Properties

Label 825.4.a.l.1.2
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $1$
Dimension $2$
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] = [825,4,Mod(1,825)]
 
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, 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("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.42443\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.42443 q^{2} +3.00000 q^{3} +11.5756 q^{4} +13.2733 q^{6} -31.6977 q^{7} +15.8199 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.42443 q^{2} +3.00000 q^{3} +11.5756 q^{4} +13.2733 q^{6} -31.6977 q^{7} +15.8199 q^{8} +9.00000 q^{9} -11.0000 q^{11} +34.7267 q^{12} -5.15114 q^{13} -140.244 q^{14} -22.6107 q^{16} -121.942 q^{17} +39.8199 q^{18} +34.8489 q^{19} -95.0931 q^{21} -48.6687 q^{22} -116.244 q^{23} +47.4596 q^{24} -22.7909 q^{26} +27.0000 q^{27} -366.919 q^{28} -69.4534 q^{29} +140.605 q^{31} -226.598 q^{32} -33.0000 q^{33} -539.524 q^{34} +104.180 q^{36} +420.070 q^{37} +154.186 q^{38} -15.4534 q^{39} -322.058 q^{41} -420.733 q^{42} -321.035 q^{43} -127.331 q^{44} -514.315 q^{46} +231.408 q^{47} -67.8322 q^{48} +661.745 q^{49} -365.826 q^{51} -59.6274 q^{52} -4.91916 q^{53} +119.460 q^{54} -501.453 q^{56} +104.547 q^{57} -307.292 q^{58} +406.443 q^{59} -556.431 q^{61} +622.095 q^{62} -285.279 q^{63} -821.683 q^{64} -146.006 q^{66} -84.7452 q^{67} -1411.55 q^{68} -348.733 q^{69} +49.0808 q^{71} +142.379 q^{72} -785.884 q^{73} +1858.57 q^{74} +403.395 q^{76} +348.675 q^{77} -68.3726 q^{78} -383.118 q^{79} +81.0000 q^{81} -1424.92 q^{82} +930.211 q^{83} -1100.76 q^{84} -1420.40 q^{86} -208.360 q^{87} -174.018 q^{88} -732.559 q^{89} +163.279 q^{91} -1345.59 q^{92} +421.814 q^{93} +1023.85 q^{94} -679.795 q^{96} +1171.49 q^{97} +2927.84 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 6 q^{3} + 33 q^{4} - 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 6 q^{3} + 33 q^{4} - 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9} - 22 q^{11} + 99 q^{12} - 30 q^{13} - 182 q^{14} + 201 q^{16} - 106 q^{17} - 9 q^{18} + 50 q^{19} - 72 q^{21} + 11 q^{22} - 134 q^{23} - 171 q^{24} + 112 q^{26} + 54 q^{27} - 202 q^{28} - 198 q^{29} + 360 q^{31} - 857 q^{32} - 66 q^{33} - 626 q^{34} + 297 q^{36} + 328 q^{37} + 72 q^{38} - 90 q^{39} - 782 q^{41} - 546 q^{42} - 386 q^{43} - 363 q^{44} - 418 q^{46} - 266 q^{47} + 603 q^{48} + 378 q^{49} - 318 q^{51} - 592 q^{52} + 522 q^{53} - 27 q^{54} - 1062 q^{56} + 150 q^{57} + 390 q^{58} - 172 q^{59} - 778 q^{61} - 568 q^{62} - 216 q^{63} + 809 q^{64} + 33 q^{66} + 776 q^{67} - 1070 q^{68} - 402 q^{69} + 630 q^{71} - 513 q^{72} - 1296 q^{73} + 2358 q^{74} + 728 q^{76} + 264 q^{77} + 336 q^{78} + 652 q^{79} + 162 q^{81} + 1070 q^{82} + 324 q^{83} - 606 q^{84} - 1068 q^{86} - 594 q^{87} + 627 q^{88} - 756 q^{89} - 28 q^{91} - 1726 q^{92} + 1080 q^{93} + 3722 q^{94} - 2571 q^{96} + 452 q^{97} + 4467 q^{98} - 198 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.42443 1.56427 0.782136 0.623108i \(-0.214130\pi\)
0.782136 + 0.623108i \(0.214130\pi\)
\(3\) 3.00000 0.577350
\(4\) 11.5756 1.44695
\(5\) 0 0
\(6\) 13.2733 0.903133
\(7\) −31.6977 −1.71152 −0.855758 0.517377i \(-0.826909\pi\)
−0.855758 + 0.517377i \(0.826909\pi\)
\(8\) 15.8199 0.699146
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 34.7267 0.835395
\(13\) −5.15114 −0.109898 −0.0549488 0.998489i \(-0.517500\pi\)
−0.0549488 + 0.998489i \(0.517500\pi\)
\(14\) −140.244 −2.67728
\(15\) 0 0
\(16\) −22.6107 −0.353293
\(17\) −121.942 −1.73972 −0.869861 0.493297i \(-0.835792\pi\)
−0.869861 + 0.493297i \(0.835792\pi\)
\(18\) 39.8199 0.521424
\(19\) 34.8489 0.420783 0.210391 0.977617i \(-0.432526\pi\)
0.210391 + 0.977617i \(0.432526\pi\)
\(20\) 0 0
\(21\) −95.0931 −0.988144
\(22\) −48.6687 −0.471646
\(23\) −116.244 −1.05385 −0.526926 0.849911i \(-0.676656\pi\)
−0.526926 + 0.849911i \(0.676656\pi\)
\(24\) 47.4596 0.403652
\(25\) 0 0
\(26\) −22.7909 −0.171910
\(27\) 27.0000 0.192450
\(28\) −366.919 −2.47647
\(29\) −69.4534 −0.444730 −0.222365 0.974963i \(-0.571378\pi\)
−0.222365 + 0.974963i \(0.571378\pi\)
\(30\) 0 0
\(31\) 140.605 0.814623 0.407312 0.913289i \(-0.366466\pi\)
0.407312 + 0.913289i \(0.366466\pi\)
\(32\) −226.598 −1.25179
\(33\) −33.0000 −0.174078
\(34\) −539.524 −2.72140
\(35\) 0 0
\(36\) 104.180 0.482315
\(37\) 420.070 1.86646 0.933232 0.359276i \(-0.116976\pi\)
0.933232 + 0.359276i \(0.116976\pi\)
\(38\) 154.186 0.658219
\(39\) −15.4534 −0.0634495
\(40\) 0 0
\(41\) −322.058 −1.22676 −0.613378 0.789789i \(-0.710190\pi\)
−0.613378 + 0.789789i \(0.710190\pi\)
\(42\) −420.733 −1.54573
\(43\) −321.035 −1.13854 −0.569272 0.822149i \(-0.692775\pi\)
−0.569272 + 0.822149i \(0.692775\pi\)
\(44\) −127.331 −0.436271
\(45\) 0 0
\(46\) −514.315 −1.64851
\(47\) 231.408 0.718176 0.359088 0.933304i \(-0.383088\pi\)
0.359088 + 0.933304i \(0.383088\pi\)
\(48\) −67.8322 −0.203974
\(49\) 661.745 1.92929
