Properties

Label 5225.2.a.m.1.7
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
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] = [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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-2.40300\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+2.40300 q^{2} +1.17935 q^{3} +3.77440 q^{4} +2.83397 q^{6} +2.15598 q^{7} +4.26389 q^{8} -1.60914 q^{9} +O(q^{10})\) \(q+2.40300 q^{2} +1.17935 q^{3} +3.77440 q^{4} +2.83397 q^{6} +2.15598 q^{7} +4.26389 q^{8} -1.60914 q^{9} +1.00000 q^{11} +4.45134 q^{12} +3.54733 q^{13} +5.18083 q^{14} +2.69732 q^{16} +0.622194 q^{17} -3.86675 q^{18} -1.00000 q^{19} +2.54266 q^{21} +2.40300 q^{22} +5.67863 q^{23} +5.02862 q^{24} +8.52423 q^{26} -5.43578 q^{27} +8.13756 q^{28} +7.32397 q^{29} -7.89640 q^{31} -2.04612 q^{32} +1.17935 q^{33} +1.49513 q^{34} -6.07353 q^{36} +9.19792 q^{37} -2.40300 q^{38} +4.18354 q^{39} +2.29509 q^{41} +6.11000 q^{42} -0.194427 q^{43} +3.77440 q^{44} +13.6457 q^{46} -0.340863 q^{47} +3.18108 q^{48} -2.35173 q^{49} +0.733784 q^{51} +13.3891 q^{52} +6.94545 q^{53} -13.0622 q^{54} +9.19289 q^{56} -1.17935 q^{57} +17.5995 q^{58} +8.99709 q^{59} -13.7381 q^{61} -18.9750 q^{62} -3.46927 q^{63} -10.3115 q^{64} +2.83397 q^{66} +0.226658 q^{67} +2.34841 q^{68} +6.69709 q^{69} +0.343554 q^{71} -6.86119 q^{72} -9.18704 q^{73} +22.1026 q^{74} -3.77440 q^{76} +2.15598 q^{77} +10.0530 q^{78} +14.4980 q^{79} -1.58327 q^{81} +5.51510 q^{82} -3.15514 q^{83} +9.59702 q^{84} -0.467207 q^{86} +8.63752 q^{87} +4.26389 q^{88} +5.44854 q^{89} +7.64799 q^{91} +21.4335 q^{92} -9.31261 q^{93} -0.819094 q^{94} -2.41309 q^{96} -13.7041 q^{97} -5.65121 q^{98} -1.60914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} - 3 q^{3} + 7 q^{4} + 8 q^{6} + q^{7} - 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} - 3 q^{3} + 7 q^{4} + 8 q^{6} + q^{7} - 3 q^{8} + 2 q^{9} + 7 q^{11} - 13 q^{12} - q^{13} + 12 q^{14} + 3 q^{16} - q^{17} - 7 q^{18} - 7 q^{19} + 5 q^{21} - q^{22} + 8 q^{23} + 25 q^{24} - 12 q^{27} - 4 q^{28} + 11 q^{29} + 7 q^{31} - 12 q^{32} - 3 q^{33} - 14 q^{34} + 7 q^{36} + 17 q^{37} + q^{38} + 30 q^{39} + 17 q^{41} - 33 q^{42} + 3 q^{43} + 7 q^{44} + 18 q^{46} - 14 q^{47} + 12 q^{48} + 6 q^{49} + 8 q^{51} + 17 q^{52} - 7 q^{53} - 27 q^{54} + 36 q^{56} + 3 q^{57} + 15 q^{58} + 35 q^{59} + 17 q^{61} - 46 q^{62} + 22 q^{63} + 5 