Properties

Label 825.4.c.n.199.1
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
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] = [825,4,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.181494784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 12x^{3} + 49x^{2} - 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.51966 + 1.51966i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.n.199.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.03932i q^{2} -3.00000i q^{3} -8.31608 q^{4} -12.1180 q^{6} +6.49923i q^{7} +1.27677i q^{8} -9.00000 q^{9} +11.0000 q^{11} +24.9483i q^{12} +30.5385i q^{13} +26.2524 q^{14} -61.3714 q^{16} +43.1314i q^{17} +36.3539i q^{18} +6.46044 q^{19} +19.4977 q^{21} -44.4325i q^{22} +108.104i q^{23} +3.83030 q^{24} +123.355 q^{26} +27.0000i q^{27} -54.0481i q^{28} +274.857 q^{29} -68.5519 q^{31} +258.113i q^{32} -33.0000i q^{33} +174.221 q^{34} +74.8448 q^{36} +402.200i q^{37} -26.0958i q^{38} +91.6156 q^{39} -268.450 q^{41} -78.7573i q^{42} -30.5334i q^{43} -91.4769 q^{44} +436.666 q^{46} -31.5850i q^{47} +184.114i q^{48} +300.760 q^{49} +129.394 q^{51} -253.961i q^{52} +252.497i q^{53} +109.062 q^{54} -8.29800 q^{56} -19.3813i q^{57} -1110.24i q^{58} +558.331 q^{59} +335.270 q^{61} +276.903i q^{62} -58.4930i q^{63} +551.628 q^{64} -133.297 q^{66} -28.7521i q^{67} -358.684i q^{68} +324.312 q^{69} +89.0081 q^{71} -11.4909i q^{72} -717.622i q^{73} +1624.61 q^{74} -53.7256 q^{76} +71.4915i q^{77} -370.065i q^{78} -200.702 q^{79} +81.0000 q^{81} +1084.36i q^{82} +243.681i q^{83} -162.144 q^{84} -123.334 q^{86} -824.571i q^{87} +14.0444i q^{88} +312.832 q^{89} -198.477 q^{91} -899.002i q^{92} +205.656i q^{93} -127.582 q^{94} +774.338 q^{96} -27.9996i q^{97} -1214.87i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 14 q^{4} - 6 q^{6} - 54 q^{9} + 66 q^{11} + 232 q^{14} - 170 q^{16} + 338 q^{19} - 96 q^{21} - 18 q^{24} + 334 q^{26} + 554 q^{29} - 346 q^{31} + 292 q^{34} + 126 q^{36} + 270 q^{39} + 88 q^{41} - 154 q^{44}+ \cdots - 594 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.03932i − 1.42811i −0.700087 0.714057i \(-0.746856\pi\)
0.700087 0.714057i \(-0.253144\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −8.31608 −1.03951
\(5\) 0 0
\(6\) −12.1180 −0.824522
\(7\) 6.49923i 0.350925i 0.984486 + 0.175463i \(0.0561421\pi\)
−0.984486 + 0.175463i \(0.943858\pi\)
\(8\) 1.27677i 0.0564257i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 24.9483i 0.600162i
\(13\) 30.5385i 0.651528i 0.945451 + 0.325764i \(0.105622\pi\)
−0.945451 + 0.325764i \(0.894378\pi\)
\(14\) 26.2524 0.501161
\(15\) 0 0
\(16\) −61.3714 −0.958928
\(17\) 43.1314i 0.615347i 0.951492 + 0.307674i \(0.0995505\pi\)
−0.951492 + 0.307674i \(0.900450\pi\)
\(18\) 36.3539i 0.476038i
\(19\) 6.46044 0.0780066 0.0390033 0.999239i \(-0.487582\pi\)
0.0390033 + 0.999239i \(0.487582\pi\)
\(20\) 0 0
\(21\) 19.4977 0.202607
\(22\) − 44.4325i − 0.430593i
\(23\) 108.104i 0.980054i 0.871707 + 0.490027i \(0.163013\pi\)
−0.871707 + 0.490027i \(0.836987\pi\)
\(24\) 3.83030 0.0325774
\(25\) 0 0
\(26\) 123.355 0.930457
\(27\) 27.0000i 0.192450i
\(28\) − 54.0481i − 0.364791i
\(29\) 274.857 1.75999 0.879995 0.474984i \(-0.157546\pi\)
0.879995 + 0.474984i \(0.157546\pi\)
\(30\) 0 0
\(31\) −68.5519 −0.397170 −0.198585 0.980084i \(-0.563635\pi\)
−0.198585 + 0.980084i \(0.563635\pi\)
\(32\) 258.113i 1.42588i
\(33\) − 33.0000i − 0.174078i
\(34\) 174.221 0.878786
\(35\) 0 0
\(36\) 74.8448 0.346504
\(37\) 402.200i 1.78706i 0.449004 + 0.893530i \(0.351779\pi\)
−0.449004 + 0.893530i \(0.648221\pi\)
\(38\) − 26.0958i − 0.111402i
\(39\) 91.6156 0.376160
\(40\) 0 0
\(41\) −268.450 −1.02256 −0.511280 0.859414i \(-0.670828\pi\)
−0.511280 + 0.859414i \(0.670828\pi\)
\(42\) − 78.7573i − 0.289346i
\(43\) − 30.5334i − 0.108286i −0.998533 0.0541431i \(-0.982757\pi\)
0.998533 0.0541431i \(-0.0172427\pi\)
\(44\) −91.4769 −0.313424
\(45\) 0 0
\(46\) 436.666 1.39963
\(47\) − 31.5850i − 0.0980243i −0.998798 0.0490121i \(-0.984393\pi\)
0.998798 0.0490121i \(-0.0156073\pi\)
\(48\) 184.114i 0.553638i
\(49\) 300.760 0.876851
\(50\) 0 0
\(51\) 129.394 0.355271
\(52\) − 253.961i − 0.677271i
\(53\) 252.497i 0.654398i 0.944956 + 0.327199i \(0.106105\pi\)
−0.944956 + 0.327199i \(0.893895\pi\)
\(54\) 109.062 0.274841
\(55\) 0 0
\(56\) −8.29800 −0.0198012
\(57\) − 19.3813i − 0.0450372i
\(58\) − 1110.24i − 2.51347i
\(59\) 558.331 1.23201 0.616004 0.787743i \(-0.288750\pi\)
0.616004 + 0.787743i \(0.288750\pi\)
\(60\) 0 0
\(61\) 335.270 0.703719 0.351860 0.936053i \(-0.385549\pi\)
0.351860 + 0.936053i \(0.385549\pi\)
\(62\) 276.903i 0.567205i
\(63\) − 58.4930i − 0.116975i
\(64\) 551.628 1.07740
\(65\) 0 0
\(66\) −133.297 −0.248603
\(67\) − 28.7521i − 0.0524273i −0.999656 0.0262136i \(-0.991655\pi\)
0.999656 0.0262136i \(-0.00834502\pi\)
\(68\) − 358.684i − 0.639660i
\(69\) 324.312 0.565834
\(70\) 0 0
\(71\) 89.0081 0.148779 0.0743895 0.997229i \(-0.476299\pi\)