\(50\) 0 0
\(51\) −365.826 −1.00443
\(52\) −59.6274 −0.159016
\(53\) −4.91916 −0.0127490 −0.00637452 0.999980i \(-0.502029\pi\)
−0.00637452 + 0.999980i \(0.502029\pi\)
\(54\) 119.460 0.301044
\(55\) 0 0
\(56\) −501.453 −1.19660
\(57\) 104.547 0.242939
\(58\) −307.292 −0.695679
\(59\) 406.443 0.896854 0.448427 0.893820i \(-0.351984\pi\)
0.448427 + 0.893820i \(0.351984\pi\)
\(60\) 0 0
\(61\) −556.431 −1.16793 −0.583964 0.811779i \(-0.698499\pi\)
−0.583964 + 0.811779i \(0.698499\pi\)
\(62\) 622.095 1.27429
\(63\) −285.279 −0.570505
\(64\) −821.683 −1.60485
\(65\) 0 0
\(66\) −146.006 −0.272305
\(67\) −84.7452 −0.154526 −0.0772632 0.997011i \(-0.524618\pi\)
−0.0772632 + 0.997011i \(0.524618\pi\)
\(68\) −1411.55 −2.51728
\(69\) −348.733 −0.608442
\(70\) 0 0
\(71\) 49.0808 0.0820398 0.0410199 0.999158i \(-0.486939\pi\)
0.0410199 + 0.999158i \(0.486939\pi\)
\(72\) 142.379 0.233049
\(73\) −785.884 −1.26001 −0.630005 0.776591i \(-0.716947\pi\)
−0.630005 + 0.776591i \(0.716947\pi\)
\(74\) 1858.57 2.91966
\(75\) 0 0
\(76\) 403.395 0.608850
\(77\) 348.675 0.516041
\(78\) −68.3726 −0.0992522
\(79\) −383.118 −0.545622 −0.272811 0.962068i \(-0.587953\pi\)
−0.272811 + 0.962068i \(0.587953\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −1424.92 −1.91898
\(83\) 930.211 1.23017 0.615084 0.788462i \(-0.289122\pi\)
0.615084 + 0.788462i \(0.289122\pi\)
\(84\) −1100.76 −1.42979
\(85\) 0 0
\(86\) −1420.40 −1.78099
\(87\) −208.360 −0.256765
\(88\) −174.018 −0.210800
\(89\) −732.559 −0.872484 −0.436242 0.899829i \(-0.643691\pi\)
−0.436242 + 0.899829i \(0.643691\pi\)
\(90\) 0 0
\(91\) 163.279 0.188092
\(92\) −1345.59 −1.52487
\(93\) 421.814 0.470323
\(94\) 1023.85 1.12342
\(95\) 0 0
\(96\) −679.795 −0.722722
\(97\) 1171.49 1.22626 0.613128 0.789984i \(-0.289911\pi\)
0.613128 + 0.789984i \(0.289911\pi\)
\(98\) 2927.84 3.01793
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) −1221.27 −1.20318 −0.601589 0.798806i \(-0.705465\pi\)
−0.601589 + 0.798806i \(0.705465\pi\)
\(102\) −1618.57 −1.57120
\(103\) −516.745 −0.494334 −0.247167 0.968973i \(-0.579500\pi\)
−0.247167 + 0.968973i \(0.579500\pi\)
\(104\) −81.4903 −0.0768345
\(105\) 0 0
\(106\) −21.7645 −0.0199430
\(107\) 152.025 0.137353 0.0686765 0.997639i \(-0.478122\pi\)
0.0686765 + 0.997639i \(0.478122\pi\)
\(108\) 312.540 0.278465
\(109\) 2170.32 1.90714 0.953572 0.301164i \(-0.0973752\pi\)
0.953572 + 0.301164i \(0.0973752\pi\)
\(110\) 0 0
\(111\) 1260.21 1.07760
\(112\) 716.708 0.604666
\(113\) 646.397 0.538123 0.269062 0.963123i \(-0.413286\pi\)
0.269062 + 0.963123i \(0.413286\pi\)
\(114\) 462.559 0.380023
\(115\) 0 0
\(116\) −803.963 −0.643501
\(117\) −46.3603 −0.0366326
\(118\) 1798.28 1.40292
\(119\) 3865.28 2.97756
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −2461.89 −1.82696
\(123\) −966.174 −0.708268
\(124\) 1627.58 1.17872
\(125\) 0 0
\(126\) −1262.20 −0.892425
\(127\) 993.304 0.694027 0.347014 0.937860i \(-0.387196\pi\)
0.347014 + 0.937860i \(0.387196\pi\)
\(128\) −1822.69 −1.25863
\(129\) −963.105 −0.657339
\(130\) 0 0
\(131\) 385.814 0.257318 0.128659 0.991689i \(-0.458933\pi\)
0.128659 + 0.991689i \(0.458933\pi\)
\(132\) −381.994 −0.251881
\(133\) −1104.63 −0.720177
\(134\) −374.949 −0.241721
\(135\) 0 0
\(136\) −1929.11 −1.21632
\(137\) −884.840 −0.551803 −0.275901 0.961186i \(-0.588976\pi\)
−0.275901 + 0.961186i \(0.588976\pi\)
\(138\) −1542.94 −0.951769
\(139\) −1091.94 −0.666312 −0.333156 0.942872i \(-0.608114\pi\)
−0.333156 + 0.942872i \(0.608114\pi\)
\(140\) 0 0
\(141\) 694.223 0.414639
\(142\) 217.155 0.128333
\(143\) 56.6626 0.0331354
\(144\) −203.497 −0.117764
\(145\) 0 0
\(146\) −3477.09 −1.97100
\(147\) 1985.24 1.11387
\(148\) 4862.55 2.70067
\(149\) 297.014 0.163304 0.0816522 0.996661i \(-0.473980\pi\)
0.0816522 + 0.996661i \(0.473980\pi\)
\(150\) 0 0
\(151\) −1887.86 −1.01743 −0.508716 0.860935i \(-0.669880\pi\)
−0.508716 + 0.860935i \(0.669880\pi\)
\(152\) 551.304 0.294189
\(153\) −1097.48 −0.579907
\(154\) 1542.69 0.807229
\(155\) 0 0
\(156\) −178.882 −0.0918080
\(157\) 56.5343 0.0287384 0.0143692 0.999897i \(-0.495426\pi\)
0.0143692 + 0.999897i \(0.495426\pi\)
\(158\) −1695.08 −0.853501
\(159\) −14.7575 −0.00736066
\(160\) 0 0
\(161\) 3684.68 1.80369
\(162\) 358.379 0.173808
\(163\) 49.2338 0.0236582 0.0118291 0.999930i \(-0.496235\pi\)
0.0118291 + 0.999930i \(0.496235\pi\)
\(164\) −3728.01 −1.77505
\(165\) 0 0
\(166\) 4115.65 1.92432
\(167\) −2068.75 −0.958589 −0.479294 0.877654i \(-0.659107\pi\)
−0.479294 + 0.877654i \(0.659107\pi\)
\(168\) −1504.36 −0.690857
\(169\) −2170.47 −0.987923
\(170\) 0 0
\(171\) 313.640 0.140261
\(172\) −3716.17 −1.64741
\(173\) 604.012 0.265446 0.132723 0.991153i \(-0.457628\pi\)
0.132723 + 0.991153i \(0.457628\pi\)
\(174\) −921.875 −0.401650
\(175\) 0 0
\(176\) 248.718 0.106522
\(177\) 1219.33 0.517799
\(178\) −3241.15 −1.36480
\(179\) −2132.02 −0.890251 −0.445126 0.895468i \(-0.646841\pi\)
