Properties

Label 2175.2.a.ba.1.2
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
Analytic rank $0$
Dimension $7$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.26695\) of defining polynomial
Character \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.26695 q^{2} -1.00000 q^{3} +3.13907 q^{4} +2.26695 q^{6} -0.159887 q^{7} -2.58223 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.26695 q^{2} -1.00000 q^{3} +3.13907 q^{4} +2.26695 q^{6} -0.159887 q^{7} -2.58223 q^{8} +1.00000 q^{9} -3.24930 q^{11} -3.13907 q^{12} -7.05679 q^{13} +0.362456 q^{14} -0.424361 q^{16} -7.63808 q^{17} -2.26695 q^{18} -3.99028 q^{19} +0.159887 q^{21} +7.36601 q^{22} -2.06402 q^{23} +2.58223 q^{24} +15.9974 q^{26} -1.00000 q^{27} -0.501897 q^{28} -1.00000 q^{29} +10.2044 q^{31} +6.12646 q^{32} +3.24930 q^{33} +17.3152 q^{34} +3.13907 q^{36} +4.27815 q^{37} +9.04576 q^{38} +7.05679 q^{39} -5.12330 q^{41} -0.362456 q^{42} +3.94180 q^{43} -10.1998 q^{44} +4.67903 q^{46} -5.61626 q^{47} +0.424361 q^{48} -6.97444 q^{49} +7.63808 q^{51} -22.1518 q^{52} -12.0822 q^{53} +2.26695 q^{54} +0.412864 q^{56} +3.99028 q^{57} +2.26695 q^{58} -7.06781 q^{59} +11.4310 q^{61} -23.1329 q^{62} -0.159887 q^{63} -13.0397 q^{64} -7.36601 q^{66} +14.9512 q^{67} -23.9765 q^{68} +2.06402 q^{69} +3.43347 q^{71} -2.58223 q^{72} +12.0772 q^{73} -9.69836 q^{74} -12.5258 q^{76} +0.519521 q^{77} -15.9974 q^{78} -12.7441 q^{79} +1.00000 q^{81} +11.6143 q^{82} +2.59792 q^{83} +0.501897 q^{84} -8.93586 q^{86} +1.00000 q^{87} +8.39044 q^{88} +4.33165 q^{89} +1.12829 q^{91} -6.47910 q^{92} -10.2044 q^{93} +12.7318 q^{94} -6.12646 q^{96} +3.88355 q^{97} +15.8107 q^{98} -3.24930 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{2} - 7 q^{3} + 10 q^{4} + 2 q^{6} + q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 2 q^{2} - 7 q^{3} + 10 q^{4} + 2 q^{6} + q^{7} - 3 q^{8} + 7 q^{9} + 4 q^{11} - 10 q^{12} + q^{13} + 15 q^{14} + 12 q^{16} - 8 q^{17} - 2 q^{18} + 15 q^{19} - q^{21} - 3 q^{22} - 14 q^{23} + 3 q^{24} + 6 q^{26} - 7 q^{27} + 24 q^{28} - 7 q^{29} + 5 q^{31} - 18 q^{32} - 4 q^{33} + 7 q^{34} + 10 q^{36} + 6 q^{37} + 18 q^{38} - q^{39} + 22 q^{41} - 15 q^{42} + 19 q^{43} + 15 q^{44} - 4 q^{46} - 22 q^{47} - 12 q^{48} + 12 q^{49} + 8 q^{51} - 11 q^{52} - 10 q^{53} + 2 q^{54} + 14 q^{56} - 15 q^{57} + 2 q^{58} + 6 q^{59} + 23 q^{61} - 40 q^{62} + q^{63} + 5 q^{64} + 3 q^{66} + 13 q^{67} + 7 q^{68} + 14 q^{69} + 26 q^{71} - 3 q^{72} + 24 q^{73} - 10 q^{74} + 46 q^{76} - 4 q^{77} - 6 q^{78} + 14 q^{79} + 7 q^{81} - 16 q^{82} - 10 q^{83} - 24 q^{84} + 44 q^{86} + 7 q^{87} + 66 q^{88} + 14 q^{89} + 13 q^{91} - 58 q^{92} - 5 q^{93} - 3 q^{94} + 18 q^{96} + 31 q^{97} + 59 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.26695 −1.60298 −0.801489 0.598010i \(-0.795958\pi\)
−0.801489 + 0.598010i \(0.795958\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.13907 1.56954
\(5\) 0 0
\(6\) 2.26695 0.925480
\(7\) −0.159887 −0.0604316 −0.0302158 0.999543i \(-0.509619\pi\)
−0.0302158 + 0.999543i \(0.509619\pi\)
\(8\) −2.58223 −0.912955
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.24930 −0.979701 −0.489851 0.871806i \(-0.662949\pi\)
−0.489851 + 0.871806i \(0.662949\pi\)
\(12\) −3.13907 −0.906173
\(13\) −7.05679 −1.95720 −0.978600 0.205770i \(-0.934030\pi\)
−0.978600 + 0.205770i \(0.934030\pi\)
\(14\) 0.362456 0.0968704
\(15\) 0 0
\(16\) −0.424361 −0.106090
\(17\) −7.63808 −1.85251 −0.926254 0.376901i \(-0.876990\pi\)
−0.926254 + 0.376901i \(0.876990\pi\)
\(18\) −2.26695 −0.534326
\(19\) −3.99028 −0.915432 −0.457716 0.889098i \(-0.651332\pi\)
−0.457716 + 0.889098i \(0.651332\pi\)
\(20\) 0 0
\(21\) 0.159887 0.0348902
\(22\) 7.36601 1.57044
\(23\) −2.06402 −0.430377 −0.215189 0.976572i \(-0.569037\pi\)
−0.215189 + 0.976572i \(0.569037\pi\)
\(24\) 2.58223 0.527095
\(25\) 0 0
\(26\) 15.9974 3.13735
\(27\) −1.00000 −0.192450
\(28\) −0.501897 −0.0948496
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 10.2044 1.83277 0.916383 0.400303i \(-0.131095\pi\)
0.916383 + 0.400303i \(0.131095\pi\)
\(32\) 6.12646 1.08302
\(33\) 3.24930 0.565631
\(34\) 17.3152 2.96953
\(35\) 0 0
\(36\) 3.13907 0.523179
\(37\) 4.27815 0.703323 0.351662 0.936127i \(-0.385617\pi\)
0.351662 + 0.936127i \(0.385617\pi\)
\(38\) 9.04576 1.46742
\(39\) 7.05679 1.12999
\(40\) 0 0
\(41\) −5.12330 −0.800125 −0.400063 0.916488i \(-0.631012\pi\)
−0.400063 + 0.916488i \(0.631012\pi\)
\(42\) −0.362456 −0.0559282
\(43\) 3.94180 0.601118 0.300559 0.953763i \(-0.402827\pi\)
0.300559 + 0.953763i \(0.402827\pi\)
\(44\) −10.1998 −1.53768
\(45\) 0 0
\(46\) 4.67903 0.689885
\(47\) −5.61626 −0.819215 −0.409608 0.912262i \(-0.634334\pi\)
−0.409608 + 0.912262i \(0.634334\pi\)
\(48\) 0.424361 0.0612513
\(49\) −6.97444 −0.996348
\(50\) 0 0
\(51\) 7.63808 1.06955
\(52\) −22.1518 −3.07190
\(53\) −12.0822 −1.65962 −0.829808 0.558049i \(-0.811550\pi\)
