Properties

Label 825.4.c.h.199.3
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{97})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 49x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(4.42443i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.h.199.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.42443i q^{2} -3.00000i q^{3} -11.5756 q^{4} +13.2733 q^{6} -31.6977i q^{7} -15.8199i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+4.42443i q^{2} -3.00000i q^{3} -11.5756 q^{4} +13.2733 q^{6} -31.6977i q^{7} -15.8199i q^{8} -9.00000 q^{9} -11.0000 q^{11} +34.7267i q^{12} +5.15114i q^{13} +140.244 q^{14} -22.6107 q^{16} -121.942i q^{17} -39.8199i q^{18} -34.8489 q^{19} -95.0931 q^{21} -48.6687i q^{22} +116.244i q^{23} -47.4596 q^{24} -22.7909 q^{26} +27.0000i q^{27} +366.919i q^{28} +69.4534 q^{29} +140.605 q^{31} -226.598i q^{32} +33.0000i q^{33} +539.524 q^{34} +104.180 q^{36} +420.070i q^{37} -154.186i q^{38} +15.4534 q^{39} -322.058 q^{41} -420.733i q^{42} +321.035i q^{43} +127.331 q^{44} -514.315 q^{46} +231.408i q^{47} +67.8322i q^{48} -661.745 q^{49} -365.826 q^{51} -59.6274i q^{52} +4.91916i q^{53} -119.460 q^{54} -501.453 q^{56} +104.547i q^{57} +307.292i q^{58} -406.443 q^{59} -556.431 q^{61} +622.095i q^{62} +285.279i q^{63} +821.683 q^{64} -146.006 q^{66} -84.7452i q^{67} +1411.55i q^{68} +348.733 q^{69} +49.0808 q^{71} +142.379i q^{72} +785.884i q^{73} -1858.57 q^{74} +403.395 q^{76} +348.675i q^{77} +68.3726i q^{78} +383.118 q^{79} +81.0000 q^{81} -1424.92i q^{82} -930.211i q^{83} +1100.76 q^{84} -1420.40 q^{86} -208.360i q^{87} +174.018i q^{88} +732.559 q^{89} +163.279 q^{91} -1345.59i q^{92} -421.814i q^{93} -1023.85 q^{94} -679.795 q^{96} +1171.49i q^{97} -2927.84i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 66 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 66 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 364 q^{14} + 402 q^{16} - 100 q^{19} - 144 q^{21} + 342 q^{24} + 224 q^{26} + 396 q^{29} + 720 q^{31} + 1252 q^{34} + 594 q^{36} + 180 q^{39} - 1564 q^{41} + 726 q^{44} - 836 q^{46} - 756 q^{49} - 636 q^{51} + 54 q^{54} - 2124 q^{56} + 344 q^{59} - 1556 q^{61} - 1618 q^{64} + 66 q^{66} + 804 q^{69} + 1260 q^{71} - 4716 q^{74} + 1456 q^{76} - 1304 q^{79} + 324 q^{81} + 1212 q^{84} - 2136 q^{86} + 1512 q^{89} - 56 q^{91} - 7444 q^{94} - 5142 q^{96} + 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.42443i 1.56427i 0.623108 + 0.782136i \(0.285870\pi\)
−0.623108 + 0.782136i \(0.714130\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −11.5756 −1.44695
\(5\) 0 0
\(6\) 13.2733 0.903133
\(7\) − 31.6977i − 1.71152i −0.517377 0.855758i \(-0.673091\pi\)
0.517377 0.855758i \(-0.326909\pi\)
\(8\) − 15.8199i − 0.699146i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 34.7267i 0.835395i
\(13\) 5.15114i 0.109898i 0.998489 + 0.0549488i \(0.0174996\pi\)
−0.998489 + 0.0549488i \(0.982500\pi\)
\(14\) 140.244 2.67728
\(15\) 0 0
\(16\) −22.6107 −0.353293
\(17\) − 121.942i − 1.73972i −0.493297 0.869861i \(-0.664208\pi\)
0.493297 0.869861i \(-0.335792\pi\)
\(18\) − 39.8199i − 0.521424i
\(19\) −34.8489 −0.420783 −0.210391 0.977617i \(-0.567474\pi\)
−0.210391 + 0.977617i \(0.567474\pi\)
\(20\) 0 0
\(21\) −95.0931 −0.988144
\(22\) − 48.6687i − 0.471646i
\(23\) 116.244i 1.05385i 0.849911 + 0.526926i \(0.176656\pi\)
−0.849911 + 0.526926i \(0.823344\pi\)
\(24\) −47.4596 −0.403652
\(25\) 0 0
\(26\) −22.7909 −0.171910
\(27\) 27.0000i 0.192450i
\(28\) 366.919i 2.47647i
\(29\) 69.4534 0.444730 0.222365 0.974963i \(-0.428622\pi\)
0.222365 + 0.974963i \(0.428622\pi\)
\(30\) 0 0
\(31\) 140.605 0.814623 0.407312 0.913289i \(-0.366466\pi\)
0.407312 + 0.913289i \(0.366466\pi\)
\(32\) − 226.598i − 1.25179i
\(33\) 33.0000i 0.174078i
\(34\) 539.524 2.72140
\(35\) 0 0
\(36\) 104.180 0.482315
\(37\) 420.070i 1.86646i 0.359276 + 0.933232i \(0.383024\pi\)
−0.359276 + 0.933232i \(0.616976\pi\)
\(38\) − 154.186i − 0.658219i
\(39\) 15.4534 0.0634495
\(40\) 0 0
\(41\) −322.058 −1.22676 −0.613378 0.789789i \(-0.710190\pi\)
−0.613378 + 0.789789i \(0.710190\pi\)
\(42\) − 420.733i − 1.54573i
\(43\) 321.035i 1.13854i 0.822149 + 0.569272i \(0.192775\pi\)
−0.822149 + 0.569272i \(0.807225\pi\)
\(44\) 127.331 0.436271
\(45\) 0 0
\(46\) −514.315 −1.64851
\(47\) 231.408i 0.718176i 0.933304 + 0.359088i \(0.116912\pi\)
−0.933304 + 0.359088i \(0.883088\pi\)
\(48\) 67.8322i 0.203974i
\(49\) −661.745 −1.92929
\(50\) 0 0
\(51\) −365.826 −1.00443
\(52\) − 59.6274i − 0.159016i
\(53\) 4.91916i 0.0127490i 0.999980 + 0.00637452i \(0.00202909\pi\)
−0.999980 + 0.00637452i \(0.997971\pi\)
\(54\) −119.460 −0.301044
\(55\) 0 0
\(56\) −501.453 −1.19660
\(57\) 104.547i 0.242939i
\(58\) 307.292i 0.695679i
\(59\) −406.443 −0.896854 −0.448427 0.893820i \(-0.648016\pi\)
−0.448427 + 0.893820i \(0.648016\pi\)
\(60\) 0 0
\(61\) −556.431 −1.16793 −0.583964 0.811779i \(-0.698499\pi\)
−0.583964 + 0.811779i \(0.698499\pi\)
\(62\) 622.095i 1.27429i
\(63\) 285.279i 0.570505i
\(64\) 821.683 1.60485