−0.445126 + 0.895468i \(0.646841\pi\)
\(180\) 0 0
\(181\) −589.371 −0.242031 −0.121015 0.992651i \(-0.538615\pi\)
−0.121015 + 0.992651i \(0.538615\pi\)
\(182\) 722.418 0.294226
\(183\) −1669.29 −0.674304
\(184\) −1838.97 −0.736796
\(185\) 0 0
\(186\) 1866.28 0.735713
\(187\) 1341.36 0.524546
\(188\) 2678.68 1.03916
\(189\) −855.838 −0.329381
\(190\) 0 0
\(191\) −2160.90 −0.818624 −0.409312 0.912395i \(-0.634231\pi\)
−0.409312 + 0.912395i \(0.634231\pi\)
\(192\) −2465.05 −0.926560
\(193\) 1490.91 0.556052 0.278026 0.960574i \(-0.410320\pi\)
0.278026 + 0.960574i \(0.410320\pi\)
\(194\) 5183.18 1.91820
\(195\) 0 0
\(196\) 7660.08 2.79157
\(197\) 230.529 0.0833732 0.0416866 0.999131i \(-0.486727\pi\)
0.0416866 + 0.999131i \(0.486727\pi\)
\(198\) −438.018 −0.157215
\(199\) 22.4007 0.00797963 0.00398982 0.999992i \(-0.498730\pi\)
0.00398982 + 0.999992i \(0.498730\pi\)
\(200\) 0 0
\(201\) −254.236 −0.0892159
\(202\) −5403.43 −1.88210
\(203\) 2201.51 0.761163
\(204\) −4234.65 −1.45336
\(205\) 0 0
\(206\) −2286.30 −0.773273
\(207\) −1046.20 −0.351284
\(208\) 116.471 0.0388260
\(209\) −383.337 −0.126871
\(210\) 0 0
\(211\) −1051.64 −0.343117 −0.171558 0.985174i \(-0.554880\pi\)
−0.171558 + 0.985174i \(0.554880\pi\)
\(212\) −56.9421 −0.0184472
\(213\) 147.243 0.0473657
\(214\) 672.622 0.214857
\(215\) 0 0
\(216\) 427.136 0.134551
\(217\) −4456.84 −1.39424
\(218\) 9602.42 2.98329
\(219\) −2357.65 −0.727467
\(220\) 0 0
\(221\) 628.141 0.191191
\(222\) 5575.71 1.68566
\(223\) −3861.80 −1.15966 −0.579832 0.814736i \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(224\) 7182.65 2.14246
\(225\) 0 0
\(226\) 2859.94 0.841771
\(227\) 872.721 0.255174 0.127587 0.991827i \(-0.459277\pi\)
0.127587 + 0.991827i \(0.459277\pi\)
\(228\) 1210.19 0.351520
\(229\) 1841.72 0.531459 0.265730 0.964048i \(-0.414387\pi\)
0.265730 + 0.964048i \(0.414387\pi\)
\(230\) 0 0
\(231\) 1046.02 0.297937
\(232\) −1098.74 −0.310931
\(233\) −3932.14 −1.10559 −0.552796 0.833317i \(-0.686439\pi\)
−0.552796 + 0.833317i \(0.686439\pi\)
\(234\) −205.118 −0.0573033
\(235\) 0 0
\(236\) 4704.81 1.29770
\(237\) −1149.35 −0.315015
\(238\) 17101.7 4.65772
\(239\) 4772.10 1.29155 0.645777 0.763526i \(-0.276534\pi\)
0.645777 + 0.763526i \(0.276534\pi\)
\(240\) 0 0
\(241\) 3988.84 1.06616 0.533078 0.846066i \(-0.321035\pi\)
0.533078 + 0.846066i \(0.321035\pi\)
\(242\) 535.356 0.142207
\(243\) 243.000 0.0641500
\(244\) −6441.00 −1.68993
\(245\) 0 0
\(246\) −4274.77 −1.10792
\(247\) −179.511 −0.0462431
\(248\) 2224.34 0.569540
\(249\) 2790.63 0.710238
\(250\) 0 0
\(251\) −5474.22 −1.37661 −0.688306 0.725421i \(-0.741645\pi\)
−0.688306 + 0.725421i \(0.741645\pi\)
\(252\) −3302.27 −0.825491
\(253\) 1278.69 0.317749
\(254\) 4394.80 1.08565
\(255\) 0 0
\(256\) −1490.90 −0.363989
\(257\) 6434.01 1.56164 0.780822 0.624754i \(-0.214801\pi\)
0.780822 + 0.624754i \(0.214801\pi\)
\(258\) −4261.19 −1.02826
\(259\) −13315.3 −3.19448
\(260\) 0 0
\(261\) −625.081 −0.148243
\(262\) 1707.01 0.402516
\(263\) −7589.00 −1.77931 −0.889654 0.456636i \(-0.849054\pi\)
−0.889654 + 0.456636i \(0.849054\pi\)
\(264\) −522.055 −0.121706
\(265\) 0 0
\(266\) −4887.35 −1.12655
\(267\) −2197.68 −0.503729
\(268\) −980.974 −0.223591
\(269\) 478.178 0.108383 0.0541914 0.998531i \(-0.482742\pi\)
0.0541914 + 0.998531i \(0.482742\pi\)
\(270\) 0 0
\(271\) −122.323 −0.0274192 −0.0137096 0.999906i \(-0.504364\pi\)
−0.0137096 + 0.999906i \(0.504364\pi\)
\(272\) 2757.20 0.614631
\(273\) 489.838 0.108595
\(274\) −3914.91 −0.863170
\(275\) 0 0
\(276\) −4036.78 −0.880383
\(277\) −8199.41 −1.77854 −0.889269 0.457385i \(-0.848786\pi\)
−0.889269 + 0.457385i \(0.848786\pi\)
\(278\) −4831.22 −1.04229
\(279\) 1265.44 0.271541
\(280\) 0 0
\(281\) 6943.79 1.47413 0.737067 0.675820i \(-0.236210\pi\)
0.737067 + 0.675820i \(0.236210\pi\)
\(282\) 3071.54 0.648609
\(283\) −1035.14 −0.217429 −0.108715 0.994073i \(-0.534673\pi\)
−0.108715 + 0.994073i \(0.534673\pi\)
\(284\) 568.139 0.118707
\(285\) 0 0
\(286\) 250.699 0.0518328
\(287\) 10208.5 2.09961
\(288\) −2039.39 −0.417264
\(289\) 9956.85 2.02663
\(290\) 0 0
\(291\) 3514.47 0.707979
\(292\) −9097.06 −1.82317
\(293\) 6144.81 1.22520 0.612600 0.790393i \(-0.290124\pi\)
0.612600 + 0.790393i \(0.290124\pi\)
\(294\) 8783.53 1.74240
\(295\) 0 0
\(296\) 6645.45 1.30493
\(297\) −297.000 −0.0580259
\(298\) 1314.12 0.255452
\(299\) 598.791 0.115816
\(300\) 0 0
\(301\) 10176.1 1.94864
\(302\) −8352.72 −1.59154
\(303\) −3663.81 −0.694655
\(304\) −787.958 −0.148659
\(305\) 0 0
\(306\) −4855.71 −0.907133
\(307\) 2186.09 0.406406 0.203203 0.979137i \(-0.434865\pi\)
0.203203 + 0.979137i \(0.434865\pi\)
\(308\) 4036.11 0.746684
\(309\) −1550.24 −0.285404
\(310\) 0 0
\(311\) −7484.83 −1.36471 −0.682357 0.731019i \(-0.739045\pi\)
−0.682357 + 0.731019i \(0.739045\pi\)
\(312\) −244.471 −0.0443604
\(313\) 6833.33 1.23400 0.617001 0.786962i \(-0.288347\pi\)