−0.829808 + 0.558049i \(0.811550\pi\)
\(54\) 2.26695 0.308493
\(55\) 0 0
\(56\) 0.412864 0.0551713
\(57\) 3.99028 0.528525
\(58\) 2.26695 0.297665
\(59\) −7.06781 −0.920151 −0.460075 0.887880i \(-0.652178\pi\)
−0.460075 + 0.887880i \(0.652178\pi\)
\(60\) 0 0
\(61\) 11.4310 1.46359 0.731796 0.681524i \(-0.238682\pi\)
0.731796 + 0.681524i \(0.238682\pi\)
\(62\) −23.1329 −2.93788
\(63\) −0.159887 −0.0201439
\(64\) −13.0397 −1.62996
\(65\) 0 0
\(66\) −7.36601 −0.906694
\(67\) 14.9512 1.82658 0.913289 0.407311i \(-0.133534\pi\)
0.913289 + 0.407311i \(0.133534\pi\)
\(68\) −23.9765 −2.90758
\(69\) 2.06402 0.248478
\(70\) 0 0
\(71\) 3.43347 0.407478 0.203739 0.979025i \(-0.434691\pi\)
0.203739 + 0.979025i \(0.434691\pi\)
\(72\) −2.58223 −0.304318
\(73\) 12.0772 1.41353 0.706764 0.707449i \(-0.250154\pi\)
0.706764 + 0.707449i \(0.250154\pi\)
\(74\) −9.69836 −1.12741
\(75\) 0 0
\(76\) −12.5258 −1.43680
\(77\) 0.519521 0.0592049
\(78\) −15.9974 −1.81135
\(79\) −12.7441 −1.43383 −0.716913 0.697163i \(-0.754446\pi\)
−0.716913 + 0.697163i \(0.754446\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 11.6143 1.28258
\(83\) 2.59792 0.285159 0.142580 0.989783i \(-0.454460\pi\)
0.142580 + 0.989783i \(0.454460\pi\)
\(84\) 0.501897 0.0547614
\(85\) 0 0
\(86\) −8.93586 −0.963579
\(87\) 1.00000 0.107211
\(88\) 8.39044 0.894423
\(89\) 4.33165 0.459154 0.229577 0.973290i \(-0.426266\pi\)
0.229577 + 0.973290i \(0.426266\pi\)
\(90\) 0 0
\(91\) 1.12829 0.118277
\(92\) −6.47910 −0.675493
\(93\) −10.2044 −1.05815
\(94\) 12.7318 1.31318
\(95\) 0 0
\(96\) −6.12646 −0.625279
\(97\) 3.88355 0.394315 0.197158 0.980372i \(-0.436829\pi\)
0.197158 + 0.980372i \(0.436829\pi\)
\(98\) 15.8107 1.59712
\(99\) −3.24930 −0.326567
\(100\) 0 0
\(101\) 15.4045 1.53280 0.766400 0.642363i \(-0.222046\pi\)
0.766400 + 0.642363i \(0.222046\pi\)
\(102\) −17.3152 −1.71446
\(103\) −7.68484 −0.757210 −0.378605 0.925558i \(-0.623596\pi\)
−0.378605 + 0.925558i \(0.623596\pi\)
\(104\) 18.2222 1.78684
\(105\) 0 0
\(106\) 27.3897 2.66033
\(107\) 4.60924 0.445592 0.222796 0.974865i \(-0.428482\pi\)
0.222796 + 0.974865i \(0.428482\pi\)
\(108\) −3.13907 −0.302058
\(109\) −14.4509 −1.38415 −0.692073 0.721827i \(-0.743302\pi\)
−0.692073 + 0.721827i \(0.743302\pi\)
\(110\) 0 0
\(111\) −4.27815 −0.406064
\(112\) 0.0678498 0.00641120
\(113\) 4.08614 0.384392 0.192196 0.981357i \(-0.438439\pi\)
0.192196 + 0.981357i \(0.438439\pi\)
\(114\) −9.04576 −0.847213
\(115\) 0 0
\(116\) −3.13907 −0.291456
\(117\) −7.05679 −0.652400
\(118\) 16.0224 1.47498
\(119\) 1.22123 0.111950
\(120\) 0 0
\(121\) −0.442038 −0.0401853
\(122\) −25.9136 −2.34610
\(123\) 5.12330 0.461952
\(124\) 32.0324 2.87659
\(125\) 0 0
\(126\) 0.362456 0.0322901
\(127\) 3.69650 0.328011 0.164006 0.986459i \(-0.447559\pi\)
0.164006 + 0.986459i \(0.447559\pi\)
\(128\) 17.3074 1.52977
\(129\) −3.94180 −0.347056
\(130\) 0 0
\(131\) −9.92811 −0.867423 −0.433712 0.901052i \(-0.642796\pi\)
−0.433712 + 0.901052i \(0.642796\pi\)
\(132\) 10.1998 0.887779
\(133\) 0.637993 0.0553210
\(134\) −33.8936 −2.92797
\(135\) 0 0
\(136\) 19.7233 1.69126
\(137\) −3.82810 −0.327057 −0.163528 0.986539i \(-0.552288\pi\)
−0.163528 + 0.986539i \(0.552288\pi\)
\(138\) −4.67903 −0.398305
\(139\) −2.06987 −0.175564 −0.0877822 0.996140i \(-0.527978\pi\)
−0.0877822 + 0.996140i \(0.527978\pi\)
\(140\) 0 0
\(141\) 5.61626 0.472974
\(142\) −7.78351 −0.653177
\(143\) 22.9296 1.91747
\(144\) −0.424361 −0.0353634
\(145\) 0 0
\(146\) −27.3784 −2.26585
\(147\) 6.97444 0.575242
\(148\) 13.4294 1.10389
\(149\) −1.46609 −0.120107 −0.0600536 0.998195i \(-0.519127\pi\)
−0.0600536 + 0.998195i \(0.519127\pi\)
\(150\) 0 0
\(151\) −20.1279 −1.63799 −0.818995 0.573801i \(-0.805468\pi\)
−0.818995 + 0.573801i \(0.805468\pi\)
\(152\) 10.3038 0.835748
\(153\) −7.63808 −0.617502
\(154\) −1.17773 −0.0949041
\(155\) 0 0
\(156\) 22.1518 1.77356
\(157\) −7.09106 −0.565928 −0.282964 0.959131i \(-0.591318\pi\)
−0.282964 + 0.959131i \(0.591318\pi\)
\(158\) 28.8903 2.29839
\(159\) 12.0822 0.958180
\(160\) 0 0
\(161\) 0.330009 0.0260084
\(162\) −2.26695 −0.178109
\(163\) 6.58715 0.515946 0.257973 0.966152i \(-0.416945\pi\)
0.257973 + 0.966152i \(0.416945\pi\)
\(164\) −16.0824 −1.25583
\(165\) 0 0
\(166\) −5.88937 −0.457104
\(167\) 10.6638 0.825192 0.412596 0.910914i \(-0.364622\pi\)
0.412596 + 0.910914i \(0.364622\pi\)
\(168\) −0.412864 −0.0318532
\(169\) 36.7983 2.83063
\(170\) 0 0
\(171\) −3.99028 −0.305144
\(172\) 12.3736 0.943477
\(173\) 13.8799 1.05527 0.527633 0.849472i \(-0.323080\pi\)
0.527633 + 0.849472i \(0.323080\pi\)
\(174\) −2.26695 −0.171857
\(175\) 0 0
\(176\) 1.37888 0.103937
\(177\) 7.06781 0.531249
\(178\) −9.81965 −0.736014
\(179\) −11.1105 −0.830439 −0.415219 0.909721i \(-0.636295\pi\)
−0.415219 + 0.909721i \(0.636295\pi\)
\(180\) 0 0