\(65\) 0 0
\(66\) −146.006 −0.272305
\(67\) − 84.7452i − 0.154526i −0.997011 0.0772632i \(-0.975382\pi\)
0.997011 0.0772632i \(-0.0246182\pi\)
\(68\) 1411.55i 2.51728i
\(69\) 348.733 0.608442
\(70\) 0 0
\(71\) 49.0808 0.0820398 0.0410199 0.999158i \(-0.486939\pi\)
0.0410199 + 0.999158i \(0.486939\pi\)
\(72\) 142.379i 0.233049i
\(73\) 785.884i 1.26001i 0.776591 + 0.630005i \(0.216947\pi\)
−0.776591 + 0.630005i \(0.783053\pi\)
\(74\) −1858.57 −2.91966
\(75\) 0 0
\(76\) 403.395 0.608850
\(77\) 348.675i 0.516041i
\(78\) 68.3726i 0.0992522i
\(79\) 383.118 0.545622 0.272811 0.962068i \(-0.412047\pi\)
0.272811 + 0.962068i \(0.412047\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 1424.92i − 1.91898i
\(83\) − 930.211i − 1.23017i −0.788462 0.615084i \(-0.789122\pi\)
0.788462 0.615084i \(-0.210878\pi\)
\(84\) 1100.76 1.42979
\(85\) 0 0
\(86\) −1420.40 −1.78099
\(87\) − 208.360i − 0.256765i
\(88\) 174.018i 0.210800i
\(89\) 732.559 0.872484 0.436242 0.899829i \(-0.356309\pi\)
0.436242 + 0.899829i \(0.356309\pi\)
\(90\) 0 0
\(91\) 163.279 0.188092
\(92\) − 1345.59i − 1.52487i
\(93\) − 421.814i − 0.470323i
\(94\) −1023.85 −1.12342
\(95\) 0 0
\(96\) −679.795 −0.722722
\(97\) 1171.49i 1.22626i 0.789984 + 0.613128i \(0.210089\pi\)
−0.789984 + 0.613128i \(0.789911\pi\)
\(98\) − 2927.84i − 3.01793i
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) −1221.27 −1.20318 −0.601589 0.798806i \(-0.705465\pi\)
−0.601589 + 0.798806i \(0.705465\pi\)
\(102\) − 1618.57i − 1.57120i
\(103\) 516.745i 0.494334i 0.968973 + 0.247167i \(0.0794996\pi\)
−0.968973 + 0.247167i \(0.920500\pi\)
\(104\) 81.4903 0.0768345
\(105\) 0 0
\(106\) −21.7645 −0.0199430
\(107\) 152.025i 0.137353i 0.997639 + 0.0686765i \(0.0218776\pi\)
−0.997639 + 0.0686765i \(0.978122\pi\)
\(108\) − 312.540i − 0.278465i
\(109\) −2170.32 −1.90714 −0.953572 0.301164i \(-0.902625\pi\)
−0.953572 + 0.301164i \(0.902625\pi\)
\(110\) 0 0
\(111\) 1260.21 1.07760
\(112\) 716.708i 0.604666i
\(113\) − 646.397i − 0.538123i −0.963123 0.269062i \(-0.913286\pi\)
0.963123 0.269062i \(-0.0867135\pi\)
\(114\) −462.559 −0.380023
\(115\) 0 0
\(116\) −803.963 −0.643501
\(117\) − 46.3603i − 0.0366326i
\(118\) − 1798.28i − 1.40292i
\(119\) −3865.28 −2.97756
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) − 2461.89i − 1.82696i
\(123\) 966.174i 0.708268i
\(124\) −1627.58 −1.17872
\(125\) 0 0
\(126\) −1262.20 −0.892425
\(127\) 993.304i 0.694027i 0.937860 + 0.347014i \(0.112804\pi\)
−0.937860 + 0.347014i \(0.887196\pi\)
\(128\) 1822.69i 1.25863i
\(129\) 963.105 0.657339
\(130\) 0 0
\(131\) 385.814 0.257318 0.128659 0.991689i \(-0.458933\pi\)
0.128659 + 0.991689i \(0.458933\pi\)
\(132\) − 381.994i − 0.251881i
\(133\) 1104.63i 0.720177i
\(134\) 374.949 0.241721
\(135\) 0 0
\(136\) −1929.11 −1.21632
\(137\) − 884.840i − 0.551803i −0.961186 0.275901i \(-0.911024\pi\)
0.961186 0.275901i \(-0.0889763\pi\)
\(138\) 1542.94i 0.951769i
\(139\) 1091.94 0.666312 0.333156 0.942872i \(-0.391886\pi\)
0.333156 + 0.942872i \(0.391886\pi\)
\(140\) 0 0
\(141\) 694.223 0.414639
\(142\) 217.155i 0.128333i
\(143\) − 56.6626i − 0.0331354i
\(144\) 203.497 0.117764
\(145\) 0 0
\(146\) −3477.09 −1.97100
\(147\) 1985.24i 1.11387i
\(148\) − 4862.55i − 2.70067i
\(149\) −297.014 −0.163304 −0.0816522 0.996661i \(-0.526020\pi\)
−0.0816522 + 0.996661i \(0.526020\pi\)
\(150\) 0 0
\(151\) −1887.86 −1.01743 −0.508716 0.860935i \(-0.669880\pi\)
−0.508716 + 0.860935i \(0.669880\pi\)
\(152\) 551.304i 0.294189i
\(153\) 1097.48i 0.579907i
\(154\) −1542.69 −0.807229
\(155\) 0 0
\(156\) −178.882 −0.0918080
\(157\) 56.5343i 0.0287384i 0.999897 + 0.0143692i \(0.00457401\pi\)
−0.999897 + 0.0143692i \(0.995426\pi\)
\(158\) 1695.08i 0.853501i
\(159\) 14.7575 0.00736066
\(160\) 0 0
\(161\) 3684.68 1.80369
\(162\) 358.379i 0.173808i
\(163\) − 49.2338i − 0.0236582i −0.999930 0.0118291i \(-0.996235\pi\)
0.999930 0.0118291i \(-0.00376541\pi\)
\(164\) 3728.01 1.77505
\(165\) 0 0
\(166\) 4115.65 1.92432
\(167\) − 2068.75i − 0.958589i −0.877654 0.479294i \(-0.840893\pi\)
0.877654 0.479294i \(-0.159107\pi\)
\(168\) 1504.36i 0.690857i
\(169\) 2170.47 0.987923
\(170\) 0 0
\(171\) 313.640 0.140261
\(172\) − 3716.17i − 1.64741i
\(173\) − 604.012i − 0.265446i −0.991153 0.132723i \(-0.957628\pi\)
0.991153 0.132723i \(-0.0423721\pi\)
\(174\) 921.875 0.401650
\(175\) 0 0
\(176\) 248.718 0.106522
\(177\) 1219.33i 0.517799i
\(178\) 3241.15i 1.36480i
\(179\) 2132.02 0.890251 0.445126 0.895468i \(-0.353159\pi\)
0.445126 + 0.895468i \(0.353159\pi\)
\(180\) 0 0
\(181\) −589.371 −0.242031 −0.121015 0.992651i \(-0.538615\pi\)
−0.121015 + 0.992651i \(0.538615\pi\)
\(182\) 722.418i 0.294226i
\(183\) 1669.29i 0.674304i
\(184\) 1838.97 0.736796
\(185\) 0 0
\(186\) 1866.28 0.735713
\(187\) 1341.36i 0.524546i
\(188\) − 2678.68i − 1.03916i
\(189\) 855.838 0.329381
\(190\) 0 0
\(191\) −2160.90 −0.818624 −0.409312 0.912395i \(-0.634231\pi\)
−0.409312 + 0.912395i \(0.634231\pi\)
\(192\) − 2465.05i − 0.926560i