0.617001 + 0.786962i \(0.288347\pi\)
\(314\) 250.132 0.0449546
\(315\) 0 0
\(316\) −4434.81 −0.789485
\(317\) −924.265 −0.163760 −0.0818800 0.996642i \(-0.526092\pi\)
−0.0818800 + 0.996642i \(0.526092\pi\)
\(318\) −65.2934 −0.0115141
\(319\) 763.988 0.134091
\(320\) 0 0
\(321\) 456.074 0.0793008
\(322\) 16302.6 2.82145
\(323\) −4249.54 −0.732046
\(324\) 937.621 0.160772
\(325\) 0 0
\(326\) 217.831 0.0370078
\(327\) 6510.95 1.10109
\(328\) −5094.91 −0.857681
\(329\) −7335.10 −1.22917
\(330\) 0 0
\(331\) −9820.46 −1.63076 −0.815380 0.578927i \(-0.803472\pi\)
−0.815380 + 0.578927i \(0.803472\pi\)
\(332\) 10767.7 1.77999
\(333\) 3780.63 0.622154
\(334\) −9153.02 −1.49949
\(335\) 0 0
\(336\) 2150.12 0.349104
\(337\) −600.808 −0.0971161 −0.0485580 0.998820i \(-0.515463\pi\)
−0.0485580 + 0.998820i \(0.515463\pi\)
\(338\) −9603.07 −1.54538
\(339\) 1939.19 0.310686
\(340\) 0 0
\(341\) −1546.65 −0.245618
\(342\) 1387.68 0.219406
\(343\) −10103.5 −1.59049
\(344\) −5078.73 −0.796008
\(345\) 0 0
\(346\) 2672.41 0.415230
\(347\) 3143.41 0.486303 0.243152 0.969988i \(-0.421819\pi\)
0.243152 + 0.969988i \(0.421819\pi\)
\(348\) −2411.89 −0.371525
\(349\) 720.663 0.110533 0.0552667 0.998472i \(-0.482399\pi\)
0.0552667 + 0.998472i \(0.482399\pi\)
\(350\) 0 0
\(351\) −139.081 −0.0211498
\(352\) 2492.58 0.377429
\(353\) −1207.12 −0.182007 −0.0910034 0.995851i \(-0.529007\pi\)
−0.0910034 + 0.995851i \(0.529007\pi\)
\(354\) 5394.83 0.809978
\(355\) 0 0
\(356\) −8479.79 −1.26244
\(357\) 11595.8 1.71910
\(358\) −9432.99 −1.39260
\(359\) 8748.31 1.28612 0.643062 0.765814i \(-0.277664\pi\)
0.643062 + 0.765814i \(0.277664\pi\)
\(360\) 0 0
\(361\) −5644.56 −0.822942
\(362\) −2607.63 −0.378602
\(363\) 363.000 0.0524864
\(364\) 1890.05 0.272158
\(365\) 0 0
\(366\) −7385.66 −1.05479
\(367\) 6730.45 0.957293 0.478647 0.878008i \(-0.341128\pi\)
0.478647 + 0.878008i \(0.341128\pi\)
\(368\) 2628.37 0.372318
\(369\) −2898.52 −0.408919
\(370\) 0 0
\(371\) 155.926 0.0218202
\(372\) 4882.73 0.680532
\(373\) 227.394 0.0315657 0.0157828 0.999875i \(-0.494976\pi\)
0.0157828 + 0.999875i \(0.494976\pi\)
\(374\) 5934.76 0.820533
\(375\) 0 0
\(376\) 3660.84 0.502110
\(377\) 357.764 0.0488748
\(378\) −3786.60 −0.515242
\(379\) 11356.2 1.53913 0.769565 0.638568i \(-0.220473\pi\)
0.769565 + 0.638568i \(0.220473\pi\)
\(380\) 0 0
\(381\) 2979.91 0.400697
\(382\) −9560.74 −1.28055
\(383\) −10753.6 −1.43468 −0.717338 0.696725i \(-0.754640\pi\)
−0.717338 + 0.696725i \(0.754640\pi\)
\(384\) −5468.07 −0.726670
\(385\) 0 0
\(386\) 6596.43 0.869817
\(387\) −2889.32 −0.379515
\(388\) 13560.7 1.77433
\(389\) −11727.1 −1.52850 −0.764252 0.644918i \(-0.776891\pi\)
−0.764252 + 0.644918i \(0.776891\pi\)
\(390\) 0 0
\(391\) 14175.1 1.83341
\(392\) 10468.7 1.34885
\(393\) 1157.44 0.148563
\(394\) 1019.96 0.130418
\(395\) 0 0
\(396\) −1145.98 −0.145424
\(397\) 359.905 0.0454990 0.0227495 0.999741i \(-0.492758\pi\)
0.0227495 + 0.999741i \(0.492758\pi\)
\(398\) 99.1105 0.0124823
\(399\) −3313.89 −0.415794
\(400\) 0 0
\(401\) −4066.71 −0.506438 −0.253219 0.967409i \(-0.581489\pi\)
−0.253219 + 0.967409i \(0.581489\pi\)
\(402\) −1124.85 −0.139558
\(403\) −724.274 −0.0895252
\(404\) −14136.9 −1.74093
\(405\) 0 0
\(406\) 9740.45 1.19067
\(407\) −4620.77 −0.562760
\(408\) −5787.32 −0.702242
\(409\) −13488.8 −1.63076 −0.815379 0.578927i \(-0.803472\pi\)
−0.815379 + 0.578927i \(0.803472\pi\)
\(410\) 0 0
\(411\) −2654.52 −0.318584
\(412\) −5981.62 −0.715275
\(413\) −12883.3 −1.53498
\(414\) −4628.83 −0.549504
\(415\) 0 0
\(416\) 1167.24 0.137569
\(417\) −3275.83 −0.384695
\(418\) −1696.05 −0.198460
\(419\) −7040.12 −0.820841 −0.410420 0.911896i \(-0.634618\pi\)
−0.410420 + 0.911896i \(0.634618\pi\)
\(420\) 0 0
\(421\) 9171.74 1.06177 0.530883 0.847445i \(-0.321860\pi\)
0.530883 + 0.847445i \(0.321860\pi\)
\(422\) −4652.89 −0.536728
\(423\) 2082.67 0.239392
\(424\) −77.8204 −0.00891343
\(425\) 0 0
\(426\) 651.464 0.0740928
\(427\) 17637.6 1.99893
\(428\) 1759.77 0.198742
\(429\) 169.988 0.0191307
\(430\) 0 0
\(431\) 992.995 0.110976 0.0554882 0.998459i \(-0.482328\pi\)
0.0554882 + 0.998459i \(0.482328\pi\)
\(432\) −610.490 −0.0679912
\(433\) −3790.21 −0.420660 −0.210330 0.977630i \(-0.567454\pi\)
−0.210330 + 0.977630i \(0.567454\pi\)
\(434\) −19719.0 −2.18097
\(435\) 0 0
\(436\) 25122.7 2.75954
\(437\) −4050.98 −0.443443
\(438\) −10431.3 −1.13796
\(439\) −5136.97 −0.558483 −0.279242 0.960221i \(-0.590083\pi\)
−0.279242 + 0.960221i \(0.590083\pi\)
\(440\) 0 0
\(441\) 5955.71 0.643095
\(442\) 2779.16 0.299075
\(443\) −10676.8 −1.14508 −0.572541 0.819876i \(-0.694042\pi\)
−0.572541 + 0.819876i \(0.694042\pi\)
\(444\) 14587.7 1.55923
\(445\) 0 0
\(446\) −17086.2 −1.81403
\(447\) 891.042 0.0942838
\(448\) 26045.5 2.74672
\(449\) 10529.9 1.10676 0.553379 0.832929i \(-0.313338\pi\)
0.553379 + 0.832929i \(0.313338\pi\)
\(450\) 0 0
\(451\) 3542.64 0.369881
