Properties

Label 9025.2.a.bt.1.2
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4227136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{4} + 7x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.15904\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.862781 q^{2} -3.07914 q^{3} -1.25561 q^{4} +2.65662 q^{6} +0.566520 q^{7} +2.80888 q^{8} +6.48108 q^{9} +O(q^{10})\) \(q-0.862781 q^{2} -3.07914 q^{3} -1.25561 q^{4} +2.65662 q^{6} +0.566520 q^{7} +2.80888 q^{8} +6.48108 q^{9} -1.91223 q^{11} +3.86619 q^{12} -0.194531 q^{13} -0.488783 q^{14} +0.0877708 q^{16} +5.29549 q^{17} -5.59175 q^{18} -1.74439 q^{21} +1.64984 q^{22} +3.37540 q^{23} -8.64892 q^{24} +0.167838 q^{26} -10.7187 q^{27} -0.711327 q^{28} +8.73669 q^{29} -5.65662 q^{31} -5.69348 q^{32} +5.88801 q^{33} -4.56885 q^{34} -8.13770 q^{36} -0.955582 q^{37} +0.598988 q^{39} -10.0499 q^{41} +1.50503 q^{42} -4.93243 q^{43} +2.40101 q^{44} -2.91223 q^{46} +8.83942 q^{47} -0.270258 q^{48} -6.67906 q^{49} -16.3055 q^{51} +0.244255 q^{52} +8.20610 q^{53} +9.24791 q^{54} +1.59128 q^{56} -7.53785 q^{58} -3.71425 q^{59} -3.51122 q^{61} +4.88043 q^{62} +3.67166 q^{63} +4.73669 q^{64} -5.08007 q^{66} -4.04365 q^{67} -6.64906 q^{68} -10.3933 q^{69} -5.19533 q^{71} +18.2046 q^{72} +8.60409 q^{73} +0.824458 q^{74} -1.08332 q^{77} -0.516796 q^{78} -6.62648 q^{79} +13.5611 q^{81} +8.67089 q^{82} -4.51737 q^{83} +2.19027 q^{84} +4.25561 q^{86} -26.9014 q^{87} -5.37122 q^{88} -3.37352 q^{89} -0.110206 q^{91} -4.23818 q^{92} +17.4175 q^{93} -7.62648 q^{94} +17.5310 q^{96} -15.1922 q^{97} +5.76256 q^{98} -12.3933 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9} + 2 q^{11} - 22 q^{14} + 14 q^{16} - 20 q^{21} + 2 q^{24} - 22 q^{26} + 12 q^{29} - 30 q^{31} - 10 q^{34} - 14 q^{36} - 2 q^{39} - 12 q^{41} + 20 q^{44} - 4 q^{46} + 2 q^{49} - 40 q^{51} - 4 q^{54} - 46 q^{56} - 20 q^{59} - 2 q^{61} - 12 q^{64} + 6 q^{66} - 18 q^{69} + 2 q^{71} - 22 q^{74} - 24 q^{79} + 14 q^{81} - 48 q^{84} + 16 q^{86} - 36 q^{89} + 24 q^{91} - 30 q^{94} + 26 q^{96} - 30 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.862781 −0.610078 −0.305039 0.952340i \(-0.598670\pi\)
−0.305039 + 0.952340i \(0.598670\pi\)
\(3\) −3.07914 −1.77774 −0.888870 0.458159i \(-0.848509\pi\)
−0.888870 + 0.458159i \(0.848509\pi\)
\(4\) −1.25561 −0.627804
\(5\) 0 0
\(6\) 2.65662 1.08456
\(7\) 0.566520 0.214124 0.107062 0.994252i \(-0.465856\pi\)
0.107062 + 0.994252i \(0.465856\pi\)
\(8\) 2.80888 0.993088
\(9\) 6.48108 2.16036
\(10\) 0 0
\(11\) −1.91223 −0.576559 −0.288279 0.957546i \(-0.593083\pi\)
−0.288279 + 0.957546i \(0.593083\pi\)
\(12\) 3.86619 1.11607
\(13\) −0.194531 −0.0539533 −0.0269766 0.999636i \(-0.508588\pi\)
−0.0269766 + 0.999636i \(0.508588\pi\)
\(14\) −0.488783 −0.130633
\(15\) 0 0
\(16\) 0.0877708 0.0219427
\(17\) 5.29549 1.28435 0.642173 0.766560i \(-0.278033\pi\)
0.642173 + 0.766560i \(0.278033\pi\)
\(18\) −5.59175 −1.31799
\(19\) 0 0
\(20\) 0 0
\(21\) −1.74439 −0.380657
\(22\) 1.64984 0.351746
\(23\) 3.37540 0.703819 0.351910 0.936034i \(-0.385532\pi\)
0.351910 + 0.936034i \(0.385532\pi\)
\(24\) −8.64892 −1.76545
\(25\) 0 0
\(26\) 0.167838 0.0329157
\(27\) −10.7187 −2.06282
\(28\) −0.711327 −0.134428
\(29\) 8.73669 1.62236 0.811181 0.584795i \(-0.198825\pi\)
0.811181 + 0.584795i \(0.198825\pi\)
\(30\) 0 0
\(31\) −5.65662 −1.01596 −0.507980 0.861369i \(-0.669607\pi\)
−0.507980 + 0.861369i \(0.669607\pi\)
\(32\) −5.69348 −1.00648
\(33\) 5.88801 1.02497
\(34\) −4.56885 −0.783551
\(35\) 0 0
\(36\) −8.13770 −1.35628
\(37\) −0.955582 −0.157097 −0.0785484 0.996910i \(-0.525029\pi\)
−0.0785484 + 0.996910i \(0.525029\pi\)
\(38\) 0 0
\(39\) 0.598988 0.0959149
\(40\) 0 0
\(41\) −10.0499 −1.56954 −0.784768 0.619790i \(-0.787218\pi\)
−0.784768 + 0.619790i \(0.787218\pi\)
\(42\) 1.50503 0.232231
\(43\) −4.93243 −0.752189 −0.376094 0.926581i \(-0.622733\pi\)
−0.376094 + 0.926581i \(0.622733\pi\)
\(44\) 2.40101 0.361966
\(45\) 0 0
\(46\) −2.91223 −0.429385
\(47\) 8.83942 1.28936 0.644681 0.764452i \(-0.276990\pi\)
0.644681 + 0.764452i \(0.276990\pi\)
\(48\) −0.270258 −0.0390084
\(49\) −6.67906 −0.954151
\(50\) 0 0
\(51\) −16.3055 −2.28323
\(52\) 0.244255 0.0338721
\(53\) 8.20610 1.12719 0.563597 0.826050i \(-0.309417\pi\)
0.563597 + 0.826050i \(0.309417\pi\)
\(54\) 9.24791 1.25848
\(55\) 0 0
\(56\) 1.59128 0.212644
\(57\) 0 0
\(58\) −7.53785 −0.989768
\(59\) −3.71425 −0.483554 −0.241777 0.970332i \(-0.577730\pi\)
−0.241777 + 0.970332i \(0.577730\pi\)
\(60\) 0 0
\(61\) −3.51122 −0.449565 −0.224783 0.974409i \(-0.572167\pi\)
−0.224783 + 0.974409i \(0.572167\pi\)
\(62\) 4.88043 0.619815
\(63\) 3.67166 0.462586
\(64\) 4.73669 0.592086
\(65\) 0 0
\(66\) −5.08007 −0.625313
\(67\) −4.04365 −0.494010 −0.247005 0.969014i \(-0.579446\pi\)
−0.247005 + 0.969014i \(0.579446\pi\)
\(68\) −6.64906 −0.806318