0.0743895 + 0.997229i \(0.476299\pi\)
\(72\) − 11.4909i − 0.0188086i
\(73\) − 717.622i − 1.15057i −0.817954 0.575283i \(-0.804892\pi\)
0.817954 0.575283i \(-0.195108\pi\)
\(74\) 1624.61 2.55213
\(75\) 0 0
\(76\) −53.7256 −0.0810887
\(77\) 71.4915i 0.105808i
\(78\) − 370.065i − 0.537200i
\(79\) −200.702 −0.285832 −0.142916 0.989735i \(-0.545648\pi\)
−0.142916 + 0.989735i \(0.545648\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1084.36i 1.46033i
\(83\) 243.681i 0.322258i 0.986933 + 0.161129i \(0.0515136\pi\)
−0.986933 + 0.161129i \(0.948486\pi\)
\(84\) −162.144 −0.210612
\(85\) 0 0
\(86\) −123.334 −0.154645
\(87\) − 824.571i − 1.01613i
\(88\) 14.0444i 0.0170130i
\(89\) 312.832 0.372585 0.186293 0.982494i \(-0.440353\pi\)
0.186293 + 0.982494i \(0.440353\pi\)
\(90\) 0 0
\(91\) −198.477 −0.228638
\(92\) − 899.002i − 1.01878i
\(93\) 205.656i 0.229306i
\(94\) −127.582 −0.139990
\(95\) 0 0
\(96\) 774.338 0.823235
\(97\) − 27.9996i − 0.0293085i −0.999893 0.0146543i \(-0.995335\pi\)
0.999893 0.0146543i \(-0.00466476\pi\)
\(98\) − 1214.87i − 1.25224i
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) −228.278 −0.224896 −0.112448 0.993658i \(-0.535869\pi\)
−0.112448 + 0.993658i \(0.535869\pi\)
\(102\) − 522.664i − 0.507367i
\(103\) − 800.328i − 0.765618i −0.923827 0.382809i \(-0.874957\pi\)
0.923827 0.382809i \(-0.125043\pi\)
\(104\) −38.9906 −0.0367629
\(105\) 0 0
\(106\) 1019.91 0.934555
\(107\) 1046.89i 0.945857i 0.881101 + 0.472929i \(0.156803\pi\)
−0.881101 + 0.472929i \(0.843197\pi\)
\(108\) − 224.534i − 0.200054i
\(109\) −317.023 −0.278581 −0.139291 0.990252i \(-0.544482\pi\)
−0.139291 + 0.990252i \(0.544482\pi\)
\(110\) 0 0
\(111\) 1206.60 1.03176
\(112\) − 398.867i − 0.336512i
\(113\) 1333.82i 1.11040i 0.831718 + 0.555198i \(0.187358\pi\)
−0.831718 + 0.555198i \(0.812642\pi\)
\(114\) −78.2873 −0.0643182
\(115\) 0 0
\(116\) −2285.74 −1.82953
\(117\) − 274.847i − 0.217176i
\(118\) − 2255.28i − 1.75945i
\(119\) −280.321 −0.215941
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) − 1354.26i − 1.00499i
\(123\) 805.351i 0.590375i
\(124\) 570.083 0.412863
\(125\) 0 0
\(126\) −236.272 −0.167054
\(127\) − 564.888i − 0.394691i −0.980334 0.197345i \(-0.936768\pi\)
0.980334 0.197345i \(-0.0632320\pi\)
\(128\) − 163.299i − 0.112763i
\(129\) −91.6003 −0.0625190
\(130\) 0 0
\(131\) −214.082 −0.142782 −0.0713908 0.997448i \(-0.522744\pi\)
−0.0713908 + 0.997448i \(0.522744\pi\)
\(132\) 274.431i 0.180956i
\(133\) 41.9879i 0.0273745i
\(134\) −116.139 −0.0748721
\(135\) 0 0
\(136\) −55.0688 −0.0347214
\(137\) 65.2288i 0.0406779i 0.999793 + 0.0203389i \(0.00647453\pi\)
−0.999793 + 0.0203389i \(0.993525\pi\)
\(138\) − 1310.00i − 0.808076i
\(139\) 407.084 0.248406 0.124203 0.992257i \(-0.460363\pi\)
0.124203 + 0.992257i \(0.460363\pi\)
\(140\) 0 0
\(141\) −94.7549 −0.0565943
\(142\) − 359.532i − 0.212474i
\(143\) 335.924i 0.196443i
\(144\) 552.343 0.319643
\(145\) 0 0
\(146\) −2898.70 −1.64314
\(147\) − 902.280i − 0.506250i
\(148\) − 3344.73i − 1.85767i
\(149\) −792.092 −0.435508 −0.217754 0.976004i \(-0.569873\pi\)
−0.217754 + 0.976004i \(0.569873\pi\)
\(150\) 0 0
\(151\) −2861.80 −1.54232 −0.771159 0.636642i \(-0.780323\pi\)
−0.771159 + 0.636642i \(0.780323\pi\)
\(152\) 8.24848i 0.00440158i
\(153\) − 388.183i − 0.205116i
\(154\) 288.777 0.151106
\(155\) 0 0
\(156\) −761.883 −0.391022
\(157\) − 2991.55i − 1.52071i −0.649507 0.760355i \(-0.725025\pi\)
0.649507 0.760355i \(-0.274975\pi\)
\(158\) 810.698i 0.408201i
\(159\) 757.490 0.377817
\(160\) 0 0
\(161\) −702.592 −0.343926
\(162\) − 327.185i − 0.158679i
\(163\) 2907.60i 1.39718i 0.715520 + 0.698592i \(0.246190\pi\)
−0.715520 + 0.698592i \(0.753810\pi\)
\(164\) 2232.46 1.06296
\(165\) 0 0
\(166\) 984.304 0.460222
\(167\) − 1100.39i − 0.509885i −0.966956 0.254942i \(-0.917944\pi\)
0.966956 0.254942i \(-0.0820565\pi\)
\(168\) 24.8940i 0.0114322i
\(169\) 1264.40 0.575511
\(170\) 0 0
\(171\) −58.1439 −0.0260022
\(172\) 253.919i 0.112565i
\(173\) 3754.75i 1.65011i 0.565054 + 0.825054i \(0.308855\pi\)
−0.565054 + 0.825054i \(0.691145\pi\)
\(174\) −3330.71 −1.45115
\(175\) 0 0
\(176\) −675.086 −0.289128
\(177\) − 1674.99i − 0.711301i
\(178\) − 1263.63i − 0.532094i
\(179\) 3218.79 1.34404 0.672021 0.740532i \(-0.265426\pi\)
0.672021 + 0.740532i \(0.265426\pi\)
\(180\) 0 0
\(181\) −456.885 −0.187624 −0.0938121 0.995590i \(-0.529905\pi\)
−0.0938121 + 0.995590i \(0.529905\pi\)
\(182\) 801.711i 0.326521i
\(183\) − 1005.81i − 0.406292i
\(184\) −138.024 −0.0553002
\(185\) 0 0
\(186\) 830.709 0.327476
\(187\) 474.445i 0.185534i
\(188\) 262.663i 0.101897i
\(189\) −175.479 −0.0675356
\(190\) 0 0
\(191\) −4907.11 −1.85899 −0.929493 0.368841i \(-0.879755\pi\)
−0.929493 + 0.368841i \(0.879755\pi\)
\(192\) − 1654.88i − 0.622036i
\(193\) 1462.17i 0.545333i 0.962109 + 0.272667i \(0.0879056\pi\)
−0.962109 + 0.272667i \(0.912094\pi\)
\(194\) −113.099 −0.0418559