\(193\) − 1490.91i − 0.556052i −0.960574 0.278026i \(-0.910320\pi\)
0.960574 0.278026i \(-0.0896802\pi\)
\(194\) −5183.18 −1.91820
\(195\) 0 0
\(196\) 7660.08 2.79157
\(197\) 230.529i 0.0833732i 0.999131 + 0.0416866i \(0.0132731\pi\)
−0.999131 + 0.0416866i \(0.986727\pi\)
\(198\) 438.018i 0.157215i
\(199\) −22.4007 −0.00797963 −0.00398982 0.999992i \(-0.501270\pi\)
−0.00398982 + 0.999992i \(0.501270\pi\)
\(200\) 0 0
\(201\) −254.236 −0.0892159
\(202\) − 5403.43i − 1.88210i
\(203\) − 2201.51i − 0.761163i
\(204\) 4234.65 1.45336
\(205\) 0 0
\(206\) −2286.30 −0.773273
\(207\) − 1046.20i − 0.351284i
\(208\) − 116.471i − 0.0388260i
\(209\) 383.337 0.126871
\(210\) 0 0
\(211\) −1051.64 −0.343117 −0.171558 0.985174i \(-0.554880\pi\)
−0.171558 + 0.985174i \(0.554880\pi\)
\(212\) − 56.9421i − 0.0184472i
\(213\) − 147.243i − 0.0473657i
\(214\) −672.622 −0.214857
\(215\) 0 0
\(216\) 427.136 0.134551
\(217\) − 4456.84i − 1.39424i
\(218\) − 9602.42i − 2.98329i
\(219\) 2357.65 0.727467
\(220\) 0 0
\(221\) 628.141 0.191191
\(222\) 5575.71i 1.68566i
\(223\) 3861.80i 1.15966i 0.814736 + 0.579832i \(0.196882\pi\)
−0.814736 + 0.579832i \(0.803118\pi\)
\(224\) −7182.65 −2.14246
\(225\) 0 0
\(226\) 2859.94 0.841771
\(227\) 872.721i 0.255174i 0.991827 + 0.127587i \(0.0407232\pi\)
−0.991827 + 0.127587i \(0.959277\pi\)
\(228\) − 1210.19i − 0.351520i
\(229\) −1841.72 −0.531459 −0.265730 0.964048i \(-0.585613\pi\)
−0.265730 + 0.964048i \(0.585613\pi\)
\(230\) 0 0
\(231\) 1046.02 0.297937
\(232\) − 1098.74i − 0.310931i
\(233\) 3932.14i 1.10559i 0.833317 + 0.552796i \(0.186439\pi\)
−0.833317 + 0.552796i \(0.813561\pi\)
\(234\) 205.118 0.0573033
\(235\) 0 0
\(236\) 4704.81 1.29770
\(237\) − 1149.35i − 0.315015i
\(238\) − 17101.7i − 4.65772i
\(239\) −4772.10 −1.29155 −0.645777 0.763526i \(-0.723466\pi\)
−0.645777 + 0.763526i \(0.723466\pi\)
\(240\) 0 0
\(241\) 3988.84 1.06616 0.533078 0.846066i \(-0.321035\pi\)
0.533078 + 0.846066i \(0.321035\pi\)
\(242\) 535.356i 0.142207i
\(243\) − 243.000i − 0.0641500i
\(244\) 6441.00 1.68993
\(245\) 0 0
\(246\) −4274.77 −1.10792
\(247\) − 179.511i − 0.0462431i
\(248\) − 2224.34i − 0.569540i
\(249\) −2790.63 −0.710238
\(250\) 0 0
\(251\) −5474.22 −1.37661 −0.688306 0.725421i \(-0.741645\pi\)
−0.688306 + 0.725421i \(0.741645\pi\)
\(252\) − 3302.27i − 0.825491i
\(253\) − 1278.69i − 0.317749i
\(254\) −4394.80 −1.08565
\(255\) 0 0
\(256\) −1490.90 −0.363989
\(257\) 6434.01i 1.56164i 0.624754 + 0.780822i \(0.285199\pi\)
−0.624754 + 0.780822i \(0.714801\pi\)
\(258\) 4261.19i 1.02826i
\(259\) 13315.3 3.19448
\(260\) 0 0
\(261\) −625.081 −0.148243
\(262\) 1707.01i 0.402516i
\(263\) 7589.00i 1.77931i 0.456636 + 0.889654i \(0.349054\pi\)
−0.456636 + 0.889654i \(0.650946\pi\)
\(264\) 522.055 0.121706
\(265\) 0 0
\(266\) −4887.35 −1.12655
\(267\) − 2197.68i − 0.503729i
\(268\) 980.974i 0.223591i
\(269\) −478.178 −0.108383 −0.0541914 0.998531i \(-0.517258\pi\)
−0.0541914 + 0.998531i \(0.517258\pi\)
\(270\) 0 0
\(271\) −122.323 −0.0274192 −0.0137096 0.999906i \(-0.504364\pi\)
−0.0137096 + 0.999906i \(0.504364\pi\)
\(272\) 2757.20i 0.614631i
\(273\) − 489.838i − 0.108595i
\(274\) 3914.91 0.863170
\(275\) 0 0
\(276\) −4036.78 −0.880383
\(277\) − 8199.41i − 1.77854i −0.457385 0.889269i \(-0.651214\pi\)
0.457385 0.889269i \(-0.348786\pi\)
\(278\) 4831.22i 1.04229i
\(279\) −1265.44 −0.271541
\(280\) 0 0
\(281\) 6943.79 1.47413 0.737067 0.675820i \(-0.236210\pi\)
0.737067 + 0.675820i \(0.236210\pi\)
\(282\) 3071.54i 0.648609i
\(283\) 1035.14i 0.217429i 0.994073 + 0.108715i \(0.0346735\pi\)
−0.994073 + 0.108715i \(0.965327\pi\)
\(284\) −568.139 −0.118707
\(285\) 0 0
\(286\) 250.699 0.0518328
\(287\) 10208.5i 2.09961i
\(288\) 2039.39i 0.417264i
\(289\) −9956.85 −2.02663
\(290\) 0 0
\(291\) 3514.47 0.707979
\(292\) − 9097.06i − 1.82317i
\(293\) − 6144.81i − 1.22520i −0.790393 0.612600i \(-0.790124\pi\)
0.790393 0.612600i \(-0.209876\pi\)
\(294\) −8783.53 −1.74240
\(295\) 0 0
\(296\) 6645.45 1.30493
\(297\) − 297.000i − 0.0580259i
\(298\) − 1314.12i − 0.255452i
\(299\) −598.791 −0.115816
\(300\) 0 0
\(301\) 10176.1 1.94864
\(302\) − 8352.72i − 1.59154i
\(303\) 3663.81i 0.694655i
\(304\) 787.958 0.148659
\(305\) 0 0
\(306\) −4855.71 −0.907133
\(307\) 2186.09i 0.406406i 0.979137 + 0.203203i \(0.0651351\pi\)
−0.979137 + 0.203203i \(0.934865\pi\)
\(308\) − 4036.11i − 0.746684i
\(309\) 1550.24 0.285404
\(310\) 0 0
\(311\) −7484.83 −1.36471 −0.682357 0.731019i \(-0.739045\pi\)
−0.682357 + 0.731019i \(0.739045\pi\)
\(312\) − 244.471i − 0.0443604i
\(313\) − 6833.33i − 1.23400i −0.786962 0.617001i \(-0.788347\pi\)
0.786962 0.617001i \(-0.211653\pi\)
\(314\) −250.132 −0.0449546
\(315\) 0 0
\(316\) −4434.81 −0.789485
\(317\) − 924.265i − 0.163760i −0.996642 0.0818800i \(-0.973908\pi\)
0.996642 0.0818800i \(-0.0260924\pi\)
\(318\) 65.2934i 0.0115141i
\(319\) −763.988 −0.134091
\(320\) 0 0
\(321\) 456.074 0.0793008
\(322\) 16302.6i 2.82145i
\(323\) 4249.54i 0.732046i
\(324\) −937.621 −0.160772
\(325\) 0 0
\(326\) 217.831 0.0370078
