Properties

Label 5225.2.a.bc.1.15
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.202376 q^{2} -2.47875 q^{3} -1.95904 q^{4} +0.501638 q^{6} +5.06314 q^{7} +0.801215 q^{8} +3.14418 q^{9} +O(q^{10})\) \(q-0.202376 q^{2} -2.47875 q^{3} -1.95904 q^{4} +0.501638 q^{6} +5.06314 q^{7} +0.801215 q^{8} +3.14418 q^{9} +1.00000 q^{11} +4.85597 q^{12} -1.72877 q^{13} -1.02466 q^{14} +3.75594 q^{16} +2.65088 q^{17} -0.636306 q^{18} +1.00000 q^{19} -12.5502 q^{21} -0.202376 q^{22} +6.75421 q^{23} -1.98601 q^{24} +0.349861 q^{26} -0.357381 q^{27} -9.91891 q^{28} +10.1690 q^{29} +6.13793 q^{31} -2.36254 q^{32} -2.47875 q^{33} -0.536473 q^{34} -6.15958 q^{36} +7.01000 q^{37} -0.202376 q^{38} +4.28518 q^{39} -3.72063 q^{41} +2.53986 q^{42} +0.935858 q^{43} -1.95904 q^{44} -1.36689 q^{46} +3.29300 q^{47} -9.31002 q^{48} +18.6354 q^{49} -6.57085 q^{51} +3.38674 q^{52} -0.668027 q^{53} +0.0723253 q^{54} +4.05666 q^{56} -2.47875 q^{57} -2.05796 q^{58} -3.35364 q^{59} +13.2751 q^{61} -1.24217 q^{62} +15.9194 q^{63} -7.03376 q^{64} +0.501638 q^{66} -12.8646 q^{67} -5.19318 q^{68} -16.7420 q^{69} +0.833169 q^{71} +2.51916 q^{72} -1.38130 q^{73} -1.41865 q^{74} -1.95904 q^{76} +5.06314 q^{77} -0.867217 q^{78} +11.3823 q^{79} -8.54668 q^{81} +0.752965 q^{82} -14.3628 q^{83} +24.5865 q^{84} -0.189395 q^{86} -25.2064 q^{87} +0.801215 q^{88} -1.51234 q^{89} -8.75300 q^{91} -13.2318 q^{92} -15.2144 q^{93} -0.666425 q^{94} +5.85614 q^{96} -14.9617 q^{97} -3.77135 q^{98} +3.14418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46} + 60 q^{49} + 64 q^{51} + 62 q^{54} - 32 q^{56} - 14 q^{59} + 78 q^{61} + 90 q^{64} + 12 q^{66} - 28 q^{69} + 20 q^{71} + 42 q^{74} + 42 q^{76} + 102 q^{79} + 42 q^{81} + 98 q^{84} - 52 q^{86} - 8 q^{89} + 56 q^{91} + 40 q^{94} - 74 q^{96} + 40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.202376 −0.143101 −0.0715507 0.997437i \(-0.522795\pi\)
−0.0715507 + 0.997437i \(0.522795\pi\)
\(3\) −2.47875 −1.43110 −0.715552 0.698559i \(-0.753825\pi\)
−0.715552 + 0.698559i \(0.753825\pi\)
\(4\) −1.95904 −0.979522
\(5\) 0 0
\(6\) 0.501638 0.204793
\(7\) 5.06314 1.91369 0.956843 0.290605i \(-0.0938565\pi\)
0.956843 + 0.290605i \(0.0938565\pi\)
\(8\) 0.801215 0.283272
\(9\) 3.14418 1.04806
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 4.85597 1.40180
\(13\) −1.72877 −0.479474 −0.239737 0.970838i \(-0.577061\pi\)
−0.239737 + 0.970838i \(0.577061\pi\)
\(14\) −1.02466 −0.273851
\(15\) 0 0
\(16\) 3.75594 0.938985
\(17\) 2.65088 0.642932 0.321466 0.946921i \(-0.395824\pi\)
0.321466 + 0.946921i \(0.395824\pi\)
\(18\) −0.636306 −0.149979
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −12.5502 −2.73868
\(22\) −0.202376 −0.0431467
\(23\) 6.75421 1.40835 0.704175 0.710027i \(-0.251317\pi\)
0.704175 + 0.710027i \(0.251317\pi\)
\(24\) −1.98601 −0.405392
\(25\) 0 0
\(26\) 0.349861 0.0686134
\(27\) −0.357381 −0.0687780
\(28\) −9.91891 −1.87450
\(29\) 10.1690 1.88834 0.944169 0.329462i \(-0.106867\pi\)
0.944169 + 0.329462i \(0.106867\pi\)
\(30\) 0 0
\(31\) 6.13793 1.10240 0.551202 0.834372i \(-0.314169\pi\)
0.551202 + 0.834372i \(0.314169\pi\)
\(32\) −2.36254 −0.417642
\(33\) −2.47875 −0.431494
\(34\) −0.536473 −0.0920044
\(35\) 0 0
\(36\) −6.15958 −1.02660
\(37\) 7.01000 1.15244 0.576218 0.817296i \(-0.304528\pi\)
0.576218 + 0.817296i \(0.304528\pi\)
\(38\) −0.202376 −0.0328297
\(39\) 4.28518 0.686178
\(40\) 0 0
\(41\) −3.72063 −0.581064 −0.290532 0.956865i \(-0.593832\pi\)
−0.290532 + 0.956865i \(0.593832\pi\)
\(42\) 2.53986 0.391909
\(43\) 0.935858 0.142717 0.0713585 0.997451i \(-0.477267\pi\)
0.0713585 + 0.997451i \(0.477267\pi\)
\(44\) −1.95904 −0.295337
\(45\) 0 0
\(46\) −1.36689 −0.201537
\(47\) 3.29300 0.480334 0.240167 0.970732i \(-0.422798\pi\)
0.240167 + 0.970732i \(0.422798\pi\)
\(48\) −9.31002 −1.34379
\(49\) 18.6354 2.66220
\(50\) 0 0
\(51\) −6.57085 −0.920103
\(52\) 3.38674 0.469656
\(53\) −0.668027 −0.0917606 −0.0458803 0.998947i \(-0.514609\pi\)
−0.0458803 + 0.998947i \(0.514609\pi\)
\(54\) 0.0723253 0.00984222
\(55\) 0 0
\(56\) 4.05666 0.542094
\(57\) −2.47875 −0.328318
\(58\) −2.05796 −0.270224
\(59\) −3.35364 −0.436606 −0.218303 0.975881i \(-0.570052\pi\)
−0.218303 + 0.975881i \(0.570052\pi\)
\(60\) 0 0
\(61\) 13.2751 1.69970 0.849850 0.527024i \(-0.176692\pi\)
0.849850 + 0.527024i \(0.176692\pi\)
\(62\) −1.24217 −0.157755
\(63\) 15.9194 2.00566
\(64\) −7.03376 −0.879220
\(65\) 0 0
\(66\) 0.501638 0.0617474
\(67\) −12.8646 −1.57166 −0.785829 0.618444i \(-0.787764\pi\)
−0.785829 + 0.618444i \(0.787764\pi\)
\(68\) −5.19318 −0.629766
\(69\) −16.7420 −2.01550
\(70\) 0 0
\(71\) 0.833169 0.0988790 0.0494395 0.998777i \(-0.484257\pi\)
0.0494395 + 0.998777i \(0.484257\pi\)
\(72\) 2.51916 0.296886