\(69\) −10.3933 −1.25121
\(70\) 0 0
\(71\) −5.19533 −0.616572 −0.308286 0.951294i \(-0.599755\pi\)
−0.308286 + 0.951294i \(0.599755\pi\)
\(72\) 18.2046 2.14543
\(73\) 8.60409 1.00703 0.503516 0.863986i \(-0.332039\pi\)
0.503516 + 0.863986i \(0.332039\pi\)
\(74\) 0.824458 0.0958413
\(75\) 0 0
\(76\) 0 0
\(77\) −1.08332 −0.123455
\(78\) −0.516796 −0.0585156
\(79\) −6.62648 −0.745537 −0.372769 0.927924i \(-0.621592\pi\)
−0.372769 + 0.927924i \(0.621592\pi\)
\(80\) 0 0
\(81\) 13.5611 1.50679
\(82\) 8.67089 0.957539
\(83\) −4.51737 −0.495845 −0.247923 0.968780i \(-0.579748\pi\)
−0.247923 + 0.968780i \(0.579748\pi\)
\(84\) 2.19027 0.238978
\(85\) 0 0
\(86\) 4.25561 0.458894
\(87\) −26.9014 −2.88414
\(88\) −5.37122 −0.572574
\(89\) −3.37352 −0.357592 −0.178796 0.983886i \(-0.557220\pi\)
−0.178796 + 0.983886i \(0.557220\pi\)
\(90\) 0 0
\(91\) −0.110206 −0.0115527
\(92\) −4.23818 −0.441861
\(93\) 17.4175 1.80611
\(94\) −7.62648 −0.786612
\(95\) 0 0
\(96\) 17.5310 1.78925
\(97\) −15.1922 −1.54254 −0.771268 0.636510i \(-0.780377\pi\)
−0.771268 + 0.636510i \(0.780377\pi\)
\(98\) 5.76256 0.582107
\(99\) −12.3933 −1.24557
\(100\) 0 0
\(101\) 3.54136 0.352378 0.176189 0.984356i \(-0.443623\pi\)
0.176189 + 0.984356i \(0.443623\pi\)
\(102\) 14.0681 1.39295
\(103\) −15.6919 −1.54617 −0.773086 0.634301i \(-0.781288\pi\)
−0.773086 + 0.634301i \(0.781288\pi\)
\(104\) −0.546415 −0.0535804
\(105\) 0 0
\(106\) −7.08007 −0.687677
\(107\) −1.05731 −0.102214 −0.0511071 0.998693i \(-0.516275\pi\)
−0.0511071 + 0.998693i \(0.516275\pi\)
\(108\) 13.4585 1.29505
\(109\) 6.74439 0.645996 0.322998 0.946400i \(-0.395309\pi\)
0.322998 + 0.946400i \(0.395309\pi\)
\(110\) 0 0
\(111\) 2.94237 0.279277
\(112\) 0.0497239 0.00469847
\(113\) 7.90091 0.743255 0.371627 0.928382i \(-0.378800\pi\)
0.371627 + 0.928382i \(0.378800\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −10.9699 −1.01853
\(117\) −1.26077 −0.116558
\(118\) 3.20459 0.295006
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −7.34338 −0.667580
\(122\) 3.02941 0.274270
\(123\) 30.9451 2.79023
\(124\) 7.10250 0.637824
\(125\) 0 0
\(126\) −3.16784 −0.282213
\(127\) 8.47636 0.752155 0.376078 0.926588i \(-0.377273\pi\)
0.376078 + 0.926588i \(0.377273\pi\)
\(128\) 7.30024 0.645256
\(129\) 15.1876 1.33720
\(130\) 0 0
\(131\) 1.87439 0.163766 0.0818830 0.996642i \(-0.473907\pi\)
0.0818830 + 0.996642i \(0.473907\pi\)
\(132\) −7.39304 −0.643482
\(133\) 0 0
\(134\) 3.48878 0.301385
\(135\) 0 0
\(136\) 14.8744 1.27547
\(137\) 12.4614 1.06465 0.532323 0.846542i \(-0.321319\pi\)
0.532323 + 0.846542i \(0.321319\pi\)
\(138\) 8.96715 0.763334
\(139\) −0.313241 −0.0265687 −0.0132844 0.999912i \(-0.504229\pi\)
−0.0132844 + 0.999912i \(0.504229\pi\)
\(140\) 0 0
\(141\) −27.2178 −2.29215
\(142\) 4.48243 0.376157
\(143\) 0.371988 0.0311072
\(144\) 0.568850 0.0474041
\(145\) 0 0
\(146\) −7.42345 −0.614369
\(147\) 20.5657 1.69623
\(148\) 1.19984 0.0986260
\(149\) 4.36581 0.357661 0.178831 0.983880i \(-0.442769\pi\)
0.178831 + 0.983880i \(0.442769\pi\)
\(150\) 0 0
\(151\) −0.197977 −0.0161111 −0.00805555 0.999968i \(-0.502564\pi\)
−0.00805555 + 0.999968i \(0.502564\pi\)
\(152\) 0 0
\(153\) 34.3205 2.77465
\(154\) 0.934664 0.0753174
\(155\) 0 0
\(156\) −0.752095 −0.0602158
\(157\) 15.7700 1.25858 0.629290 0.777171i \(-0.283346\pi\)
0.629290 + 0.777171i \(0.283346\pi\)
\(158\) 5.71720 0.454836
\(159\) −25.2677 −2.00386
\(160\) 0 0
\(161\) 1.91223 0.150705
\(162\) −11.7003 −0.919262
\(163\) 9.18768 0.719635 0.359817 0.933023i \(-0.382839\pi\)
0.359817 + 0.933023i \(0.382839\pi\)
\(164\) 12.6188 0.985361
\(165\) 0 0
\(166\) 3.89750 0.302505
\(167\) 3.07021 0.237580 0.118790 0.992919i \(-0.462099\pi\)
0.118790 + 0.992919i \(0.462099\pi\)
\(168\) −4.89978 −0.378026
\(169\) −12.9622 −0.997089
\(170\) 0 0
\(171\) 0 0
\(172\) 6.19320 0.472227
\(173\) −10.4135 −0.791726 −0.395863 0.918310i \(-0.629554\pi\)
−0.395863 + 0.918310i \(0.629554\pi\)
\(174\) 23.2101 1.75955
\(175\) 0 0
\(176\) −0.167838 −0.0126513
\(177\) 11.4367 0.859634
\(178\) 2.91061 0.218159
\(179\) 13.4432 1.00479 0.502397 0.864637i \(-0.332451\pi\)
0.502397 + 0.864637i \(0.332451\pi\)
\(180\) 0 0
\(181\) −7.96216 −0.591823 −0.295911 0.955215i \(-0.595623\pi\)
−0.295911 + 0.955215i \(0.595623\pi\)
\(182\) 0.0950835 0.00704806
\(183\) 10.8115 0.799210
\(184\) 9.48108 0.698954
\(185\) 0 0
\(186\) −15.0275 −1.10187
\(187\) −10.1262 −0.740501
\(188\) −11.0988 −0.809467
\(189\) −6.07236 −0.441699
\(190\) 0 0
\(191\) −14.0999 −1.02023 −0.510115 0.860106i \(-0.670397\pi\)
−0.510115 + 0.860106i \(0.670397\pi\)
\(192\) −14.5849 −1.05257
\(193\) −3.97685 −0.286260 −0.143130 0.989704i \(-0.545717\pi\)
−0.143130 + 0.989704i \(0.545717\pi\)
\(194\) 13.1076 0.941068
\(195\) 0 0
\(196\) 8.38628 0.599020
\(197\) 18.3494 1.30734 0.653669 0.756781i \(-0.273229\pi\)