\(181\) 10.9453 0.813557 0.406778 0.913527i \(-0.366652\pi\)
0.406778 + 0.913527i \(0.366652\pi\)
\(182\) −2.55777 −0.189595
\(183\) −11.4310 −0.845005
\(184\) 5.32976 0.392915
\(185\) 0 0
\(186\) 23.1329 1.69619
\(187\) 24.8184 1.81490
\(188\) −17.6298 −1.28579
\(189\) 0.159887 0.0116301
\(190\) 0 0
\(191\) 18.1472 1.31309 0.656544 0.754287i \(-0.272017\pi\)
0.656544 + 0.754287i \(0.272017\pi\)
\(192\) 13.0397 0.941058
\(193\) 17.3305 1.24747 0.623737 0.781634i \(-0.285614\pi\)
0.623737 + 0.781634i \(0.285614\pi\)
\(194\) −8.80383 −0.632078
\(195\) 0 0
\(196\) −21.8933 −1.56381
\(197\) 10.4273 0.742915 0.371458 0.928450i \(-0.378858\pi\)
0.371458 + 0.928450i \(0.378858\pi\)
\(198\) 7.36601 0.523480
\(199\) 11.5570 0.819256 0.409628 0.912253i \(-0.365659\pi\)
0.409628 + 0.912253i \(0.365659\pi\)
\(200\) 0 0
\(201\) −14.9512 −1.05458
\(202\) −34.9212 −2.45704
\(203\) 0.159887 0.0112219
\(204\) 23.9765 1.67869
\(205\) 0 0
\(206\) 17.4212 1.21379
\(207\) −2.06402 −0.143459
\(208\) 2.99463 0.207640
\(209\) 12.9656 0.896850
\(210\) 0 0
\(211\) −21.1616 −1.45682 −0.728412 0.685140i \(-0.759741\pi\)
−0.728412 + 0.685140i \(0.759741\pi\)
\(212\) −37.9269 −2.60483
\(213\) −3.43347 −0.235257
\(214\) −10.4489 −0.714274
\(215\) 0 0
\(216\) 2.58223 0.175698
\(217\) −1.63155 −0.110757
\(218\) 32.7595 2.21876
\(219\) −12.0772 −0.816101
\(220\) 0 0
\(221\) 53.9003 3.62573
\(222\) 9.69836 0.650911
\(223\) −9.46925 −0.634108 −0.317054 0.948408i \(-0.602694\pi\)
−0.317054 + 0.948408i \(0.602694\pi\)
\(224\) −0.979541 −0.0654483
\(225\) 0 0
\(226\) −9.26310 −0.616172
\(227\) −22.4030 −1.48694 −0.743471 0.668768i \(-0.766822\pi\)
−0.743471 + 0.668768i \(0.766822\pi\)
\(228\) 12.5258 0.829539
\(229\) 2.03117 0.134223 0.0671117 0.997745i \(-0.478622\pi\)
0.0671117 + 0.997745i \(0.478622\pi\)
\(230\) 0 0
\(231\) −0.519521 −0.0341820
\(232\) 2.58223 0.169532
\(233\) −11.0704 −0.725249 −0.362624 0.931935i \(-0.618119\pi\)
−0.362624 + 0.931935i \(0.618119\pi\)
\(234\) 15.9974 1.04578
\(235\) 0 0
\(236\) −22.1864 −1.44421
\(237\) 12.7441 0.827820
\(238\) −2.76847 −0.179453
\(239\) −2.24098 −0.144957 −0.0724786 0.997370i \(-0.523091\pi\)
−0.0724786 + 0.997370i \(0.523091\pi\)
\(240\) 0 0
\(241\) 14.4838 0.932986 0.466493 0.884525i \(-0.345517\pi\)
0.466493 + 0.884525i \(0.345517\pi\)
\(242\) 1.00208 0.0644161
\(243\) −1.00000 −0.0641500
\(244\) 35.8828 2.29716
\(245\) 0 0
\(246\) −11.6143 −0.740499
\(247\) 28.1585 1.79168
\(248\) −26.3501 −1.67323
\(249\) −2.59792 −0.164637
\(250\) 0 0
\(251\) −16.2044 −1.02281 −0.511407 0.859339i \(-0.670875\pi\)
−0.511407 + 0.859339i \(0.670875\pi\)
\(252\) −0.501897 −0.0316165
\(253\) 6.70661 0.421641
\(254\) −8.37978 −0.525794
\(255\) 0 0
\(256\) −13.1557 −0.822232
\(257\) 3.36942 0.210178 0.105089 0.994463i \(-0.466487\pi\)
0.105089 + 0.994463i \(0.466487\pi\)
\(258\) 8.93586 0.556323
\(259\) −0.684020 −0.0425029
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 22.5066 1.39046
\(263\) −1.79671 −0.110790 −0.0553950 0.998465i \(-0.517642\pi\)
−0.0553950 + 0.998465i \(0.517642\pi\)
\(264\) −8.39044 −0.516396
\(265\) 0 0
\(266\) −1.44630 −0.0886783
\(267\) −4.33165 −0.265093
\(268\) 46.9329 2.86688
\(269\) −0.117055 −0.00713699 −0.00356849 0.999994i \(-0.501136\pi\)
−0.00356849 + 0.999994i \(0.501136\pi\)
\(270\) 0 0
\(271\) −4.06294 −0.246806 −0.123403 0.992357i \(-0.539381\pi\)
−0.123403 + 0.992357i \(0.539381\pi\)
\(272\) 3.24131 0.196533
\(273\) −1.12829 −0.0682871
\(274\) 8.67813 0.524265
\(275\) 0 0
\(276\) 6.47910 0.389996
\(277\) 12.5436 0.753674 0.376837 0.926279i \(-0.377012\pi\)
0.376837 + 0.926279i \(0.377012\pi\)
\(278\) 4.69231 0.281426
\(279\) 10.2044 0.610922
\(280\) 0 0
\(281\) −2.83163 −0.168921 −0.0844605 0.996427i \(-0.526917\pi\)
−0.0844605 + 0.996427i \(0.526917\pi\)
\(282\) −12.7318 −0.758167
\(283\) −28.2249 −1.67780 −0.838898 0.544288i \(-0.816800\pi\)
−0.838898 + 0.544288i \(0.816800\pi\)
\(284\) 10.7779 0.639551
\(285\) 0 0
\(286\) −51.9804 −3.07366
\(287\) 0.819148 0.0483528
\(288\) 6.12646 0.361005
\(289\) 41.3403 2.43178
\(290\) 0 0
\(291\) −3.88355 −0.227658
\(292\) 37.9112 2.21859
\(293\) −21.9337 −1.28138 −0.640691 0.767799i \(-0.721352\pi\)
−0.640691 + 0.767799i \(0.721352\pi\)
\(294\) −15.8107 −0.922100
\(295\) 0 0
\(296\) −11.0472 −0.642103
\(297\) 3.24930 0.188544
\(298\) 3.32357 0.192529
\(299\) 14.5653 0.842335
\(300\) 0 0
\(301\) −0.630241 −0.0363265
\(302\) 45.6291 2.62566
\(303\) −15.4045 −0.884963
\(304\) 1.69332 0.0971185
\(305\) 0 0
\(306\) 17.3152 0.989843
\(307\) 0.461574 0.0263434 0.0131717 0.999913i \(-0.495807\pi\)
0.0131717 + 0.999913i \(0.495807\pi\)
\(308\) 1.63081 0.0929243
\(309\) 7.68484 0.437175
\(310\) 0 0
\(311\) 27.5507 1.56226 0.781128 0.624371i \(-0.214645\pi\)
0.781128 + 0.624371i \(0.214645\pi\)
\(312\) −18.2222 −1.03163
\(313\) −14.9121 −0.842882 −0.421441 0.906856i \(-0.638476\pi\)