\(73\) −1.38130 −0.161669 −0.0808345 0.996728i \(-0.525759\pi\)
−0.0808345 + 0.996728i \(0.525759\pi\)
\(74\) −1.41865 −0.164915
\(75\) 0 0
\(76\) −1.95904 −0.224718
\(77\) 5.06314 0.576998
\(78\) −0.867217 −0.0981930
\(79\) 11.3823 1.28061 0.640306 0.768120i \(-0.278807\pi\)
0.640306 + 0.768120i \(0.278807\pi\)
\(80\) 0 0
\(81\) −8.54668 −0.949631
\(82\) 0.752965 0.0831511
\(83\) −14.3628 −1.57652 −0.788261 0.615341i \(-0.789018\pi\)
−0.788261 + 0.615341i \(0.789018\pi\)
\(84\) 24.5865 2.68260
\(85\) 0 0
\(86\) −0.189395 −0.0204230
\(87\) −25.2064 −2.70241
\(88\) 0.801215 0.0854098
\(89\) −1.51234 −0.160308 −0.0801540 0.996782i \(-0.525541\pi\)
−0.0801540 + 0.996782i \(0.525541\pi\)
\(90\) 0 0
\(91\) −8.75300 −0.917564
\(92\) −13.2318 −1.37951
\(93\) −15.2144 −1.57765
\(94\) −0.666425 −0.0687364
\(95\) 0 0
\(96\) 5.85614 0.597690
\(97\) −14.9617 −1.51913 −0.759563 0.650434i \(-0.774587\pi\)
−0.759563 + 0.650434i \(0.774587\pi\)
\(98\) −3.77135 −0.380964
\(99\) 3.14418 0.316002
\(100\) 0 0
\(101\) 6.11490 0.608455 0.304227 0.952599i \(-0.401602\pi\)
0.304227 + 0.952599i \(0.401602\pi\)
\(102\) 1.32978 0.131668
\(103\) −18.1934 −1.79265 −0.896327 0.443394i \(-0.853774\pi\)
−0.896327 + 0.443394i \(0.853774\pi\)
\(104\) −1.38512 −0.135822
\(105\) 0 0
\(106\) 0.135193 0.0131311
\(107\) 1.35532 0.131023 0.0655117 0.997852i \(-0.479132\pi\)
0.0655117 + 0.997852i \(0.479132\pi\)
\(108\) 0.700125 0.0673696
\(109\) −5.01210 −0.480072 −0.240036 0.970764i \(-0.577159\pi\)
−0.240036 + 0.970764i \(0.577159\pi\)
\(110\) 0 0
\(111\) −17.3760 −1.64926
\(112\) 19.0169 1.79692
\(113\) 8.91136 0.838310 0.419155 0.907915i \(-0.362326\pi\)
0.419155 + 0.907915i \(0.362326\pi\)
\(114\) 0.501638 0.0469827
\(115\) 0 0
\(116\) −19.9215 −1.84967
\(117\) −5.43556 −0.502518
\(118\) 0.678695 0.0624789
\(119\) 13.4218 1.23037
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.68656 −0.243229
\(123\) 9.22249 0.831564
\(124\) −12.0245 −1.07983
\(125\) 0 0
\(126\) −3.22170 −0.287012
\(127\) 6.60871 0.586428 0.293214 0.956047i \(-0.405275\pi\)
0.293214 + 0.956047i \(0.405275\pi\)
\(128\) 6.14855 0.543460
\(129\) −2.31975 −0.204243
\(130\) 0 0
\(131\) 5.78122 0.505108 0.252554 0.967583i \(-0.418729\pi\)
0.252554 + 0.967583i \(0.418729\pi\)
\(132\) 4.85597 0.422658
\(133\) 5.06314 0.439030
\(134\) 2.60348 0.224906
\(135\) 0 0
\(136\) 2.12392 0.182125
\(137\) 9.55287 0.816156 0.408078 0.912947i \(-0.366199\pi\)
0.408078 + 0.912947i \(0.366199\pi\)
\(138\) 3.38817 0.288420
\(139\) −8.39417 −0.711984 −0.355992 0.934489i \(-0.615857\pi\)
−0.355992 + 0.934489i \(0.615857\pi\)
\(140\) 0 0
\(141\) −8.16252 −0.687408
\(142\) −0.168613 −0.0141497
\(143\) −1.72877 −0.144567
\(144\) 11.8093 0.984112
\(145\) 0 0
\(146\) 0.279542 0.0231350
\(147\) −46.1923 −3.80988
\(148\) −13.7329 −1.12884
\(149\) −9.09038 −0.744713 −0.372356 0.928090i \(-0.621450\pi\)
−0.372356 + 0.928090i \(0.621450\pi\)
\(150\) 0 0
\(151\) 11.6838 0.950815 0.475407 0.879766i \(-0.342301\pi\)
0.475407 + 0.879766i \(0.342301\pi\)
\(152\) 0.801215 0.0649871
\(153\) 8.33483 0.673831
\(154\) −1.02466 −0.0825692
\(155\) 0 0
\(156\) −8.39486 −0.672126
\(157\) −13.3380 −1.06449 −0.532246 0.846590i \(-0.678652\pi\)
−0.532246 + 0.846590i \(0.678652\pi\)
\(158\) −2.30351 −0.183257
\(159\) 1.65587 0.131319
\(160\) 0 0
\(161\) 34.1975 2.69514
\(162\) 1.72964 0.135893
\(163\) 2.77342 0.217231 0.108616 0.994084i \(-0.465358\pi\)
0.108616 + 0.994084i \(0.465358\pi\)
\(164\) 7.28887 0.569165
\(165\) 0 0
\(166\) 2.90668 0.225602
\(167\) −9.90398 −0.766393 −0.383196 0.923667i \(-0.625177\pi\)
−0.383196 + 0.923667i \(0.625177\pi\)
\(168\) −10.0554 −0.775793
\(169\) −10.0114 −0.770104
\(170\) 0 0
\(171\) 3.14418 0.240441
\(172\) −1.83339 −0.139795
\(173\) −4.99846 −0.380026 −0.190013 0.981782i \(-0.560853\pi\)
−0.190013 + 0.981782i \(0.560853\pi\)
\(174\) 5.10116 0.386718
\(175\) 0 0
\(176\) 3.75594 0.283115
\(177\) 8.31281 0.624829
\(178\) 0.306061 0.0229403
\(179\) −12.8738 −0.962234 −0.481117 0.876656i \(-0.659769\pi\)
−0.481117 + 0.876656i \(0.659769\pi\)
\(180\) 0 0
\(181\) 10.3518 0.769445 0.384723 0.923032i \(-0.374297\pi\)
0.384723 + 0.923032i \(0.374297\pi\)
\(182\) 1.77140 0.131305
\(183\) −32.9056 −2.43245
\(184\) 5.41157 0.398946
\(185\) 0 0
\(186\) 3.07902 0.225764
\(187\) 2.65088 0.193851
\(188\) −6.45114 −0.470498
\(189\) −1.80947 −0.131620
\(190\) 0 0
\(191\) 20.5833 1.48936 0.744678 0.667424i \(-0.232603\pi\)
0.744678 + 0.667424i \(0.232603\pi\)
\(192\) 17.4349 1.25826
\(193\) 8.48648 0.610870 0.305435 0.952213i \(-0.401198\pi\)
0.305435 + 0.952213i \(0.401198\pi\)
\(194\) 3.02788 0.217389
\(195\) 0 0
\(196\) −36.5075 −2.60768
\(197\) 3.74247 0.266640 0.133320 0.991073i \(-0.457436\pi\)