q^{64} + 8 q^{66} - 4 q^{67} + 35 q^{68} - 4 q^{69} + 10 q^{71} - 12 q^{72} - 22 q^{73} - 11 q^{74} - 7 q^{76} + q^{77} + 41 q^{78} + 11 q^{79} - 21 q^{81} + 14 q^{82} - 39 q^{83} + 21 q^{84} - 24 q^{86} + 2 q^{87} - 3 q^{88} + 18 q^{89} - 22 q^{91} + 51 q^{92} - 10 q^{93} + 14 q^{94} - 11 q^{96} + 4 q^{97} + 26 q^{98} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.40300 1.69918 0.849589 0.527446i \(-0.176850\pi\)
0.849589 + 0.527446i \(0.176850\pi\)
\(3\) 1.17935 0.680897 0.340449 0.940263i \(-0.389421\pi\)
0.340449 + 0.940263i \(0.389421\pi\)
\(4\) 3.77440 1.88720
\(5\) 0 0
\(6\) 2.83397 1.15697
\(7\) 2.15598 0.814885 0.407443 0.913231i \(-0.366421\pi\)
0.407443 + 0.913231i \(0.366421\pi\)
\(8\) 4.26389 1.50751
\(9\) −1.60914 −0.536379
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 4.45134 1.28499
\(13\) 3.54733 0.983852 0.491926 0.870637i \(-0.336293\pi\)
0.491926 + 0.870637i \(0.336293\pi\)
\(14\) 5.18083 1.38463
\(15\) 0 0
\(16\) 2.69732 0.674331
\(17\) 0.622194 0.150904 0.0754521 0.997149i \(-0.475960\pi\)
0.0754521 + 0.997149i \(0.475960\pi\)
\(18\) −3.86675 −0.911403
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.54266 0.554853
\(22\) 2.40300 0.512321
\(23\) 5.67863 1.18408 0.592038 0.805910i \(-0.298323\pi\)
0.592038 + 0.805910i \(0.298323\pi\)
\(24\) 5.02862 1.02646
\(25\) 0 0
\(26\) 8.52423 1.67174
\(27\) −5.43578 −1.04612
\(28\) 8.13756 1.53785
\(29\) 7.32397 1.36003 0.680014 0.733199i \(-0.261974\pi\)
0.680014 + 0.733199i \(0.261974\pi\)
\(30\) 0 0
\(31\) −7.89640 −1.41824 −0.709118 0.705090i \(-0.750907\pi\)
−0.709118 + 0.705090i \(0.750907\pi\)
\(32\) −2.04612 −0.361707
\(33\) 1.17935 0.205298
\(34\) 1.49513 0.256413
\(35\) 0 0
\(36\) −6.07353 −1.01226
\(37\) 9.19792 1.51213 0.756065 0.654497i \(-0.227119\pi\)
0.756065 + 0.654497i \(0.227119\pi\)
\(38\) −2.40300 −0.389818
\(39\) 4.18354 0.669902
\(40\) 0 0
\(41\) 2.29509 0.358433 0.179216 0.983810i \(-0.442644\pi\)
0.179216 + 0.983810i \(0.442644\pi\)
\(42\) 6.11000 0.942794
\(43\) −0.194427 −0.0296498 −0.0148249 0.999890i \(-0.504719\pi\)
−0.0148249 + 0.999890i \(0.504719\pi\)
\(44\) 3.77440 0.569013
\(45\) 0 0
\(46\) 13.6457 2.01196
\(47\) −0.340863 −0.0497200 −0.0248600 0.999691i \(-0.507914\pi\)
−0.0248600 + 0.999691i \(0.507914\pi\)
\(48\) 3.18108 0.459150
\(49\) −2.35173 −0.335962
\(50\) 0 0
\(51\) 0.733784 0.102750
\(52\) 13.3891 1.85673