\(327\) 6510.95i 1.10109i
\(328\) 5094.91i 0.857681i
\(329\) 7335.10 1.22917
\(330\) 0 0
\(331\) −9820.46 −1.63076 −0.815380 0.578927i \(-0.803472\pi\)
−0.815380 + 0.578927i \(0.803472\pi\)
\(332\) 10767.7i 1.77999i
\(333\) − 3780.63i − 0.622154i
\(334\) 9153.02 1.49949
\(335\) 0 0
\(336\) 2150.12 0.349104
\(337\) − 600.808i − 0.0971161i −0.998820 0.0485580i \(-0.984537\pi\)
0.998820 0.0485580i \(-0.0154626\pi\)
\(338\) 9603.07i 1.54538i
\(339\) −1939.19 −0.310686
\(340\) 0 0
\(341\) −1546.65 −0.245618
\(342\) 1387.68i 0.219406i
\(343\) 10103.5i 1.59049i
\(344\) 5078.73 0.796008
\(345\) 0 0
\(346\) 2672.41 0.415230
\(347\) 3143.41i 0.486303i 0.969988 + 0.243152i \(0.0781813\pi\)
−0.969988 + 0.243152i \(0.921819\pi\)
\(348\) 2411.89i 0.371525i
\(349\) −720.663 −0.110533 −0.0552667 0.998472i \(-0.517601\pi\)
−0.0552667 + 0.998472i \(0.517601\pi\)
\(350\) 0 0
\(351\) −139.081 −0.0211498
\(352\) 2492.58i 0.377429i
\(353\) 1207.12i 0.182007i 0.995851 + 0.0910034i \(0.0290074\pi\)
−0.995851 + 0.0910034i \(0.970993\pi\)
\(354\) −5394.83 −0.809978
\(355\) 0 0
\(356\) −8479.79 −1.26244
\(357\) 11595.8i 1.71910i
\(358\) 9432.99i 1.39260i
\(359\) −8748.31 −1.28612 −0.643062 0.765814i \(-0.722336\pi\)
−0.643062 + 0.765814i \(0.722336\pi\)
\(360\) 0 0
\(361\) −5644.56 −0.822942
\(362\) − 2607.63i − 0.378602i
\(363\) − 363.000i − 0.0524864i
\(364\) −1890.05 −0.272158
\(365\) 0 0
\(366\) −7385.66 −1.05479
\(367\) 6730.45i 0.957293i 0.878008 + 0.478647i \(0.158872\pi\)
−0.878008 + 0.478647i \(0.841128\pi\)
\(368\) − 2628.37i − 0.372318i
\(369\) 2898.52 0.408919
\(370\) 0 0
\(371\) 155.926 0.0218202
\(372\) 4882.73i 0.680532i
\(373\) − 227.394i − 0.0315657i −0.999875 0.0157828i \(-0.994976\pi\)
0.999875 0.0157828i \(-0.00502404\pi\)
\(374\) −5934.76 −0.820533
\(375\) 0 0
\(376\) 3660.84 0.502110
\(377\) 357.764i 0.0488748i
\(378\) 3786.60i 0.515242i
\(379\) −11356.2 −1.53913 −0.769565 0.638568i \(-0.779527\pi\)
−0.769565 + 0.638568i \(0.779527\pi\)
\(380\) 0 0
\(381\) 2979.91 0.400697
\(382\) − 9560.74i − 1.28055i
\(383\) 10753.6i 1.43468i 0.696725 + 0.717338i \(0.254640\pi\)
−0.696725 + 0.717338i \(0.745360\pi\)
\(384\) 5468.07 0.726670
\(385\) 0 0
\(386\) 6596.43 0.869817
\(387\) − 2889.32i − 0.379515i
\(388\) − 13560.7i − 1.77433i
\(389\) 11727.1 1.52850 0.764252 0.644918i \(-0.223109\pi\)
0.764252 + 0.644918i \(0.223109\pi\)
\(390\) 0 0
\(391\) 14175.1 1.83341
\(392\) 10468.7i 1.34885i
\(393\) − 1157.44i − 0.148563i
\(394\) −1019.96 −0.130418
\(395\) 0 0
\(396\) −1145.98 −0.145424
\(397\) 359.905i 0.0454990i 0.999741 + 0.0227495i \(0.00724202\pi\)
−0.999741 + 0.0227495i \(0.992758\pi\)
\(398\) − 99.1105i − 0.0124823i
\(399\) 3313.89 0.415794
\(400\) 0 0
\(401\) −4066.71 −0.506438 −0.253219 0.967409i \(-0.581489\pi\)
−0.253219 + 0.967409i \(0.581489\pi\)
\(402\) − 1124.85i − 0.139558i
\(403\) 724.274i 0.0895252i
\(404\) 14136.9 1.74093
\(405\) 0 0
\(406\) 9740.45 1.19067
\(407\) − 4620.77i − 0.562760i
\(408\) 5787.32i 0.702242i
\(409\) 13488.8 1.63076 0.815379 0.578927i \(-0.196528\pi\)
0.815379 + 0.578927i \(0.196528\pi\)
\(410\) 0 0
\(411\) −2654.52 −0.318584
\(412\) − 5981.62i − 0.715275i
\(413\) 12883.3i 1.53498i
\(414\) 4628.83 0.549504
\(415\) 0 0
\(416\) 1167.24 0.137569
\(417\) − 3275.83i − 0.384695i
\(418\) 1696.05i 0.198460i
\(419\) 7040.12 0.820841 0.410420 0.911896i \(-0.365382\pi\)
0.410420 + 0.911896i \(0.365382\pi\)
\(420\) 0 0
\(421\) 9171.74 1.06177 0.530883 0.847445i \(-0.321860\pi\)
0.530883 + 0.847445i \(0.321860\pi\)
\(422\) − 4652.89i − 0.536728i
\(423\) − 2082.67i − 0.239392i
\(424\) 77.8204 0.00891343
\(425\) 0 0
\(426\) 651.464 0.0740928
\(427\) 17637.6i 1.99893i
\(428\) − 1759.77i − 0.198742i
\(429\) −169.988 −0.0191307
\(430\) 0 0
\(431\) 992.995 0.110976 0.0554882 0.998459i \(-0.482328\pi\)
0.0554882 + 0.998459i \(0.482328\pi\)
\(432\) − 610.490i − 0.0679912i
\(433\) 3790.21i 0.420660i 0.977630 + 0.210330i \(0.0674538\pi\)
−0.977630 + 0.210330i \(0.932546\pi\)
\(434\) 19719.0 2.18097
\(435\) 0 0
\(436\) 25122.7 2.75954
\(437\) − 4050.98i − 0.443443i
\(438\) 10431.3i 1.13796i
\(439\) 5136.97 0.558483 0.279242 0.960221i \(-0.409917\pi\)
0.279242 + 0.960221i \(0.409917\pi\)
\(440\) 0 0
\(441\) 5955.71 0.643095
\(442\) 2779.16i 0.299075i
\(443\) 10676.8i 1.14508i 0.819876 + 0.572541i \(0.194042\pi\)
−0.819876 + 0.572541i \(0.805958\pi\)
\(444\) −14587.7 −1.55923
\(445\) 0 0
\(446\) −17086.2 −1.81403
\(447\) 891.042i 0.0942838i
\(448\) − 26045.5i − 2.74672i
\(449\) −10529.9 −1.10676 −0.553379 0.832929i \(-0.686662\pi\)
−0.553379 + 0.832929i \(0.686662\pi\)
\(450\) 0 0
\(451\) 3542.64 0.369881
\(452\) 7482.42i 0.778636i
\(453\) 5663.59i 0.587414i
\(454\) −3861.29 −0.399162
\(455\) 0 0
\(456\) 1653.91 0.169850
\(457\) 14072.5i 1.44045i 0.693741 + 0.720225i \(0.255961\pi\)
−0.693741 + 0.720225i \(0.744039\pi\)
\(458\) − 8148.55i − 0.831347i
\(459\) 3292.43 0.334810
\(460\) 0 0
\(461\) −30.8173 −0.00311346 −0.00155673 0.999999i \(-0.500496\pi\)
−0.00155673 + 0.999999i \(0.500496\pi\)