0.653669 + 0.756781i \(0.273229\pi\)
\(198\) 10.6927 0.759898
\(199\) −1.60669 −0.113895 −0.0569477 0.998377i \(-0.518137\pi\)
−0.0569477 + 0.998377i \(0.518137\pi\)
\(200\) 0 0
\(201\) 12.4509 0.878222
\(202\) −3.05542 −0.214978
\(203\) 4.94951 0.347387
\(204\) 20.4734 1.43342
\(205\) 0 0
\(206\) 13.5387 0.943287
\(207\) 21.8762 1.52050
\(208\) −0.0170742 −0.00118388
\(209\) 0 0
\(210\) 0 0
\(211\) 6.09986 0.419931 0.209966 0.977709i \(-0.432665\pi\)
0.209966 + 0.977709i \(0.432665\pi\)
\(212\) −10.3036 −0.707658
\(213\) 15.9971 1.09611
\(214\) 0.912229 0.0623587
\(215\) 0 0
\(216\) −30.1076 −2.04856
\(217\) −3.20459 −0.217542
\(218\) −5.81893 −0.394108
\(219\) −26.4932 −1.79024
\(220\) 0 0
\(221\) −1.03014 −0.0692946
\(222\) −2.53862 −0.170381
\(223\) −17.2253 −1.15349 −0.576744 0.816925i \(-0.695677\pi\)
−0.576744 + 0.816925i \(0.695677\pi\)
\(224\) −3.22547 −0.215511
\(225\) 0 0
\(226\) −6.81675 −0.453444
\(227\) −1.18505 −0.0786542 −0.0393271 0.999226i \(-0.512521\pi\)
−0.0393271 + 0.999226i \(0.512521\pi\)
\(228\) 0 0
\(229\) 6.24791 0.412873 0.206437 0.978460i \(-0.433813\pi\)
0.206437 + 0.978460i \(0.433813\pi\)
\(230\) 0 0
\(231\) 3.33568 0.219471
\(232\) 24.5403 1.61115
\(233\) −2.75687 −0.180609 −0.0903043 0.995914i \(-0.528784\pi\)
−0.0903043 + 0.995914i \(0.528784\pi\)
\(234\) 1.08777 0.0711098
\(235\) 0 0
\(236\) 4.66365 0.303578
\(237\) 20.4038 1.32537
\(238\) −2.58834 −0.167777
\(239\) 17.3055 1.11940 0.559701 0.828695i \(-0.310916\pi\)
0.559701 + 0.828695i \(0.310916\pi\)
\(240\) 0 0
\(241\) 4.78662 0.308333 0.154167 0.988045i \(-0.450731\pi\)
0.154167 + 0.988045i \(0.450731\pi\)
\(242\) 6.33573 0.407276
\(243\) −9.60047 −0.615870
\(244\) 4.40872 0.282239
\(245\) 0 0
\(246\) −26.6988 −1.70226
\(247\) 0 0
\(248\) −15.8888 −1.00894
\(249\) 13.9096 0.881484
\(250\) 0 0
\(251\) −26.2479 −1.65675 −0.828377 0.560172i \(-0.810735\pi\)
−0.828377 + 0.560172i \(0.810735\pi\)
\(252\) −4.61017 −0.290413
\(253\) −6.45453 −0.405793
\(254\) −7.31324 −0.458874
\(255\) 0 0
\(256\) −15.7719 −0.985743
\(257\) 22.2505 1.38795 0.693974 0.720000i \(-0.255858\pi\)
0.693974 + 0.720000i \(0.255858\pi\)
\(258\) −13.1036 −0.815794
\(259\) −0.541356 −0.0336382
\(260\) 0 0
\(261\) 56.6232 3.50489
\(262\) −1.61719 −0.0999101
\(263\) 8.81341 0.543458 0.271729 0.962374i \(-0.412405\pi\)
0.271729 + 0.962374i \(0.412405\pi\)
\(264\) 16.5387 1.01789
\(265\) 0 0
\(266\) 0 0
\(267\) 10.3875 0.635706
\(268\) 5.07724 0.310142
\(269\) 4.76418 0.290477 0.145239 0.989397i \(-0.453605\pi\)
0.145239 + 0.989397i \(0.453605\pi\)
\(270\) 0 0
\(271\) 3.51892 0.213759 0.106880 0.994272i \(-0.465914\pi\)
0.106880 + 0.994272i \(0.465914\pi\)
\(272\) 0.464790 0.0281820
\(273\) 0.339339 0.0205377
\(274\) −10.7514 −0.649517
\(275\) 0 0
\(276\) 13.0499 0.785513
\(277\) 32.7724 1.96910 0.984551 0.175098i \(-0.0560242\pi\)
0.984551 + 0.175098i \(0.0560242\pi\)
\(278\) 0.270258 0.0162090
\(279\) −36.6610 −2.19484
\(280\) 0 0
\(281\) −9.91993 −0.591774 −0.295887 0.955223i \(-0.595615\pi\)
−0.295887 + 0.955223i \(0.595615\pi\)
\(282\) 23.4830 1.39839
\(283\) −2.47182 −0.146935 −0.0734673 0.997298i \(-0.523406\pi\)
−0.0734673 + 0.997298i \(0.523406\pi\)
\(284\) 6.52330 0.387087
\(285\) 0 0
\(286\) −0.320945 −0.0189779
\(287\) −5.69348 −0.336076
\(288\) −36.8999 −2.17435
\(289\) 11.0422 0.649543
\(290\) 0 0
\(291\) 46.7789 2.74223
\(292\) −10.8034 −0.632219
\(293\) −17.0284 −0.994812 −0.497406 0.867518i \(-0.665714\pi\)
−0.497406 + 0.867518i \(0.665714\pi\)
\(294\) −17.7437 −1.03483
\(295\) 0 0
\(296\) −2.68411 −0.156011
\(297\) 20.4966 1.18934
\(298\) −3.76674 −0.218202
\(299\) −0.656620 −0.0379733
\(300\) 0 0
\(301\) −2.79432 −0.161062
\(302\) 0.170810 0.00982904
\(303\) −10.9043 −0.626437
\(304\) 0 0
\(305\) 0 0
\(306\) −29.6111 −1.69275
\(307\) 20.9793 1.19735 0.598676 0.800992i \(-0.295694\pi\)
0.598676 + 0.800992i \(0.295694\pi\)
\(308\) 1.36022 0.0775058
\(309\) 48.3176 2.74869
\(310\) 0 0
\(311\) −3.14805 −0.178509 −0.0892547 0.996009i \(-0.528449\pi\)
−0.0892547 + 0.996009i \(0.528449\pi\)
\(312\) 1.68248 0.0952520
\(313\) −12.2928 −0.694831 −0.347416 0.937711i \(-0.612941\pi\)
−0.347416 + 0.937711i \(0.612941\pi\)
\(314\) −13.6060 −0.767832
\(315\) 0 0
\(316\) 8.32027 0.468052
\(317\) −11.4619 −0.643765 −0.321882 0.946780i \(-0.604316\pi\)
−0.321882 + 0.946780i \(0.604316\pi\)
\(318\) 21.8005 1.22251
\(319\) −16.7065 −0.935387
\(320\) 0 0
\(321\) 3.25561 0.181710
\(322\) −1.64984 −0.0919417
\(323\) 0 0
\(324\) −17.0275 −0.945972
\(325\) 0 0
\(326\) −7.92696 −0.439034
\(327\) −20.7669 −1.14841
\(328\) −28.2290 −1.55869
\(329\) 5.00770 0.276084
\(330\) 0 0
\(331\) −5.85724 −0.321943 −0.160972 0.986959i \(-0.551463\pi\)
−0.160972 + 0.986959i \(0.551463\pi\)
\(332\) 5.67204 0.311294
\(333\) −6.19320 −0.339385
\(334\) −2.64892 −0.144942