\(452\) 7482.42 0.778636
\(453\) −5663.59 −0.587414
\(454\) 3861.29 0.399162
\(455\) 0 0
\(456\) 1653.91 0.169850
\(457\) 14072.5 1.44045 0.720225 0.693741i \(-0.244039\pi\)
0.720225 + 0.693741i \(0.244039\pi\)
\(458\) 8148.55 0.831347
\(459\) −3292.43 −0.334810
\(460\) 0 0
\(461\) −30.8173 −0.00311346 −0.00155673 0.999999i \(-0.500496\pi\)
−0.00155673 + 0.999999i \(0.500496\pi\)
\(462\) 4628.06 0.466054
\(463\) −17591.3 −1.76573 −0.882867 0.469622i \(-0.844390\pi\)
−0.882867 + 0.469622i \(0.844390\pi\)
\(464\) 1570.39 0.157120
\(465\) 0 0
\(466\) −17397.5 −1.72945
\(467\) −13273.1 −1.31522 −0.657609 0.753360i \(-0.728432\pi\)
−0.657609 + 0.753360i \(0.728432\pi\)
\(468\) −536.647 −0.0530053
\(469\) 2686.23 0.264474
\(470\) 0 0
\(471\) 169.603 0.0165921
\(472\) 6429.87 0.627031
\(473\) 3531.39 0.343284
\(474\) −5085.23 −0.492769
\(475\) 0 0
\(476\) 44742.9 4.30837
\(477\) −44.2724 −0.00424968
\(478\) 21113.8 2.02034
\(479\) 2496.68 0.238155 0.119077 0.992885i \(-0.462006\pi\)
0.119077 + 0.992885i \(0.462006\pi\)
\(480\) 0 0
\(481\) −2163.84 −0.205120
\(482\) 17648.3 1.66776
\(483\) 11054.0 1.04136
\(484\) 1400.64 0.131541
\(485\) 0 0
\(486\) 1075.14 0.100348
\(487\) 3464.42 0.322357 0.161178 0.986925i \(-0.448471\pi\)
0.161178 + 0.986925i \(0.448471\pi\)
\(488\) −8802.65 −0.816552
\(489\) 147.701 0.0136591
\(490\) 0 0
\(491\) −16224.6 −1.49125 −0.745625 0.666366i \(-0.767849\pi\)
−0.745625 + 0.666366i \(0.767849\pi\)
\(492\) −11184.0 −1.02483
\(493\) 8469.29 0.773707
\(494\) −794.236 −0.0723367
\(495\) 0 0
\(496\) −3179.17 −0.287800
\(497\) −1555.75 −0.140412
\(498\) 12347.0 1.11100
\(499\) 9993.81 0.896562 0.448281 0.893893i \(-0.352036\pi\)
0.448281 + 0.893893i \(0.352036\pi\)
\(500\) 0 0
\(501\) −6206.24 −0.553441
\(502\) −24220.3 −2.15340
\(503\) 15334.8 1.35933 0.679667 0.733520i \(-0.262124\pi\)
0.679667 + 0.733520i \(0.262124\pi\)
\(504\) −4513.08 −0.398866
\(505\) 0 0
\(506\) 5657.46 0.497045
\(507\) −6511.40 −0.570377
\(508\) 11498.1 1.00422
\(509\) −7291.23 −0.634927 −0.317464 0.948270i \(-0.602831\pi\)
−0.317464 + 0.948270i \(0.602831\pi\)
\(510\) 0 0
\(511\) 24910.7 2.15653
\(512\) 7985.14 0.689251
\(513\) 940.919 0.0809797
\(514\) 28466.8 2.44283
\(515\) 0 0
\(516\) −11148.5 −0.951134
\(517\) −2545.49 −0.216538
\(518\) −58912.5 −4.99704
\(519\) 1812.04 0.153255
\(520\) 0 0
\(521\) 16794.3 1.41223 0.706114 0.708098i \(-0.250447\pi\)
0.706114 + 0.708098i \(0.250447\pi\)
\(522\) −2765.63 −0.231893
\(523\) 21009.4 1.75655 0.878275 0.478157i \(-0.158695\pi\)
0.878275 + 0.478157i \(0.158695\pi\)
\(524\) 4466.01 0.372326
\(525\) 0 0
\(526\) −33577.0 −2.78332
\(527\) −17145.6 −1.41722
\(528\) 746.154 0.0615003
\(529\) 1345.73 0.110605
\(530\) 0 0
\(531\) 3657.99 0.298951
\(532\) −12786.7 −1.04206
\(533\) 1658.97 0.134818
\(534\) −9723.46 −0.787969
\(535\) 0 0
\(536\) −1340.66 −0.108036
\(537\) −6396.07 −0.513987
\(538\) 2115.66 0.169540
\(539\) −7279.20 −0.581702
\(540\) 0 0
\(541\) −16802.8 −1.33532 −0.667662 0.744464i \(-0.732705\pi\)
−0.667662 + 0.744464i \(0.732705\pi\)
\(542\) −541.211 −0.0428911
\(543\) −1768.11 −0.139737
\(544\) 27631.9 2.17777
\(545\) 0 0
\(546\) 2167.25 0.169872
\(547\) −16784.5 −1.31198 −0.655990 0.754770i \(-0.727749\pi\)
−0.655990 + 0.754770i \(0.727749\pi\)
\(548\) −10242.5 −0.798429
\(549\) −5007.88 −0.389309
\(550\) 0 0
\(551\) −2420.37 −0.187135
\(552\) −5516.91 −0.425390
\(553\) 12144.0 0.933840
\(554\) −36277.7 −2.78212
\(555\) 0 0
\(556\) −12639.9 −0.964117
\(557\) −18127.0 −1.37893 −0.689467 0.724317i \(-0.742155\pi\)
−0.689467 + 0.724317i \(0.742155\pi\)
\(558\) 5598.85 0.424764
\(559\) 1653.70 0.125123
\(560\) 0 0
\(561\) 4024.09 0.302847
\(562\) 30722.3 2.30595
\(563\) −2090.88 −0.156518 −0.0782592 0.996933i \(-0.524936\pi\)
−0.0782592 + 0.996933i \(0.524936\pi\)
\(564\) 8036.03 0.599961
\(565\) 0 0
\(566\) −4579.89 −0.340119
\(567\) −2567.51 −0.190168
\(568\) 776.452 0.0573578
\(569\) 6249.23 0.460424 0.230212 0.973140i \(-0.426058\pi\)
0.230212 + 0.973140i \(0.426058\pi\)
\(570\) 0 0
\(571\) 6048.79 0.443317 0.221659 0.975124i \(-0.428853\pi\)
0.221659 + 0.975124i \(0.428853\pi\)
\(572\) 655.902 0.0479451
\(573\) −6482.69 −0.472633
\(574\) 45166.8 3.28437
\(575\) 0 0
\(576\) −7395.15 −0.534950
\(577\) 15729.1 1.13486 0.567429 0.823423i \(-0.307938\pi\)
0.567429 + 0.823423i \(0.307938\pi\)
\(578\) 44053.4 3.17021
\(579\) 4472.73 0.321037
\(580\) 0 0
\(581\) −29485.6 −2.10545
\(582\) 15549.5 1.10747
\(583\) 54.1108 0.00384398
\(584\) −12432.6 −0.880931
\(585\) 0 0
\(586\) 27187.3 1.91655
\(587\) −15620.5 −1.09835 −0.549173 0.835709i \(-0.685057\pi\)
−0.549173 + 0.835709i \(0.685057\pi\)
\(588\) 22980.2 1.61172
\(589\) 4899.91 0.342780
\(590\) 0 0
\(591\) 691.587 0.0481355
\(592\) −9498.09 −0.659407
\(593\) 493.541 0.0341776 0.0170888 0.999854i \(-0.494560\pi\)
0.0170888 + 0.999854i \(0.494560\pi\)
\(594\) −1314.06 −0.0907683
\(595\) 0 0