\(53\) 6.94545 0.954031 0.477015 0.878895i \(-0.341719\pi\)
0.477015 + 0.878895i \(0.341719\pi\)
\(54\) −13.0622 −1.77754
\(55\) 0 0
\(56\) 9.19289 1.22845
\(57\) −1.17935 −0.156209
\(58\) 17.5995 2.31093
\(59\) 8.99709 1.17132 0.585660 0.810557i \(-0.300835\pi\)
0.585660 + 0.810557i \(0.300835\pi\)
\(60\) 0 0
\(61\) −13.7381 −1.75899 −0.879494 0.475910i \(-0.842119\pi\)
−0.879494 + 0.475910i \(0.842119\pi\)
\(62\) −18.9750 −2.40983
\(63\) −3.46927 −0.437087
\(64\) −10.3115 −1.28893
\(65\) 0 0
\(66\) 2.83397 0.348838
\(67\) 0.226658 0.0276907 0.0138453 0.999904i \(-0.495593\pi\)
0.0138453 + 0.999904i \(0.495593\pi\)
\(68\) 2.34841 0.284787
\(69\) 6.69709 0.806234
\(70\) 0 0
\(71\) 0.343554 0.0407724 0.0203862 0.999792i \(-0.493510\pi\)
0.0203862 + 0.999792i \(0.493510\pi\)
\(72\) −6.86119 −0.808599
\(73\) −9.18704 −1.07526 −0.537631 0.843180i \(-0.680681\pi\)
−0.537631 + 0.843180i \(0.680681\pi\)
\(74\) 22.1026 2.56938
\(75\) 0 0
\(76\) −3.77440 −0.432954
\(77\) 2.15598 0.245697
\(78\) 10.0530 1.13828
\(79\) 14.4980 1.63115 0.815574 0.578653i \(-0.196422\pi\)
0.815574 + 0.578653i \(0.196422\pi\)
\(80\) 0 0
\(81\) −1.58327 −0.175919
\(82\) 5.51510 0.609041
\(83\) −3.15514 −0.346321 −0.173161 0.984894i \(-0.555398\pi\)
−0.173161 + 0.984894i \(0.555398\pi\)
\(84\) 9.59702 1.04712
\(85\) 0 0
\(86\) −0.467207 −0.0503802
\(87\) 8.63752 0.926039
\(88\) 4.26389 0.454533
\(89\) 5.44854 0.577544 0.288772 0.957398i \(-0.406753\pi\)
0.288772 + 0.957398i \(0.406753\pi\)
\(90\) 0 0
\(91\) 7.64799 0.801727
\(92\) 21.4335 2.23459
\(93\) −9.31261 −0.965673
\(94\) −0.819094 −0.0844831
\(95\) 0 0
\(96\) −2.41309 −0.246285
\(97\) −13.7041 −1.39144 −0.695722 0.718311i \(-0.744916\pi\)
−0.695722 + 0.718311i \(0.744916\pi\)
\(98\) −5.65121 −0.570858
\(99\) −1.60914 −0.161724
\(100\) 0 0
\(101\) −4.70463 −0.468128 −0.234064 0.972221i \(-0.575203\pi\)
−0.234064 + 0.972221i \(0.575203\pi\)
\(102\) 1.76328 0.174591
\(103\) −13.3333 −1.31377 −0.656885 0.753991i \(-0.728126\pi\)
−0.656885 + 0.753991i \(0.728126\pi\)
\(104\) 15.1254 1.48317
\(105\) 0 0
\(106\) 16.6899 1.62107
\(107\) 7.07306 0.683779 0.341889 0.939740i \(-0.388933\pi\)
0.341889 + 0.939740i \(0.388933\pi\)
\(108\) −20.5168 −1.97423
\(109\) 8.93287 0.855614 0.427807 0.903870i \(-0.359286\pi\)
0.427807 + 0.903870i \(0.359286\pi\)
\(110\) 0 0
\(111\) 10.8476 1.02960
\(112\) 5.81539 0.549502