0.133320 + 0.991073i \(0.457436\pi\)
\(198\) −0.636306 −0.0452203
\(199\) 1.18891 0.0842795 0.0421397 0.999112i \(-0.486583\pi\)
0.0421397 + 0.999112i \(0.486583\pi\)
\(200\) 0 0
\(201\) 31.8880 2.24921
\(202\) −1.23751 −0.0870707
\(203\) 51.4871 3.61369
\(204\) 12.8726 0.901261
\(205\) 0 0
\(206\) 3.68191 0.256531
\(207\) 21.2364 1.47603
\(208\) −6.49316 −0.450219
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 10.8311 0.745643 0.372822 0.927903i \(-0.378390\pi\)
0.372822 + 0.927903i \(0.378390\pi\)
\(212\) 1.30869 0.0898815
\(213\) −2.06521 −0.141506
\(214\) −0.274283 −0.0187496
\(215\) 0 0
\(216\) −0.286339 −0.0194829
\(217\) 31.0772 2.10966
\(218\) 1.01433 0.0686990
\(219\) 3.42389 0.231365
\(220\) 0 0
\(221\) −4.58275 −0.308269
\(222\) 3.51648 0.236011
\(223\) −3.32216 −0.222468 −0.111234 0.993794i \(-0.535480\pi\)
−0.111234 + 0.993794i \(0.535480\pi\)
\(224\) −11.9619 −0.799236
\(225\) 0 0
\(226\) −1.80344 −0.119963
\(227\) −24.5086 −1.62669 −0.813345 0.581781i \(-0.802356\pi\)
−0.813345 + 0.581781i \(0.802356\pi\)
\(228\) 4.85597 0.321595
\(229\) 14.9309 0.986664 0.493332 0.869841i \(-0.335779\pi\)
0.493332 + 0.869841i \(0.335779\pi\)
\(230\) 0 0
\(231\) −12.5502 −0.825744
\(232\) 8.14756 0.534914
\(233\) −27.0720 −1.77355 −0.886773 0.462204i \(-0.847059\pi\)
−0.886773 + 0.462204i \(0.847059\pi\)
\(234\) 1.10003 0.0719109
\(235\) 0 0
\(236\) 6.56992 0.427665
\(237\) −28.2139 −1.83269
\(238\) −2.71624 −0.176068
\(239\) 19.9281 1.28904 0.644519 0.764588i \(-0.277058\pi\)
0.644519 + 0.764588i \(0.277058\pi\)
\(240\) 0 0
\(241\) 11.8613 0.764051 0.382026 0.924152i \(-0.375227\pi\)
0.382026 + 0.924152i \(0.375227\pi\)
\(242\) −0.202376 −0.0130092
\(243\) 22.2572 1.42780
\(244\) −26.0065 −1.66489
\(245\) 0 0
\(246\) −1.86641 −0.118998
\(247\) −1.72877 −0.109999
\(248\) 4.91780 0.312280
\(249\) 35.6017 2.25617
\(250\) 0 0
\(251\) 12.8036 0.808158 0.404079 0.914724i \(-0.367592\pi\)
0.404079 + 0.914724i \(0.367592\pi\)
\(252\) −31.1868 −1.96459
\(253\) 6.75421 0.424633
\(254\) −1.33744 −0.0839187
\(255\) 0 0
\(256\) 12.8232 0.801450
\(257\) 7.75124 0.483509 0.241754 0.970337i \(-0.422277\pi\)
0.241754 + 0.970337i \(0.422277\pi\)
\(258\) 0.469462 0.0292274
\(259\) 35.4926 2.20540
\(260\) 0 0
\(261\) 31.9732 1.97909
\(262\) −1.16998 −0.0722816
\(263\) −16.9790 −1.04697 −0.523485 0.852035i \(-0.675369\pi\)
−0.523485 + 0.852035i \(0.675369\pi\)
\(264\) −1.98601 −0.122230
\(265\) 0 0
\(266\) −1.02466 −0.0628257
\(267\) 3.74871 0.229417
\(268\) 25.2023 1.53947
\(269\) −24.5098 −1.49439 −0.747193 0.664607i \(-0.768599\pi\)
−0.747193 + 0.664607i \(0.768599\pi\)
\(270\) 0 0
\(271\) 4.34679 0.264049 0.132024 0.991246i \(-0.457852\pi\)
0.132024 + 0.991246i \(0.457852\pi\)
\(272\) 9.95654 0.603704
\(273\) 21.6965 1.31313
\(274\) −1.93327 −0.116793
\(275\) 0 0
\(276\) 32.7982 1.97422
\(277\) −15.9348 −0.957432 −0.478716 0.877970i \(-0.658898\pi\)
−0.478716 + 0.877970i \(0.658898\pi\)
\(278\) 1.69878 0.101886
\(279\) 19.2987 1.15538
\(280\) 0 0
\(281\) −9.70532 −0.578971 −0.289485 0.957182i \(-0.593484\pi\)
−0.289485 + 0.957182i \(0.593484\pi\)
\(282\) 1.65190 0.0983690
\(283\) 19.9487 1.18583 0.592913 0.805266i \(-0.297978\pi\)
0.592913 + 0.805266i \(0.297978\pi\)
\(284\) −1.63221 −0.0968541
\(285\) 0 0
\(286\) 0.349861 0.0206877
\(287\) −18.8380 −1.11197
\(288\) −7.42825 −0.437714
\(289\) −9.97286 −0.586639
\(290\) 0 0
\(291\) 37.0861 2.17403
\(292\) 2.70603 0.158358
\(293\) −8.45190 −0.493765 −0.246883 0.969045i \(-0.579406\pi\)
−0.246883 + 0.969045i \(0.579406\pi\)
\(294\) 9.34821 0.545199
\(295\) 0 0
\(296\) 5.61651 0.326453
\(297\) −0.357381 −0.0207374
\(298\) 1.83967 0.106569
\(299\) −11.6765 −0.675268
\(300\) 0 0
\(301\) 4.73838 0.273116
\(302\) −2.36452 −0.136063
\(303\) −15.1573 −0.870762
\(304\) 3.75594 0.215418
\(305\) 0 0
\(306\) −1.68677 −0.0964261
\(307\) −14.1923 −0.809996 −0.404998 0.914318i \(-0.632728\pi\)
−0.404998 + 0.914318i \(0.632728\pi\)
\(308\) −9.91891 −0.565182
\(309\) 45.0969 2.56547
\(310\) 0 0
\(311\) 7.59809 0.430848 0.215424 0.976521i \(-0.430887\pi\)
0.215424 + 0.976521i \(0.430887\pi\)
\(312\) 3.43335 0.194375
\(313\) 31.8101 1.79801 0.899005 0.437937i \(-0.144291\pi\)
0.899005 + 0.437937i \(0.144291\pi\)
\(314\) 2.69930 0.152330
\(315\) 0 0
\(316\) −22.2985 −1.25439
\(317\) 6.90608 0.387884 0.193942 0.981013i \(-0.437873\pi\)
0.193942 + 0.981013i \(0.437873\pi\)
\(318\) −0.335108 −0.0187919
\(319\) 10.1690 0.569355
\(320\) 0 0
\(321\) −3.35948 −0.187508
\(322\) −6.92074 −0.385678
\(323\) 2.65088 0.147499
\(324\) 16.7433 0.930184
\(325\) 0 0
\(326\) −0.561274 −0.0310861
\(327\) 12.4237 0.687034
\(328\) −2.98102 −0.164599
\(329\) 16.6729 0.919209
\(330\) 0 0
\(331\) −31.0030 −1.70408 −0.852038 0.523480i \(-0.824634\pi\)
