992
\(975\) 10355.2 0.340136
\(976\) 8125.58 0.266489
\(977\) 5543.33 0.181522 0.0907610 0.995873i \(-0.471070\pi\)
0.0907610 + 0.995873i \(0.471070\pi\)
\(978\) −19750.6 −0.645762
\(979\) 1061.36 0.0346487
\(980\) −20886.4 −0.680809
\(981\) −10891.7 −0.354482
\(982\) −15693.2 −0.509971
\(983\) −48179.0 −1.56325 −0.781624 0.623750i \(-0.785608\pi\)
−0.781624 + 0.623750i \(0.785608\pi\)
\(984\) −198.308 −0.00642461
\(985\) 32807.3 1.06125
\(986\) −1513.11 −0.0488713
\(987\) 44910.7 1.44835
\(988\) 26521.4 0.854006
\(989\) −17566.2 −0.564784
\(990\) −6059.42 −0.194526
\(991\) −13100.7 −0.419936 −0.209968 0.977708i \(-0.567336\pi\)
−0.209968 + 0.977708i \(0.567336\pi\)
\(992\) −1510.48 −0.0483445
\(993\) −63.5881 −0.00203213
\(994\) 3101.76 0.0989756
\(995\) −2440.22 −0.0777490
\(996\) −12207.1 −0.388349
\(997\) 31014.5 0.985195 0.492598 0.870257i \(-0.336047\pi\)
0.492598 + 0.870257i \(0.336047\pi\)
\(998\) −40004.9 −1.26887
\(999\) 7362.05 0.233158
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 354.4.a.e.1.1 3
3.2 odd 2 1062.4.a.j.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
354.4.a.e.1.1 3 1.1 even 1 trivial
1062.4.a.j.1.3 3 3.2 odd 2