−0.421441 + 0.906856i \(0.638476\pi\)
\(314\) 16.0751 0.907170
\(315\) 0 0
\(316\) −40.0048 −2.25044
\(317\) 22.9266 1.28769 0.643843 0.765157i \(-0.277339\pi\)
0.643843 + 0.765157i \(0.277339\pi\)
\(318\) −27.3897 −1.53594
\(319\) 3.24930 0.181926
\(320\) 0 0
\(321\) −4.60924 −0.257263
\(322\) −0.748115 −0.0416908
\(323\) 30.4781 1.69584
\(324\) 3.13907 0.174393
\(325\) 0 0
\(326\) −14.9328 −0.827049
\(327\) 14.4509 0.799137
\(328\) 13.2295 0.730478
\(329\) 0.897966 0.0495065
\(330\) 0 0
\(331\) −3.23147 −0.177618 −0.0888088 0.996049i \(-0.528306\pi\)
−0.0888088 + 0.996049i \(0.528306\pi\)
\(332\) 8.15507 0.447568
\(333\) 4.27815 0.234441
\(334\) −24.1744 −1.32277
\(335\) 0 0
\(336\) −0.0678498 −0.00370151
\(337\) −28.8026 −1.56898 −0.784489 0.620143i \(-0.787075\pi\)
−0.784489 + 0.620143i \(0.787075\pi\)
\(338\) −83.4199 −4.53744
\(339\) −4.08614 −0.221929
\(340\) 0 0
\(341\) −33.1572 −1.79556
\(342\) 9.04576 0.489139
\(343\) 2.23433 0.120642
\(344\) −10.1786 −0.548794
\(345\) 0 0
\(346\) −31.4650 −1.69157
\(347\) 16.4175 0.881337 0.440669 0.897670i \(-0.354741\pi\)
0.440669 + 0.897670i \(0.354741\pi\)
\(348\) 3.13907 0.168272
\(349\) −18.2107 −0.974796 −0.487398 0.873180i \(-0.662054\pi\)
−0.487398 + 0.873180i \(0.662054\pi\)
\(350\) 0 0
\(351\) 7.05679 0.376663
\(352\) −19.9067 −1.06103
\(353\) 18.8790 1.00483 0.502414 0.864627i \(-0.332445\pi\)
0.502414 + 0.864627i \(0.332445\pi\)
\(354\) −16.0224 −0.851581
\(355\) 0 0
\(356\) 13.5974 0.720660
\(357\) −1.22123 −0.0646343
\(358\) 25.1870 1.33117
\(359\) 14.4500 0.762640 0.381320 0.924443i \(-0.375470\pi\)
0.381320 + 0.924443i \(0.375470\pi\)
\(360\) 0 0
\(361\) −3.07770 −0.161984
\(362\) −24.8125 −1.30411
\(363\) 0.442038 0.0232010
\(364\) 3.54178 0.185640
\(365\) 0 0
\(366\) 25.9136 1.35452
\(367\) 21.1171 1.10231 0.551153 0.834404i \(-0.314188\pi\)
0.551153 + 0.834404i \(0.314188\pi\)
\(368\) 0.875889 0.0456589
\(369\) −5.12330 −0.266708
\(370\) 0 0
\(371\) 1.93178 0.100293
\(372\) −32.0324 −1.66080
\(373\) 17.1185 0.886361 0.443181 0.896432i \(-0.353850\pi\)
0.443181 + 0.896432i \(0.353850\pi\)
\(374\) −56.2622 −2.90925
\(375\) 0 0
\(376\) 14.5025 0.747907
\(377\) 7.05679 0.363443
\(378\) −0.362456 −0.0186427
\(379\) −0.206688 −0.0106169 −0.00530844 0.999986i \(-0.501690\pi\)
−0.00530844 + 0.999986i \(0.501690\pi\)
\(380\) 0 0
\(381\) −3.69650 −0.189377
\(382\) −41.1389 −2.10485
\(383\) −16.3701 −0.836474 −0.418237 0.908338i \(-0.637352\pi\)
−0.418237 + 0.908338i \(0.637352\pi\)
\(384\) −17.3074 −0.883215
\(385\) 0 0
\(386\) −39.2873 −1.99967
\(387\) 3.94180 0.200373
\(388\) 12.1908 0.618892
\(389\) 11.7627 0.596393 0.298197 0.954504i \(-0.403615\pi\)
0.298197 + 0.954504i \(0.403615\pi\)
\(390\) 0 0
\(391\) 15.7651 0.797277
\(392\) 18.0096 0.909621
\(393\) 9.92811 0.500807
\(394\) −23.6382 −1.19088
\(395\) 0 0
\(396\) −10.1998 −0.512559
\(397\) 4.96134 0.249002 0.124501 0.992219i \(-0.460267\pi\)
0.124501 + 0.992219i \(0.460267\pi\)
\(398\) −26.1992 −1.31325
\(399\) −0.637993 −0.0319396
\(400\) 0 0
\(401\) 36.1967 1.80758 0.903789 0.427979i \(-0.140774\pi\)
0.903789 + 0.427979i \(0.140774\pi\)
\(402\) 33.8936 1.69046
\(403\) −72.0103 −3.58709
\(404\) 48.3557 2.40579
\(405\) 0 0
\(406\) −0.362456 −0.0179884
\(407\) −13.9010 −0.689047
\(408\) −19.7233 −0.976447
\(409\) −16.1607 −0.799094 −0.399547 0.916713i \(-0.630832\pi\)
−0.399547 + 0.916713i \(0.630832\pi\)
\(410\) 0 0
\(411\) 3.82810 0.188826
\(412\) −24.1233 −1.18847
\(413\) 1.13005 0.0556061
\(414\) 4.67903 0.229962
\(415\) 0 0
\(416\) −43.2331 −2.11968
\(417\) 2.06987 0.101362
\(418\) −29.3924 −1.43763
\(419\) −33.5544 −1.63924 −0.819621 0.572906i \(-0.805816\pi\)
−0.819621 + 0.572906i \(0.805816\pi\)
\(420\) 0 0
\(421\) 26.9847 1.31515 0.657577 0.753388i \(-0.271582\pi\)
0.657577 + 0.753388i \(0.271582\pi\)
\(422\) 47.9723 2.33526
\(423\) −5.61626 −0.273072
\(424\) 31.1990 1.51515
\(425\) 0 0
\(426\) 7.78351 0.377112
\(427\) −1.82767 −0.0884471
\(428\) 14.4687 0.699373
\(429\) −22.9296 −1.10705
\(430\) 0 0
\(431\) 0.832366 0.0400937 0.0200468 0.999799i \(-0.493618\pi\)
0.0200468 + 0.999799i \(0.493618\pi\)
\(432\) 0.424361 0.0204171
\(433\) 22.5647 1.08439 0.542195 0.840253i \(-0.317593\pi\)
0.542195 + 0.840253i \(0.317593\pi\)
\(434\) 3.69865 0.177541
\(435\) 0 0
\(436\) −45.3625 −2.17247
\(437\) 8.23600 0.393981
\(438\) 27.3784 1.30819
\(439\) 23.9583 1.14347 0.571734 0.820439i \(-0.306271\pi\)
0.571734 + 0.820439i \(0.306271\pi\)
\(440\) 0 0
\(441\) −6.97444 −0.332116
\(442\) −122.190 −5.81196
\(443\) −17.4925 −0.831092 −0.415546 0.909572i \(-0.636409\pi\)
−0.415546 + 0.909572i \(0.636409\pi\)
\(444\) −13.4294 −0.637332
\(445\) 0 0
\(446\) 21.4663 1.01646
\(447\) 1.46609 0.0693439
\(448\) 2.08487 0.0985010
\(449\) 4.22200 0.199249 0.0996243 0.995025i \(-0.468236\pi\)
0.0996243 + 0.995025i \(0.468236\pi\)