−0.852038 + 0.523480i \(0.824634\pi\)
\(332\) 28.1373 1.54424
\(333\) 22.0407 1.20782
\(334\) 2.00433 0.109672
\(335\) 0 0
\(336\) −47.1379 −2.57158
\(337\) 30.7881 1.67714 0.838568 0.544797i \(-0.183393\pi\)
0.838568 + 0.544797i \(0.183393\pi\)
\(338\) 2.02606 0.110203
\(339\) −22.0890 −1.19971
\(340\) 0 0
\(341\) 6.13793 0.332387
\(342\) −0.636306 −0.0344075
\(343\) 58.9115 3.18092
\(344\) 0.749824 0.0404278
\(345\) 0 0
\(346\) 1.01157 0.0543822
\(347\) 8.35559 0.448551 0.224276 0.974526i \(-0.427998\pi\)
0.224276 + 0.974526i \(0.427998\pi\)
\(348\) 49.3804 2.64707
\(349\) −16.7466 −0.896425 −0.448213 0.893927i \(-0.647939\pi\)
−0.448213 + 0.893927i \(0.647939\pi\)
\(350\) 0 0
\(351\) 0.617829 0.0329773
\(352\) −2.36254 −0.125924
\(353\) 28.1372 1.49759 0.748797 0.662799i \(-0.230632\pi\)
0.748797 + 0.662799i \(0.230632\pi\)
\(354\) −1.68231 −0.0894138
\(355\) 0 0
\(356\) 2.96274 0.157025
\(357\) −33.2691 −1.76079
\(358\) 2.60535 0.137697
\(359\) −29.6897 −1.56696 −0.783480 0.621417i \(-0.786558\pi\)
−0.783480 + 0.621417i \(0.786558\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −2.09496 −0.110109
\(363\) −2.47875 −0.130100
\(364\) 17.1475 0.898774
\(365\) 0 0
\(366\) 6.65929 0.348087
\(367\) −12.5412 −0.654645 −0.327322 0.944913i \(-0.606146\pi\)
−0.327322 + 0.944913i \(0.606146\pi\)
\(368\) 25.3684 1.32242
\(369\) −11.6983 −0.608990
\(370\) 0 0
\(371\) −3.38231 −0.175601
\(372\) 29.8056 1.54535
\(373\) −13.7844 −0.713727 −0.356864 0.934157i \(-0.616154\pi\)
−0.356864 + 0.934157i \(0.616154\pi\)
\(374\) −0.536473 −0.0277404
\(375\) 0 0
\(376\) 2.63840 0.136065
\(377\) −17.5799 −0.905410
\(378\) 0.366193 0.0188349
\(379\) 20.8733 1.07219 0.536094 0.844158i \(-0.319899\pi\)
0.536094 + 0.844158i \(0.319899\pi\)
\(380\) 0 0
\(381\) −16.3813 −0.839240
\(382\) −4.16556 −0.213129
\(383\) −21.9844 −1.12335 −0.561676 0.827357i \(-0.689843\pi\)
−0.561676 + 0.827357i \(0.689843\pi\)
\(384\) −15.2407 −0.777748
\(385\) 0 0
\(386\) −1.71746 −0.0874163
\(387\) 2.94251 0.149576
\(388\) 29.3105 1.48802
\(389\) −10.7856 −0.546854 −0.273427 0.961893i \(-0.588157\pi\)
−0.273427 + 0.961893i \(0.588157\pi\)
\(390\) 0 0
\(391\) 17.9046 0.905473
\(392\) 14.9309 0.754126
\(393\) −14.3302 −0.722862
\(394\) −0.757386 −0.0381565
\(395\) 0 0
\(396\) −6.15958 −0.309531
\(397\) 3.66005 0.183692 0.0918462 0.995773i \(-0.470723\pi\)
0.0918462 + 0.995773i \(0.470723\pi\)
\(398\) −0.240606 −0.0120605
\(399\) −12.5502 −0.628297
\(400\) 0 0
\(401\) −2.69857 −0.134760 −0.0673801 0.997727i \(-0.521464\pi\)
−0.0673801 + 0.997727i \(0.521464\pi\)
\(402\) −6.45336 −0.321864
\(403\) −10.6111 −0.528574
\(404\) −11.9793 −0.595995
\(405\) 0 0
\(406\) −10.4197 −0.517123
\(407\) 7.01000 0.347473
\(408\) −5.26466 −0.260639
\(409\) −0.352613 −0.0174356 −0.00871780 0.999962i \(-0.502775\pi\)
−0.00871780 + 0.999962i \(0.502775\pi\)
\(410\) 0 0
\(411\) −23.6791 −1.16800
\(412\) 35.6418 1.75594
\(413\) −16.9799 −0.835527
\(414\) −4.29774 −0.211222
\(415\) 0 0
\(416\) 4.08429 0.200249
\(417\) 20.8070 1.01892
\(418\) −0.202376 −0.00989852
\(419\) −13.7803 −0.673214 −0.336607 0.941645i \(-0.609279\pi\)
−0.336607 + 0.941645i \(0.609279\pi\)
\(420\) 0 0
\(421\) −24.9174 −1.21440 −0.607200 0.794549i \(-0.707707\pi\)
−0.607200 + 0.794549i \(0.707707\pi\)
\(422\) −2.19195 −0.106703
\(423\) 10.3538 0.503419
\(424\) −0.535233 −0.0259932
\(425\) 0 0
\(426\) 0.417949 0.0202497
\(427\) 67.2136 3.25269
\(428\) −2.65512 −0.128340
\(429\) 4.28518 0.206890
\(430\) 0 0
\(431\) −33.8426 −1.63014 −0.815070 0.579362i \(-0.803302\pi\)
−0.815070 + 0.579362i \(0.803302\pi\)
\(432\) −1.34230 −0.0645815
\(433\) −25.4495 −1.22302 −0.611512 0.791235i \(-0.709438\pi\)
−0.611512 + 0.791235i \(0.709438\pi\)
\(434\) −6.28927 −0.301894
\(435\) 0 0
\(436\) 9.81893 0.470241
\(437\) 6.75421 0.323098
\(438\) −0.692913 −0.0331086
\(439\) 33.8978 1.61785 0.808927 0.587910i \(-0.200049\pi\)
0.808927 + 0.587910i \(0.200049\pi\)
\(440\) 0 0
\(441\) 58.5929 2.79014
\(442\) 0.927439 0.0441138
\(443\) 29.9900 1.42487 0.712433 0.701740i \(-0.247593\pi\)
0.712433 + 0.701740i \(0.247593\pi\)
\(444\) 34.0403 1.61548
\(445\) 0 0
\(446\) 0.672325 0.0318355
\(447\) 22.5327 1.06576
\(448\) −35.6129 −1.68255
\(449\) 25.6195 1.20906 0.604530 0.796583i \(-0.293361\pi\)
0.604530 + 0.796583i \(0.293361\pi\)
\(450\) 0 0
\(451\) −3.72063 −0.175197
\(452\) −17.4577 −0.821143
\(453\) −28.9612 −1.36071
\(454\) 4.95994 0.232782
\(455\) 0 0
\(456\) −1.98601 −0.0930033
\(457\) 33.0797 1.54740 0.773702 0.633550i \(-0.218403\pi\)
0.773702 + 0.633550i \(0.218403\pi\)
\(458\) −3.02166 −0.141193
\(459\) −0.947373 −0.0442196
\(460\) 0 0
\(461\) 42.3987 1.97470 0.987352 0.158540i \(-0.0506788\pi\)