\(335\) 0 0
\(336\) −0.153107 −0.00835265
\(337\) −8.60995 −0.469014 −0.234507 0.972114i \(-0.575348\pi\)
−0.234507 + 0.972114i \(0.575348\pi\)
\(338\) 11.1835 0.608302
\(339\) −24.3280 −1.32131
\(340\) 0 0
\(341\) 10.8168 0.585760
\(342\) 0 0
\(343\) −7.74945 −0.418431
\(344\) −13.8546 −0.746990
\(345\) 0 0
\(346\) 8.98459 0.483015
\(347\) −12.0901 −0.649033 −0.324517 0.945880i \(-0.605202\pi\)
−0.324517 + 0.945880i \(0.605202\pi\)
\(348\) 33.7777 1.81067
\(349\) 18.9819 1.01608 0.508040 0.861333i \(-0.330370\pi\)
0.508040 + 0.861333i \(0.330370\pi\)
\(350\) 0 0
\(351\) 2.08513 0.111296
\(352\) 10.8872 0.580292
\(353\) −7.71759 −0.410766 −0.205383 0.978682i \(-0.565844\pi\)
−0.205383 + 0.978682i \(0.565844\pi\)
\(354\) −9.86736 −0.524444
\(355\) 0 0
\(356\) 4.23582 0.224498
\(357\) −9.23741 −0.488895
\(358\) −11.5986 −0.613003
\(359\) −7.03014 −0.371037 −0.185518 0.982641i \(-0.559396\pi\)
−0.185518 + 0.982641i \(0.559396\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 6.86960 0.361058
\(363\) 22.6113 1.18678
\(364\) 0.138375 0.00725284
\(365\) 0 0
\(366\) −9.32797 −0.487581
\(367\) −33.7109 −1.75969 −0.879847 0.475257i \(-0.842355\pi\)
−0.879847 + 0.475257i \(0.842355\pi\)
\(368\) 0.296261 0.0154437
\(369\) −65.1344 −3.39076
\(370\) 0 0
\(371\) 4.64892 0.241360
\(372\) −21.8696 −1.13388
\(373\) −10.0097 −0.518281 −0.259141 0.965840i \(-0.583439\pi\)
−0.259141 + 0.965840i \(0.583439\pi\)
\(374\) 8.73669 0.451763
\(375\) 0 0
\(376\) 24.8288 1.28045
\(377\) −1.69956 −0.0875317
\(378\) 5.23912 0.269471
\(379\) 35.8064 1.83925 0.919626 0.392796i \(-0.128492\pi\)
0.919626 + 0.392796i \(0.128492\pi\)
\(380\) 0 0
\(381\) −26.0999 −1.33714
\(382\) 12.1651 0.622420
\(383\) 16.2495 0.830312 0.415156 0.909750i \(-0.363727\pi\)
0.415156 + 0.909750i \(0.363727\pi\)
\(384\) −22.4784 −1.14710
\(385\) 0 0
\(386\) 3.43115 0.174641
\(387\) −31.9675 −1.62500
\(388\) 19.0755 0.968411
\(389\) −19.3856 −0.982889 −0.491445 0.870909i \(-0.663531\pi\)
−0.491445 + 0.870909i \(0.663531\pi\)
\(390\) 0 0
\(391\) 17.8744 0.903947
\(392\) −18.7607 −0.947556
\(393\) −5.77149 −0.291133
\(394\) −15.8315 −0.797579
\(395\) 0 0
\(396\) 15.5611 0.781977
\(397\) 5.35414 0.268717 0.134358 0.990933i \(-0.457103\pi\)
0.134358 + 0.990933i \(0.457103\pi\)
\(398\) 1.38622 0.0694851
\(399\) 0 0
\(400\) 0 0
\(401\) −8.33832 −0.416396 −0.208198 0.978087i \(-0.566760\pi\)
−0.208198 + 0.978087i \(0.566760\pi\)
\(402\) −10.7424 −0.535784
\(403\) 1.10039 0.0548143
\(404\) −4.44656 −0.221225
\(405\) 0 0
\(406\) −4.27034 −0.211933
\(407\) 1.82729 0.0905755
\(408\) −45.8003 −2.26745
\(409\) 35.3227 1.74659 0.873297 0.487188i \(-0.161977\pi\)
0.873297 + 0.487188i \(0.161977\pi\)
\(410\) 0 0
\(411\) −38.3702 −1.89266
\(412\) 19.7029 0.970694
\(413\) −2.10420 −0.103541
\(414\) −18.8744 −0.927626
\(415\) 0 0
\(416\) 1.10756 0.0543026
\(417\) 0.964511 0.0472323
\(418\) 0 0
\(419\) 21.2453 1.03790 0.518949 0.854805i \(-0.326323\pi\)
0.518949 + 0.854805i \(0.326323\pi\)
\(420\) 0 0
\(421\) 14.8744 0.724933 0.362467 0.931997i \(-0.381935\pi\)
0.362467 + 0.931997i \(0.381935\pi\)
\(422\) −5.26284 −0.256191
\(423\) 57.2890 2.78548
\(424\) 23.0499 1.11940
\(425\) 0 0
\(426\) −13.8020 −0.668710
\(427\) −1.98917 −0.0962629
\(428\) 1.32757 0.0641706
\(429\) −1.14540 −0.0553006
\(430\) 0 0
\(431\) 16.9518 0.816540 0.408270 0.912861i \(-0.366132\pi\)
0.408270 + 0.912861i \(0.366132\pi\)
\(432\) −0.940790 −0.0452638
\(433\) −23.1459 −1.11232 −0.556161 0.831075i \(-0.687726\pi\)
−0.556161 + 0.831075i \(0.687726\pi\)
\(434\) 2.76486 0.132717
\(435\) 0 0
\(436\) −8.46832 −0.405559
\(437\) 0 0
\(438\) 22.8578 1.09219
\(439\) −33.5457 −1.60105 −0.800525 0.599299i \(-0.795446\pi\)
−0.800525 + 0.599299i \(0.795446\pi\)
\(440\) 0 0
\(441\) −43.2875 −2.06131
\(442\) 0.888784 0.0422752
\(443\) −31.3579 −1.48986 −0.744929 0.667144i \(-0.767517\pi\)
−0.744929 + 0.667144i \(0.767517\pi\)
\(444\) −3.69446 −0.175331
\(445\) 0 0
\(446\) 14.8616 0.703718
\(447\) −13.4429 −0.635829
\(448\) 2.68343 0.126780
\(449\) 16.0301 0.756509 0.378255 0.925702i \(-0.376524\pi\)
0.378255 + 0.925702i \(0.376524\pi\)
\(450\) 0 0
\(451\) 19.2178 0.904929
\(452\) −9.92045 −0.466619
\(453\) 0.609597 0.0286414
\(454\) 1.02243 0.0479853
\(455\) 0 0
\(456\) 0 0
\(457\) −15.3980 −0.720286 −0.360143 0.932897i \(-0.617272\pi\)
−0.360143 + 0.932897i \(0.617272\pi\)
\(458\) −5.39057 −0.251885
\(459\) −56.7609 −2.64937
\(460\) 0 0
\(461\) −5.81411 −0.270790 −0.135395 0.990792i \(-0.543230\pi\)
−0.135395 + 0.990792i \(0.543230\pi\)
\(462\) −2.87796 −0.133895
\(463\) −32.6788 −1.51871 −0.759356 0.650675i \(-0.774486\pi\)
−0.759356 + 0.650675i \(0.774486\pi\)
\(464\) 0.766826 0.0355990
\(465\) 0 0
\(466\) 2.37858 0.110185
\(467\) 6.59041 0.304968 0.152484 0.988306i \(-0.451273\pi\)
0.152484 + 0.988306i \(0.451273\pi\)