\(462\) 4628.06i 0.466054i
\(463\) 17591.3i 1.76573i 0.469622 + 0.882867i \(0.344390\pi\)
−0.469622 + 0.882867i \(0.655610\pi\)
\(464\) −1570.39 −0.157120
\(465\) 0 0
\(466\) −17397.5 −1.72945
\(467\) − 13273.1i − 1.31522i −0.753360 0.657609i \(-0.771568\pi\)
0.753360 0.657609i \(-0.228432\pi\)
\(468\) 536.647i 0.0530053i
\(469\) −2686.23 −0.264474
\(470\) 0 0
\(471\) 169.603 0.0165921
\(472\) 6429.87i 0.627031i
\(473\) − 3531.39i − 0.343284i
\(474\) 5085.23 0.492769
\(475\) 0 0
\(476\) 44742.9 4.30837
\(477\) − 44.2724i − 0.00424968i
\(478\) − 21113.8i − 2.02034i
\(479\) −2496.68 −0.238155 −0.119077 0.992885i \(-0.537994\pi\)
−0.119077 + 0.992885i \(0.537994\pi\)
\(480\) 0 0
\(481\) −2163.84 −0.205120
\(482\) 17648.3i 1.66776i
\(483\) − 11054.0i − 1.04136i
\(484\) −1400.64 −0.131541
\(485\) 0 0
\(486\) 1075.14 0.100348
\(487\) 3464.42i 0.322357i 0.986925 + 0.161178i \(0.0515295\pi\)
−0.986925 + 0.161178i \(0.948471\pi\)
\(488\) 8802.65i 0.816552i
\(489\) −147.701 −0.0136591
\(490\) 0 0
\(491\) −16224.6 −1.49125 −0.745625 0.666366i \(-0.767849\pi\)
−0.745625 + 0.666366i \(0.767849\pi\)
\(492\) − 11184.0i − 1.02483i
\(493\) − 8469.29i − 0.773707i
\(494\) 794.236 0.0723367
\(495\) 0 0
\(496\) −3179.17 −0.287800
\(497\) − 1555.75i − 0.140412i
\(498\) − 12347.0i − 1.11100i
\(499\) −9993.81 −0.896562 −0.448281 0.893893i \(-0.647964\pi\)
−0.448281 + 0.893893i \(0.647964\pi\)
\(500\) 0 0
\(501\) −6206.24 −0.553441
\(502\) − 24220.3i − 2.15340i
\(503\) − 15334.8i − 1.35933i −0.733520 0.679667i \(-0.762124\pi\)
0.733520 0.679667i \(-0.237876\pi\)
\(504\) 4513.08 0.398866
\(505\) 0 0
\(506\) 5657.46 0.497045
\(507\) − 6511.40i − 0.570377i
\(508\) − 11498.1i − 1.00422i
\(509\) 7291.23 0.634927 0.317464 0.948270i \(-0.397169\pi\)
0.317464 + 0.948270i \(0.397169\pi\)
\(510\) 0 0
\(511\) 24910.7 2.15653
\(512\) 7985.14i 0.689251i
\(513\) − 940.919i − 0.0809797i
\(514\) −28466.8 −2.44283
\(515\) 0 0
\(516\) −11148.5 −0.951134
\(517\) − 2545.49i − 0.216538i
\(518\) 58912.5i 4.99704i
\(519\) −1812.04 −0.153255
\(520\) 0 0
\(521\) 16794.3 1.41223 0.706114 0.708098i \(-0.250447\pi\)
0.706114 + 0.708098i \(0.250447\pi\)
\(522\) − 2765.63i − 0.231893i
\(523\) − 21009.4i − 1.75655i −0.478157 0.878275i \(-0.658695\pi\)
0.478157 0.878275i \(-0.341305\pi\)
\(524\) −4466.01 −0.372326
\(525\) 0 0
\(526\) −33577.0 −2.78332
\(527\) − 17145.6i − 1.41722i
\(528\) − 746.154i − 0.0615003i
\(529\) −1345.73 −0.110605
\(530\) 0 0
\(531\) 3657.99 0.298951
\(532\) − 12786.7i − 1.04206i
\(533\) − 1658.97i − 0.134818i
\(534\) 9723.46 0.787969
\(535\) 0 0
\(536\) −1340.66 −0.108036
\(537\) − 6396.07i − 0.513987i
\(538\) − 2115.66i − 0.169540i
\(539\) 7279.20 0.581702
\(540\) 0 0
\(541\) −16802.8 −1.33532 −0.667662 0.744464i \(-0.732705\pi\)
−0.667662 + 0.744464i \(0.732705\pi\)
\(542\) − 541.211i − 0.0428911i
\(543\) 1768.11i 0.139737i
\(544\) −27631.9 −2.17777
\(545\) 0 0
\(546\) 2167.25 0.169872
\(547\) − 16784.5i − 1.31198i −0.754770 0.655990i \(-0.772251\pi\)
0.754770 0.655990i \(-0.227749\pi\)
\(548\) 10242.5i 0.798429i
\(549\) 5007.88 0.389309
\(550\) 0 0
\(551\) −2420.37 −0.187135
\(552\) − 5516.91i − 0.425390i
\(553\) − 12144.0i − 0.933840i
\(554\) 36277.7 2.78212
\(555\) 0 0
\(556\) −12639.9 −0.964117
\(557\) − 18127.0i − 1.37893i −0.724317 0.689467i \(-0.757845\pi\)
0.724317 0.689467i \(-0.242155\pi\)
\(558\) − 5598.85i − 0.424764i
\(559\) −1653.70 −0.125123
\(560\) 0 0
\(561\) 4024.09 0.302847
\(562\) 30722.3i 2.30595i
\(563\) 2090.88i 0.156518i 0.996933 + 0.0782592i \(0.0249362\pi\)
−0.996933 + 0.0782592i \(0.975064\pi\)
\(564\) −8036.03 −0.599961
\(565\) 0 0
\(566\) −4579.89 −0.340119
\(567\) − 2567.51i − 0.190168i
\(568\) − 776.452i − 0.0573578i
\(569\) −6249.23 −0.460424 −0.230212 0.973140i \(-0.573942\pi\)
−0.230212 + 0.973140i \(0.573942\pi\)
\(570\) 0 0
\(571\) 6048.79 0.443317 0.221659 0.975124i \(-0.428853\pi\)
0.221659 + 0.975124i \(0.428853\pi\)
\(572\) 655.902i 0.0479451i
\(573\) 6482.69i 0.472633i
\(574\) −45166.8 −3.28437
\(575\) 0 0
\(576\) −7395.15 −0.534950
\(577\) 15729.1i 1.13486i 0.823423 + 0.567429i \(0.192062\pi\)
−0.823423 + 0.567429i \(0.807938\pi\)
\(578\) − 44053.4i − 3.17021i
\(579\) −4472.73 −0.321037
\(580\) 0 0
\(581\) −29485.6 −2.10545
\(582\) 15549.5i 1.10747i
\(583\) − 54.1108i − 0.00384398i
\(584\) 12432.6 0.880931
\(585\) 0 0
\(586\) 27187.3 1.91655
\(587\) − 15620.5i − 1.09835i −0.835709 0.549173i \(-0.814943\pi\)
0.835709 0.549173i \(-0.185057\pi\)
\(588\) − 22980.2i − 1.61172i
\(589\) −4899.91 −0.342780
\(590\) 0 0
\(591\) 691.587 0.0481355
\(592\) − 9498.09i − 0.659407i
\(593\) − 493.541i − 0.0341776i −0.999854 0.0170888i \(-0.994560\pi\)
0.999854 0.0170888i \(-0.00543980\pi\)
\(594\) 1314.06 0.0907683
\(595\) 0 0
\(596\) 3438.11 0.236293
\(597\) 67.2022i 0.00460704i
\(598\) − 2649.31i − 0.181168i
\(599\) 12455.1 0.849585 0.424793 0.905291i \(-0.360347\pi\)
0.424793 + 0.905291i \(0.360347\pi\)
\(600\) 0 0
\(601\) 12454.8 0.845329 0.422664 0.906286i \(-0.361095\pi\)