0.987352 + 0.158540i \(0.0506788\pi\)
\(462\) 2.53986 0.118165
\(463\) 22.5364 1.04736 0.523678 0.851916i \(-0.324560\pi\)
0.523678 + 0.851916i \(0.324560\pi\)
\(464\) 38.1942 1.77312
\(465\) 0 0
\(466\) 5.47872 0.253797
\(467\) 9.25897 0.428454 0.214227 0.976784i \(-0.431277\pi\)
0.214227 + 0.976784i \(0.431277\pi\)
\(468\) 10.6485 0.492227
\(469\) −65.1351 −3.00766
\(470\) 0 0
\(471\) 33.0616 1.52340
\(472\) −2.68698 −0.123678
\(473\) 0.935858 0.0430308
\(474\) 5.70981 0.262260
\(475\) 0 0
\(476\) −26.2938 −1.20517
\(477\) −2.10040 −0.0961706
\(478\) −4.03296 −0.184463
\(479\) 7.87129 0.359649 0.179824 0.983699i \(-0.442447\pi\)
0.179824 + 0.983699i \(0.442447\pi\)
\(480\) 0 0
\(481\) −12.1187 −0.552564
\(482\) −2.40043 −0.109337
\(483\) −84.7669 −3.85703
\(484\) −1.95904 −0.0890475
\(485\) 0 0
\(486\) −4.50432 −0.204320
\(487\) 8.22682 0.372793 0.186396 0.982475i \(-0.440319\pi\)
0.186396 + 0.982475i \(0.440319\pi\)
\(488\) 10.6362 0.481478
\(489\) −6.87461 −0.310881
\(490\) 0 0
\(491\) −10.7584 −0.485518 −0.242759 0.970087i \(-0.578052\pi\)
−0.242759 + 0.970087i \(0.578052\pi\)
\(492\) −18.0673 −0.814535
\(493\) 26.9568 1.21407
\(494\) 0.349861 0.0157410
\(495\) 0 0
\(496\) 23.0537 1.03514
\(497\) 4.21845 0.189223
\(498\) −7.20492 −0.322860
\(499\) 1.56834 0.0702085 0.0351043 0.999384i \(-0.488824\pi\)
0.0351043 + 0.999384i \(0.488824\pi\)
\(500\) 0 0
\(501\) 24.5494 1.09679
\(502\) −2.59114 −0.115648
\(503\) −29.2811 −1.30558 −0.652790 0.757539i \(-0.726401\pi\)
−0.652790 + 0.757539i \(0.726401\pi\)
\(504\) 12.7549 0.568147
\(505\) 0 0
\(506\) −1.36689 −0.0607656
\(507\) 24.8156 1.10210
\(508\) −12.9468 −0.574419
\(509\) −10.8216 −0.479659 −0.239830 0.970815i \(-0.577092\pi\)
−0.239830 + 0.970815i \(0.577092\pi\)
\(510\) 0 0
\(511\) −6.99371 −0.309384
\(512\) −14.8922 −0.658148
\(513\) −0.357381 −0.0157788
\(514\) −1.56866 −0.0691908
\(515\) 0 0
\(516\) 4.54450 0.200061
\(517\) 3.29300 0.144826
\(518\) −7.18284 −0.315596
\(519\) 12.3899 0.543857
\(520\) 0 0
\(521\) −33.0597 −1.44837 −0.724186 0.689605i \(-0.757784\pi\)
−0.724186 + 0.689605i \(0.757784\pi\)
\(522\) −6.47060 −0.283210
\(523\) 3.69508 0.161574 0.0807872 0.996731i \(-0.474257\pi\)
0.0807872 + 0.996731i \(0.474257\pi\)
\(524\) −11.3257 −0.494764
\(525\) 0 0
\(526\) 3.43614 0.149823
\(527\) 16.2709 0.708771
\(528\) −9.31002 −0.405167
\(529\) 22.6193 0.983449
\(530\) 0 0
\(531\) −10.5444 −0.457589
\(532\) −9.91891 −0.430039
\(533\) 6.43211 0.278605
\(534\) −0.758648 −0.0328299
\(535\) 0 0
\(536\) −10.3073 −0.445207
\(537\) 31.9109 1.37706
\(538\) 4.96018 0.213849
\(539\) 18.6354 0.802682
\(540\) 0 0
\(541\) 7.84255 0.337178 0.168589 0.985686i \(-0.446079\pi\)
0.168589 + 0.985686i \(0.446079\pi\)
\(542\) −0.879685 −0.0377857
\(543\) −25.6595 −1.10116
\(544\) −6.26280 −0.268516
\(545\) 0 0
\(546\) −4.39084 −0.187911
\(547\) −33.9101 −1.44989 −0.724946 0.688806i \(-0.758135\pi\)
−0.724946 + 0.688806i \(0.758135\pi\)
\(548\) −18.7145 −0.799443
\(549\) 41.7392 1.78139
\(550\) 0 0
\(551\) 10.1690 0.433214
\(552\) −13.4139 −0.570934
\(553\) 57.6303 2.45069
\(554\) 3.22483 0.137010
\(555\) 0 0
\(556\) 16.4445 0.697404
\(557\) −24.5959 −1.04216 −0.521081 0.853508i \(-0.674471\pi\)
−0.521081 + 0.853508i \(0.674471\pi\)
\(558\) −3.90560 −0.165337
\(559\) −1.61788 −0.0684292
\(560\) 0 0
\(561\) −6.57085 −0.277421
\(562\) 1.96412 0.0828515
\(563\) 0.155899 0.00657035 0.00328517 0.999995i \(-0.498954\pi\)
0.00328517 + 0.999995i \(0.498954\pi\)
\(564\) 15.9907 0.673331
\(565\) 0 0
\(566\) −4.03713 −0.169693
\(567\) −43.2730 −1.81730
\(568\) 0.667547 0.0280097
\(569\) 12.1528 0.509472 0.254736 0.967011i \(-0.418011\pi\)
0.254736 + 0.967011i \(0.418011\pi\)
\(570\) 0 0
\(571\) 21.6646 0.906636 0.453318 0.891349i \(-0.350240\pi\)
0.453318 + 0.891349i \(0.350240\pi\)
\(572\) 3.38674 0.141607
\(573\) −51.0208 −2.13142
\(574\) 3.81237 0.159125
\(575\) 0 0
\(576\) −22.1154 −0.921475
\(577\) 5.43970 0.226458 0.113229 0.993569i \(-0.463881\pi\)
0.113229 + 0.993569i \(0.463881\pi\)
\(578\) 2.01826 0.0839488
\(579\) −21.0358 −0.874219
\(580\) 0 0
\(581\) −72.7208 −3.01697
\(582\) −7.50533 −0.311106
\(583\) −0.668027 −0.0276669
\(584\) −1.10672 −0.0457963
\(585\) 0 0
\(586\) 1.71046 0.0706584
\(587\) −7.30061 −0.301329 −0.150664 0.988585i \(-0.548141\pi\)
−0.150664 + 0.988585i \(0.548141\pi\)
\(588\) 90.4928 3.73186
\(589\) 6.13793 0.252909
\(590\) 0 0
\(591\) −9.27663 −0.381590
\(592\) 26.3291 1.08212
\(593\) −5.19172 −0.213198 −0.106599 0.994302i \(-0.533996\pi\)
−0.106599 + 0.994302i \(0.533996\pi\)
\(594\) 0.0723253 0.00296754
\(595\) 0 0
\(596\) 17.8085 0.729463
\(597\) −2.94700 −0.120613
\(598\) 2.36304 0.0966317
\(599\) 26.5360 1.08423 0.542116 0.840303i \(-0.317623\pi\)