\(596\) 3438.11 0.236293
\(597\) 67.2022 0.00460704
\(598\) 2649.31 0.181168
\(599\) −12455.1 −0.849585 −0.424793 0.905291i \(-0.639653\pi\)
−0.424793 + 0.905291i \(0.639653\pi\)
\(600\) 0 0
\(601\) 12454.8 0.845329 0.422664 0.906286i \(-0.361095\pi\)
0.422664 + 0.906286i \(0.361095\pi\)
\(602\) 45023.3 3.04820
\(603\) −762.707 −0.0515088
\(604\) −21853.1 −1.47217
\(605\) 0 0
\(606\) −16210.3 −1.08663
\(607\) 4243.19 0.283733 0.141867 0.989886i \(-0.454690\pi\)
0.141867 + 0.989886i \(0.454690\pi\)
\(608\) −7896.70 −0.526732
\(609\) 6604.54 0.439458
\(610\) 0 0
\(611\) −1192.01 −0.0789259
\(612\) −12703.9 −0.839095
\(613\) −5733.14 −0.377748 −0.188874 0.982001i \(-0.560484\pi\)
−0.188874 + 0.982001i \(0.560484\pi\)
\(614\) 9672.18 0.635729
\(615\) 0 0
\(616\) 5515.99 0.360788
\(617\) −15642.1 −1.02063 −0.510314 0.859988i \(-0.670471\pi\)
−0.510314 + 0.859988i \(0.670471\pi\)
\(618\) −6858.91 −0.446449
\(619\) −7467.40 −0.484879 −0.242440 0.970167i \(-0.577948\pi\)
−0.242440 + 0.970167i \(0.577948\pi\)
\(620\) 0 0
\(621\) −3138.60 −0.202814
\(622\) −33116.1 −2.13478
\(623\) 23220.4 1.49327
\(624\) 349.413 0.0224162
\(625\) 0 0
\(626\) 30233.6 1.93031
\(627\) −1150.01 −0.0732489
\(628\) 654.416 0.0415829
\(629\) −51224.2 −3.24713
\(630\) 0 0
\(631\) −1486.38 −0.0937745 −0.0468872 0.998900i \(-0.514930\pi\)
−0.0468872 + 0.998900i \(0.514930\pi\)
\(632\) −6060.87 −0.381469
\(633\) −3154.91 −0.198099
\(634\) −4089.35 −0.256165
\(635\) 0 0
\(636\) −170.826 −0.0106505
\(637\) −3408.74 −0.212024
\(638\) 3380.21 0.209755
\(639\) 441.728 0.0273466
\(640\) 0 0
\(641\) 12386.0 0.763211 0.381606 0.924325i \(-0.375371\pi\)
0.381606 + 0.924325i \(0.375371\pi\)
\(642\) 2017.87 0.124048
\(643\) 14458.1 0.886737 0.443369 0.896339i \(-0.353783\pi\)
0.443369 + 0.896339i \(0.353783\pi\)
\(644\) 42652.3 2.60984
\(645\) 0 0
\(646\) −18801.8 −1.14512
\(647\) −15792.8 −0.959625 −0.479813 0.877371i \(-0.659295\pi\)
−0.479813 + 0.877371i \(0.659295\pi\)
\(648\) 1281.41 0.0776828
\(649\) −4470.87 −0.270412
\(650\) 0 0
\(651\) −13370.5 −0.804965
\(652\) 569.909 0.0342321
\(653\) 3179.93 0.190567 0.0952837 0.995450i \(-0.469624\pi\)
0.0952837 + 0.995450i \(0.469624\pi\)
\(654\) 28807.3 1.72240
\(655\) 0 0
\(656\) 7281.96 0.433404
\(657\) −7072.96 −0.420003
\(658\) −32453.6 −1.92276
\(659\) 11593.5 0.685308 0.342654 0.939462i \(-0.388674\pi\)
0.342654 + 0.939462i \(0.388674\pi\)
\(660\) 0 0
\(661\) 3233.88 0.190293 0.0951464 0.995463i \(-0.469668\pi\)
0.0951464 + 0.995463i \(0.469668\pi\)
\(662\) −43449.9 −2.55095
\(663\) 1884.42 0.110384
\(664\) 14715.8 0.860066
\(665\) 0 0
\(666\) 16727.1 0.973219
\(667\) 8073.56 0.468680
\(668\) −23946.9 −1.38703
\(669\) −11585.4 −0.669532
\(670\) 0 0
\(671\) 6120.74 0.352144
\(672\) 21548.0 1.23695
\(673\) 5495.72 0.314776 0.157388 0.987537i \(-0.449693\pi\)
0.157388 + 0.987537i \(0.449693\pi\)
\(674\) −2658.23 −0.151916
\(675\) 0 0
\(676\) −25124.4 −1.42947
\(677\) −33836.7 −1.92090 −0.960451 0.278448i \(-0.910180\pi\)
−0.960451 + 0.278448i \(0.910180\pi\)
\(678\) 8579.82 0.485997
\(679\) −37133.6 −2.09876
\(680\) 0 0
\(681\) 2618.16 0.147325
\(682\) −6843.04 −0.384214
\(683\) 21080.3 1.18099 0.590493 0.807043i \(-0.298933\pi\)
0.590493 + 0.807043i \(0.298933\pi\)
\(684\) 3630.56 0.202950
\(685\) 0 0
\(686\) −44702.2 −2.48796
\(687\) 5525.16 0.306838
\(688\) 7258.84 0.402239
\(689\) 25.3393 0.00140109
\(690\) 0 0
\(691\) 11811.3 0.650253 0.325127 0.945671i \(-0.394593\pi\)
0.325127 + 0.945671i \(0.394593\pi\)
\(692\) 6991.79 0.384087
\(693\) 3138.07 0.172014
\(694\) 13907.8 0.760711
\(695\) 0 0
\(696\) −3296.23 −0.179516
\(697\) 39272.4 2.13422
\(698\) 3188.52 0.172904
\(699\) −11796.4 −0.638313
\(700\) 0 0
\(701\) 4244.99 0.228718 0.114359 0.993440i \(-0.463519\pi\)
0.114359 + 0.993440i \(0.463519\pi\)
\(702\) −615.353 −0.0330841
\(703\) 14639.0 0.785376
\(704\) 9038.51 0.483880
\(705\) 0 0
\(706\) −5340.81 −0.284708
\(707\) 38711.5 2.05926
\(708\) 14114.4 0.749227
\(709\) −898.822 −0.0476107 −0.0238053 0.999717i \(-0.507578\pi\)
−0.0238053 + 0.999717i \(0.507578\pi\)
\(710\) 0 0
\(711\) −3448.06 −0.181874
\(712\) −11589.0 −0.609993
\(713\) −16344.5 −0.858493
\(714\) 51305.0 2.68913
\(715\) 0 0
\(716\) −24679.4 −1.28815
\(717\) 14316.3 0.745679
\(718\) 38706.3 2.01185
\(719\) 10741.8 0.557165 0.278582 0.960412i \(-0.410135\pi\)
0.278582 + 0.960412i \(0.410135\pi\)
\(720\) 0 0
\(721\) 16379.6 0.846061
\(722\) −24973.9 −1.28730
\(723\) 11966.5 0.615546
\(724\) −6822.30 −0.350206
\(725\) 0 0
\(726\) 1606.07 0.0821030
\(727\) −16794.2 −0.856758 −0.428379 0.903599i \(-0.640915\pi\)
−0.428379 + 0.903599i \(0.640915\pi\)
\(728\) 2583.06 0.131503
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 39147.7 1.98075
\(732\) −19323.0 −0.975681
\(733\) −8659.40 −0.436347 −0.218173 0.975910i \(-0.570010\pi\)
−0.218173 + 0.975910i \(0.570010\pi\)
\(734\) 29778.4 1.49747
\(735\) 0 0
\(736\) 26340.8 1.31920