\(113\) −8.90054 −0.837293 −0.418646 0.908149i \(-0.637495\pi\)
−0.418646 + 0.908149i \(0.637495\pi\)
\(114\) −2.83397 −0.265426
\(115\) 0 0
\(116\) 27.6436 2.56665
\(117\) −5.70814 −0.527718
\(118\) 21.6200 1.99028
\(119\) 1.34144 0.122970
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −33.0127 −2.98883
\(123\) 2.70671 0.244056
\(124\) −29.8042 −2.67650
\(125\) 0 0
\(126\) −8.33666 −0.742689
\(127\) −14.9809 −1.32934 −0.664672 0.747135i \(-0.731429\pi\)
−0.664672 + 0.747135i \(0.731429\pi\)
\(128\) −20.6862 −1.82842
\(129\) −0.229297 −0.0201885
\(130\) 0 0
\(131\) 7.81438 0.682746 0.341373 0.939928i \(-0.389108\pi\)
0.341373 + 0.939928i \(0.389108\pi\)
\(132\) 4.45134 0.387439
\(133\) −2.15598 −0.186948
\(134\) 0.544659 0.0470513
\(135\) 0 0
\(136\) 2.65297 0.227490
\(137\) −14.1390 −1.20798 −0.603988 0.796994i \(-0.706422\pi\)
−0.603988 + 0.796994i \(0.706422\pi\)
\(138\) 16.0931 1.36994
\(139\) 12.5953 1.06832 0.534158 0.845385i \(-0.320629\pi\)
0.534158 + 0.845385i \(0.320629\pi\)
\(140\) 0 0
\(141\) −0.401997 −0.0338542
\(142\) 0.825561 0.0692795
\(143\) 3.54733 0.296643
\(144\) −4.34036 −0.361697
\(145\) 0 0
\(146\) −22.0764 −1.82706
\(147\) −2.77351 −0.228755
\(148\) 34.7167 2.85369
\(149\) 4.77915 0.391523 0.195761 0.980652i \(-0.437282\pi\)
0.195761 + 0.980652i \(0.437282\pi\)
\(150\) 0 0
\(151\) −24.0449 −1.95674 −0.978371 0.206856i \(-0.933677\pi\)
−0.978371 + 0.206856i \(0.933677\pi\)
\(152\) −4.26389 −0.345847
\(153\) −1.00119 −0.0809418
\(154\) 5.18083 0.417483
\(155\) 0 0
\(156\) 15.7904 1.26424
\(157\) 12.9931 1.03696 0.518481 0.855089i \(-0.326498\pi\)
0.518481 + 0.855089i \(0.326498\pi\)
\(158\) 34.8386 2.77161
\(159\) 8.19110 0.649597
\(160\) 0 0
\(161\) 12.2430 0.964887
\(162\) −3.80459 −0.298917
\(163\) 19.7251 1.54499 0.772495 0.635021i \(-0.219008\pi\)
0.772495 + 0.635021i \(0.219008\pi\)
\(164\) 8.66260 0.676435
\(165\) 0 0
\(166\) −7.58180 −0.588461
\(167\) −9.91934 −0.767582 −0.383791 0.923420i \(-0.625382\pi\)
−0.383791 + 0.923420i \(0.625382\pi\)
\(168\) 10.8416 0.836449
\(169\) −0.416450 −0.0320346
\(170\) 0 0
\(171\) 1.60914 0.123054
\(172\) −0.733845 −0.0559552
\(173\) −10.7461 −0.817009 −0.408504 0.912756i \(-0.633950\pi\)
−0.408504 + 0.912756i \(0.633950\pi\)
\(174\) 20.7559 1.57350
\(175\) 0 0
\(176\) 2.69732 0.203318
\(177\) 10.6107 0.797549
\(178\) 13.0928 0.981350
\(179\) 9.08355 0.678936 0.339468 0.940618i \(-0.389753\pi\)