0.542116 + 0.840303i \(0.317623\pi\)
\(600\) 0 0
\(601\) −17.3856 −0.709175 −0.354587 0.935023i \(-0.615379\pi\)
−0.354587 + 0.935023i \(0.615379\pi\)
\(602\) −0.958934 −0.0390832
\(603\) −40.4485 −1.64719
\(604\) −22.8891 −0.931344
\(605\) 0 0
\(606\) 3.06746 0.124607
\(607\) −8.31498 −0.337494 −0.168747 0.985659i \(-0.553972\pi\)
−0.168747 + 0.985659i \(0.553972\pi\)
\(608\) −2.36254 −0.0958137
\(609\) −127.623 −5.17156
\(610\) 0 0
\(611\) −5.69285 −0.230308
\(612\) −16.3283 −0.660032
\(613\) 5.99839 0.242273 0.121136 0.992636i \(-0.461346\pi\)
0.121136 + 0.992636i \(0.461346\pi\)
\(614\) 2.87217 0.115912
\(615\) 0 0
\(616\) 4.05666 0.163448
\(617\) −39.3159 −1.58280 −0.791400 0.611299i \(-0.790647\pi\)
−0.791400 + 0.611299i \(0.790647\pi\)
\(618\) −9.12652 −0.367123
\(619\) 23.1425 0.930177 0.465089 0.885264i \(-0.346022\pi\)
0.465089 + 0.885264i \(0.346022\pi\)
\(620\) 0 0
\(621\) −2.41383 −0.0968635
\(622\) −1.53767 −0.0616549
\(623\) −7.65720 −0.306779
\(624\) 16.0949 0.644311
\(625\) 0 0
\(626\) −6.43759 −0.257298
\(627\) −2.47875 −0.0989916
\(628\) 26.1298 1.04269
\(629\) 18.5826 0.740938
\(630\) 0 0
\(631\) 12.3052 0.489862 0.244931 0.969540i \(-0.421235\pi\)
0.244931 + 0.969540i \(0.421235\pi\)
\(632\) 9.11970 0.362762
\(633\) −26.8475 −1.06709
\(634\) −1.39762 −0.0555067
\(635\) 0 0
\(636\) −3.24392 −0.128630
\(637\) −32.2163 −1.27645
\(638\) −2.05796 −0.0814755
\(639\) 2.61963 0.103631
\(640\) 0 0
\(641\) −11.6616 −0.460604 −0.230302 0.973119i \(-0.573971\pi\)
−0.230302 + 0.973119i \(0.573971\pi\)
\(642\) 0.679878 0.0268327
\(643\) −9.76053 −0.384918 −0.192459 0.981305i \(-0.561646\pi\)
−0.192459 + 0.981305i \(0.561646\pi\)
\(644\) −66.9944 −2.63995
\(645\) 0 0
\(646\) −0.536473 −0.0211073
\(647\) 6.56903 0.258255 0.129128 0.991628i \(-0.458782\pi\)
0.129128 + 0.991628i \(0.458782\pi\)
\(648\) −6.84772 −0.269004
\(649\) −3.35364 −0.131642
\(650\) 0 0
\(651\) −77.0324 −3.01914
\(652\) −5.43326 −0.212783
\(653\) 9.88177 0.386704 0.193352 0.981129i \(-0.438064\pi\)
0.193352 + 0.981129i \(0.438064\pi\)
\(654\) −2.51426 −0.0983154
\(655\) 0 0
\(656\) −13.9745 −0.545611
\(657\) −4.34305 −0.169439
\(658\) −3.37420 −0.131540
\(659\) 20.7690 0.809047 0.404523 0.914528i \(-0.367437\pi\)
0.404523 + 0.914528i \(0.367437\pi\)
\(660\) 0 0
\(661\) −27.5222 −1.07049 −0.535244 0.844697i \(-0.679781\pi\)
−0.535244 + 0.844697i \(0.679781\pi\)
\(662\) 6.27425 0.243856
\(663\) 11.3595 0.441166
\(664\) −11.5077 −0.446585
\(665\) 0 0
\(666\) −4.46050 −0.172841
\(667\) 68.6836 2.65944
\(668\) 19.4023 0.750699
\(669\) 8.23479 0.318375
\(670\) 0 0
\(671\) 13.2751 0.512479
\(672\) 29.6504 1.14379
\(673\) 14.9080 0.574660 0.287330 0.957832i \(-0.407232\pi\)
0.287330 + 0.957832i \(0.407232\pi\)
\(674\) −6.23077 −0.240000
\(675\) 0 0
\(676\) 19.6127 0.754334
\(677\) 21.3317 0.819842 0.409921 0.912121i \(-0.365556\pi\)
0.409921 + 0.912121i \(0.365556\pi\)
\(678\) 4.47028 0.171680
\(679\) −75.7529 −2.90713
\(680\) 0 0
\(681\) 60.7505 2.32796
\(682\) −1.24217 −0.0475651
\(683\) −2.64869 −0.101349 −0.0506747 0.998715i \(-0.516137\pi\)
−0.0506747 + 0.998715i \(0.516137\pi\)
\(684\) −6.15958 −0.235518
\(685\) 0 0
\(686\) −11.9223 −0.455194
\(687\) −37.0100 −1.41202
\(688\) 3.51503 0.134009
\(689\) 1.15487 0.0439969
\(690\) 0 0
\(691\) −25.2356 −0.960009 −0.480005 0.877266i \(-0.659365\pi\)
−0.480005 + 0.877266i \(0.659365\pi\)
\(692\) 9.79220 0.372244
\(693\) 15.9194 0.604728
\(694\) −1.69097 −0.0641883
\(695\) 0 0
\(696\) −20.1957 −0.765517
\(697\) −9.86292 −0.373585
\(698\) 3.38911 0.128280
\(699\) 67.1046 2.53813
\(700\) 0 0
\(701\) −25.1770 −0.950924 −0.475462 0.879736i \(-0.657719\pi\)
−0.475462 + 0.879736i \(0.657719\pi\)
\(702\) −0.125034 −0.00471909
\(703\) 7.01000 0.264387
\(704\) −7.03376 −0.265095
\(705\) 0 0
\(706\) −5.69430 −0.214308
\(707\) 30.9606 1.16439
\(708\) −16.2852 −0.612034
\(709\) −38.3670 −1.44090 −0.720451 0.693506i \(-0.756065\pi\)
−0.720451 + 0.693506i \(0.756065\pi\)
\(710\) 0 0
\(711\) 35.7881 1.34216
\(712\) −1.21171 −0.0454108
\(713\) 41.4568 1.55257
\(714\) 6.73286 0.251971
\(715\) 0 0
\(716\) 25.2204 0.942529
\(717\) −49.3966 −1.84475
\(718\) 6.00847 0.224234
\(719\) 38.1597 1.42312 0.711559 0.702626i \(-0.247989\pi\)
0.711559 + 0.702626i \(0.247989\pi\)
\(720\) 0 0
\(721\) −92.1159 −3.43058
\(722\) −0.202376 −0.00753165
\(723\) −29.4011 −1.09344
\(724\) −20.2797 −0.753689
\(725\) 0 0
\(726\) 0.501638 0.0186175
\(727\) −4.76411 −0.176691 −0.0883455 0.996090i \(-0.528158\pi\)
−0.0883455 + 0.996090i \(0.528158\pi\)
\(728\) −7.01303 −0.259920
\(729\) −29.5298 −1.09370
\(730\) 0 0
\(731\) 2.48084 0.0917574
\(732\) 64.4634 2.38264
\(733\) −25.4352 −0.939472 −0.469736 0.882807i \(-0.655651\pi\)
−0.469736 + 0.882807i \(0.655651\pi\)