\(195\) 0 0
\(196\) −2501.15 −0.911496
\(197\) − 2210.10i − 0.799307i −0.916666 0.399653i \(-0.869131\pi\)
0.916666 0.399653i \(-0.130869\pi\)
\(198\) 399.892i 0.143531i
\(199\) 395.272 0.140804 0.0704022 0.997519i \(-0.477572\pi\)
0.0704022 + 0.997519i \(0.477572\pi\)
\(200\) 0 0
\(201\) −86.2563 −0.0302689
\(202\) 922.086i 0.321177i
\(203\) 1786.36i 0.617625i
\(204\) −1076.05 −0.369308
\(205\) 0 0
\(206\) −3232.78 −1.09339
\(207\) − 972.936i − 0.326685i
\(208\) − 1874.19i − 0.624769i
\(209\) 71.0648 0.0235199
\(210\) 0 0
\(211\) 1559.98 0.508974 0.254487 0.967076i \(-0.418093\pi\)
0.254487 + 0.967076i \(0.418093\pi\)
\(212\) − 2099.78i − 0.680253i
\(213\) − 267.024i − 0.0858976i
\(214\) 4228.72 1.35079
\(215\) 0 0
\(216\) −34.4727 −0.0108591
\(217\) − 445.534i − 0.139377i
\(218\) 1280.56i 0.397846i
\(219\) −2152.87 −0.664280
\(220\) 0 0
\(221\) −1317.17 −0.400916
\(222\) − 4873.84i − 1.47347i
\(223\) 756.215i 0.227085i 0.993533 + 0.113542i \(0.0362198\pi\)
−0.993533 + 0.113542i \(0.963780\pi\)
\(224\) −1677.53 −0.500379
\(225\) 0 0
\(226\) 5387.70 1.58577
\(227\) 5758.69i 1.68378i 0.539649 + 0.841890i \(0.318557\pi\)
−0.539649 + 0.841890i \(0.681443\pi\)
\(228\) 161.177i 0.0468166i
\(229\) 1912.63 0.551921 0.275961 0.961169i \(-0.411004\pi\)
0.275961 + 0.961169i \(0.411004\pi\)
\(230\) 0 0
\(231\) 214.474 0.0610882
\(232\) 350.929i 0.0993086i
\(233\) 1034.86i 0.290970i 0.989360 + 0.145485i \(0.0464742\pi\)
−0.989360 + 0.145485i \(0.953526\pi\)
\(234\) −1110.19 −0.310152
\(235\) 0 0
\(236\) −4643.13 −1.28069
\(237\) 602.105i 0.165025i
\(238\) 1132.30i 0.308388i
\(239\) −2647.23 −0.716465 −0.358232 0.933632i \(-0.616620\pi\)
−0.358232 + 0.933632i \(0.616620\pi\)
\(240\) 0 0
\(241\) 6734.00 1.79990 0.899948 0.435996i \(-0.143604\pi\)
0.899948 + 0.435996i \(0.143604\pi\)
\(242\) − 488.757i − 0.129829i
\(243\) − 243.000i − 0.0641500i
\(244\) −2788.13 −0.731523
\(245\) 0 0
\(246\) 3253.07 0.843123
\(247\) 197.292i 0.0508235i
\(248\) − 87.5248i − 0.0224106i
\(249\) 731.043 0.186056
\(250\) 0 0
\(251\) −5377.57 −1.35231 −0.676154 0.736761i \(-0.736354\pi\)
−0.676154 + 0.736761i \(0.736354\pi\)
\(252\) 486.433i 0.121597i
\(253\) 1189.14i 0.295497i
\(254\) −2281.76 −0.563663
\(255\) 0 0
\(256\) 3753.41 0.916360
\(257\) − 5882.33i − 1.42774i −0.700278 0.713871i \(-0.746940\pi\)
0.700278 0.713871i \(-0.253060\pi\)
\(258\) 370.003i 0.0892843i
\(259\) −2613.99 −0.627124
\(260\) 0 0
\(261\) −2473.71 −0.586663
\(262\) 864.744i 0.203909i
\(263\) − 4001.67i − 0.938227i −0.883138 0.469113i \(-0.844574\pi\)
0.883138 0.469113i \(-0.155426\pi\)
\(264\) 42.1333 0.00982245
\(265\) 0 0
\(266\) 169.602 0.0390939
\(267\) − 938.495i − 0.215112i
\(268\) 239.105i 0.0544987i
\(269\) 1758.29 0.398530 0.199265 0.979946i \(-0.436144\pi\)
0.199265 + 0.979946i \(0.436144\pi\)
\(270\) 0 0
\(271\) 6018.44 1.34906 0.674528 0.738249i \(-0.264347\pi\)
0.674528 + 0.738249i \(0.264347\pi\)
\(272\) − 2647.03i − 0.590074i
\(273\) 595.431i 0.132004i
\(274\) 263.480 0.0580927
\(275\) 0 0
\(276\) −2697.01 −0.588191
\(277\) − 2910.30i − 0.631275i −0.948880 0.315637i \(-0.897782\pi\)
0.948880 0.315637i \(-0.102218\pi\)
\(278\) − 1644.34i − 0.354752i
\(279\) 616.967 0.132390
\(280\) 0 0
\(281\) 6893.19 1.46339 0.731697 0.681631i \(-0.238729\pi\)
0.731697 + 0.681631i \(0.238729\pi\)
\(282\) 382.745i 0.0808232i
\(283\) 3120.16i 0.655385i 0.944784 + 0.327693i \(0.106271\pi\)
−0.944784 + 0.327693i \(0.893729\pi\)
\(284\) −740.199 −0.154657
\(285\) 0 0
\(286\) 1356.90 0.280543
\(287\) − 1744.72i − 0.358842i
\(288\) − 2323.01i − 0.475295i
\(289\) 3052.68 0.621348
\(290\) 0 0
\(291\) −83.9988 −0.0169213
\(292\) 5967.81i 1.19603i
\(293\) 8869.62i 1.76849i 0.467020 + 0.884247i \(0.345328\pi\)
−0.467020 + 0.884247i \(0.654672\pi\)
\(294\) −3644.60 −0.722983
\(295\) 0 0
\(296\) −513.515 −0.100836
\(297\) 297.000i 0.0580259i
\(298\) 3199.51i 0.621955i
\(299\) −3301.34 −0.638533
\(300\) 0 0
\(301\) 198.444 0.0380003
\(302\) 11559.7i 2.20261i
\(303\) 684.833i 0.129844i
\(304\) −396.486 −0.0748028
\(305\) 0 0
\(306\) −1567.99 −0.292929
\(307\) 5713.42i 1.06216i 0.847323 + 0.531078i \(0.178213\pi\)
−0.847323 + 0.531078i \(0.821787\pi\)
\(308\) − 594.529i − 0.109988i
\(309\) −2400.98 −0.442030
\(310\) 0 0
\(311\) 1984.01 0.361746 0.180873 0.983506i \(-0.442108\pi\)
0.180873 + 0.983506i \(0.442108\pi\)
\(312\) 116.972i 0.0212251i
\(313\) − 2675.50i − 0.483156i −0.970381 0.241578i \(-0.922335\pi\)
0.970381 0.241578i \(-0.0776650\pi\)
\(314\) −12083.8 −2.17175
\(315\) 0 0
\(316\) 1669.05 0.297125
\(317\) 7083.76i 1.25509i 0.778580 + 0.627545i \(0.215940\pi\)
−0.778580 + 0.627545i \(0.784060\pi\)
\(318\) − 3059.74i − 0.539565i
\(319\) 3023.43 0.530657
\(320\) 0 0
\(321\) 3140.67 0.546091
\(322\) 2837.99i 0.491165i
\(323\) 278.648i 0.0480012i
\(324\) −673.603 −0.115501
\(325\) 0 0
\(326\) 11744.7 1.99534
\(327\) 951.070i 0.160839i
\(328\) − 342.749i − 0.0576986i