\(468\) 1.58304 0.0731759
\(469\) −2.29081 −0.105780
\(470\) 0 0
\(471\) −48.5578 −2.23743
\(472\) −10.4329 −0.480212
\(473\) 9.43194 0.433681
\(474\) −17.6040 −0.808581
\(475\) 0 0
\(476\) −3.76683 −0.172652
\(477\) 53.1844 2.43515
\(478\) −14.9309 −0.682923
\(479\) −36.8064 −1.68173 −0.840864 0.541247i \(-0.817952\pi\)
−0.840864 + 0.541247i \(0.817952\pi\)
\(480\) 0 0
\(481\) 0.185891 0.00847588
\(482\) −4.12980 −0.188107
\(483\) −5.88801 −0.267914
\(484\) 9.22041 0.419110
\(485\) 0 0
\(486\) 8.28310 0.375729
\(487\) 2.45475 0.111235 0.0556176 0.998452i \(-0.482287\pi\)
0.0556176 + 0.998452i \(0.482287\pi\)
\(488\) −9.86258 −0.446458
\(489\) −28.2901 −1.27932
\(490\) 0 0
\(491\) 17.7943 0.803046 0.401523 0.915849i \(-0.368481\pi\)
0.401523 + 0.915849i \(0.368481\pi\)
\(492\) −38.8549 −1.75172
\(493\) 46.2651 2.08367
\(494\) 0 0
\(495\) 0 0
\(496\) −0.496486 −0.0222929
\(497\) −2.94326 −0.132023
\(498\) −12.0009 −0.537774
\(499\) −29.2798 −1.31074 −0.655371 0.755307i \(-0.727488\pi\)
−0.655371 + 0.755307i \(0.727488\pi\)
\(500\) 0 0
\(501\) −9.45359 −0.422355
\(502\) 22.6462 1.01075
\(503\) 29.4378 1.31257 0.656283 0.754515i \(-0.272128\pi\)
0.656283 + 0.754515i \(0.272128\pi\)
\(504\) 10.3132 0.459388
\(505\) 0 0
\(506\) 5.56885 0.247566
\(507\) 39.9122 1.77257
\(508\) −10.6430 −0.472206
\(509\) −6.69446 −0.296727 −0.148363 0.988933i \(-0.547401\pi\)
−0.148363 + 0.988933i \(0.547401\pi\)
\(510\) 0 0
\(511\) 4.87439 0.215630
\(512\) −0.992797 −0.0438758
\(513\) 0 0
\(514\) −19.1973 −0.846757
\(515\) 0 0
\(516\) −19.0697 −0.839498
\(517\) −16.9030 −0.743393
\(518\) 0.467072 0.0205220
\(519\) 32.0647 1.40748
\(520\) 0 0
\(521\) −20.0801 −0.879724 −0.439862 0.898065i \(-0.644973\pi\)
−0.439862 + 0.898065i \(0.644973\pi\)
\(522\) −48.8534 −2.13825
\(523\) 36.4440 1.59359 0.796793 0.604252i \(-0.206528\pi\)
0.796793 + 0.604252i \(0.206528\pi\)
\(524\) −2.35350 −0.102813
\(525\) 0 0
\(526\) −7.60405 −0.331552
\(527\) −29.9546 −1.30484
\(528\) 0.516796 0.0224907
\(529\) −11.6067 −0.504639
\(530\) 0 0
\(531\) −24.0724 −1.04465
\(532\) 0 0
\(533\) 1.95503 0.0846816
\(534\) −8.96216 −0.387830
\(535\) 0 0
\(536\) −11.3581 −0.490596
\(537\) −41.3936 −1.78626
\(538\) −4.11045 −0.177214
\(539\) 12.7719 0.550124
\(540\) 0 0
\(541\) −14.6291 −0.628955 −0.314478 0.949265i \(-0.601829\pi\)
−0.314478 + 0.949265i \(0.601829\pi\)
\(542\) −3.03606 −0.130410
\(543\) 24.5166 1.05211
\(544\) −30.1498 −1.29266
\(545\) 0 0
\(546\) −0.292775 −0.0125296
\(547\) −20.4186 −0.873038 −0.436519 0.899695i \(-0.643789\pi\)
−0.436519 + 0.899695i \(0.643789\pi\)
\(548\) −15.6466 −0.668389
\(549\) −22.7565 −0.971223
\(550\) 0 0
\(551\) 0 0
\(552\) −29.1935 −1.24256
\(553\) −3.75403 −0.159638
\(554\) −28.2754 −1.20131
\(555\) 0 0
\(556\) 0.393308 0.0166800
\(557\) 35.8798 1.52028 0.760138 0.649761i \(-0.225131\pi\)
0.760138 + 0.649761i \(0.225131\pi\)
\(558\) 31.6304 1.33902
\(559\) 0.959512 0.0405830
\(560\) 0 0
\(561\) 31.1799 1.31642
\(562\) 8.55873 0.361028
\(563\) −37.7708 −1.59185 −0.795925 0.605395i \(-0.793015\pi\)
−0.795925 + 0.605395i \(0.793015\pi\)
\(564\) 34.1749 1.43902
\(565\) 0 0
\(566\) 2.13264 0.0896416
\(567\) 7.68266 0.322641
\(568\) −14.5931 −0.612311
\(569\) −28.0844 −1.17736 −0.588681 0.808366i \(-0.700352\pi\)
−0.588681 + 0.808366i \(0.700352\pi\)
\(570\) 0 0
\(571\) 20.6040 0.862253 0.431126 0.902292i \(-0.358116\pi\)
0.431126 + 0.902292i \(0.358116\pi\)
\(572\) −0.467072 −0.0195293
\(573\) 43.4154 1.81370
\(574\) 4.91223 0.205032
\(575\) 0 0
\(576\) 30.6988 1.27912
\(577\) −43.8371 −1.82496 −0.912481 0.409119i \(-0.865836\pi\)
−0.912481 + 0.409119i \(0.865836\pi\)
\(578\) −9.52702 −0.396272
\(579\) 12.2453 0.508896
\(580\) 0 0
\(581\) −2.55918 −0.106173
\(582\) −40.3600 −1.67297
\(583\) −15.6919 −0.649894
\(584\) 24.1678 1.00007
\(585\) 0 0
\(586\) 14.6918 0.606913
\(587\) −37.9809 −1.56764 −0.783821 0.620987i \(-0.786732\pi\)
−0.783821 + 0.620987i \(0.786732\pi\)
\(588\) −25.8225 −1.06490
\(589\) 0 0
\(590\) 0 0
\(591\) −56.5002 −2.32411
\(592\) −0.0838722 −0.00344713
\(593\) 5.38015 0.220936 0.110468 0.993880i \(-0.464765\pi\)
0.110468 + 0.993880i \(0.464765\pi\)
\(594\) −17.6841 −0.725588
\(595\) 0 0
\(596\) −5.48175 −0.224541
\(597\) 4.94722 0.202476
\(598\) 0.566520 0.0231667
\(599\) −22.4285 −0.916404 −0.458202 0.888848i \(-0.651506\pi\)
−0.458202 + 0.888848i \(0.651506\pi\)
\(600\) 0 0
\(601\) 29.5732 1.20632 0.603159 0.797621i \(-0.293909\pi\)
0.603159 + 0.797621i \(0.293909\pi\)
\(602\) 2.41089 0.0982604
\(603\) −26.2072 −1.06724
\(604\) 0.248581 0.0101146
\(605\) 0 0
\(606\) 9.40804 0.382175
\(607\) 24.5915 0.998139 0.499069 0.866562i \(-0.333675\pi\)
0.499069 + 0.866562i \(0.333675\pi\)
\(608\) 0 0
\(609\) −15.2402 −0.617564
\(610\) 0 0
\(611\) −1.71954 −0.0695653
\(612\) −43.0931 −1.74194