0.422664 + 0.906286i \(0.361095\pi\)
\(602\) 45023.3i 3.04820i
\(603\) 762.707i 0.0515088i
\(604\) 21853.1 1.47217
\(605\) 0 0
\(606\) −16210.3 −1.08663
\(607\) 4243.19i 0.283733i 0.989886 + 0.141867i \(0.0453104\pi\)
−0.989886 + 0.141867i \(0.954690\pi\)
\(608\) 7896.70i 0.526732i
\(609\) −6604.54 −0.439458
\(610\) 0 0
\(611\) −1192.01 −0.0789259
\(612\) − 12703.9i − 0.839095i
\(613\) 5733.14i 0.377748i 0.982001 + 0.188874i \(0.0604838\pi\)
−0.982001 + 0.188874i \(0.939516\pi\)
\(614\) −9672.18 −0.635729
\(615\) 0 0
\(616\) 5515.99 0.360788
\(617\) − 15642.1i − 1.02063i −0.859988 0.510314i \(-0.829529\pi\)
0.859988 0.510314i \(-0.170471\pi\)
\(618\) 6858.91i 0.446449i
\(619\) 7467.40 0.484879 0.242440 0.970167i \(-0.422052\pi\)
0.242440 + 0.970167i \(0.422052\pi\)
\(620\) 0 0
\(621\) −3138.60 −0.202814
\(622\) − 33116.1i − 2.13478i
\(623\) − 23220.4i − 1.49327i
\(624\) −349.413 −0.0224162
\(625\) 0 0
\(626\) 30233.6 1.93031
\(627\) − 1150.01i − 0.0732489i
\(628\) − 654.416i − 0.0415829i
\(629\) 51224.2 3.24713
\(630\) 0 0
\(631\) −1486.38 −0.0937745 −0.0468872 0.998900i \(-0.514930\pi\)
−0.0468872 + 0.998900i \(0.514930\pi\)
\(632\) − 6060.87i − 0.381469i
\(633\) 3154.91i 0.198099i
\(634\) 4089.35 0.256165
\(635\) 0 0
\(636\) −170.826 −0.0106505
\(637\) − 3408.74i − 0.212024i
\(638\) − 3380.21i − 0.209755i
\(639\) −441.728 −0.0273466
\(640\) 0 0
\(641\) 12386.0 0.763211 0.381606 0.924325i \(-0.375371\pi\)
0.381606 + 0.924325i \(0.375371\pi\)
\(642\) 2017.87i 0.124048i
\(643\) − 14458.1i − 0.886737i −0.896339 0.443369i \(-0.853783\pi\)
0.896339 0.443369i \(-0.146217\pi\)
\(644\) −42652.3 −2.60984
\(645\) 0 0
\(646\) −18801.8 −1.14512
\(647\) − 15792.8i − 0.959625i −0.877371 0.479813i \(-0.840705\pi\)
0.877371 0.479813i \(-0.159295\pi\)
\(648\) − 1281.41i − 0.0776828i
\(649\) 4470.87 0.270412
\(650\) 0 0
\(651\) −13370.5 −0.804965
\(652\) 569.909i 0.0342321i
\(653\) − 3179.93i − 0.190567i −0.995450 0.0952837i \(-0.969624\pi\)
0.995450 0.0952837i \(-0.0303758\pi\)
\(654\) −28807.3 −1.72240
\(655\) 0 0
\(656\) 7281.96 0.433404
\(657\) − 7072.96i − 0.420003i
\(658\) 32453.6i 1.92276i
\(659\) −11593.5 −0.685308 −0.342654 0.939462i \(-0.611326\pi\)
−0.342654 + 0.939462i \(0.611326\pi\)
\(660\) 0 0
\(661\) 3233.88 0.190293 0.0951464 0.995463i \(-0.469668\pi\)
0.0951464 + 0.995463i \(0.469668\pi\)
\(662\) − 43449.9i − 2.55095i
\(663\) − 1884.42i − 0.110384i
\(664\) −14715.8 −0.860066
\(665\) 0 0
\(666\) 16727.1 0.973219
\(667\) 8073.56i 0.468680i
\(668\) 23946.9i 1.38703i
\(669\) 11585.4 0.669532
\(670\) 0 0
\(671\) 6120.74 0.352144
\(672\) 21548.0i 1.23695i
\(673\) − 5495.72i − 0.314776i −0.987537 0.157388i \(-0.949693\pi\)
0.987537 0.157388i \(-0.0503074\pi\)
\(674\) 2658.23 0.151916
\(675\) 0 0
\(676\) −25124.4 −1.42947
\(677\) − 33836.7i − 1.92090i −0.278448 0.960451i \(-0.589820\pi\)
0.278448 0.960451i \(-0.410180\pi\)
\(678\) − 8579.82i − 0.485997i
\(679\) 37133.6 2.09876
\(680\) 0 0
\(681\) 2618.16 0.147325
\(682\) − 6843.04i − 0.384214i
\(683\) − 21080.3i − 1.18099i −0.807043 0.590493i \(-0.798933\pi\)
0.807043 0.590493i \(-0.201067\pi\)
\(684\) −3630.56 −0.202950
\(685\) 0 0
\(686\) −44702.2 −2.48796
\(687\) 5525.16i 0.306838i
\(688\) − 7258.84i − 0.402239i
\(689\) −25.3393 −0.00140109
\(690\) 0 0
\(691\) 11811.3 0.650253 0.325127 0.945671i \(-0.394593\pi\)
0.325127 + 0.945671i \(0.394593\pi\)
\(692\) 6991.79i 0.384087i
\(693\) − 3138.07i − 0.172014i
\(694\) −13907.8 −0.760711
\(695\) 0 0
\(696\) −3296.23 −0.179516
\(697\) 39272.4i 2.13422i
\(698\) − 3188.52i − 0.172904i
\(699\) 11796.4 0.638313
\(700\) 0 0
\(701\) 4244.99 0.228718 0.114359 0.993440i \(-0.463519\pi\)
0.114359 + 0.993440i \(0.463519\pi\)
\(702\) − 615.353i − 0.0330841i
\(703\) − 14639.0i − 0.785376i
\(704\) −9038.51 −0.483880
\(705\) 0 0
\(706\) −5340.81 −0.284708
\(707\) 38711.5i 2.05926i
\(708\) − 14114.4i − 0.749227i
\(709\) 898.822 0.0476107 0.0238053 0.999717i \(-0.492422\pi\)
0.0238053 + 0.999717i \(0.492422\pi\)
\(710\) 0 0
\(711\) −3448.06 −0.181874
\(712\) − 11589.0i − 0.609993i
\(713\) 16344.5i 0.858493i
\(714\) −51305.0 −2.68913
\(715\) 0 0
\(716\) −24679.4 −1.28815
\(717\) 14316.3i 0.745679i
\(718\) − 38706.3i − 2.01185i
\(719\) −10741.8 −0.557165 −0.278582 0.960412i \(-0.589865\pi\)
−0.278582 + 0.960412i \(0.589865\pi\)
\(720\) 0 0
\(721\) 16379.6 0.846061
\(722\) − 24973.9i − 1.28730i
\(723\) − 11966.5i − 0.615546i
\(724\) 6822.30 0.350206
\(725\) 0 0
\(726\) 1606.07 0.0821030
\(727\) − 16794.2i − 0.856758i −0.903599 0.428379i \(-0.859085\pi\)
0.903599 0.428379i \(-0.140915\pi\)
\(728\) − 2583.06i − 0.131503i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 39147.7 1.98075
\(732\) − 19323.0i − 0.975681i
\(733\) 8659.40i 0.436347i 0.975910 + 0.218173i \(0.0700098\pi\)
−0.975910 + 0.218173i \(0.929990\pi\)
\(734\) −29778.4 −1.49747
\(735\) 0 0
\(736\) 26340.8 1.31920
\(737\) 932.197i 0.0465915i
\(738\) 12824.3i 0.639660i
\(739\) −16705.7 −0.831567 −0.415783 0.909464i \(-0.636493\pi\)
−0.415783 + 0.909464i \(0.636493\pi\)