\(329\) 205.278 0.0343992
\(330\) 0 0
\(331\) 7812.39 1.29730 0.648652 0.761085i \(-0.275333\pi\)
0.648652 + 0.761085i \(0.275333\pi\)
\(332\) − 2026.47i − 0.334991i
\(333\) − 3619.80i − 0.595686i
\(334\) −4444.82 −0.728173
\(335\) 0 0
\(336\) −1196.60 −0.194285
\(337\) 11866.4i 1.91811i 0.283218 + 0.959056i \(0.408598\pi\)
−0.283218 + 0.959056i \(0.591402\pi\)
\(338\) − 5107.30i − 0.821895i
\(339\) 4001.45 0.641088
\(340\) 0 0
\(341\) −754.071 −0.119751
\(342\) 234.862i 0.0371341i
\(343\) 4183.94i 0.658635i
\(344\) 38.9841 0.00611012
\(345\) 0 0
\(346\) 15166.6 2.35654
\(347\) − 3485.50i − 0.539225i −0.962969 0.269613i \(-0.913104\pi\)
0.962969 0.269613i \(-0.0868957\pi\)
\(348\) 6857.21i 1.05628i
\(349\) 6366.76 0.976518 0.488259 0.872699i \(-0.337632\pi\)
0.488259 + 0.872699i \(0.337632\pi\)
\(350\) 0 0
\(351\) −824.541 −0.125387
\(352\) 2839.24i 0.429920i
\(353\) 8223.15i 1.23987i 0.784653 + 0.619935i \(0.212841\pi\)
−0.784653 + 0.619935i \(0.787159\pi\)
\(354\) −6765.83 −1.01582
\(355\) 0 0
\(356\) −2601.54 −0.387306
\(357\) 840.962i 0.124674i
\(358\) − 13001.7i − 1.91945i
\(359\) 9377.54 1.37863 0.689314 0.724462i \(-0.257912\pi\)
0.689314 + 0.724462i \(0.257912\pi\)
\(360\) 0 0
\(361\) −6817.26 −0.993915
\(362\) 1845.50i 0.267949i
\(363\) − 363.000i − 0.0524864i
\(364\) 1650.55 0.237671
\(365\) 0 0
\(366\) −4062.78 −0.580232
\(367\) 310.895i 0.0442196i 0.999756 + 0.0221098i \(0.00703834\pi\)
−0.999756 + 0.0221098i \(0.992962\pi\)
\(368\) − 6634.49i − 0.939801i
\(369\) 2416.05 0.340853
\(370\) 0 0
\(371\) −1641.03 −0.229645
\(372\) − 1710.25i − 0.238366i
\(373\) − 4738.51i − 0.657777i −0.944369 0.328889i \(-0.893326\pi\)
0.944369 0.328889i \(-0.106674\pi\)
\(374\) 1916.44 0.264964
\(375\) 0 0
\(376\) 40.3267 0.00553109
\(377\) 8393.74i 1.14668i
\(378\) 708.816i 0.0964486i
\(379\) 4735.47 0.641806 0.320903 0.947112i \(-0.396014\pi\)
0.320903 + 0.947112i \(0.396014\pi\)
\(380\) 0 0
\(381\) −1694.66 −0.227875
\(382\) 19821.4i 2.65484i
\(383\) 3632.14i 0.484579i 0.970204 + 0.242289i \(0.0778984\pi\)
−0.970204 + 0.242289i \(0.922102\pi\)
\(384\) −489.896 −0.0651039
\(385\) 0 0
\(386\) 5906.17 0.778798
\(387\) 274.801i 0.0360954i
\(388\) 232.847i 0.0304665i
\(389\) −6557.50 −0.854701 −0.427350 0.904086i \(-0.640553\pi\)
−0.427350 + 0.904086i \(0.640553\pi\)
\(390\) 0 0
\(391\) −4662.68 −0.603073
\(392\) 384.001i 0.0494769i
\(393\) 642.245i 0.0824350i
\(394\) −8927.31 −1.14150
\(395\) 0 0
\(396\) 823.292 0.104475
\(397\) − 4084.97i − 0.516420i −0.966089 0.258210i \(-0.916867\pi\)
0.966089 0.258210i \(-0.0831327\pi\)
\(398\) − 1596.63i − 0.201085i
\(399\) 125.964 0.0158047
\(400\) 0 0
\(401\) −6676.04 −0.831386 −0.415693 0.909505i \(-0.636461\pi\)
−0.415693 + 0.909505i \(0.636461\pi\)
\(402\) 348.417i 0.0432275i
\(403\) − 2093.47i − 0.258768i
\(404\) 1898.38 0.233782
\(405\) 0 0
\(406\) 7215.67 0.882039
\(407\) 4424.20i 0.538819i
\(408\) 165.206i 0.0200464i
\(409\) −1718.56 −0.207769 −0.103884 0.994589i \(-0.533127\pi\)
−0.103884 + 0.994589i \(0.533127\pi\)
\(410\) 0 0
\(411\) 195.686 0.0234854
\(412\) 6655.60i 0.795868i
\(413\) 3628.72i 0.432343i
\(414\) −3930.00 −0.466543
\(415\) 0 0
\(416\) −7882.39 −0.929004
\(417\) − 1221.25i − 0.143417i
\(418\) − 287.053i − 0.0335891i
\(419\) −5732.23 −0.668348 −0.334174 0.942511i \(-0.608457\pi\)
−0.334174 + 0.942511i \(0.608457\pi\)
\(420\) 0 0
\(421\) −9647.38 −1.11683 −0.558414 0.829563i \(-0.688590\pi\)
−0.558414 + 0.829563i \(0.688590\pi\)
\(422\) − 6301.26i − 0.726873i
\(423\) 284.265i 0.0326748i
\(424\) −322.379 −0.0369248
\(425\) 0 0
\(426\) −1078.60 −0.122672
\(427\) 2178.99i 0.246953i
\(428\) − 8706.03i − 0.983229i
\(429\) 1007.77 0.113417
\(430\) 0 0
\(431\) −3054.57 −0.341376 −0.170688 0.985325i \(-0.554599\pi\)
−0.170688 + 0.985325i \(0.554599\pi\)
\(432\) − 1657.03i − 0.184546i
\(433\) 11022.8i 1.22337i 0.791100 + 0.611686i \(0.209509\pi\)
−0.791100 + 0.611686i \(0.790491\pi\)
\(434\) −1799.65 −0.199046
\(435\) 0 0
\(436\) 2636.39 0.289588
\(437\) 698.399i 0.0764507i
\(438\) 8696.11i 0.948667i
\(439\) 7589.30 0.825097 0.412548 0.910936i \(-0.364639\pi\)
0.412548 + 0.910936i \(0.364639\pi\)
\(440\) 0 0
\(441\) −2706.84 −0.292284
\(442\) 5320.47i 0.572554i
\(443\) 4043.30i 0.433641i 0.976211 + 0.216821i \(0.0695687\pi\)
−0.976211 + 0.216821i \(0.930431\pi\)
\(444\) −10034.2 −1.07252
\(445\) 0 0
\(446\) 3054.59 0.324303
\(447\) 2376.28i 0.251441i
\(448\) 3585.16i 0.378086i
\(449\) −14619.3 −1.53658 −0.768291 0.640101i \(-0.778893\pi\)
−0.768291 + 0.640101i \(0.778893\pi\)
\(450\) 0 0
\(451\) −2952.96 −0.308313
\(452\) − 11092.1i − 1.15427i
\(453\) 8585.40i 0.890458i
\(454\) 23261.2 2.40463
\(455\) 0 0
\(456\) 24.7454 0.00254125
\(457\) 10910.8i 1.11682i 0.829565 + 0.558411i \(0.188589\pi\)
−0.829565 + 0.558411i \(0.811411\pi\)
\(458\) − 7725.71i − 0.788207i
\(459\) −1164.55 −0.118424
\(460\) 0 0
\(461\) 16371.0 1.65395 0.826977 0.562236i \(-0.190059\pi\)