0.339468 + 0.940618i \(0.389753\pi\)
\(180\) 0 0
\(181\) −6.00048 −0.446012 −0.223006 0.974817i \(-0.571587\pi\)
−0.223006 + 0.974817i \(0.571587\pi\)
\(182\) 18.3781 1.36228
\(183\) −16.2020 −1.19769
\(184\) 24.2131 1.78501
\(185\) 0 0
\(186\) −22.3782 −1.64085
\(187\) 0.622194 0.0454993
\(188\) −1.28656 −0.0938318
\(189\) −11.7195 −0.852465
\(190\) 0 0
\(191\) −12.6793 −0.917443 −0.458721 0.888580i \(-0.651692\pi\)
−0.458721 + 0.888580i \(0.651692\pi\)
\(192\) −12.1608 −0.877632
\(193\) −14.5221 −1.04533 −0.522664 0.852539i \(-0.675062\pi\)
−0.522664 + 0.852539i \(0.675062\pi\)
\(194\) −32.9310 −2.36431
\(195\) 0 0
\(196\) −8.87639 −0.634028
\(197\) 1.83384 0.130656 0.0653278 0.997864i \(-0.479191\pi\)
0.0653278 + 0.997864i \(0.479191\pi\)
\(198\) −3.86675 −0.274798
\(199\) −12.9881 −0.920701 −0.460350 0.887737i \(-0.652276\pi\)
−0.460350 + 0.887737i \(0.652276\pi\)
\(200\) 0 0
\(201\) 0.267309 0.0188545
\(202\) −11.3052 −0.795433
\(203\) 15.7904 1.10827
\(204\) 2.76960 0.193911
\(205\) 0 0
\(206\) −32.0399 −2.23233
\(207\) −9.13769 −0.635114
\(208\) 9.56830 0.663442
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 9.51881 0.655302 0.327651 0.944799i \(-0.393743\pi\)
0.327651 + 0.944799i \(0.393743\pi\)
\(212\) 26.2149 1.80045
\(213\) 0.405170 0.0277618
\(214\) 16.9966 1.16186
\(215\) 0 0
\(216\) −23.1776 −1.57703
\(217\) −17.0245 −1.15570
\(218\) 21.4657 1.45384
\(219\) −10.8347 −0.732143
\(220\) 0 0
\(221\) 2.20713 0.148467
\(222\) 26.0667 1.74948
\(223\) −0.494807 −0.0331348 −0.0165674 0.999863i \(-0.505274\pi\)
−0.0165674 + 0.999863i \(0.505274\pi\)
\(224\) −4.41141 −0.294750
\(225\) 0 0
\(226\) −21.3880 −1.42271
\(227\) −27.7697 −1.84314 −0.921571 0.388210i \(-0.873094\pi\)
−0.921571 + 0.388210i \(0.873094\pi\)
\(228\) −4.45134 −0.294797
\(229\) −2.65545 −0.175477 −0.0877386 0.996144i \(-0.527964\pi\)
−0.0877386 + 0.996144i \(0.527964\pi\)
\(230\) 0 0
\(231\) 2.54266 0.167295
\(232\) 31.2286 2.05026
\(233\) 19.8348 1.29942 0.649711 0.760182i \(-0.274890\pi\)
0.649711 + 0.760182i \(0.274890\pi\)
\(234\) −13.7167 −0.896686
\(235\) 0 0
\(236\) 33.9586 2.21052
\(237\) 17.0981 1.11064
\(238\) 3.22348 0.208947
\(239\) 9.34316 0.604359 0.302179 0.953251i \(-0.402286\pi\)
0.302179 + 0.953251i \(0.402286\pi\)
\(240\) 0 0
\(241\) −14.6621 −0.944470 −0.472235 0.881473i \(-0.656553\pi\)
−0.472235 + 0.881473i \(0.656553\pi\)