\(740\) 0 0
\(741\) −538.534 −0.0266984
\(742\) 689.884i 0.0341327i
\(743\) 1292.12i 0.0637996i 0.999491 + 0.0318998i \(0.0101558\pi\)
−0.999491 + 0.0318998i \(0.989844\pi\)
\(744\) −6673.03 −0.328824
\(745\) 0 0
\(746\) 1006.09 0.0493773
\(747\) 8371.90i 0.410056i
\(748\) − 15527.0i − 0.758990i
\(749\) 4818.83 0.235082
\(750\) 0 0
\(751\) −14980.4 −0.727886 −0.363943 0.931421i \(-0.618570\pi\)
−0.363943 + 0.931421i \(0.618570\pi\)
\(752\) − 5232.30i − 0.253726i
\(753\) 16422.7i 0.794787i
\(754\) −1582.90 −0.0764535
\(755\) 0 0
\(756\) −9906.82 −0.476597
\(757\) − 3003.41i − 0.144202i −0.997397 0.0721010i \(-0.977030\pi\)
0.997397 0.0721010i \(-0.0229704\pi\)
\(758\) − 50244.8i − 2.40762i
\(759\) −3836.06 −0.183452
\(760\) 0 0
\(761\) −20375.0 −0.970555 −0.485277 0.874360i \(-0.661281\pi\)
−0.485277 + 0.874360i \(0.661281\pi\)
\(762\) 13184.4i 0.626799i
\(763\) 68794.1i 3.26411i
\(764\) 25013.6 1.18450
\(765\) 0 0
\(766\) −47578.4 −2.24422
\(767\) − 2093.65i − 0.0985621i
\(768\) 4472.70i 0.210149i
\(769\) 12372.4 0.580184 0.290092 0.956999i \(-0.406314\pi\)
0.290092 + 0.956999i \(0.406314\pi\)
\(770\) 0 0
\(771\) 19302.0 0.901615
\(772\) 17258.1i 0.804578i
\(773\) 21023.6i 0.978225i 0.872221 + 0.489113i \(0.162679\pi\)
−0.872221 + 0.489113i \(0.837321\pi\)
\(774\) 12783.6 0.593664
\(775\) 0 0
\(776\) 18532.8 0.857331
\(777\) − 39945.8i − 1.84433i
\(778\) 51885.7i 2.39099i
\(779\) 11223.4 0.516198
\(780\) 0 0
\(781\) −539.889 −0.0247359
\(782\) 62716.6i 2.86795i
\(783\) 1875.24i 0.0855884i
\(784\) 14962.5 0.681602
\(785\) 0 0
\(786\) 5121.02 0.232393
\(787\) − 30286.2i − 1.37177i −0.727709 0.685886i \(-0.759415\pi\)
0.727709 0.685886i \(-0.240585\pi\)
\(788\) − 2668.51i − 0.120637i
\(789\) 22767.0 1.02728
\(790\) 0 0
\(791\) −20489.3 −0.921007
\(792\) − 1566.17i − 0.0702668i
\(793\) − 2866.25i − 0.128353i
\(794\) −1592.37 −0.0711729
\(795\) 0 0
\(796\) 259.301 0.0115461
\(797\) − 32337.8i − 1.43722i −0.695413 0.718610i \(-0.744779\pi\)
0.695413 0.718610i \(-0.255221\pi\)
\(798\) 14662.1i 0.650415i
\(799\) 28218.3 1.24943
\(800\) 0 0
\(801\) −6593.03 −0.290828
\(802\) − 17992.9i − 0.792207i
\(803\) − 8644.72i − 0.379907i
\(804\) 2942.92 0.129091
\(805\) 0 0
\(806\) −3204.50 −0.140042
\(807\) 1434.53i 0.0625749i
\(808\) 19320.3i 0.841197i
\(809\) 891.707 0.0387525 0.0193762 0.999812i \(-0.493832\pi\)
0.0193762 + 0.999812i \(0.493832\pi\)
\(810\) 0 0
\(811\) −10114.9 −0.437957 −0.218978 0.975730i \(-0.570272\pi\)
−0.218978 + 0.975730i \(0.570272\pi\)
\(812\) 25483.8i 1.10136i
\(813\) 366.970i 0.0158305i
\(814\) 20444.3 0.880309
\(815\) 0 0
\(816\) 8271.59 0.354857
\(817\) − 11187.7i − 0.479080i
\(818\) 59680.4i 2.55095i
\(819\) −1469.51 −0.0626972
\(820\) 0 0
\(821\) 10833.5 0.460525 0.230262 0.973129i \(-0.426042\pi\)
0.230262 + 0.973129i \(0.426042\pi\)
\(822\) − 11744.7i − 0.498351i
\(823\) 31958.5i 1.35359i 0.736173 + 0.676794i \(0.236631\pi\)
−0.736173 + 0.676794i \(0.763369\pi\)
\(824\) 8174.84 0.345612
\(825\) 0 0
\(826\) −57001.3 −2.40112
\(827\) − 34847.3i − 1.46525i −0.680634 0.732624i \(-0.738296\pi\)
0.680634 0.732624i \(-0.261704\pi\)
\(828\) 12110.3i 0.508289i
\(829\) −6537.91 −0.273910 −0.136955 0.990577i \(-0.543732\pi\)
−0.136955 + 0.990577i \(0.543732\pi\)
\(830\) 0 0
\(831\) −24598.2 −1.02684
\(832\) 4232.60i 0.176369i
\(833\) 80694.5i 3.35642i
\(834\) 14493.7 0.601768
\(835\) 0 0
\(836\) −4437.35 −0.183575
\(837\) 3796.32i 0.156774i
\(838\) 31148.5i 1.28402i
\(839\) −2710.34 −0.111527 −0.0557635 0.998444i \(-0.517759\pi\)
−0.0557635 + 0.998444i \(0.517759\pi\)
\(840\) 0 0
\(841\) −19565.2 −0.802215
\(842\) 40579.7i 1.66089i
\(843\) − 20831.4i − 0.851092i
\(844\) 12173.3 0.496471
\(845\) 0 0
\(846\) 9214.62 0.374474
\(847\) − 3835.42i − 0.155592i
\(848\) − 111.226i − 0.00450414i
\(849\) 3105.41 0.125533
\(850\) 0 0
\(851\) −48830.8 −1.96698
\(852\) 1704.42i 0.0685356i
\(853\) 9759.32i 0.391738i 0.980630 + 0.195869i \(0.0627528\pi\)
−0.980630 + 0.195869i \(0.937247\pi\)
\(854\) −78036.2 −3.12687
\(855\) 0 0
\(856\) 2405.01 0.0960298
\(857\) 13649.8i 0.544072i 0.962287 + 0.272036i \(0.0876970\pi\)
−0.962287 + 0.272036i \(0.912303\pi\)
\(858\) − 752.098i − 0.0299257i
\(859\) −7796.42 −0.309674 −0.154837 0.987940i \(-0.549485\pi\)
−0.154837 + 0.987940i \(0.549485\pi\)
\(860\) 0 0
\(861\) 30625.5 1.21221
\(862\) 4393.43i 0.173597i
\(863\) 7183.57i 0.283350i 0.989913 + 0.141675i \(0.0452489\pi\)
−0.989913 + 0.141675i \(0.954751\pi\)
\(864\) 6118.16 0.240907
\(865\) 0 0
\(866\) −16769.5 −0.658026
\(867\) 29870.6i 1.17008i
\(868\) 51590.5i 2.01739i
\(869\) −4214.30 −0.164511
\(870\) 0 0
\(871\) 436.534 0.0169821
\(872\) 34334.1i 1.33337i
\(873\) − 10543.4i − 0.408752i
\(874\) 17923.3 0.693666
\(875\) 0 0
\(876\) −27291.2 −1.05261
\(877\) − 17063.1i − 0.656991i −0.944506 0.328495i \(-0.893458\pi\)
0.944506 0.328495i \(-0.106542\pi\)
\(878\) 22728.2i 0.873620i
\(879\) −18434.4 −0.707369
\(880\) 0 0
\(881\) −32174.9 −1.23042 −0.615210 0.788363i \(-0.710929\pi\)