\(450\) 0 0
\(451\) 16.6471 0.783884
\(452\) 12.8267 0.603318
\(453\) 20.1279 0.945694
\(454\) 50.7866 2.38353
\(455\) 0 0
\(456\) −10.3038 −0.482520
\(457\) 1.28562 0.0601389 0.0300694 0.999548i \(-0.490427\pi\)
0.0300694 + 0.999548i \(0.490427\pi\)
\(458\) −4.60456 −0.215157
\(459\) 7.63808 0.356515
\(460\) 0 0
\(461\) −23.8534 −1.11097 −0.555483 0.831528i \(-0.687466\pi\)
−0.555483 + 0.831528i \(0.687466\pi\)
\(462\) 1.17773 0.0547929
\(463\) −14.1402 −0.657152 −0.328576 0.944478i \(-0.606569\pi\)
−0.328576 + 0.944478i \(0.606569\pi\)
\(464\) 0.424361 0.0197005
\(465\) 0 0
\(466\) 25.0962 1.16256
\(467\) −7.06487 −0.326923 −0.163462 0.986550i \(-0.552266\pi\)
−0.163462 + 0.986550i \(0.552266\pi\)
\(468\) −22.1518 −1.02397
\(469\) −2.39050 −0.110383
\(470\) 0 0
\(471\) 7.09106 0.326739
\(472\) 18.2507 0.840056
\(473\) −12.8081 −0.588916
\(474\) −28.8903 −1.32698
\(475\) 0 0
\(476\) 3.83353 0.175710
\(477\) −12.0822 −0.553205
\(478\) 5.08020 0.232363
\(479\) 22.9561 1.04889 0.524444 0.851445i \(-0.324273\pi\)
0.524444 + 0.851445i \(0.324273\pi\)
\(480\) 0 0
\(481\) −30.1900 −1.37654
\(482\) −32.8342 −1.49556
\(483\) −0.330009 −0.0150159
\(484\) −1.38759 −0.0630723
\(485\) 0 0
\(486\) 2.26695 0.102831
\(487\) 27.0319 1.22493 0.612467 0.790496i \(-0.290177\pi\)
0.612467 + 0.790496i \(0.290177\pi\)
\(488\) −29.5175 −1.33619
\(489\) −6.58715 −0.297881
\(490\) 0 0
\(491\) −11.5498 −0.521235 −0.260617 0.965442i \(-0.583926\pi\)
−0.260617 + 0.965442i \(0.583926\pi\)
\(492\) 16.0824 0.725051
\(493\) 7.63808 0.344002
\(494\) −63.8340 −2.87203
\(495\) 0 0
\(496\) −4.33036 −0.194439
\(497\) −0.548966 −0.0246245
\(498\) 5.88937 0.263909
\(499\) 3.77942 0.169190 0.0845950 0.996415i \(-0.473040\pi\)
0.0845950 + 0.996415i \(0.473040\pi\)
\(500\) 0 0
\(501\) −10.6638 −0.476425
\(502\) 36.7346 1.63955
\(503\) 14.7684 0.658489 0.329244 0.944245i \(-0.393206\pi\)
0.329244 + 0.944245i \(0.393206\pi\)
\(504\) 0.412864 0.0183904
\(505\) 0 0
\(506\) −15.2036 −0.675881
\(507\) −36.7983 −1.63427
\(508\) 11.6036 0.514826
\(509\) −42.4182 −1.88015 −0.940077 0.340962i \(-0.889247\pi\)
−0.940077 + 0.340962i \(0.889247\pi\)
\(510\) 0 0
\(511\) −1.93098 −0.0854217
\(512\) −4.79143 −0.211753
\(513\) 3.99028 0.176175
\(514\) −7.63831 −0.336911
\(515\) 0 0
\(516\) −12.3736 −0.544717
\(517\) 18.2489 0.802586
\(518\) 1.55064 0.0681312
\(519\) −13.8799 −0.609258
\(520\) 0 0
\(521\) 20.3409 0.891152 0.445576 0.895244i \(-0.352999\pi\)
0.445576 + 0.895244i \(0.352999\pi\)
\(522\) 2.26695 0.0992218
\(523\) −1.52190 −0.0665481 −0.0332740 0.999446i \(-0.510593\pi\)
−0.0332740 + 0.999446i \(0.510593\pi\)
\(524\) −31.1651 −1.36145
\(525\) 0 0
\(526\) 4.07306 0.177594
\(527\) −77.9421 −3.39521
\(528\) −1.37888 −0.0600080
\(529\) −18.7398 −0.814775
\(530\) 0 0
\(531\) −7.06781 −0.306717
\(532\) 2.00271 0.0868283
\(533\) 36.1540 1.56601
\(534\) 9.81965 0.424938
\(535\) 0 0
\(536\) −38.6074 −1.66758
\(537\) 11.1105 0.479454
\(538\) 0.265359 0.0114404
\(539\) 22.6620 0.976123
\(540\) 0 0
\(541\) −9.40679 −0.404430 −0.202215 0.979341i \(-0.564814\pi\)
−0.202215 + 0.979341i \(0.564814\pi\)
\(542\) 9.21049 0.395624
\(543\) −10.9453 −0.469707
\(544\) −46.7944 −2.00629
\(545\) 0 0
\(546\) 2.55777 0.109463
\(547\) −19.7232 −0.843304 −0.421652 0.906758i \(-0.638550\pi\)
−0.421652 + 0.906758i \(0.638550\pi\)
\(548\) −12.0167 −0.513328
\(549\) 11.4310 0.487864
\(550\) 0 0
\(551\) 3.99028 0.169991
\(552\) −5.32976 −0.226850
\(553\) 2.03762 0.0866483
\(554\) −28.4358 −1.20812
\(555\) 0 0
\(556\) −6.49749 −0.275555
\(557\) −9.01986 −0.382184 −0.191092 0.981572i \(-0.561203\pi\)
−0.191092 + 0.981572i \(0.561203\pi\)
\(558\) −23.1329 −0.979294
\(559\) −27.8164 −1.17651
\(560\) 0 0
\(561\) −24.8184 −1.04784
\(562\) 6.41918 0.270777
\(563\) −17.5283 −0.738728 −0.369364 0.929285i \(-0.620424\pi\)
−0.369364 + 0.929285i \(0.620424\pi\)
\(564\) 17.6298 0.742350
\(565\) 0 0
\(566\) 63.9845 2.68947
\(567\) −0.159887 −0.00671462
\(568\) −8.86599 −0.372009
\(569\) 32.3981 1.35820 0.679099 0.734046i \(-0.262371\pi\)
0.679099 + 0.734046i \(0.262371\pi\)
\(570\) 0 0
\(571\) −11.8393 −0.495458 −0.247729 0.968829i \(-0.579684\pi\)
−0.247729 + 0.968829i \(0.579684\pi\)
\(572\) 71.9778 3.00954
\(573\) −18.1472 −0.758112
\(574\) −1.85697 −0.0775085
\(575\) 0 0
\(576\) −13.0397 −0.543320
\(577\) 0.192778 0.00802545 0.00401273 0.999992i \(-0.498723\pi\)
0.00401273 + 0.999992i \(0.498723\pi\)
\(578\) −93.7165 −3.89809
\(579\) −17.3305 −0.720229
\(580\) 0 0
\(581\) −0.415374 −0.0172326
\(582\) 8.80383 0.364931
\(583\) 39.2587 1.62593
\(584\) −31.1861 −1.29049
\(585\) 0 0
\(586\) 49.7227 2.05403
\(587\) −34.5089 −1.42433 −0.712167 0.702010i \(-0.752286\pi\)
−0.712167 + 0.702010i \(0.752286\pi\)
\(588\) 21.8933 0.902863
\(589\) −40.7184 −1.67777
\(590\) 0 0
\(591\) −10.4273 −0.428922