\(242\) 2.40300 0.154471
\(243\) 14.4401 0.926334
\(244\) −51.8533 −3.31957
\(245\) 0 0
\(246\) 6.50422 0.414694
\(247\) −3.54733 −0.225711
\(248\) −33.6694 −2.13801
\(249\) −3.72101 −0.235809
\(250\) 0 0
\(251\) 18.0162 1.13717 0.568586 0.822624i \(-0.307491\pi\)
0.568586 + 0.822624i \(0.307491\pi\)
\(252\) −13.0944 −0.824872
\(253\) 5.67863 0.357013
\(254\) −35.9992 −2.25879
\(255\) 0 0
\(256\) −29.0860 −1.81788
\(257\) 17.0567 1.06397 0.531984 0.846754i \(-0.321446\pi\)
0.531984 + 0.846754i \(0.321446\pi\)
\(258\) −0.551000 −0.0343038
\(259\) 19.8306 1.23221
\(260\) 0 0
\(261\) −11.7853 −0.729490
\(262\) 18.7780 1.16011
\(263\) −2.21160 −0.136373 −0.0681866 0.997673i \(-0.521721\pi\)
−0.0681866 + 0.997673i \(0.521721\pi\)
\(264\) 5.02862 0.309490
\(265\) 0 0
\(266\) −5.18083 −0.317657
\(267\) 6.42573 0.393248
\(268\) 0.855498 0.0522579
\(269\) −27.1402 −1.65477 −0.827384 0.561637i \(-0.810172\pi\)
−0.827384 + 0.561637i \(0.810172\pi\)
\(270\) 0 0
\(271\) 21.4161 1.30094 0.650468 0.759534i \(-0.274573\pi\)
0.650468 + 0.759534i \(0.274573\pi\)
\(272\) 1.67826 0.101759
\(273\) 9.01964 0.545894
\(274\) −33.9760 −2.05256
\(275\) 0 0
\(276\) 25.2775 1.52153
\(277\) −11.4909 −0.690424 −0.345212 0.938525i \(-0.612193\pi\)
−0.345212 + 0.938525i \(0.612193\pi\)
\(278\) 30.2664 1.81526
\(279\) 12.7064 0.760712
\(280\) 0 0
\(281\) −27.5824 −1.64543 −0.822715 0.568454i \(-0.807542\pi\)
−0.822715 + 0.568454i \(0.807542\pi\)
\(282\) −0.965998 −0.0575243
\(283\) 14.5401 0.864318 0.432159 0.901797i \(-0.357752\pi\)
0.432159 + 0.901797i \(0.357752\pi\)
\(284\) 1.29671 0.0769457
\(285\) 0 0
\(286\) 8.52423 0.504048
\(287\) 4.94818 0.292082
\(288\) 3.29249 0.194012
\(289\) −16.6129 −0.977228
\(290\) 0 0
\(291\) −16.1620 −0.947431
\(292\) −34.6756 −2.02924
\(293\) −15.1038 −0.882373 −0.441186 0.897416i \(-0.645442\pi\)
−0.441186 + 0.897416i \(0.645442\pi\)
\(294\) −6.66475 −0.388696
\(295\) 0 0
\(296\) 39.2190 2.27956
\(297\) −5.43578 −0.315416
\(298\) 11.4843 0.665267
\(299\) 20.1440 1.16496
\(300\) 0 0
\(301\) −0.419181 −0.0241612
\(302\) −57.7798 −3.32485
\(303\) −5.54840 −0.318747
\(304\) −2.69732 −0.154702
\(305\) 0 0
\(306\) −2.40587 −0.137534
\(307\) −14.1979 −0.810318 −0.405159 0.914246i \(-0.632784\pi\)
−0.405159 + 0.914246i \(0.632784\pi\)
\(308\) 8.13756 0.463680
\(309\) −15.7246 −0.894543
\(310\) 0 0
\(311\) 11.4136 0.647203 0.323602 0.946193i \(-0.395106\pi\)