\(613\) −29.8901 −1.20725 −0.603624 0.797269i \(-0.706277\pi\)
−0.603624 + 0.797269i \(0.706277\pi\)
\(614\) −18.1005 −0.730478
\(615\) 0 0
\(616\) −3.04290 −0.122602
\(617\) −40.7141 −1.63909 −0.819544 0.573017i \(-0.805773\pi\)
−0.819544 + 0.573017i \(0.805773\pi\)
\(618\) −41.6875 −1.67692
\(619\) −28.4784 −1.14464 −0.572322 0.820029i \(-0.693957\pi\)
−0.572322 + 0.820029i \(0.693957\pi\)
\(620\) 0 0
\(621\) −36.1799 −1.45185
\(622\) 2.71608 0.108905
\(623\) −1.91116 −0.0765692
\(624\) 0.0525737 0.00210463
\(625\) 0 0
\(626\) 10.6060 0.423902
\(627\) 0 0
\(628\) −19.8009 −0.790141
\(629\) −5.06028 −0.201766
\(630\) 0 0
\(631\) 42.4201 1.68872 0.844359 0.535777i \(-0.179981\pi\)
0.844359 + 0.535777i \(0.179981\pi\)
\(632\) −18.6130 −0.740384
\(633\) −18.7823 −0.746529
\(634\) 9.88912 0.392747
\(635\) 0 0
\(636\) 31.7263 1.25803
\(637\) 1.29929 0.0514796
\(638\) 14.4141 0.570659
\(639\) −33.6714 −1.33202
\(640\) 0 0
\(641\) 29.5259 1.16620 0.583102 0.812399i \(-0.301839\pi\)
0.583102 + 0.812399i \(0.301839\pi\)
\(642\) −2.80888 −0.110858
\(643\) 21.5970 0.851704 0.425852 0.904793i \(-0.359974\pi\)
0.425852 + 0.904793i \(0.359974\pi\)
\(644\) −2.40101 −0.0946131
\(645\) 0 0
\(646\) 0 0
\(647\) −12.6128 −0.495861 −0.247930 0.968778i \(-0.579750\pi\)
−0.247930 + 0.968778i \(0.579750\pi\)
\(648\) 38.0916 1.49638
\(649\) 7.10250 0.278798
\(650\) 0 0
\(651\) 9.86736 0.386732
\(652\) −11.5361 −0.451790
\(653\) −26.6312 −1.04216 −0.521080 0.853508i \(-0.674471\pi\)
−0.521080 + 0.853508i \(0.674471\pi\)
\(654\) 17.9173 0.700621
\(655\) 0 0
\(656\) −0.882090 −0.0344398
\(657\) 55.7638 2.17555
\(658\) −4.32055 −0.168433
\(659\) −39.9545 −1.55640 −0.778202 0.628014i \(-0.783868\pi\)
−0.778202 + 0.628014i \(0.783868\pi\)
\(660\) 0 0
\(661\) 13.5233 0.525996 0.262998 0.964796i \(-0.415289\pi\)
0.262998 + 0.964796i \(0.415289\pi\)
\(662\) 5.05352 0.196411
\(663\) 3.17194 0.123188
\(664\) −12.6887 −0.492418
\(665\) 0 0
\(666\) 5.34338 0.207052
\(667\) 29.4898 1.14185
\(668\) −3.85498 −0.149154
\(669\) 53.0389 2.05060
\(670\) 0 0
\(671\) 6.71425 0.259201
\(672\) 9.93166 0.383122
\(673\) −34.4110 −1.32645 −0.663224 0.748421i \(-0.730812\pi\)
−0.663224 + 0.748421i \(0.730812\pi\)
\(674\) 7.42851 0.286135
\(675\) 0 0
\(676\) 16.2754 0.625977
\(677\) 29.6650 1.14012 0.570059 0.821604i \(-0.306920\pi\)
0.570059 + 0.821604i \(0.306920\pi\)
\(678\) 20.9897 0.806105
\(679\) −8.60669 −0.330295
\(680\) 0 0
\(681\) 3.64892 0.139827
\(682\) −9.33249 −0.357360
\(683\) 27.8978 1.06748 0.533740 0.845648i \(-0.320786\pi\)
0.533740 + 0.845648i \(0.320786\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 6.68608 0.255276
\(687\) −19.2381 −0.733981
\(688\) −0.432924 −0.0165051
\(689\) −1.59634 −0.0608158
\(690\) 0 0
\(691\) −49.6386 −1.88834 −0.944170 0.329459i \(-0.893134\pi\)
−0.944170 + 0.329459i \(0.893134\pi\)
\(692\) 13.0753 0.497049
\(693\) −7.02105 −0.266708
\(694\) 10.4312 0.395961
\(695\) 0 0
\(696\) −75.5629 −2.86420
\(697\) −53.2193 −2.01582
\(698\) −16.3773 −0.619889
\(699\) 8.48878 0.321075
\(700\) 0 0
\(701\) −49.9813 −1.88777 −0.943883 0.330279i \(-0.892857\pi\)
−0.943883 + 0.330279i \(0.892857\pi\)
\(702\) −1.79901 −0.0678991
\(703\) 0 0
\(704\) −9.05763 −0.341372
\(705\) 0 0
\(706\) 6.65859 0.250599
\(707\) 2.00625 0.0754527
\(708\) −14.3600 −0.539682
\(709\) 25.1876 0.945941 0.472971 0.881078i \(-0.343182\pi\)
0.472971 + 0.881078i \(0.343182\pi\)
\(710\) 0 0
\(711\) −42.9468 −1.61063
\(712\) −9.47580 −0.355121
\(713\) −19.0933 −0.715051
\(714\) 7.96986 0.298265
\(715\) 0 0
\(716\) −16.8794 −0.630814
\(717\) −53.2861 −1.99001
\(718\) 6.06547 0.226361
\(719\) 17.1755 0.640540 0.320270 0.947326i \(-0.396226\pi\)
0.320270 + 0.947326i \(0.396226\pi\)
\(720\) 0 0
\(721\) −8.88979 −0.331073
\(722\) 0 0
\(723\) −14.7386 −0.548136
\(724\) 9.99735 0.371549
\(725\) 0 0
\(726\) −19.5086 −0.724031
\(727\) −1.05653 −0.0391845 −0.0195922 0.999808i \(-0.506237\pi\)
−0.0195922 + 0.999808i \(0.506237\pi\)
\(728\) −0.309555 −0.0114729
\(729\) −11.1223 −0.411937
\(730\) 0 0
\(731\) −26.1196 −0.966070
\(732\) −13.5750 −0.501748
\(733\) 20.5025 0.757277 0.378639 0.925545i \(-0.376392\pi\)
0.378639 + 0.925545i \(0.376392\pi\)
\(734\) 29.0851 1.07355
\(735\) 0 0
\(736\) −19.2178 −0.708376
\(737\) 7.73238 0.284826
\(738\) 56.1967 2.06863
\(739\) 24.9096 0.916314 0.458157 0.888871i \(-0.348510\pi\)
0.458157 + 0.888871i \(0.348510\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.01100 −0.147248
\(743\) −13.2573 −0.486364 −0.243182 0.969981i \(-0.578191\pi\)
−0.243182 + 0.969981i \(0.578191\pi\)
\(744\) 48.9236 1.79363
\(745\) 0 0
\(746\) 8.63615 0.316192
\(747\) −29.2774 −1.07120
\(748\) 12.7145 0.464889
\(749\) −0.598988 −0.0218866
\(750\) 0 0
\(751\) −24.8365 −0.906298 −0.453149 0.891435i \(-0.649700\pi\)
−0.453149 + 0.891435i \(0.649700\pi\)