0.826977 + 0.562236i \(0.190059\pi\)
\(462\) − 866.331i − 0.0872410i
\(463\) − 16023.5i − 1.60837i −0.594377 0.804186i \(-0.702601\pi\)
0.594377 0.804186i \(-0.297399\pi\)
\(464\) −16868.4 −1.68770
\(465\) 0 0
\(466\) 4180.13 0.415538
\(467\) 15097.3i 1.49597i 0.663716 + 0.747985i \(0.268978\pi\)
−0.663716 + 0.747985i \(0.731022\pi\)
\(468\) 2285.65i 0.225757i
\(469\) 186.866 0.0183981
\(470\) 0 0
\(471\) −8974.65 −0.877983
\(472\) 712.859i 0.0695170i
\(473\) − 335.868i − 0.0326495i
\(474\) 2432.09 0.235675
\(475\) 0 0
\(476\) 2331.17 0.224473
\(477\) − 2272.47i − 0.218133i
\(478\) 10693.0i 1.02319i
\(479\) 10459.0 0.997667 0.498834 0.866698i \(-0.333762\pi\)
0.498834 + 0.866698i \(0.333762\pi\)
\(480\) 0 0
\(481\) −12282.6 −1.16432
\(482\) − 27200.8i − 2.57046i
\(483\) 2107.78i 0.198566i
\(484\) −1006.25 −0.0945010
\(485\) 0 0
\(486\) −981.554 −0.0916136
\(487\) 7939.86i 0.738788i 0.929273 + 0.369394i \(0.120435\pi\)
−0.929273 + 0.369394i \(0.879565\pi\)
\(488\) 428.061i 0.0397078i
\(489\) 8722.81 0.806665
\(490\) 0 0
\(491\) 11492.7 1.05633 0.528167 0.849141i \(-0.322880\pi\)
0.528167 + 0.849141i \(0.322880\pi\)
\(492\) − 6697.37i − 0.613701i
\(493\) 11855.0i 1.08300i
\(494\) 796.927 0.0725818
\(495\) 0 0
\(496\) 4207.13 0.380858
\(497\) 578.484i 0.0522103i
\(498\) − 2952.91i − 0.265709i
\(499\) −4813.54 −0.431831 −0.215915 0.976412i \(-0.569274\pi\)
−0.215915 + 0.976412i \(0.569274\pi\)
\(500\) 0 0
\(501\) −3301.17 −0.294382
\(502\) 21721.7i 1.93125i
\(503\) 17756.3i 1.57399i 0.616962 + 0.786993i \(0.288363\pi\)
−0.616962 + 0.786993i \(0.711637\pi\)
\(504\) 74.6820 0.00660040
\(505\) 0 0
\(506\) 4803.33 0.422004
\(507\) − 3793.19i − 0.332271i
\(508\) 4697.66i 0.410285i
\(509\) −3183.05 −0.277183 −0.138592 0.990350i \(-0.544258\pi\)
−0.138592 + 0.990350i \(0.544258\pi\)
\(510\) 0 0
\(511\) 4663.99 0.403763
\(512\) − 16467.6i − 1.42143i
\(513\) 174.432i 0.0150124i
\(514\) −23760.6 −2.03898
\(515\) 0 0
\(516\) 761.756 0.0649892
\(517\) − 347.435i − 0.0295554i
\(518\) 10558.7i 0.895605i
\(519\) 11264.3 0.952690
\(520\) 0 0
\(521\) 7246.60 0.609365 0.304683 0.952454i \(-0.401450\pi\)
0.304683 + 0.952454i \(0.401450\pi\)
\(522\) 9992.12i 0.837822i
\(523\) − 17985.0i − 1.50369i −0.659340 0.751845i \(-0.729164\pi\)
0.659340 0.751845i \(-0.270836\pi\)
\(524\) 1780.32 0.148423
\(525\) 0 0
\(526\) −16164.0 −1.33990
\(527\) − 2956.74i − 0.244398i
\(528\) 2025.26i 0.166928i
\(529\) 480.530 0.0394945
\(530\) 0 0
\(531\) −5024.98 −0.410670
\(532\) − 349.175i − 0.0284561i
\(533\) − 8198.09i − 0.666226i
\(534\) −3790.88 −0.307205
\(535\) 0 0
\(536\) 36.7097 0.00295825
\(537\) − 9656.36i − 0.775983i
\(538\) − 7102.28i − 0.569147i
\(539\) 3308.36 0.264381
\(540\) 0 0
\(541\) −14549.3 −1.15624 −0.578119 0.815952i \(-0.696213\pi\)
−0.578119 + 0.815952i \(0.696213\pi\)
\(542\) − 24310.4i − 1.92661i
\(543\) 1370.65i 0.108325i
\(544\) −11132.8 −0.877414
\(545\) 0 0
\(546\) 2405.13 0.188517
\(547\) − 5487.29i − 0.428920i −0.976733 0.214460i \(-0.931201\pi\)
0.976733 0.214460i \(-0.0687992\pi\)
\(548\) − 542.448i − 0.0422851i
\(549\) −3017.43 −0.234573
\(550\) 0 0
\(551\) 1775.70 0.137291
\(552\) 414.071i 0.0319276i
\(553\) − 1304.41i − 0.100306i
\(554\) −11755.6 −0.901532
\(555\) 0 0
\(556\) −3385.35 −0.258221
\(557\) 7366.60i 0.560382i 0.959944 + 0.280191i \(0.0903978\pi\)
−0.959944 + 0.280191i \(0.909602\pi\)
\(558\) − 2492.13i − 0.189068i
\(559\) 932.446 0.0705515
\(560\) 0 0
\(561\) 1423.34 0.107118
\(562\) − 27843.8i − 2.08989i
\(563\) − 4516.14i − 0.338068i −0.985610 0.169034i \(-0.945935\pi\)
0.985610 0.169034i \(-0.0540648\pi\)
\(564\) 787.990 0.0588304
\(565\) 0 0
\(566\) 12603.3 0.935965
\(567\) 526.437i 0.0389917i
\(568\) 113.643i 0.00839496i
\(569\) −2174.35 −0.160200 −0.0800999 0.996787i \(-0.525524\pi\)
−0.0800999 + 0.996787i \(0.525524\pi\)
\(570\) 0 0
\(571\) −25339.7 −1.85715 −0.928575 0.371145i \(-0.878965\pi\)
−0.928575 + 0.371145i \(0.878965\pi\)
\(572\) − 2793.57i − 0.204205i
\(573\) 14721.3i 1.07329i
\(574\) −7047.48 −0.512467
\(575\) 0 0
\(576\) −4964.65 −0.359133
\(577\) 21909.1i 1.58074i 0.612630 + 0.790370i \(0.290112\pi\)
−0.612630 + 0.790370i \(0.709888\pi\)
\(578\) − 12330.8i − 0.887356i
\(579\) 4386.51 0.314848
\(580\) 0 0
\(581\) −1583.74 −0.113089
\(582\) 339.298i 0.0241655i
\(583\) 2777.46i 0.197308i
\(584\) 916.237 0.0649215
\(585\) 0 0
\(586\) 35827.2 2.52561
\(587\) 922.235i 0.0648462i 0.999474 + 0.0324231i \(0.0103224\pi\)
−0.999474 + 0.0324231i \(0.989678\pi\)
\(588\) 7503.44i 0.526253i
\(589\) −442.875 −0.0309819
\(590\) 0 0
\(591\) −6630.31 −0.461480
\(592\) − 24683.6i − 1.71366i
\(593\) − 9075.86i − 0.628501i −0.949340 0.314250i \(-0.898247\pi\)
0.949340 0.314250i \(-0.101753\pi\)
\(594\) 1199.68 0.0828676
\(595\) 0 0
\(596\) 6587.10 0.452715
\(597\) − 1185.82i − 0.0812935i
\(598\) 13335.2i 0.911898i
\(599\) −19987.0 −1.36335 −0.681675 0.731655i \(-0.738748\pi\)