0.323602 + 0.946193i \(0.395106\pi\)
\(312\) 17.8382 1.00989
\(313\) 17.5462 0.991773 0.495886 0.868387i \(-0.334843\pi\)
0.495886 + 0.868387i \(0.334843\pi\)
\(314\) 31.2224 1.76198
\(315\) 0 0
\(316\) 54.7211 3.07830
\(317\) −9.50403 −0.533800 −0.266900 0.963724i \(-0.585999\pi\)
−0.266900 + 0.963724i \(0.585999\pi\)
\(318\) 19.6832 1.10378
\(319\) 7.32397 0.410064
\(320\) 0 0
\(321\) 8.34160 0.465583
\(322\) 29.4200 1.63951
\(323\) −0.622194 −0.0346198
\(324\) −5.97590 −0.331994
\(325\) 0 0
\(326\) 47.3994 2.62521
\(327\) 10.5350 0.582585
\(328\) 9.78602 0.540343
\(329\) −0.734896 −0.0405161
\(330\) 0 0
\(331\) 18.8296 1.03497 0.517483 0.855693i \(-0.326869\pi\)
0.517483 + 0.855693i \(0.326869\pi\)
\(332\) −11.9088 −0.653579
\(333\) −14.8007 −0.811074
\(334\) −23.8362 −1.30426
\(335\) 0 0
\(336\) 6.85837 0.374155
\(337\) 3.44511 0.187667 0.0938336 0.995588i \(-0.470088\pi\)
0.0938336 + 0.995588i \(0.470088\pi\)
\(338\) −1.00073 −0.0544324
\(339\) −10.4968 −0.570110
\(340\) 0 0
\(341\) −7.89640 −0.427614
\(342\) 3.86675 0.209090
\(343\) −20.1622 −1.08866
\(344\) −0.829015 −0.0446975
\(345\) 0 0
\(346\) −25.8228 −1.38824
\(347\) −13.2512 −0.711361 −0.355681 0.934608i \(-0.615751\pi\)
−0.355681 + 0.934608i \(0.615751\pi\)
\(348\) 32.6015 1.74762
\(349\) 14.8843 0.796737 0.398369 0.917225i \(-0.369576\pi\)
0.398369 + 0.917225i \(0.369576\pi\)
\(350\) 0 0
\(351\) −19.2825 −1.02922
\(352\) −2.04612 −0.109059
\(353\) 26.2048 1.39474 0.697371 0.716710i \(-0.254353\pi\)
0.697371 + 0.716710i \(0.254353\pi\)
\(354\) 25.4975 1.35518
\(355\) 0 0
\(356\) 20.5650 1.08994
\(357\) 1.58203 0.0837297
\(358\) 21.8278 1.15363
\(359\) −24.3560 −1.28546 −0.642731 0.766092i \(-0.722199\pi\)
−0.642731 + 0.766092i \(0.722199\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.4191 −0.757854
\(363\) 1.17935 0.0618998
\(364\) 28.8666 1.51302
\(365\) 0 0
\(366\) −38.9335 −2.03509
\(367\) −27.2556 −1.42273 −0.711365 0.702823i \(-0.751923\pi\)
−0.711365 + 0.702823i \(0.751923\pi\)
\(368\) 15.3171 0.798459
\(369\) −3.69311 −0.192256
\(370\) 0 0
\(371\) 14.9743 0.777426
\(372\) −35.1496 −1.82242
\(373\) 28.9149 1.49716 0.748579 0.663045i \(-0.230736\pi\)
0.748579 + 0.663045i \(0.230736\pi\)
\(374\) 1.49513 0.0773114
\(375\) 0 0
\(376\) −1.45341 −0.0749537
\(377\) 25.9805 1.33807
\(378\) −28.1618 −1.44849
\(379\) −31.0010 −1.59241 −0.796207 0.605025i \(-0.793163\pi\)