\(592\) −1.81548 −0.0746158
\(593\) −2.42748 −0.0996846 −0.0498423 0.998757i \(-0.515872\pi\)
−0.0498423 + 0.998757i \(0.515872\pi\)
\(594\) −7.36601 −0.302231
\(595\) 0 0
\(596\) −4.60218 −0.188513
\(597\) −11.5570 −0.472997
\(598\) −33.0189 −1.35024
\(599\) 29.3312 1.19844 0.599220 0.800585i \(-0.295478\pi\)
0.599220 + 0.800585i \(0.295478\pi\)
\(600\) 0 0
\(601\) 29.5631 1.20590 0.602952 0.797778i \(-0.293991\pi\)
0.602952 + 0.797778i \(0.293991\pi\)
\(602\) 1.42873 0.0582306
\(603\) 14.9512 0.608860
\(604\) −63.1831 −2.57089
\(605\) 0 0
\(606\) 34.9212 1.41858
\(607\) −33.1756 −1.34656 −0.673278 0.739390i \(-0.735114\pi\)
−0.673278 + 0.739390i \(0.735114\pi\)
\(608\) −24.4463 −0.991427
\(609\) −0.159887 −0.00647894
\(610\) 0 0
\(611\) 39.6327 1.60337
\(612\) −23.9765 −0.969193
\(613\) 13.9979 0.565371 0.282686 0.959213i \(-0.408775\pi\)
0.282686 + 0.959213i \(0.408775\pi\)
\(614\) −1.04637 −0.0422279
\(615\) 0 0
\(616\) −1.34152 −0.0540514
\(617\) 29.9344 1.20511 0.602556 0.798077i \(-0.294149\pi\)
0.602556 + 0.798077i \(0.294149\pi\)
\(618\) −17.4212 −0.700782
\(619\) −14.9032 −0.599009 −0.299505 0.954095i \(-0.596821\pi\)
−0.299505 + 0.954095i \(0.596821\pi\)
\(620\) 0 0
\(621\) 2.06402 0.0828261
\(622\) −62.4561 −2.50426
\(623\) −0.692574 −0.0277474
\(624\) −2.99463 −0.119881
\(625\) 0 0
\(626\) 33.8050 1.35112
\(627\) −12.9656 −0.517796
\(628\) −22.2594 −0.888245
\(629\) −32.6769 −1.30291
\(630\) 0 0
\(631\) −28.8643 −1.14907 −0.574534 0.818480i \(-0.694817\pi\)
−0.574534 + 0.818480i \(0.694817\pi\)
\(632\) 32.9082 1.30902
\(633\) 21.1616 0.841097
\(634\) −51.9735 −2.06413
\(635\) 0 0
\(636\) 37.9269 1.50390
\(637\) 49.2171 1.95005
\(638\) −7.36601 −0.291623
\(639\) 3.43347 0.135826
\(640\) 0 0
\(641\) 44.0167 1.73856 0.869278 0.494324i \(-0.164585\pi\)
0.869278 + 0.494324i \(0.164585\pi\)
\(642\) 10.4489 0.412386
\(643\) 29.6239 1.16825 0.584125 0.811663i \(-0.301438\pi\)
0.584125 + 0.811663i \(0.301438\pi\)
\(644\) 1.03592 0.0408211
\(645\) 0 0
\(646\) −69.0923 −2.71840
\(647\) 39.3943 1.54875 0.774375 0.632727i \(-0.218064\pi\)
0.774375 + 0.632727i \(0.218064\pi\)
\(648\) −2.58223 −0.101439
\(649\) 22.9654 0.901473
\(650\) 0 0
\(651\) 1.63155 0.0639455
\(652\) 20.6776 0.809796
\(653\) −22.3518 −0.874692 −0.437346 0.899293i \(-0.644082\pi\)
−0.437346 + 0.899293i \(0.644082\pi\)
\(654\) −32.7595 −1.28100
\(655\) 0 0
\(656\) 2.17413 0.0848855
\(657\) 12.0772 0.471176
\(658\) −2.03565 −0.0793577
\(659\) 42.1979 1.64380 0.821898 0.569634i \(-0.192915\pi\)
0.821898 + 0.569634i \(0.192915\pi\)
\(660\) 0 0
\(661\) 8.38035 0.325958 0.162979 0.986630i \(-0.447890\pi\)
0.162979 + 0.986630i \(0.447890\pi\)
\(662\) 7.32559 0.284717
\(663\) −53.9003 −2.09332
\(664\) −6.70843 −0.260337
\(665\) 0 0
\(666\) −9.69836 −0.375804
\(667\) 2.06402 0.0799191
\(668\) 33.4746 1.29517
\(669\) 9.46925 0.366102
\(670\) 0 0
\(671\) −37.1428 −1.43388
\(672\) 0.979541 0.0377866
\(673\) 3.13725 0.120932 0.0604660 0.998170i \(-0.480741\pi\)
0.0604660 + 0.998170i \(0.480741\pi\)
\(674\) 65.2941 2.51504
\(675\) 0 0
\(676\) 115.512 4.44279
\(677\) −16.2776 −0.625600 −0.312800 0.949819i \(-0.601267\pi\)
−0.312800 + 0.949819i \(0.601267\pi\)
\(678\) 9.26310 0.355747
\(679\) −0.620929 −0.0238291
\(680\) 0 0
\(681\) 22.4030 0.858486
\(682\) 75.1658 2.87825
\(683\) 1.78249 0.0682050 0.0341025 0.999418i \(-0.489143\pi\)
0.0341025 + 0.999418i \(0.489143\pi\)
\(684\) −12.5258 −0.478935
\(685\) 0 0
\(686\) −5.06512 −0.193387
\(687\) −2.03117 −0.0774939
\(688\) −1.67275 −0.0637728
\(689\) 85.2614 3.24820
\(690\) 0 0
\(691\) −20.8504 −0.793187 −0.396594 0.917994i \(-0.629808\pi\)
−0.396594 + 0.917994i \(0.629808\pi\)
\(692\) 43.5699 1.65628
\(693\) 0.519521 0.0197350
\(694\) −37.2177 −1.41276
\(695\) 0 0
\(696\) −2.58223 −0.0978791
\(697\) 39.1322 1.48224
\(698\) 41.2828 1.56258
\(699\) 11.0704 0.418722
\(700\) 0 0
\(701\) 32.5504 1.22941 0.614707 0.788756i \(-0.289274\pi\)
0.614707 + 0.788756i \(0.289274\pi\)
\(702\) −15.9974 −0.603783
\(703\) −17.0710 −0.643845
\(704\) 42.3698 1.59687
\(705\) 0 0
\(706\) −42.7978 −1.61072
\(707\) −2.46297 −0.0926295
\(708\) 22.1864 0.833815
\(709\) 5.93195 0.222779 0.111390 0.993777i \(-0.464470\pi\)
0.111390 + 0.993777i \(0.464470\pi\)
\(710\) 0 0
\(711\) −12.7441 −0.477942
\(712\) −11.1853 −0.419187
\(713\) −21.0621 −0.788781
\(714\) 2.76847 0.103607
\(715\) 0 0
\(716\) −34.8767 −1.30340
\(717\) 2.24098 0.0836911
\(718\) −32.7574 −1.22250
\(719\) −13.5279 −0.504505 −0.252252 0.967661i \(-0.581171\pi\)
−0.252252 + 0.967661i \(0.581171\pi\)
\(720\) 0 0
\(721\) 1.22871 0.0457594
\(722\) 6.97701 0.259657
\(723\) −14.4838 −0.538660
\(724\) 34.3581 1.27691
\(725\) 0 0
\(726\) −1.00208 −0.0371907
\(727\) 12.7407 0.472525 0.236263 0.971689i \(-0.424077\pi\)
0.236263 + 0.971689i \(0.424077\pi\)
\(728\) −2.91350 −0.107981