−0.681675 + 0.731655i \(0.738748\pi\)
\(600\) 0 0
\(601\) 11116.7 0.754509 0.377255 0.926110i \(-0.376868\pi\)
0.377255 + 0.926110i \(0.376868\pi\)
\(602\) − 801.577i − 0.0542688i
\(603\) 258.769i 0.0174758i
\(604\) 23799.0 1.60326
\(605\) 0 0
\(606\) 2766.26 0.185432
\(607\) − 19355.0i − 1.29423i −0.762393 0.647114i \(-0.775976\pi\)
0.762393 0.647114i \(-0.224024\pi\)
\(608\) 1667.52i 0.111228i
\(609\) 5359.08 0.356586
\(610\) 0 0
\(611\) 964.559 0.0638656
\(612\) 3228.16i 0.213220i
\(613\) − 12325.5i − 0.812111i −0.913848 0.406056i \(-0.866904\pi\)
0.913848 0.406056i \(-0.133096\pi\)
\(614\) 23078.3 1.51688
\(615\) 0 0
\(616\) −91.2780 −0.00597029
\(617\) − 28207.0i − 1.84047i −0.391363 0.920236i \(-0.627996\pi\)
0.391363 0.920236i \(-0.372004\pi\)
\(618\) 9698.34i 0.631269i
\(619\) −17236.9 −1.11924 −0.559621 0.828748i \(-0.689053\pi\)
−0.559621 + 0.828748i \(0.689053\pi\)
\(620\) 0 0
\(621\) −2918.81 −0.188611
\(622\) − 8014.06i − 0.516615i
\(623\) 2033.16i 0.130750i
\(624\) −5622.58 −0.360711
\(625\) 0 0
\(626\) −10807.2 −0.690003
\(627\) − 213.194i − 0.0135792i
\(628\) 24878.0i 1.58080i
\(629\) −17347.4 −1.09966
\(630\) 0 0
\(631\) −23582.7 −1.48782 −0.743908 0.668282i \(-0.767030\pi\)
−0.743908 + 0.668282i \(0.767030\pi\)
\(632\) − 256.250i − 0.0161283i
\(633\) − 4679.94i − 0.293856i
\(634\) 28613.6 1.79241
\(635\) 0 0
\(636\) −6299.35 −0.392744
\(637\) 9184.77i 0.571294i
\(638\) − 12212.6i − 0.757839i
\(639\) −801.073 −0.0495930
\(640\) 0 0
\(641\) −13003.1 −0.801233 −0.400617 0.916246i \(-0.631204\pi\)
−0.400617 + 0.916246i \(0.631204\pi\)
\(642\) − 12686.2i − 0.779880i
\(643\) − 21269.1i − 1.30446i −0.758019 0.652232i \(-0.773833\pi\)
0.758019 0.652232i \(-0.226167\pi\)
\(644\) 5842.82 0.357514
\(645\) 0 0
\(646\) 1125.55 0.0685511
\(647\) − 29247.2i − 1.77716i −0.458717 0.888582i \(-0.651691\pi\)
0.458717 0.888582i \(-0.348309\pi\)
\(648\) 103.418i 0.00626952i
\(649\) 6141.64 0.371465
\(650\) 0 0
\(651\) −1336.60 −0.0804694
\(652\) − 24179.9i − 1.45239i
\(653\) − 1678.30i − 0.100577i −0.998735 0.0502885i \(-0.983986\pi\)
0.998735 0.0502885i \(-0.0160141\pi\)
\(654\) 3841.67 0.229696
\(655\) 0 0
\(656\) 16475.2 0.980561
\(657\) 6458.60i 0.383522i
\(658\) − 829.182i − 0.0491260i
\(659\) 7620.41 0.450454 0.225227 0.974306i \(-0.427688\pi\)
0.225227 + 0.974306i \(0.427688\pi\)
\(660\) 0 0
\(661\) −22762.2 −1.33941 −0.669703 0.742629i \(-0.733579\pi\)
−0.669703 + 0.742629i \(0.733579\pi\)
\(662\) − 31556.7i − 1.85270i
\(663\) 3951.51i 0.231469i
\(664\) −311.124 −0.0181837
\(665\) 0 0
\(666\) −14621.5 −0.850708
\(667\) 29713.2i 1.72488i
\(668\) 9150.93i 0.530030i
\(669\) 2268.65 0.131107
\(670\) 0 0
\(671\) 3687.96 0.212179
\(672\) 5032.60i 0.288894i
\(673\) 698.528i 0.0400093i 0.999800 + 0.0200047i \(0.00636811\pi\)
−0.999800 + 0.0200047i \(0.993632\pi\)
\(674\) 47932.1 2.73928
\(675\) 0 0
\(676\) −10514.8 −0.598250
\(677\) 10860.1i 0.616524i 0.951302 + 0.308262i \(0.0997473\pi\)
−0.951302 + 0.308262i \(0.900253\pi\)
\(678\) − 16163.1i − 0.915547i
\(679\) 181.976 0.0102851
\(680\) 0 0
\(681\) 17276.1 0.972131
\(682\) 3045.93i 0.171019i
\(683\) − 6534.95i − 0.366110i −0.983103 0.183055i \(-0.941401\pi\)
0.983103 0.183055i \(-0.0585986\pi\)
\(684\) 483.530 0.0270296
\(685\) 0 0
\(686\) 16900.3 0.940606
\(687\) − 5737.88i − 0.318652i
\(688\) 1873.88i 0.103839i
\(689\) −7710.88 −0.426359
\(690\) 0 0
\(691\) −10915.3 −0.600924 −0.300462 0.953794i \(-0.597141\pi\)
−0.300462 + 0.953794i \(0.597141\pi\)
\(692\) − 31224.9i − 1.71530i
\(693\) − 643.423i − 0.0352693i
\(694\) −14079.0 −0.770076
\(695\) 0 0
\(696\) 1052.79 0.0573359
\(697\) − 11578.6i − 0.629229i
\(698\) − 25717.3i − 1.39458i
\(699\) 3104.58 0.167991
\(700\) 0 0
\(701\) 18405.4 0.991674 0.495837 0.868416i \(-0.334861\pi\)
0.495837 + 0.868416i \(0.334861\pi\)
\(702\) 3330.58i 0.179067i
\(703\) 2598.39i 0.139403i
\(704\) 6067.91 0.324848
\(705\) 0 0
\(706\) 33215.9 1.77068
\(707\) − 1483.63i − 0.0789217i
\(708\) 13929.4i 0.739405i
\(709\) 21436.2 1.13547 0.567737 0.823210i \(-0.307819\pi\)
0.567737 + 0.823210i \(0.307819\pi\)
\(710\) 0 0
\(711\) 1806.32 0.0952773
\(712\) 399.413i 0.0210234i
\(713\) − 7410.73i − 0.389248i
\(714\) 3396.91 0.178048
\(715\) 0 0
\(716\) −26767.7 −1.39715
\(717\) 7941.69i 0.413651i
\(718\) − 37878.9i − 1.96884i
\(719\) −28641.7 −1.48561 −0.742807 0.669506i \(-0.766506\pi\)
−0.742807 + 0.669506i \(0.766506\pi\)
\(720\) 0 0
\(721\) 5201.51 0.268675
\(722\) 27537.1i 1.41942i
\(723\) − 20202.0i − 1.03917i
\(724\) 3799.49 0.195037
\(725\) 0 0
\(726\) −1466.27 −0.0749566
\(727\) 1723.13i 0.0879055i 0.999034 + 0.0439527i \(0.0139951\pi\)
−0.999034 + 0.0439527i \(0.986005\pi\)
\(728\) − 253.409i − 0.0129010i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 1316.95 0.0666335
\(732\) 8364.39i 0.422345i
\(733\) − 18813.4i − 0.948005i −0.880523 0.474003i \(-0.842809\pi\)
0.880523 0.474003i \(-0.157191\pi\)
\(734\) 1255.80 0.0631506