\(734\) 2.53803 0.0936806
\(735\) 0 0
\(736\) −15.9571 −0.588186
\(737\) −12.8646 −0.473873
\(738\) 2.36746 0.0871472
\(739\) −2.13812 −0.0786518 −0.0393259 0.999226i \(-0.512521\pi\)
−0.0393259 + 0.999226i \(0.512521\pi\)
\(740\) 0 0
\(741\) 4.28518 0.157420
\(742\) 0.684499 0.0251287
\(743\) 7.85670 0.288234 0.144117 0.989561i \(-0.453966\pi\)
0.144117 + 0.989561i \(0.453966\pi\)
\(744\) −12.1900 −0.446906
\(745\) 0 0
\(746\) 2.78962 0.102135
\(747\) −45.1592 −1.65229
\(748\) −5.19318 −0.189882
\(749\) 6.86215 0.250738
\(750\) 0 0
\(751\) −44.5610 −1.62605 −0.813027 0.582226i \(-0.802182\pi\)
−0.813027 + 0.582226i \(0.802182\pi\)
\(752\) 12.3683 0.451027
\(753\) −31.7369 −1.15656
\(754\) 3.55774 0.129565
\(755\) 0 0
\(756\) 3.54483 0.128924
\(757\) 10.6657 0.387652 0.193826 0.981036i \(-0.437910\pi\)
0.193826 + 0.981036i \(0.437910\pi\)
\(758\) −4.22424 −0.153431
\(759\) −16.7420 −0.607695
\(760\) 0 0
\(761\) −26.5797 −0.963514 −0.481757 0.876305i \(-0.660001\pi\)
−0.481757 + 0.876305i \(0.660001\pi\)
\(762\) 3.31518 0.120096
\(763\) −25.3770 −0.918708
\(764\) −40.3236 −1.45886
\(765\) 0 0
\(766\) 4.44912 0.160753
\(767\) 5.79766 0.209341
\(768\) −31.7855 −1.14696
\(769\) −37.5326 −1.35346 −0.676731 0.736230i \(-0.736604\pi\)
−0.676731 + 0.736230i \(0.736604\pi\)
\(770\) 0 0
\(771\) −19.2133 −0.691952
\(772\) −16.6254 −0.598361
\(773\) −41.3888 −1.48865 −0.744326 0.667816i \(-0.767229\pi\)
−0.744326 + 0.667816i \(0.767229\pi\)
\(774\) −0.595492 −0.0214045
\(775\) 0 0
\(776\) −11.9875 −0.430326
\(777\) −87.9771 −3.15616
\(778\) 2.18275 0.0782555
\(779\) −3.72063 −0.133305
\(780\) 0 0
\(781\) 0.833169 0.0298131
\(782\) −3.62345 −0.129574
\(783\) −3.63421 −0.129876
\(784\) 69.9933 2.49976
\(785\) 0 0
\(786\) 2.90008 0.103443
\(787\) −12.0371 −0.429076 −0.214538 0.976716i \(-0.568825\pi\)
−0.214538 + 0.976716i \(0.568825\pi\)
\(788\) −7.33167 −0.261180
\(789\) 42.0867 1.49832
\(790\) 0 0
\(791\) 45.1194 1.60426
\(792\) 2.51916 0.0895145
\(793\) −22.9496 −0.814963
\(794\) −0.740705 −0.0262866
\(795\) 0 0
\(796\) −2.32912 −0.0825536
\(797\) −13.1219 −0.464802 −0.232401 0.972620i \(-0.574658\pi\)
−0.232401 + 0.972620i \(0.574658\pi\)
\(798\) 2.53986 0.0899102
\(799\) 8.72935 0.308822
\(800\) 0 0
\(801\) −4.75507 −0.168012
\(802\) 0.546126 0.0192844
\(803\) −1.38130 −0.0487450
\(804\) −62.4700 −2.20315
\(805\) 0 0
\(806\) 2.14742 0.0756397
\(807\) 60.7534 2.13862
\(808\) 4.89934 0.172358
\(809\) 12.3579 0.434482 0.217241 0.976118i \(-0.430294\pi\)
0.217241 + 0.976118i \(0.430294\pi\)
\(810\) 0 0
\(811\) 28.6174 1.00489 0.502447 0.864608i \(-0.332433\pi\)
0.502447 + 0.864608i \(0.332433\pi\)
\(812\) −100.866 −3.53969
\(813\) −10.7746 −0.377881
\(814\) −1.41865 −0.0497238
\(815\) 0 0
\(816\) −24.6797 −0.863963
\(817\) 0.935858 0.0327415
\(818\) 0.0713604 0.00249506
\(819\) −27.5210 −0.961661
\(820\) 0 0
\(821\) −51.3323 −1.79151 −0.895755 0.444549i \(-0.853364\pi\)
−0.895755 + 0.444549i \(0.853364\pi\)
\(822\) 4.79208 0.167143
\(823\) 16.9859 0.592091 0.296046 0.955174i \(-0.404332\pi\)
0.296046 + 0.955174i \(0.404332\pi\)
\(824\) −14.5769 −0.507809
\(825\) 0 0
\(826\) 3.43633 0.119565
\(827\) −47.9770 −1.66832 −0.834162 0.551520i \(-0.814048\pi\)
−0.834162 + 0.551520i \(0.814048\pi\)
\(828\) −41.6031 −1.44581
\(829\) −16.9407 −0.588376 −0.294188 0.955748i \(-0.595049\pi\)
−0.294188 + 0.955748i \(0.595049\pi\)
\(830\) 0 0
\(831\) 39.4984 1.37018
\(832\) 12.1598 0.421564
\(833\) 49.4000 1.71161
\(834\) −4.21083 −0.145809
\(835\) 0 0
\(836\) −1.95904 −0.0677550
\(837\) −2.19358 −0.0758211
\(838\) 2.78881 0.0963378
\(839\) −2.36980 −0.0818145 −0.0409073 0.999163i \(-0.513025\pi\)
−0.0409073 + 0.999163i \(0.513025\pi\)
\(840\) 0 0
\(841\) 74.4088 2.56582
\(842\) 5.04268 0.173782
\(843\) 24.0570 0.828567
\(844\) −21.2186 −0.730374
\(845\) 0 0
\(846\) −2.09536 −0.0720399
\(847\) 5.06314 0.173971
\(848\) −2.50907 −0.0861619
\(849\) −49.4477 −1.69704
\(850\) 0 0
\(851\) 47.3470 1.62303
\(852\) 4.04584 0.138608
\(853\) 4.76117 0.163019 0.0815097 0.996673i \(-0.474026\pi\)
0.0815097 + 0.996673i \(0.474026\pi\)
\(854\) −13.6024 −0.465465
\(855\) 0 0
\(856\) 1.08590 0.0371153
\(857\) −15.5789 −0.532165 −0.266082 0.963950i \(-0.585729\pi\)
−0.266082 + 0.963950i \(0.585729\pi\)
\(858\) −0.867217 −0.0296063
\(859\) 39.9404 1.36275 0.681374 0.731935i \(-0.261383\pi\)
0.681374 + 0.731935i \(0.261383\pi\)
\(860\) 0 0
\(861\) 46.6947 1.59135
\(862\) 6.84892 0.233275
\(863\) 28.0580 0.955105 0.477552 0.878603i \(-0.341524\pi\)
0.477552 + 0.878603i \(0.341524\pi\)
\(864\) 0.844327 0.0287246
\(865\) 0 0
\(866\) 5.15036 0.175016
\(867\) 24.7202 0.839541
\(868\) −60.8815 −2.06645
\(869\) 11.3823 0.386119
\(870\) 0 0
\(871\) 22.2399 0.753570