−0.615210 + 0.788363i \(0.710929\pi\)
\(882\) 26350.6i 1.00598i
\(883\) 2843.68i 0.108378i 0.998531 + 0.0541889i \(0.0172573\pi\)
−0.998531 + 0.0541889i \(0.982743\pi\)
\(884\) −7271.09 −0.276644
\(885\) 0 0
\(886\) −47238.9 −1.79122
\(887\) 31417.8i 1.18930i 0.803985 + 0.594649i \(0.202709\pi\)
−0.803985 + 0.594649i \(0.797291\pi\)
\(888\) − 19936.4i − 0.753401i
\(889\) 31485.5 1.18784
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) − 44702.5i − 1.67797i
\(893\) − 8064.30i − 0.302196i
\(894\) −3942.35 −0.147485
\(895\) 0 0
\(896\) 57775.1 2.15416
\(897\) 1796.37i 0.0668664i
\(898\) − 46588.6i − 1.73127i
\(899\) 9765.47 0.362288
\(900\) 0 0
\(901\) 599.852 0.0221798
\(902\) 15674.1i 0.578594i
\(903\) − 30528.2i − 1.12505i
\(904\) −10225.9 −0.376227
\(905\) 0 0
\(906\) −25058.1 −0.918875
\(907\) − 12253.1i − 0.448573i −0.974523 0.224287i \(-0.927995\pi\)
0.974523 0.224287i \(-0.0720052\pi\)
\(908\) − 10102.2i − 0.369223i
\(909\) 10991.4 0.401059
\(910\) 0 0
\(911\) −48422.4 −1.76104 −0.880518 0.474012i \(-0.842805\pi\)
−0.880518 + 0.474012i \(0.842805\pi\)
\(912\) − 2363.87i − 0.0858286i
\(913\) 10232.3i 0.370909i
\(914\) −62262.9 −2.25326
\(915\) 0 0
\(916\) 21318.9 0.768993
\(917\) − 12229.4i − 0.440404i
\(918\) 14567.1i 0.523733i
\(919\) −5546.18 −0.199077 −0.0995385 0.995034i \(-0.531737\pi\)
−0.0995385 + 0.995034i \(0.531737\pi\)
\(920\) 0 0
\(921\) 6558.26 0.234638
\(922\) − 136.349i − 0.00487030i
\(923\) 252.822i 0.00901598i
\(924\) −12108.3 −0.431098
\(925\) 0 0
\(926\) −77831.3 −2.76209
\(927\) − 4650.71i − 0.164778i
\(928\) − 15738.0i − 0.556709i
\(929\) 35684.5 1.26025 0.630125 0.776494i \(-0.283004\pi\)
0.630125 + 0.776494i \(0.283004\pi\)
\(930\) 0 0
\(931\) 23061.1 0.811811
\(932\) − 45516.7i − 1.59973i
\(933\) 22454.5i 0.787918i
\(934\) 58726.0 2.05736
\(935\) 0 0
\(936\) −733.413 −0.0256115
\(937\) 48903.6i 1.70503i 0.522705 + 0.852514i \(0.324923\pi\)
−0.522705 + 0.852514i \(0.675077\pi\)
\(938\) − 11885.0i − 0.413710i
\(939\) −20500.0 −0.712451
\(940\) 0 0
\(941\) −23741.9 −0.822490 −0.411245 0.911525i \(-0.634906\pi\)
−0.411245 + 0.911525i \(0.634906\pi\)
\(942\) 750.396i 0.0259546i
\(943\) − 37437.4i − 1.29282i
\(944\) 9189.97 0.316852
\(945\) 0 0
\(946\) 15624.4 0.536989
\(947\) − 37612.4i − 1.29064i −0.763911 0.645321i \(-0.776724\pi\)
0.763911 0.645321i \(-0.223276\pi\)
\(948\) 13304.4i 0.455810i
\(949\) −4048.20 −0.138472
\(950\) 0 0
\(951\) −2772.80 −0.0945469
\(952\) 61148.2i 2.08175i
\(953\) − 48294.3i − 1.64156i −0.571246 0.820779i \(-0.693540\pi\)
0.571246 0.820779i \(-0.306460\pi\)
\(954\) 195.880 0.00664765
\(955\) 0 0
\(956\) 55239.8 1.86881
\(957\) 2291.96i 0.0774176i
\(958\) − 11046.4i − 0.372539i
\(959\) −28047.4 −0.944419
\(960\) 0 0
\(961\) −10021.4 −0.336389
\(962\) − 9573.76i − 0.320863i
\(963\) − 1368.22i − 0.0457843i
\(964\) −46173.1 −1.54267
\(965\) 0 0
\(966\) 48907.8 1.62897
\(967\) − 1840.92i − 0.0612204i −0.999531 0.0306102i \(-0.990255\pi\)
0.999531 0.0306102i \(-0.00974505\pi\)
\(968\) − 1914.20i − 0.0635587i
\(969\) 12748.6 0.422647
\(970\) 0 0
\(971\) 31461.8 1.03981 0.519906 0.854223i \(-0.325967\pi\)
0.519906 + 0.854223i \(0.325967\pi\)
\(972\) 2812.86i 0.0928217i
\(973\) − 34612.1i − 1.14040i
\(974\) −15328.1 −0.504254
\(975\) 0 0
\(976\) 12581.3 0.412620
\(977\) 7040.11i 0.230535i 0.993334 + 0.115268i \(0.0367726\pi\)
−0.993334 + 0.115268i \(0.963227\pi\)
\(978\) − 653.494i − 0.0213665i
\(979\) −8058.15 −0.263064
\(980\) 0 0
\(981\) 19532.9 0.635715
\(982\) − 71784.4i − 2.33272i
\(983\) − 24610.9i − 0.798541i −0.916833 0.399270i \(-0.869263\pi\)
0.916833 0.399270i \(-0.130737\pi\)
\(984\) 15284.7 0.495183
\(985\) 0 0
\(986\) 37471.8 1.21029
\(987\) − 22005.3i − 0.709662i
\(988\) 2077.95i 0.0669112i
\(989\) −37318.5 −1.19986
\(990\) 0 0
\(991\) −40003.3 −1.28229 −0.641144 0.767421i \(-0.721540\pi\)
−0.641144 + 0.767421i \(0.721540\pi\)
\(992\) − 31860.8i − 1.01974i
\(993\) 29461.4i 0.941519i
\(994\) 6883.31 0.219643
\(995\) 0 0
\(996\) 32303.2 1.02768
\(997\) 7342.61i 0.233242i 0.993176 + 0.116621i \(0.0372063\pi\)
−0.993176 + 0.116621i \(0.962794\pi\)
\(998\) − 44216.9i − 1.40247i
\(999\) −11341.9 −0.359201
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.h.199.3 4
5.2 odd 4 33.4.a.c.1.1 2
5.3 odd 4 825.4.a.l.1.2 2
5.4 even 2 inner 825.4.c.h.199.2 4
15.2 even 4 99.4.a.f.1.2 2
15.8 even 4 2475.4.a.p.1.1 2
20.7 even 4 528.4.a.p.1.2 2
35.27 even 4 1617.4.a.k.1.1 2
40.27 even 4 2112.4.a.bg.1.1 2
40.37 odd 4 2112.4.a.bn.1.1 2
55.32 even 4 363.4.a.i.1.2 2
60.47 odd 4 1584.4.a.bj.1.1 2
165.32 odd 4 1089.4.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 5.2 odd 4
99.4.a.f.1.2 2 15.2 even 4
363.4.a.i.1.2 2 55.32 even 4
528.4.a.p.1.2 2 20.7 even 4
825.4.a.l.1.2 2 5.3 odd 4
825.4.c.h.199.2 4 5.4 even 2 inner
825.4.c.h.199.3 4 1.1 even 1 trivial
1089.4.a.u.1.1 2 165.32 odd 4
1584.4.a.bj.1.1 2 60.47 odd 4
1617.4.a.k.1.1 2 35.27 even 4
2112.4.a.bg.1.1 2 40.27 even 4
2112.4.a.bn.1.1 2 40.37 odd 4
2475.4.a.p.1.1 2 15.8 even 4