\(735\) 0 0
\(736\) −27903.0 −1.39744
\(737\) − 316.273i − 0.0158074i
\(738\) − 9759.21i − 0.486777i
\(739\) 10893.3 0.542242 0.271121 0.962545i \(-0.412606\pi\)
0.271121 + 0.962545i \(0.412606\pi\)
\(740\) 0 0
\(741\) 591.877 0.0293430
\(742\) 6628.65i 0.327959i
\(743\) 11888.6i 0.587013i 0.955957 + 0.293507i \(0.0948223\pi\)
−0.955957 + 0.293507i \(0.905178\pi\)
\(744\) −262.574 −0.0129388
\(745\) 0 0
\(746\) −19140.4 −0.939381
\(747\) − 2193.13i − 0.107419i
\(748\) − 3945.53i − 0.192865i
\(749\) −6803.98 −0.331925
\(750\) 0 0
\(751\) −32134.8 −1.56140 −0.780702 0.624903i \(-0.785138\pi\)
−0.780702 + 0.624903i \(0.785138\pi\)
\(752\) 1938.41i 0.0939982i
\(753\) 16132.7i 0.780755i
\(754\) 33905.0 1.63759
\(755\) 0 0
\(756\) 1459.30 0.0702040
\(757\) − 17366.1i − 0.833794i −0.908954 0.416897i \(-0.863118\pi\)
0.908954 0.416897i \(-0.136882\pi\)
\(758\) − 19128.1i − 0.916573i
\(759\) 3567.43 0.170605
\(760\) 0 0
\(761\) −27393.9 −1.30490 −0.652450 0.757832i \(-0.726259\pi\)
−0.652450 + 0.757832i \(0.726259\pi\)
\(762\) 6845.28i 0.325431i
\(763\) − 2060.41i − 0.0977611i
\(764\) 40808.0 1.93243
\(765\) 0 0
\(766\) 14671.4 0.692034
\(767\) 17050.6i 0.802689i
\(768\) − 11260.2i − 0.529060i
\(769\) 12091.3 0.567000 0.283500 0.958972i \(-0.408504\pi\)
0.283500 + 0.958972i \(0.408504\pi\)
\(770\) 0 0
\(771\) −17647.0 −0.824307
\(772\) − 12159.5i − 0.566879i
\(773\) 18895.8i 0.879216i 0.898190 + 0.439608i \(0.144883\pi\)
−0.898190 + 0.439608i \(0.855117\pi\)
\(774\) 1110.01 0.0515483
\(775\) 0 0
\(776\) 35.7490 0.00165375
\(777\) 7841.96i 0.362070i
\(778\) 26487.8i 1.22061i
\(779\) −1734.31 −0.0797664
\(780\) 0 0
\(781\) 979.089 0.0448586
\(782\) 18834.0i 0.861258i
\(783\) 7421.14i 0.338710i
\(784\) −18458.1 −0.840838
\(785\) 0 0
\(786\) 2594.23 0.117727
\(787\) − 23705.9i − 1.07373i −0.843669 0.536863i \(-0.819609\pi\)
0.843669 0.536863i \(-0.180391\pi\)
\(788\) 18379.4i 0.830888i
\(789\) −12005.0 −0.541685
\(790\) 0 0
\(791\) −8668.77 −0.389666
\(792\) − 126.400i − 0.00567100i
\(793\) 10238.6i 0.458493i
\(794\) −16500.5 −0.737507
\(795\) 0 0
\(796\) −3287.11 −0.146368
\(797\) 37248.0i 1.65545i 0.561135 + 0.827725i \(0.310365\pi\)
−0.561135 + 0.827725i \(0.689635\pi\)
\(798\) − 508.807i − 0.0225709i
\(799\) 1362.30 0.0603189
\(800\) 0 0
\(801\) −2815.49 −0.124195
\(802\) 26966.7i 1.18731i
\(803\) − 7893.84i − 0.346909i
\(804\) 717.315 0.0314648
\(805\) 0 0
\(806\) −8456.21 −0.369550
\(807\) − 5274.86i − 0.230092i
\(808\) − 291.458i − 0.0126899i
\(809\) 24861.0 1.08043 0.540214 0.841528i \(-0.318343\pi\)
0.540214 + 0.841528i \(0.318343\pi\)
\(810\) 0 0
\(811\) 4131.24 0.178875 0.0894375 0.995992i \(-0.471493\pi\)
0.0894375 + 0.995992i \(0.471493\pi\)
\(812\) − 14855.5i − 0.642027i
\(813\) − 18055.3i − 0.778878i
\(814\) 17870.7 0.769495
\(815\) 0 0
\(816\) −7941.10 −0.340679
\(817\) − 197.259i − 0.00844704i
\(818\) 6941.81i 0.296717i
\(819\) 1786.29 0.0762126
\(820\) 0 0
\(821\) 25297.3 1.07537 0.537687 0.843145i \(-0.319298\pi\)
0.537687 + 0.843145i \(0.319298\pi\)
\(822\) − 790.439i − 0.0335398i
\(823\) − 18155.5i − 0.768969i −0.923131 0.384484i \(-0.874379\pi\)
0.923131 0.384484i \(-0.125621\pi\)
\(824\) 1021.83 0.0432005
\(825\) 0 0
\(826\) 14657.6 0.617435
\(827\) 29566.8i 1.24321i 0.783329 + 0.621607i \(0.213520\pi\)
−0.783329 + 0.621607i \(0.786480\pi\)
\(828\) 8091.02i 0.339592i
\(829\) −27249.4 −1.14163 −0.570814 0.821079i \(-0.693372\pi\)
−0.570814 + 0.821079i \(0.693372\pi\)
\(830\) 0 0
\(831\) −8730.90 −0.364467
\(832\) 16845.9i 0.701956i
\(833\) 12972.2i 0.539568i
\(834\) −4933.02 −0.204816
\(835\) 0 0
\(836\) −590.981 −0.0244492
\(837\) − 1850.90i − 0.0764355i
\(838\) 23154.3i 0.954478i
\(839\) 15506.1 0.638058 0.319029 0.947745i \(-0.396643\pi\)
0.319029 + 0.947745i \(0.396643\pi\)
\(840\) 0 0
\(841\) 51157.5 2.09756
\(842\) 38968.8i 1.59496i
\(843\) − 20679.6i − 0.844890i
\(844\) −12972.9 −0.529084
\(845\) 0 0
\(846\) 1148.24 0.0466633
\(847\) 786.406i 0.0319023i
\(848\) − 15496.1i − 0.627520i
\(849\) 9360.47 0.378387
\(850\) 0 0
\(851\) −43479.4 −1.75141
\(852\) 2220.60i 0.0892915i
\(853\) 8698.16i 0.349143i 0.984644 + 0.174572i \(0.0558541\pi\)
−0.984644 + 0.174572i \(0.944146\pi\)
\(854\) 8801.64 0.352677
\(855\) 0 0
\(856\) −1336.64 −0.0533707
\(857\) 28468.8i 1.13474i 0.823462 + 0.567372i \(0.192040\pi\)
−0.823462 + 0.567372i \(0.807960\pi\)
\(858\) − 4070.71i − 0.161972i
\(859\) 21154.3 0.840251 0.420125 0.907466i \(-0.361986\pi\)
0.420125 + 0.907466i \(0.361986\pi\)
\(860\) 0 0
\(861\) −5234.16 −0.207177
\(862\) 12338.4i 0.487525i
\(863\) − 45959.5i − 1.81284i −0.422378 0.906420i \(-0.638805\pi\)
0.422378 0.906420i \(-0.361195\pi\)
\(864\) −6969.04 −0.274412
\(865\) 0 0
\(866\) 44524.5 1.74712
\(867\) − 9158.05i − 0.358735i
\(868\) 3705.10i 0.144884i
\(869\) −2207.72 −0.0861815
\(870\) 0 0
\(871\) 878.047 0.0341579
\(872\) − 404.765i − 0.0157191i
\(873\) 251.996i 0.00976951i