\(737\) 932.197 0.0465915
\(738\) −12824.3 −0.639660
\(739\) 16705.7 0.831567 0.415783 0.909464i \(-0.363507\pi\)
0.415783 + 0.909464i \(0.363507\pi\)
\(740\) 0 0
\(741\) −538.534 −0.0266984
\(742\) 689.884 0.0341327
\(743\) −1292.12 −0.0637996 −0.0318998 0.999491i \(-0.510156\pi\)
−0.0318998 + 0.999491i \(0.510156\pi\)
\(744\) 6673.03 0.328824
\(745\) 0 0
\(746\) 1006.09 0.0493773
\(747\) 8371.90 0.410056
\(748\) 15527.0 0.758990
\(749\) −4818.83 −0.235082
\(750\) 0 0
\(751\) −14980.4 −0.727886 −0.363943 0.931421i \(-0.618570\pi\)
−0.363943 + 0.931421i \(0.618570\pi\)
\(752\) −5232.30 −0.253726
\(753\) −16422.7 −0.794787
\(754\) 1582.90 0.0764535
\(755\) 0 0
\(756\) −9906.82 −0.476597
\(757\) −3003.41 −0.144202 −0.0721010 0.997397i \(-0.522970\pi\)
−0.0721010 + 0.997397i \(0.522970\pi\)
\(758\) 50244.8 2.40762
\(759\) 3836.06 0.183452
\(760\) 0 0
\(761\) −20375.0 −0.970555 −0.485277 0.874360i \(-0.661281\pi\)
−0.485277 + 0.874360i \(0.661281\pi\)
\(762\) 13184.4 0.626799
\(763\) −68794.1 −3.26411
\(764\) −25013.6 −1.18450
\(765\) 0 0
\(766\) −47578.4 −2.24422
\(767\) −2093.65 −0.0985621
\(768\) −4472.70 −0.210149
\(769\) −12372.4 −0.580184 −0.290092 0.956999i \(-0.593686\pi\)
−0.290092 + 0.956999i \(0.593686\pi\)
\(770\) 0 0
\(771\) 19302.0 0.901615
\(772\) 17258.1 0.804578
\(773\) −21023.6 −0.978225 −0.489113 0.872221i \(-0.662679\pi\)
−0.489113 + 0.872221i \(0.662679\pi\)
\(774\) −12783.6 −0.593664
\(775\) 0 0
\(776\) 18532.8 0.857331
\(777\) −39945.8 −1.84433
\(778\) −51885.7 −2.39099
\(779\) −11223.4 −0.516198
\(780\) 0 0
\(781\) −539.889 −0.0247359
\(782\) 62716.6 2.86795
\(783\) −1875.24 −0.0855884
\(784\) −14962.5 −0.681602
\(785\) 0 0
\(786\) 5121.02 0.232393
\(787\) −30286.2 −1.37177 −0.685886 0.727709i \(-0.740585\pi\)
−0.685886 + 0.727709i \(0.740585\pi\)
\(788\) 2668.51 0.120637
\(789\) −22767.0 −1.02728
\(790\) 0 0
\(791\) −20489.3 −0.921007
\(792\) −1566.17 −0.0702668
\(793\) 2866.25 0.128353
\(794\) 1592.37 0.0711729
\(795\) 0 0
\(796\) 259.301 0.0115461
\(797\) −32337.8 −1.43722 −0.718610 0.695413i \(-0.755221\pi\)
−0.718610 + 0.695413i \(0.755221\pi\)
\(798\) −14662.1 −0.650415
\(799\) −28218.3 −1.24943
\(800\) 0 0
\(801\) −6593.03 −0.290828
\(802\) −17992.9 −0.792207
\(803\) 8644.72 0.379907
\(804\) −2942.92 −0.129091
\(805\) 0 0
\(806\) −3204.50 −0.140042
\(807\) 1434.53 0.0625749
\(808\) −19320.3 −0.841197
\(809\) −891.707 −0.0387525 −0.0193762 0.999812i \(-0.506168\pi\)
−0.0193762 + 0.999812i \(0.506168\pi\)
\(810\) 0 0
\(811\) −10114.9 −0.437957 −0.218978 0.975730i \(-0.570272\pi\)
−0.218978 + 0.975730i \(0.570272\pi\)
\(812\) 25483.8 1.10136
\(813\) −366.970 −0.0158305
\(814\) −20444.3 −0.880309
\(815\) 0 0
\(816\) 8271.59 0.354857
\(817\) −11187.7 −0.479080
\(818\) −59680.4 −2.55095
\(819\) 1469.51 0.0626972
\(820\) 0 0
\(821\) 10833.5 0.460525 0.230262 0.973129i \(-0.426042\pi\)
0.230262 + 0.973129i \(0.426042\pi\)
\(822\) −11744.7 −0.498351
\(823\) −31958.5 −1.35359 −0.676794 0.736173i \(-0.736631\pi\)
−0.676794 + 0.736173i \(0.736631\pi\)
\(824\) −8174.84 −0.345612
\(825\) 0 0
\(826\) −57001.3 −2.40112
\(827\) −34847.3 −1.46525 −0.732624 0.680634i \(-0.761704\pi\)
−0.732624 + 0.680634i \(0.761704\pi\)
\(828\) −12110.3 −0.508289
\(829\) 6537.91 0.273910 0.136955 0.990577i \(-0.456268\pi\)
0.136955 + 0.990577i \(0.456268\pi\)
\(830\) 0 0
\(831\) −24598.2 −1.02684
\(832\) 4232.60 0.176369
\(833\) −80694.5 −3.35642
\(834\) −14493.7 −0.601768
\(835\) 0 0
\(836\) −4437.35 −0.183575
\(837\) 3796.32 0.156774
\(838\) −31148.5 −1.28402
\(839\) 2710.34 0.111527 0.0557635 0.998444i \(-0.482241\pi\)
0.0557635 + 0.998444i \(0.482241\pi\)
\(840\) 0 0
\(841\) −19565.2 −0.802215
\(842\) 40579.7 1.66089
\(843\) 20831.4 0.851092
\(844\) −12173.3 −0.496471
\(845\) 0 0
\(846\) 9214.62 0.374474
\(847\) −3835.42 −0.155592
\(848\) 111.226 0.00450414
\(849\) −3105.41 −0.125533
\(850\) 0 0
\(851\) −48830.8 −1.96698
\(852\) 1704.42 0.0685356
\(853\) −9759.32 −0.391738 −0.195869 0.980630i \(-0.562753\pi\)
−0.195869 + 0.980630i \(0.562753\pi\)
\(854\) 78036.2 3.12687
\(855\) 0 0
\(856\) 2405.01 0.0960298
\(857\) 13649.8 0.544072 0.272036 0.962287i \(-0.412303\pi\)
0.272036 + 0.962287i \(0.412303\pi\)
\(858\) 752.098 0.0299257
\(859\) 7796.42 0.309674 0.154837 0.987940i \(-0.450515\pi\)
0.154837 + 0.987940i \(0.450515\pi\)
\(860\) 0 0
\(861\) 30625.5 1.21221
\(862\) 4393.43 0.173597
\(863\) −7183.57 −0.283350 −0.141675 0.989913i \(-0.545249\pi\)
−0.141675 + 0.989913i \(0.545249\pi\)
\(864\) −6118.16 −0.240907
\(865\) 0 0
\(866\) −16769.5 −0.658026
\(867\) 29870.6 1.17008
\(868\) −51590.5 −2.01739
\(869\) 4214.30 0.164511
\(870\) 0 0
\(871\) 436.534 0.0169821
\(872\) 34334.1 1.33337
\(873\) 10543.4 0.408752
\(874\) −17923.3 −0.693666
\(875\) 0 0
\(876\) −27291.2 −1.05261
\(877\) −17063.1 −0.656991 −0.328495 0.944506i \(-0.606542\pi\)
−0.328495 + 0.944506i \(0.606542\pi\)