\(752\) 0.775843 0.0282921
\(753\) 80.8209 2.94528
\(754\) 1.46635 0.0534012
\(755\) 0 0
\(756\) 7.62451 0.277301
\(757\) 48.4658 1.76152 0.880760 0.473562i \(-0.157032\pi\)
0.880760 + 0.473562i \(0.157032\pi\)
\(758\) −30.8931 −1.12209
\(759\) 19.8744 0.721395
\(760\) 0 0
\(761\) 40.3726 1.46351 0.731753 0.681570i \(-0.238702\pi\)
0.731753 + 0.681570i \(0.238702\pi\)
\(762\) 22.5185 0.815758
\(763\) 3.82083 0.138323
\(764\) 17.7039 0.640505
\(765\) 0 0
\(766\) −14.0198 −0.506556
\(767\) 0.722538 0.0260893
\(768\) 48.5638 1.75239
\(769\) −44.9545 −1.62110 −0.810550 0.585670i \(-0.800831\pi\)
−0.810550 + 0.585670i \(0.800831\pi\)
\(770\) 0 0
\(771\) −68.5123 −2.46741
\(772\) 4.99337 0.179715
\(773\) −3.68645 −0.132592 −0.0662962 0.997800i \(-0.521118\pi\)
−0.0662962 + 0.997800i \(0.521118\pi\)
\(774\) 27.5809 0.991376
\(775\) 0 0
\(776\) −42.6731 −1.53187
\(777\) 1.66691 0.0598000
\(778\) 16.7255 0.599639
\(779\) 0 0
\(780\) 0 0
\(781\) 9.93466 0.355490
\(782\) −15.4217 −0.551478
\(783\) −93.6461 −3.34664
\(784\) −0.586226 −0.0209366
\(785\) 0 0
\(786\) 4.97953 0.177614
\(787\) −45.7141 −1.62953 −0.814766 0.579790i \(-0.803135\pi\)
−0.814766 + 0.579790i \(0.803135\pi\)
\(788\) −23.0396 −0.820753
\(789\) −27.1377 −0.966128
\(790\) 0 0
\(791\) 4.47602 0.159149
\(792\) −34.8113 −1.23697
\(793\) 0.683042 0.0242555
\(794\) −4.61945 −0.163938
\(795\) 0 0
\(796\) 2.01738 0.0715040
\(797\) 30.3378 1.07462 0.537310 0.843385i \(-0.319441\pi\)
0.537310 + 0.843385i \(0.319441\pi\)
\(798\) 0 0
\(799\) 46.8091 1.65599
\(800\) 0 0
\(801\) −21.8640 −0.772528
\(802\) 7.19415 0.254034
\(803\) −16.4530 −0.580614
\(804\) −15.6335 −0.551351
\(805\) 0 0
\(806\) −0.949395 −0.0334410
\(807\) −14.6696 −0.516393
\(808\) 9.94724 0.349943
\(809\) 34.5585 1.21501 0.607506 0.794315i \(-0.292170\pi\)
0.607506 + 0.794315i \(0.292170\pi\)
\(810\) 0 0
\(811\) −33.2754 −1.16846 −0.584229 0.811589i \(-0.698603\pi\)
−0.584229 + 0.811589i \(0.698603\pi\)
\(812\) −6.21464 −0.218091
\(813\) −10.8352 −0.380008
\(814\) −1.57655 −0.0552582
\(815\) 0 0
\(816\) −1.43115 −0.0501003
\(817\) 0 0
\(818\) −30.4757 −1.06556
\(819\) −0.714253 −0.0249580
\(820\) 0 0
\(821\) 12.7013 0.443277 0.221638 0.975129i \(-0.428860\pi\)
0.221638 + 0.975129i \(0.428860\pi\)
\(822\) 33.1051 1.15467
\(823\) −12.0091 −0.418610 −0.209305 0.977850i \(-0.567120\pi\)
−0.209305 + 0.977850i \(0.567120\pi\)
\(824\) −44.0767 −1.53549
\(825\) 0 0
\(826\) 1.81546 0.0631680
\(827\) −8.54544 −0.297154 −0.148577 0.988901i \(-0.547469\pi\)
−0.148577 + 0.988901i \(0.547469\pi\)
\(828\) −27.4680 −0.954578
\(829\) 27.1042 0.941369 0.470685 0.882302i \(-0.344007\pi\)
0.470685 + 0.882302i \(0.344007\pi\)
\(830\) 0 0
\(831\) −100.911 −3.50055
\(832\) −0.921434 −0.0319450
\(833\) −35.3689 −1.22546
\(834\) −0.832162 −0.0288154
\(835\) 0 0
\(836\) 0 0
\(837\) 60.6317 2.09574
\(838\) −18.3300 −0.633200
\(839\) 36.4379 1.25798 0.628989 0.777414i \(-0.283469\pi\)
0.628989 + 0.777414i \(0.283469\pi\)
\(840\) 0 0
\(841\) 47.3297 1.63206
\(842\) −12.8333 −0.442266
\(843\) 30.5448 1.05202
\(844\) −7.65903 −0.263635
\(845\) 0 0
\(846\) −49.4278 −1.69936
\(847\) −4.16017 −0.142945
\(848\) 0.720256 0.0247337
\(849\) 7.61107 0.261211
\(850\) 0 0
\(851\) −3.22547 −0.110568
\(852\) −20.0861 −0.688140
\(853\) −18.3471 −0.628192 −0.314096 0.949391i \(-0.601701\pi\)
−0.314096 + 0.949391i \(0.601701\pi\)
\(854\) 1.71622 0.0587279
\(855\) 0 0
\(856\) −2.96986 −0.101508
\(857\) 7.85783 0.268418 0.134209 0.990953i \(-0.457151\pi\)
0.134209 + 0.990953i \(0.457151\pi\)
\(858\) 0.988232 0.0337377
\(859\) −34.3022 −1.17038 −0.585188 0.810897i \(-0.698979\pi\)
−0.585188 + 0.810897i \(0.698979\pi\)
\(860\) 0 0
\(861\) 17.5310 0.597455
\(862\) −14.6257 −0.498153
\(863\) 35.9451 1.22359 0.611793 0.791018i \(-0.290449\pi\)
0.611793 + 0.791018i \(0.290449\pi\)
\(864\) 61.0268 2.07617
\(865\) 0 0
\(866\) 19.9699 0.678604
\(867\) −34.0005 −1.15472
\(868\) 4.02371 0.136574
\(869\) 12.6714 0.429846
\(870\) 0 0
\(871\) 0.786616 0.0266535
\(872\) 18.9442 0.641531
\(873\) −98.4620 −3.33243
\(874\) 0 0
\(875\) 0 0
\(876\) 33.2650 1.12392
\(877\) 25.2516 0.852686 0.426343 0.904561i \(-0.359802\pi\)
0.426343 + 0.904561i \(0.359802\pi\)
\(878\) 28.9426 0.976766
\(879\) 52.4329 1.76852
\(880\) 0 0
\(881\) 17.7796 0.599010 0.299505 0.954095i \(-0.403179\pi\)
0.299505 + 0.954095i \(0.403179\pi\)
\(882\) 37.3476 1.25756
\(883\) 7.55342 0.254193 0.127096 0.991890i \(-0.459434\pi\)
0.127096 + 0.991890i \(0.459434\pi\)
\(884\) 1.29345 0.0435035
\(885\) 0 0
\(886\) 27.0550 0.908930
\(887\) 16.2406 0.545306 0.272653 0.962112i \(-0.412099\pi\)
0.272653 + 0.962112i \(0.412099\pi\)
\(888\) 8.26475 0.277347
\(889\) 4.80202 0.161055
\(890\) 0 0
\(891\) −25.9320 −0.868755
\(892\) 21.6282 0.724165
\(893\) 0 0
\(894\) 11.5983 0.387906
\(895\) 0 0