\(874\) 2821.06 0.109180
\(875\) 0 0
\(876\) 17903.4 0.690526
\(877\) 7315.10i 0.281657i 0.990034 + 0.140829i \(0.0449767\pi\)
−0.990034 + 0.140829i \(0.955023\pi\)
\(878\) − 30655.6i − 1.17833i
\(879\) 26608.9 1.02104
\(880\) 0 0
\(881\) 4652.19 0.177907 0.0889536 0.996036i \(-0.471648\pi\)
0.0889536 + 0.996036i \(0.471648\pi\)
\(882\) 10933.8i 0.417415i
\(883\) − 38954.3i − 1.48462i −0.670058 0.742309i \(-0.733731\pi\)
0.670058 0.742309i \(-0.266269\pi\)
\(884\) 10953.7 0.416756
\(885\) 0 0
\(886\) 16332.2 0.619289
\(887\) 16917.5i 0.640398i 0.947350 + 0.320199i \(0.103750\pi\)
−0.947350 + 0.320199i \(0.896250\pi\)
\(888\) 1540.55i 0.0582177i
\(889\) 3671.33 0.138507
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) − 6288.75i − 0.236057i
\(893\) − 204.053i − 0.00764654i
\(894\) 9598.53 0.359086
\(895\) 0 0
\(896\) 1061.32 0.0395715
\(897\) 9904.01i 0.368657i
\(898\) 59051.8i 2.19441i
\(899\) −18842.0 −0.699016
\(900\) 0 0
\(901\) −10890.5 −0.402682
\(902\) 11927.9i 0.440306i
\(903\) − 595.331i − 0.0219395i
\(904\) −1702.97 −0.0626549
\(905\) 0 0
\(906\) 34679.2 1.27168
\(907\) − 38985.4i − 1.42722i −0.700543 0.713610i \(-0.747059\pi\)
0.700543 0.713610i \(-0.252941\pi\)
\(908\) − 47889.8i − 1.75031i
\(909\) 2054.50 0.0749653
\(910\) 0 0
\(911\) 27759.8 1.00957 0.504787 0.863244i \(-0.331571\pi\)
0.504787 + 0.863244i \(0.331571\pi\)
\(912\) 1189.46i 0.0431874i
\(913\) 2680.49i 0.0971646i
\(914\) 44072.3 1.59495
\(915\) 0 0
\(916\) −15905.6 −0.573728
\(917\) − 1391.37i − 0.0501057i
\(918\) 4703.98i 0.169122i
\(919\) −29705.5 −1.06626 −0.533130 0.846033i \(-0.678984\pi\)
−0.533130 + 0.846033i \(0.678984\pi\)
\(920\) 0 0
\(921\) 17140.3 0.613236
\(922\) − 66127.6i − 2.36204i
\(923\) 2718.18i 0.0969338i
\(924\) −1783.59 −0.0635019
\(925\) 0 0
\(926\) −64724.1 −2.29694
\(927\) 7202.95i 0.255206i
\(928\) 70944.1i 2.50954i
\(929\) −2940.01 −0.103831 −0.0519153 0.998651i \(-0.516533\pi\)
−0.0519153 + 0.998651i \(0.516533\pi\)
\(930\) 0 0
\(931\) 1943.04 0.0684002
\(932\) − 8605.99i − 0.302466i
\(933\) − 5952.04i − 0.208854i
\(934\) 60982.6 2.13642
\(935\) 0 0
\(936\) 350.916 0.0122543
\(937\) 20041.9i 0.698762i 0.936981 + 0.349381i \(0.113608\pi\)
−0.936981 + 0.349381i \(0.886392\pi\)
\(938\) − 754.813i − 0.0262745i
\(939\) −8026.49 −0.278951
\(940\) 0 0
\(941\) −14446.1 −0.500457 −0.250229 0.968187i \(-0.580506\pi\)
−0.250229 + 0.968187i \(0.580506\pi\)
\(942\) 36251.4i 1.25386i
\(943\) − 29020.6i − 1.00216i
\(944\) −34265.6 −1.18141
\(945\) 0 0
\(946\) −1356.68 −0.0466272
\(947\) − 3901.33i − 0.133871i −0.997757 0.0669357i \(-0.978678\pi\)
0.997757 0.0669357i \(-0.0213223\pi\)
\(948\) − 5007.16i − 0.171545i
\(949\) 21915.1 0.749626
\(950\) 0 0
\(951\) 21251.3 0.724627
\(952\) − 357.904i − 0.0121846i
\(953\) 613.963i 0.0208691i 0.999946 + 0.0104345i \(0.00332148\pi\)
−0.999946 + 0.0104345i \(0.996679\pi\)
\(954\) −9179.22 −0.311518
\(955\) 0 0
\(956\) 22014.6 0.744773
\(957\) − 9070.29i − 0.306375i
\(958\) − 42247.1i − 1.42478i
\(959\) −423.937 −0.0142749
\(960\) 0 0
\(961\) −25091.6 −0.842256
\(962\) 49613.3i 1.66278i
\(963\) − 9422.02i − 0.315286i
\(964\) −56000.5 −1.87101
\(965\) 0 0
\(966\) 8513.98 0.283574
\(967\) 20617.4i 0.685637i 0.939402 + 0.342819i \(0.111382\pi\)
−0.939402 + 0.342819i \(0.888618\pi\)
\(968\) 154.489i 0.00512961i
\(969\) 835.943 0.0277135
\(970\) 0 0
\(971\) 22816.0 0.754069 0.377035 0.926199i \(-0.376944\pi\)
0.377035 + 0.926199i \(0.376944\pi\)
\(972\) 2020.81i 0.0666846i
\(973\) 2645.73i 0.0871719i
\(974\) 32071.6 1.05507
\(975\) 0 0
\(976\) −20576.0 −0.674816
\(977\) − 22650.2i − 0.741702i −0.928692 0.370851i \(-0.879066\pi\)
0.928692 0.370851i \(-0.120934\pi\)
\(978\) − 35234.2i − 1.15201i
\(979\) 3441.15 0.112339
\(980\) 0 0
\(981\) 2853.21 0.0928604
\(982\) − 46422.8i − 1.50857i
\(983\) − 8963.01i − 0.290820i −0.989372 0.145410i \(-0.953550\pi\)
0.989372 0.145410i \(-0.0464501\pi\)
\(984\) −1028.25 −0.0333123
\(985\) 0 0
\(986\) 47886.0 1.54665
\(987\) − 615.834i − 0.0198604i
\(988\) − 1640.70i − 0.0528316i
\(989\) 3300.78 0.106126
\(990\) 0 0
\(991\) 422.204 0.0135336 0.00676678 0.999977i \(-0.497846\pi\)
0.00676678 + 0.999977i \(0.497846\pi\)
\(992\) − 17694.1i − 0.566319i
\(993\) − 23437.2i − 0.748999i
\(994\) 2336.68 0.0745623
\(995\) 0 0
\(996\) −6079.41 −0.193407
\(997\) 12577.3i 0.399524i 0.979844 + 0.199762i \(0.0640169\pi\)
−0.979844 + 0.199762i \(0.935983\pi\)
\(998\) 19443.4i 0.616704i
\(999\) −10859.4 −0.343920
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 825.4.c.n.199.1 6
5.2 odd 4 825.4.a.q.1.3 yes 3
5.3 odd 4 825.4.a.o.1.1 3
5.4 even 2 inner 825.4.c.n.199.6 6
15.2 even 4 2475.4.a.v.1.1 3
15.8 even 4 2475.4.a.y.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.o.1.1 3 5.3 odd 4
825.4.a.q.1.3 yes 3 5.2 odd 4
825.4.c.n.199.1 6 1.1 even 1 trivial
825.4.c.n.199.6 6 5.4 even 2 inner
2475.4.a.v.1.1 3 15.2 even 4
2475.4.a.y.1.3 3 15.8 even 4