\(878\) −22728.2 −0.873620
\(879\) 18434.4 0.707369
\(880\) 0 0
\(881\) −32174.9 −1.23042 −0.615210 0.788363i \(-0.710929\pi\)
−0.615210 + 0.788363i \(0.710929\pi\)
\(882\) 26350.6 1.00598
\(883\) −2843.68 −0.108378 −0.0541889 0.998531i \(-0.517257\pi\)
−0.0541889 + 0.998531i \(0.517257\pi\)
\(884\) 7271.09 0.276644
\(885\) 0 0
\(886\) −47238.9 −1.79122
\(887\) 31417.8 1.18930 0.594649 0.803985i \(-0.297291\pi\)
0.594649 + 0.803985i \(0.297291\pi\)
\(888\) 19936.4 0.753401
\(889\) −31485.5 −1.18784
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) −44702.5 −1.67797
\(893\) 8064.30 0.302196
\(894\) 3942.35 0.147485
\(895\) 0 0
\(896\) 57775.1 2.15416
\(897\) 1796.37 0.0668664
\(898\) 46588.6 1.73127
\(899\) −9765.47 −0.362288
\(900\) 0 0
\(901\) 599.852 0.0221798
\(902\) 15674.1 0.578594
\(903\) 30528.2 1.12505
\(904\) 10225.9 0.376227
\(905\) 0 0
\(906\) −25058.1 −0.918875
\(907\) −12253.1 −0.448573 −0.224287 0.974523i \(-0.572005\pi\)
−0.224287 + 0.974523i \(0.572005\pi\)
\(908\) 10102.2 0.369223
\(909\) −10991.4 −0.401059
\(910\) 0 0
\(911\) −48422.4 −1.76104 −0.880518 0.474012i \(-0.842805\pi\)
−0.880518 + 0.474012i \(0.842805\pi\)
\(912\) −2363.87 −0.0858286
\(913\) −10232.3 −0.370909
\(914\) 62262.9 2.25326
\(915\) 0 0
\(916\) 21318.9 0.768993
\(917\) −12229.4 −0.440404
\(918\) −14567.1 −0.523733
\(919\) 5546.18 0.199077 0.0995385 0.995034i \(-0.468263\pi\)
0.0995385 + 0.995034i \(0.468263\pi\)
\(920\) 0 0
\(921\) 6558.26 0.234638
\(922\) −136.349 −0.00487030
\(923\) −252.822 −0.00901598
\(924\) 12108.3 0.431098
\(925\) 0 0
\(926\) −77831.3 −2.76209
\(927\) −4650.71 −0.164778
\(928\) 15738.0 0.556709
\(929\) −35684.5 −1.26025 −0.630125 0.776494i \(-0.716996\pi\)
−0.630125 + 0.776494i \(0.716996\pi\)
\(930\) 0 0
\(931\) 23061.1 0.811811
\(932\) −45516.7 −1.59973
\(933\) −22454.5 −0.787918
\(934\) −58726.0 −2.05736
\(935\) 0 0
\(936\) −733.413 −0.0256115
\(937\) 48903.6 1.70503 0.852514 0.522705i \(-0.175077\pi\)
0.852514 + 0.522705i \(0.175077\pi\)
\(938\) 11885.0 0.413710
\(939\) 20500.0 0.712451
\(940\) 0 0
\(941\) −23741.9 −0.822490 −0.411245 0.911525i \(-0.634906\pi\)
−0.411245 + 0.911525i \(0.634906\pi\)
\(942\) 750.396 0.0259546
\(943\) 37437.4 1.29282
\(944\) −9189.97 −0.316852
\(945\) 0 0
\(946\) 15624.4 0.536989
\(947\) −37612.4 −1.29064 −0.645321 0.763911i \(-0.723276\pi\)
−0.645321 + 0.763911i \(0.723276\pi\)
\(948\) −13304.4 −0.455810
\(949\) 4048.20 0.138472
\(950\) 0 0
\(951\) −2772.80 −0.0945469
\(952\) 61148.2 2.08175
\(953\) 48294.3 1.64156 0.820779 0.571246i \(-0.193540\pi\)
0.820779 + 0.571246i \(0.193540\pi\)
\(954\) −195.880 −0.00664765
\(955\) 0 0
\(956\) 55239.8 1.86881
\(957\) 2291.96 0.0774176
\(958\) 11046.4 0.372539
\(959\) 28047.4 0.944419
\(960\) 0 0
\(961\) −10021.4 −0.336389
\(962\) −9573.76 −0.320863
\(963\) 1368.22 0.0457843
\(964\) 46173.1 1.54267
\(965\) 0 0
\(966\) 48907.8 1.62897
\(967\) −1840.92 −0.0612204 −0.0306102 0.999531i \(-0.509745\pi\)
−0.0306102 + 0.999531i \(0.509745\pi\)
\(968\) 1914.20 0.0635587
\(969\) −12748.6 −0.422647
\(970\) 0 0
\(971\) 31461.8 1.03981 0.519906 0.854223i \(-0.325967\pi\)
0.519906 + 0.854223i \(0.325967\pi\)
\(972\) 2812.86 0.0928217
\(973\) 34612.1 1.14040
\(974\) 15328.1 0.504254
\(975\) 0 0
\(976\) 12581.3 0.412620
\(977\) 7040.11 0.230535 0.115268 0.993334i \(-0.463227\pi\)
0.115268 + 0.993334i \(0.463227\pi\)
\(978\) 653.494 0.0213665
\(979\) 8058.15 0.263064
\(980\) 0 0
\(981\) 19532.9 0.635715
\(982\) −71784.4 −2.33272
\(983\) 24610.9 0.798541 0.399270 0.916833i \(-0.369263\pi\)
0.399270 + 0.916833i \(0.369263\pi\)
\(984\) −15284.7 −0.495183
\(985\) 0 0
\(986\) 37471.8 1.21029
\(987\) −22005.3 −0.709662
\(988\) −2077.95 −0.0669112
\(989\) 37318.5 1.19986
\(990\) 0 0
\(991\) −40003.3 −1.28229 −0.641144 0.767421i \(-0.721540\pi\)
−0.641144 + 0.767421i \(0.721540\pi\)
\(992\) −31860.8 −1.01974
\(993\) −29461.4 −0.941519
\(994\) −6883.31 −0.219643
\(995\) 0 0
\(996\) 32303.2 1.02768
\(997\) 7342.61 0.233242 0.116621 0.993176i \(-0.462794\pi\)
0.116621 + 0.993176i \(0.462794\pi\)
\(998\) 44216.9 1.40247
\(999\) 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.a.l.1.2 2
3.2 odd 2 2475.4.a.p.1.1 2
5.2 odd 4 825.4.c.h.199.3 4
5.3 odd 4 825.4.c.h.199.2 4
5.4 even 2 33.4.a.c.1.1 2
15.14 odd 2 99.4.a.f.1.2 2
20.19 odd 2 528.4.a.p.1.2 2
35.34 odd 2 1617.4.a.k.1.1 2
40.19 odd 2 2112.4.a.bg.1.1 2
40.29 even 2 2112.4.a.bn.1.1 2
55.54 odd 2 363.4.a.i.1.2 2
60.59 even 2 1584.4.a.bj.1.1 2
165.164 even 2 1089.4.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 5.4 even 2
99.4.a.f.1.2 2 15.14 odd 2
363.4.a.i.1.2 2 55.54 odd 2
528.4.a.p.1.2 2 20.19 odd 2
825.4.a.l.1.2 2 1.1 even 1 trivial
825.4.c.h.199.2 4 5.3 odd 4
825.4.c.h.199.3 4 5.2 odd 4
1089.4.a.u.1.1 2 165.164 even 2
1584.4.a.bj.1.1 2 60.59 even 2
1617.4.a.k.1.1 2 35.34 odd 2
2112.4.a.bg.1.1 2 40.19 odd 2
2112.4.a.bn.1.1 2 40.29 even 2
2475.4.a.p.1.1 2 3.2 odd 2