\(872\) −4.01577 −0.135991
\(873\) −47.0421 −1.59213
\(874\) −1.36689 −0.0462357
\(875\) 0 0
\(876\) −6.70755 −0.226627
\(877\) −35.8593 −1.21088 −0.605441 0.795890i \(-0.707003\pi\)
−0.605441 + 0.795890i \(0.707003\pi\)
\(878\) −6.86009 −0.231517
\(879\) 20.9501 0.706629
\(880\) 0 0
\(881\) −4.98649 −0.167999 −0.0839996 0.996466i \(-0.526769\pi\)
−0.0839996 + 0.996466i \(0.526769\pi\)
\(882\) −11.8578 −0.399273
\(883\) 36.4602 1.22698 0.613492 0.789701i \(-0.289764\pi\)
0.613492 + 0.789701i \(0.289764\pi\)
\(884\) 8.97782 0.301957
\(885\) 0 0
\(886\) −6.06924 −0.203900
\(887\) 2.52582 0.0848089 0.0424044 0.999101i \(-0.486498\pi\)
0.0424044 + 0.999101i \(0.486498\pi\)
\(888\) −13.9219 −0.467189
\(889\) 33.4608 1.12224
\(890\) 0 0
\(891\) −8.54668 −0.286324
\(892\) 6.50826 0.217913
\(893\) 3.29300 0.110196
\(894\) −4.56008 −0.152512
\(895\) 0 0
\(896\) 31.1309 1.04001
\(897\) 28.9430 0.966378
\(898\) −5.18477 −0.173018
\(899\) 62.4166 2.08171
\(900\) 0 0
\(901\) −1.77086 −0.0589958
\(902\) 0.752965 0.0250710
\(903\) −11.7452 −0.390857
\(904\) 7.13991 0.237470
\(905\) 0 0
\(906\) 5.86104 0.194720
\(907\) −20.7383 −0.688604 −0.344302 0.938859i \(-0.611884\pi\)
−0.344302 + 0.938859i \(0.611884\pi\)
\(908\) 48.0134 1.59338
\(909\) 19.2263 0.637697
\(910\) 0 0
\(911\) 24.6244 0.815843 0.407921 0.913017i \(-0.366254\pi\)
0.407921 + 0.913017i \(0.366254\pi\)
\(912\) −9.31002 −0.308286
\(913\) −14.3628 −0.475339
\(914\) −6.69453 −0.221435
\(915\) 0 0
\(916\) −29.2504 −0.966460
\(917\) 29.2711 0.966618
\(918\) 0.191725 0.00632788
\(919\) 25.1599 0.829948 0.414974 0.909833i \(-0.363791\pi\)
0.414974 + 0.909833i \(0.363791\pi\)
\(920\) 0 0
\(921\) 35.1790 1.15919
\(922\) −8.58047 −0.282583
\(923\) −1.44036 −0.0474099
\(924\) 24.5865 0.808835
\(925\) 0 0
\(926\) −4.56082 −0.149878
\(927\) −57.2034 −1.87881
\(928\) −24.0247 −0.788650
\(929\) −10.0752 −0.330556 −0.165278 0.986247i \(-0.552852\pi\)
−0.165278 + 0.986247i \(0.552852\pi\)
\(930\) 0 0
\(931\) 18.6354 0.610749
\(932\) 53.0353 1.73723
\(933\) −18.8337 −0.616589
\(934\) −1.87379 −0.0613123
\(935\) 0 0
\(936\) −4.35505 −0.142349
\(937\) 3.95364 0.129160 0.0645800 0.997913i \(-0.479429\pi\)
0.0645800 + 0.997913i \(0.479429\pi\)
\(938\) 13.1818 0.430400
\(939\) −78.8490 −2.57314
\(940\) 0 0
\(941\) 1.92792 0.0628483 0.0314242 0.999506i \(-0.489996\pi\)
0.0314242 + 0.999506i \(0.489996\pi\)
\(942\) −6.69087 −0.218000
\(943\) −25.1299 −0.818342
\(944\) −12.5961 −0.409967
\(945\) 0 0
\(946\) −0.189395 −0.00615777
\(947\) −18.2213 −0.592113 −0.296057 0.955170i \(-0.595672\pi\)
−0.296057 + 0.955170i \(0.595672\pi\)
\(948\) 55.2723 1.79516
\(949\) 2.38795 0.0775161
\(950\) 0 0
\(951\) −17.1184 −0.555103
\(952\) 10.7537 0.348530
\(953\) −25.0122 −0.810224 −0.405112 0.914267i \(-0.632768\pi\)
−0.405112 + 0.914267i \(0.632768\pi\)
\(954\) 0.425070 0.0137621
\(955\) 0 0
\(956\) −39.0399 −1.26264
\(957\) −25.2064 −0.814807
\(958\) −1.59296 −0.0514662
\(959\) 48.3675 1.56187
\(960\) 0 0
\(961\) 6.67413 0.215294
\(962\) 2.45253 0.0790726
\(963\) 4.26136 0.137320
\(964\) −23.2367 −0.748405
\(965\) 0 0
\(966\) 17.1548 0.551945
\(967\) −27.9282 −0.898111 −0.449055 0.893504i \(-0.648239\pi\)
−0.449055 + 0.893504i \(0.648239\pi\)
\(968\) 0.801215 0.0257520
\(969\) −6.57085 −0.211086
\(970\) 0 0
\(971\) 24.1677 0.775577 0.387789 0.921748i \(-0.373239\pi\)
0.387789 + 0.921748i \(0.373239\pi\)
\(972\) −43.6028 −1.39856
\(973\) −42.5008 −1.36251
\(974\) −1.66491 −0.0533471
\(975\) 0 0
\(976\) 49.8604 1.59599
\(977\) 17.7154 0.566766 0.283383 0.959007i \(-0.408543\pi\)
0.283383 + 0.959007i \(0.408543\pi\)
\(978\) 1.39125 0.0444874
\(979\) −1.51234 −0.0483347
\(980\) 0 0
\(981\) −15.7589 −0.503144
\(982\) 2.17723 0.0694782
\(983\) 19.0942 0.609010 0.304505 0.952511i \(-0.401509\pi\)
0.304505 + 0.952511i \(0.401509\pi\)
\(984\) 7.38919 0.235559
\(985\) 0 0
\(986\) −5.45540 −0.173735
\(987\) −41.3280 −1.31548
\(988\) 3.38674 0.107746
\(989\) 6.32098 0.200996
\(990\) 0 0
\(991\) 53.1833 1.68942 0.844711 0.535222i \(-0.179772\pi\)
0.844711 + 0.535222i \(0.179772\pi\)
\(992\) −14.5011 −0.460410
\(993\) 76.8484 2.43871
\(994\) −0.853712 −0.0270781
\(995\) 0 0
\(996\) −69.7453 −2.20996
\(997\) 38.2718 1.21208 0.606039 0.795435i \(-0.292757\pi\)
0.606039 + 0.795435i \(0.292757\pi\)
\(998\) −0.317394 −0.0100469
\(999\) −2.50524 −0.0792623
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 5225.2.a.bc.1.15 30
5.2 odd 4 1045.2.b.e.419.15 30
5.3 odd 4 1045.2.b.e.419.16 yes 30
5.4 even 2 inner 5225.2.a.bc.1.16 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.e.419.15 30 5.2 odd 4
1045.2.b.e.419.16 yes 30 5.3 odd 4
5225.2.a.bc.1.15 30 1.1 even 1 trivial
5225.2.a.bc.1.16 30 5.4 even 2 inner