\(896\) 4.13573 0.138165
\(897\) 2.02182 0.0675067
\(898\) −13.8305 −0.461530
\(899\) −49.4201 −1.64825
\(900\) 0 0
\(901\) 43.4553 1.44771
\(902\) −16.5807 −0.552078
\(903\) 8.60409 0.286326
\(904\) 22.1927 0.738118
\(905\) 0 0
\(906\) −0.525949 −0.0174735
\(907\) 18.2535 0.606097 0.303049 0.952975i \(-0.401996\pi\)
0.303049 + 0.952975i \(0.401996\pi\)
\(908\) 1.48795 0.0493795
\(909\) 22.9518 0.761263
\(910\) 0 0
\(911\) −15.9019 −0.526853 −0.263426 0.964679i \(-0.584853\pi\)
−0.263426 + 0.964679i \(0.584853\pi\)
\(912\) 0 0
\(913\) 8.63824 0.285884
\(914\) 13.2851 0.439431
\(915\) 0 0
\(916\) −7.84492 −0.259204
\(917\) 1.06188 0.0350663
\(918\) 48.9722 1.61632
\(919\) −30.0748 −0.992075 −0.496038 0.868301i \(-0.665212\pi\)
−0.496038 + 0.868301i \(0.665212\pi\)
\(920\) 0 0
\(921\) −64.5981 −2.12858
\(922\) 5.01630 0.165203
\(923\) 1.01065 0.0332661
\(924\) −4.18830 −0.137785
\(925\) 0 0
\(926\) 28.1947 0.926534
\(927\) −101.701 −3.34029
\(928\) −49.7422 −1.63287
\(929\) −36.8442 −1.20882 −0.604410 0.796673i \(-0.706591\pi\)
−0.604410 + 0.796673i \(0.706591\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 3.46155 0.113387
\(933\) 9.69327 0.317343
\(934\) −5.68608 −0.186054
\(935\) 0 0
\(936\) −3.54136 −0.115753
\(937\) −43.2321 −1.41233 −0.706165 0.708047i \(-0.749576\pi\)
−0.706165 + 0.708047i \(0.749576\pi\)
\(938\) 1.97646 0.0645338
\(939\) 37.8513 1.23523
\(940\) 0 0
\(941\) 20.3203 0.662422 0.331211 0.943557i \(-0.392543\pi\)
0.331211 + 0.943557i \(0.392543\pi\)
\(942\) 41.8948 1.36501
\(943\) −33.9225 −1.10467
\(944\) −0.326003 −0.0106105
\(945\) 0 0
\(946\) −8.13770 −0.264579
\(947\) −21.5050 −0.698819 −0.349409 0.936970i \(-0.613618\pi\)
−0.349409 + 0.936970i \(0.613618\pi\)
\(948\) −25.6192 −0.832074
\(949\) −1.67376 −0.0543327
\(950\) 0 0
\(951\) 35.2928 1.14445
\(952\) 8.42663 0.273109
\(953\) −17.7301 −0.574333 −0.287166 0.957881i \(-0.592713\pi\)
−0.287166 + 0.957881i \(0.592713\pi\)
\(954\) −45.8865 −1.48563
\(955\) 0 0
\(956\) −21.7290 −0.702766
\(957\) 51.4417 1.66288
\(958\) 31.7559 1.02599
\(959\) 7.05960 0.227966
\(960\) 0 0
\(961\) 0.997355 0.0321727
\(962\) −0.160383 −0.00517095
\(963\) −6.85253 −0.220820
\(964\) −6.01012 −0.193573
\(965\) 0 0
\(966\) 5.08007 0.163448
\(967\) −13.6530 −0.439052 −0.219526 0.975607i \(-0.570451\pi\)
−0.219526 + 0.975607i \(0.570451\pi\)
\(968\) −20.6267 −0.662966
\(969\) 0 0
\(970\) 0 0
\(971\) −27.9595 −0.897263 −0.448632 0.893717i \(-0.648089\pi\)
−0.448632 + 0.893717i \(0.648089\pi\)
\(972\) 12.0544 0.386646
\(973\) −0.177457 −0.00568901
\(974\) −2.11791 −0.0678622
\(975\) 0 0
\(976\) −0.308182 −0.00986468
\(977\) −17.8540 −0.571200 −0.285600 0.958349i \(-0.592193\pi\)
−0.285600 + 0.958349i \(0.592193\pi\)
\(978\) 24.4082 0.780488
\(979\) 6.45094 0.206173
\(980\) 0 0
\(981\) 43.7109 1.39558
\(982\) −15.3526 −0.489921
\(983\) 49.0443 1.56427 0.782136 0.623108i \(-0.214130\pi\)
0.782136 + 0.623108i \(0.214130\pi\)
\(984\) 86.9210 2.77094
\(985\) 0 0
\(986\) −39.9166 −1.27120
\(987\) −15.4194 −0.490805
\(988\) 0 0
\(989\) −16.6489 −0.529405
\(990\) 0 0
\(991\) 25.0673 0.796289 0.398145 0.917323i \(-0.369654\pi\)
0.398145 + 0.917323i \(0.369654\pi\)
\(992\) 32.2059 1.02254
\(993\) 18.0352 0.572331
\(994\) 2.53939 0.0805445
\(995\) 0 0
\(996\) −17.4650 −0.553400
\(997\) 30.0222 0.950811 0.475406 0.879767i \(-0.342301\pi\)
0.475406 + 0.879767i \(0.342301\pi\)
\(998\) 25.2620 0.799656
\(999\) 10.2426 0.324062
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 9025.2.a.bt.1.2 6
5.2 odd 4 1805.2.b.g.1084.2 6
5.3 odd 4 1805.2.b.g.1084.5 6
5.4 even 2 inner 9025.2.a.bt.1.5 6
19.8 odd 6 475.2.e.g.26.2 12
19.12 odd 6 475.2.e.g.201.2 12
19.18 odd 2 9025.2.a.bu.1.5 6
95.8 even 12 95.2.i.b.64.2 yes 12
95.12 even 12 95.2.i.b.49.2 12
95.18 even 4 1805.2.b.f.1084.2 6
95.27 even 12 95.2.i.b.64.5 yes 12
95.37 even 4 1805.2.b.f.1084.5 6
95.69 odd 6 475.2.e.g.201.5 12
95.84 odd 6 475.2.e.g.26.5 12
95.88 even 12 95.2.i.b.49.5 yes 12
95.94 odd 2 9025.2.a.bu.1.2 6
285.8 odd 12 855.2.be.d.64.5 12
285.107 odd 12 855.2.be.d.334.5 12
285.122 odd 12 855.2.be.d.64.2 12
285.278 odd 12 855.2.be.d.334.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.i.b.49.2 12 95.12 even 12
95.2.i.b.49.5 yes 12 95.88 even 12
95.2.i.b.64.2 yes 12 95.8 even 12
95.2.i.b.64.5 yes 12 95.27 even 12
475.2.e.g.26.2 12 19.8 odd 6
475.2.e.g.26.5 12 95.84 odd 6
475.2.e.g.201.2 12 19.12 odd 6
475.2.e.g.201.5 12 95.69 odd 6
855.2.be.d.64.2 12 285.122 odd 12
855.2.be.d.64.5 12 285.8 odd 12
855.2.be.d.334.2 12 285.278 odd 12
855.2.be.d.334.5 12 285.107 odd 12
1805.2.b.f.1084.2 6 95.18 even 4
1805.2.b.f.1084.5 6 95.37 even 4
1805.2.b.g.1084.2 6 5.2 odd 4
1805.2.b.g.1084.5 6 5.3 odd 4
9025.2.a.bt.1.2 6 1.1 even 1 trivial
9025.2.a.bt.1.5 6 5.4 even 2 inner
9025.2.a.bu.1.2 6 95.94 odd 2
9025.2.a.bu.1.5 6 19.18 odd 2