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −30.1078 −1.11358
\(732\) −35.8828 −1.32627
\(733\) −30.9422 −1.14288 −0.571438 0.820645i \(-0.693614\pi\)
−0.571438 + 0.820645i \(0.693614\pi\)
\(734\) −47.8716 −1.76697
\(735\) 0 0
\(736\) −12.6451 −0.466105
\(737\) −48.5809 −1.78950
\(738\) 11.6143 0.427528
\(739\) 35.0047 1.28767 0.643834 0.765166i \(-0.277343\pi\)
0.643834 + 0.765166i \(0.277343\pi\)
\(740\) 0 0
\(741\) −28.1585 −1.03443
\(742\) −4.37926 −0.160768
\(743\) 12.5515 0.460468 0.230234 0.973135i \(-0.426051\pi\)
0.230234 + 0.973135i \(0.426051\pi\)
\(744\) 26.3501 0.966041
\(745\) 0 0
\(746\) −38.8068 −1.42082
\(747\) 2.59792 0.0950530
\(748\) 77.9069 2.84856
\(749\) −0.736956 −0.0269278
\(750\) 0 0
\(751\) −33.8278 −1.23439 −0.617197 0.786809i \(-0.711732\pi\)
−0.617197 + 0.786809i \(0.711732\pi\)
\(752\) 2.38332 0.0869108
\(753\) 16.2044 0.590522
\(754\) −15.9974 −0.582591
\(755\) 0 0
\(756\) 0.501897 0.0182538
\(757\) 6.60568 0.240088 0.120044 0.992769i \(-0.461697\pi\)
0.120044 + 0.992769i \(0.461697\pi\)
\(758\) 0.468553 0.0170186
\(759\) −6.70661 −0.243435
\(760\) 0 0
\(761\) −22.8638 −0.828814 −0.414407 0.910092i \(-0.636011\pi\)
−0.414407 + 0.910092i \(0.636011\pi\)
\(762\) 8.37978 0.303568
\(763\) 2.31051 0.0836461
\(764\) 56.9656 2.06094
\(765\) 0 0
\(766\) 37.1103 1.34085
\(767\) 49.8760 1.80092
\(768\) 13.1557 0.474716
\(769\) −35.0078 −1.26241 −0.631206 0.775615i \(-0.717440\pi\)
−0.631206 + 0.775615i \(0.717440\pi\)
\(770\) 0 0
\(771\) −3.36942 −0.121347
\(772\) 54.4016 1.95796
\(773\) −17.7019 −0.636691 −0.318346 0.947975i \(-0.603127\pi\)
−0.318346 + 0.947975i \(0.603127\pi\)
\(774\) −8.93586 −0.321193
\(775\) 0 0
\(776\) −10.0282 −0.359992
\(777\) 0.684020 0.0245391
\(778\) −26.6655 −0.956005
\(779\) 20.4434 0.732460
\(780\) 0 0
\(781\) −11.1564 −0.399206
\(782\) −35.7388 −1.27802
\(783\) 1.00000 0.0357371
\(784\) 2.95968 0.105703
\(785\) 0 0
\(786\) −22.5066 −0.802782
\(787\) −5.13176 −0.182928 −0.0914638 0.995808i \(-0.529155\pi\)
−0.0914638 + 0.995808i \(0.529155\pi\)
\(788\) 32.7321 1.16603
\(789\) 1.79671 0.0639647
\(790\) 0 0
\(791\) −0.653321 −0.0232294
\(792\) 8.39044 0.298141
\(793\) −80.6662 −2.86454
\(794\) −11.2471 −0.399145
\(795\) 0 0
\(796\) 36.2783 1.28585
\(797\) −27.5492 −0.975842 −0.487921 0.872888i \(-0.662245\pi\)
−0.487921 + 0.872888i \(0.662245\pi\)
\(798\) 1.44630 0.0511984
\(799\) 42.8974 1.51760
\(800\) 0 0
\(801\) 4.33165 0.153051
\(802\) −82.0562 −2.89751
\(803\) −39.2424 −1.38484
\(804\) −46.9329 −1.65520
\(805\) 0 0
\(806\) 163.244 5.75002
\(807\) 0.117055 0.00412054
\(808\) −39.7778 −1.39938
\(809\) −5.51555 −0.193917 −0.0969583 0.995288i \(-0.530911\pi\)
−0.0969583 + 0.995288i \(0.530911\pi\)
\(810\) 0 0
\(811\) −17.1711 −0.602959 −0.301479 0.953473i \(-0.597480\pi\)
−0.301479 + 0.953473i \(0.597480\pi\)
\(812\) 0.501897 0.0176131
\(813\) 4.06294 0.142493
\(814\) 31.5129 1.10453
\(815\) 0 0
\(816\) −3.24131 −0.113468
\(817\) −15.7288 −0.550283
\(818\) 36.6355 1.28093
\(819\) 1.12829 0.0394256
\(820\) 0 0
\(821\) 6.53740 0.228157 0.114078 0.993472i \(-0.463609\pi\)
0.114078 + 0.993472i \(0.463609\pi\)
\(822\) −8.67813 −0.302684
\(823\) 35.8588 1.24996 0.624979 0.780641i \(-0.285107\pi\)
0.624979 + 0.780641i \(0.285107\pi\)
\(824\) 19.8440 0.691299
\(825\) 0 0
\(826\) −2.56177 −0.0891354
\(827\) 2.36177 0.0821269 0.0410635 0.999157i \(-0.486925\pi\)
0.0410635 + 0.999157i \(0.486925\pi\)
\(828\) −6.47910 −0.225164
\(829\) 54.0332 1.87665 0.938325 0.345754i \(-0.112377\pi\)
0.938325 + 0.345754i \(0.112377\pi\)
\(830\) 0 0
\(831\) −12.5436 −0.435134
\(832\) 92.0182 3.19016
\(833\) 53.2713 1.84574
\(834\) −4.69231 −0.162481
\(835\) 0 0
\(836\) 40.7000 1.40764
\(837\) −10.2044 −0.352716
\(838\) 76.0663 2.62767
\(839\) 27.6639 0.955063 0.477532 0.878615i \(-0.341532\pi\)
0.477532 + 0.878615i \(0.341532\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −61.1730 −2.10816
\(843\) 2.83163 0.0975266
\(844\) −66.4278 −2.28654
\(845\) 0 0
\(846\) 12.7318 0.437728
\(847\) 0.0706761 0.00242846
\(848\) 5.12721 0.176069
\(849\) 28.2249 0.968676
\(850\) 0 0
\(851\) −8.83017 −0.302694
\(852\) −10.7779 −0.369245
\(853\) −16.7180 −0.572413 −0.286207 0.958168i \(-0.592394\pi\)
−0.286207 + 0.958168i \(0.592394\pi\)
\(854\) 4.14324 0.141779
\(855\) 0 0
\(856\) −11.9021 −0.406805
\(857\) 50.6304 1.72950 0.864750 0.502202i \(-0.167477\pi\)
0.864750 + 0.502202i \(0.167477\pi\)
\(858\) 51.9804 1.77458
\(859\) −23.0473 −0.786363 −0.393182 0.919461i \(-0.628626\pi\)
−0.393182 + 0.919461i \(0.628626\pi\)
\(860\) 0 0
\(861\) −0.819148 −0.0279165
\(862\) −1.88693 −0.0642692
\(863\) −45.4971 −1.54874 −0.774370 0.632733i \(-0.781933\pi\)
−0.774370 + 0.632733i \(0.781933\pi\)
\(864\) −6.12646 −0.208426
\(865\) 0 0
\(866\) −51.1531 −1.73825
\(867\) −41.3403 −1.40399
\(868\) −5.12156 −0.173837
\(869\) 41.4095 1.40472
\(870\) 0 0
\(871\) −105.507 −3.57498
\(872\) 37.3155 1.26366
\(873\) 3.88355 0.131438
\(874\) −18.6706 −0.631543
\(875\) 0 0
\(876\) −37.9112 −1.28090
\(877\) −6.44199 −0.217531 −0.108765 0.994067i \(-0.534690\pi\)
−0.108765 + 0.994067i \(0.534690\pi\)
\(878\) −54.3124 −1.83295
\(879\) 21.9337 0.739806
\(880\) 0 0
\(881\) −43.3346 −1.45998 −0.729990 0.683458i \(-0.760475\pi\)
−0.729990 + 0.683458i \(0.760475\pi\)
\(882\) 15.8107 0.532375
\(883\) 24.6736 0.830333 0.415166 0.909745i \(-0.363723\pi\)
0.415166 + 0.909745i \(0.363723\pi\)
\(884\) 169.197 5.69072
\(885\) 0 0
\(886\) 39.6546 1.33222
\(887\) −37.1527 −1.24747 −0.623733 0.781637i \(-0.714385\pi\)
−0.623733 + 0.781637i \(0.714385\pi\)
\(888\) 11.0472 0.370718
\(889\) −0.591021 −0.0198222
\(890\) 0 0
\(891\) −3.24930 −0.108856
\(892\) −29.7247 −0.995255
\(893\) 22.4104 0.749936
\(894\) −3.32357 −0.111157
\(895\) 0 0
\(896\) −2.76723 −0.0924466
\(897\) −14.5653 −0.486322
\(898\) −9.57108 −0.319391
\(899\) −10.2044 −0.340336
\(900\) 0 0
\(901\) 92.2847 3.07445
\(902\) −37.7383 −1.25655
\(903\) 0.630241 0.0209731
\(904\) −10.5514 −0.350933
\(905\) 0 0
\(906\) −45.6291 −1.51593
\(907\) −7.49457 −0.248853 −0.124427 0.992229i \(-0.539709\pi\)
−0.124427 + 0.992229i \(0.539709\pi\)
\(908\) −70.3248 −2.33381
\(909\) 15.4045 0.510934
\(910\) 0 0
\(911\) 48.6818 1.61290 0.806449 0.591303i \(-0.201386\pi\)
0.806449 + 0.591303i \(0.201386\pi\)
\(912\) −1.69332 −0.0560714
\(913\) −8.44143 −0.279371
\(914\) −2.91444 −0.0964012
\(915\) 0 0
\(916\) 6.37598 0.210668
\(917\) 1.58737 0.0524197
\(918\) −17.3152 −0.571486
\(919\) −26.1563 −0.862816 −0.431408 0.902157i \(-0.641983\pi\)
−0.431408 + 0.902157i \(0.641983\pi\)
\(920\) 0 0
\(921\) −0.461574 −0.0152094
\(922\) 54.0746 1.78085
\(923\) −24.2293 −0.797515
\(924\) −1.63081 −0.0536498
\(925\) 0 0
\(926\) 32.0552 1.05340
\(927\) −7.68484 −0.252403
\(928\) −6.12646 −0.201111
\(929\) −8.15366 −0.267513 −0.133756 0.991014i \(-0.542704\pi\)
−0.133756 + 0.991014i \(0.542704\pi\)
\(930\) 0 0
\(931\) 27.8299 0.912089
\(932\) −34.7509 −1.13830
\(933\) −27.5507 −0.901968
\(934\) 16.0157 0.524051
\(935\) 0 0
\(936\) 18.2222 0.595612
\(937\) −41.9169 −1.36936 −0.684682 0.728842i \(-0.740059\pi\)
−0.684682 + 0.728842i \(0.740059\pi\)
\(938\) 5.41915 0.176941
\(939\) 14.9121 0.486638
\(940\) 0 0
\(941\) 14.4789 0.471998 0.235999 0.971753i \(-0.424164\pi\)
0.235999 + 0.971753i \(0.424164\pi\)
\(942\) −16.0751 −0.523755
\(943\) 10.5746 0.344356
\(944\) 2.99931 0.0976191
\(945\) 0 0
\(946\) 29.0353 0.944020
\(947\) −1.83838 −0.0597392 −0.0298696 0.999554i \(-0.509509\pi\)
−0.0298696 + 0.999554i \(0.509509\pi\)
\(948\) 40.0048 1.29929
\(949\) −85.2262 −2.76656
\(950\) 0 0
\(951\) −22.9266 −0.743446
\(952\) −3.15349 −0.102205
\(953\) 49.5140 1.60391 0.801957 0.597382i \(-0.203792\pi\)
0.801957 + 0.597382i \(0.203792\pi\)
\(954\) 27.3897 0.886776
\(955\) 0 0
\(956\) −7.03461 −0.227516
\(957\) −3.24930 −0.105035
\(958\) −52.0403 −1.68135
\(959\) 0.612063 0.0197646
\(960\) 0 0
\(961\) 73.1299 2.35903
\(962\) 68.4393 2.20657
\(963\) 4.60924 0.148531
\(964\) 45.4658 1.46436
\(965\) 0 0
\(966\) 0.748115 0.0240702
\(967\) −5.90299 −0.189827 −0.0949137 0.995486i \(-0.530258\pi\)
−0.0949137 + 0.995486i \(0.530258\pi\)
\(968\) 1.14144 0.0366874
\(969\) −30.4781 −0.979096
\(970\) 0 0
\(971\) −35.7964 −1.14876 −0.574380 0.818588i \(-0.694757\pi\)
−0.574380 + 0.818588i \(0.694757\pi\)
\(972\) −3.13907 −0.100686
\(973\) 0.330946 0.0106096
\(974\) −61.2801 −1.96354
\(975\) 0 0
\(976\) −4.85088 −0.155273
\(977\) 1.79673 0.0574825 0.0287412 0.999587i \(-0.490850\pi\)
0.0287412 + 0.999587i \(0.490850\pi\)
\(978\) 14.9328 0.477497
\(979\) −14.0748 −0.449834
\(980\) 0 0
\(981\) −14.4509 −0.461382
\(982\) 26.1828 0.835528
\(983\) −6.24151 −0.199073 −0.0995365 0.995034i \(-0.531736\pi\)
−0.0995365 + 0.995034i \(0.531736\pi\)
\(984\) −13.2295 −0.421742
\(985\) 0 0
\(986\) −17.3152 −0.551427
\(987\) −0.897966 −0.0285826
\(988\) 88.3917 2.81211
\(989\) −8.13593 −0.258708
\(990\) 0 0
\(991\) 41.0410 1.30371 0.651855 0.758344i \(-0.273991\pi\)
0.651855 + 0.758344i \(0.273991\pi\)
\(992\) 62.5169 1.98491
\(993\) 3.23147 0.102548
\(994\) 1.24448 0.0394725
\(995\) 0 0
\(996\) −8.15507 −0.258403
\(997\) −29.8910 −0.946657 −0.473329 0.880886i \(-0.656948\pi\)
−0.473329 + 0.880886i \(0.656948\pi\)
\(998\) −8.56776 −0.271208
\(999\) −4.27815 −0.135355
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 2175.2.a.ba.1.2 7
3.2 odd 2 6525.2.a.bx.1.6 7
5.2 odd 4 2175.2.c.o.349.3 14
5.3 odd 4 2175.2.c.o.349.12 14
5.4 even 2 2175.2.a.bb.1.6 yes 7
15.14 odd 2 6525.2.a.bu.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.ba.1.2 7 1.1 even 1 trivial
2175.2.a.bb.1.6 yes 7 5.4 even 2
2175.2.c.o.349.3 14 5.2 odd 4
2175.2.c.o.349.12 14 5.3 odd 4
6525.2.a.bu.1.2 7 15.14 odd 2
6525.2.a.bx.1.6 7 3.2 odd 2