−0.796207 + 0.605025i \(0.793163\pi\)
\(380\) 0 0
\(381\) −17.6678 −0.905147
\(382\) −30.4684 −1.55890
\(383\) −23.3273 −1.19197 −0.595985 0.802996i \(-0.703238\pi\)
−0.595985 + 0.802996i \(0.703238\pi\)
\(384\) −24.3963 −1.24497
\(385\) 0 0
\(386\) −34.8967 −1.77620
\(387\) 0.312859 0.0159035
\(388\) −51.7250 −2.62594
\(389\) 13.0462 0.661468 0.330734 0.943724i \(-0.392704\pi\)
0.330734 + 0.943724i \(0.392704\pi\)
\(390\) 0 0
\(391\) 3.53321 0.178682
\(392\) −10.0275 −0.506467
\(393\) 9.21588 0.464880
\(394\) 4.40671 0.222007
\(395\) 0 0
\(396\) −6.07353 −0.305207
\(397\) −19.3816 −0.972733 −0.486366 0.873755i \(-0.661678\pi\)
−0.486366 + 0.873755i \(0.661678\pi\)
\(398\) −31.2104 −1.56443
\(399\) −2.54266 −0.127292
\(400\) 0 0
\(401\) −17.1713 −0.857496 −0.428748 0.903424i \(-0.641045\pi\)
−0.428748 + 0.903424i \(0.641045\pi\)
\(402\) 0.642342 0.0320371
\(403\) −28.0111 −1.39533
\(404\) −17.7572 −0.883453
\(405\) 0 0
\(406\) 37.9442 1.88314
\(407\) 9.19792 0.455924
\(408\) 3.12877 0.154897
\(409\) 32.5084 1.60744 0.803718 0.595011i \(-0.202852\pi\)
0.803718 + 0.595011i \(0.202852\pi\)
\(410\) 0 0
\(411\) −16.6748 −0.822507
\(412\) −50.3253 −2.47935
\(413\) 19.3976 0.954492
\(414\) −21.9579 −1.07917
\(415\) 0 0
\(416\) −7.25827 −0.355866
\(417\) 14.8542 0.727414
\(418\) −2.40300 −0.117535
\(419\) 0.917867 0.0448407 0.0224204 0.999749i \(-0.492863\pi\)
0.0224204 + 0.999749i \(0.492863\pi\)
\(420\) 0 0
\(421\) 14.2065 0.692382 0.346191 0.938164i \(-0.387475\pi\)
0.346191 + 0.938164i \(0.387475\pi\)
\(422\) 22.8737 1.11347
\(423\) 0.548496 0.0266688
\(424\) 29.6147 1.43821
\(425\) 0 0
\(426\) 0.973624 0.0471722
\(427\) −29.6192 −1.43337
\(428\) 26.6966 1.29043
\(429\) 4.18354 0.201983
\(430\) 0 0
\(431\) −24.3672 −1.17373 −0.586863 0.809687i \(-0.699637\pi\)
−0.586863 + 0.809687i \(0.699637\pi\)
\(432\) −14.6621 −0.705428
\(433\) 34.1956 1.64333 0.821667 0.569967i \(-0.193044\pi\)
0.821667 + 0.569967i \(0.193044\pi\)
\(434\) −40.9099 −1.96374
\(435\) 0 0
\(436\) 33.7163 1.61472
\(437\) −5.67863 −0.271646
\(438\) −26.0358 −1.24404
\(439\) −14.2883 −0.681942 −0.340971 0.940074i \(-0.610756\pi\)
−0.340971 + 0.940074i \(0.610756\pi\)
\(440\) 0 0
\(441\) 3.78426 0.180203
\(442\) 5.30372 0.252272
\(443\) −26.4567 −1.25700 −0.628498 0.777812i \(-0.716330\pi\)
−0.628498 + 0.777812i \(0.716330\pi\)
\(444\) 40.9431 1.94307
\(445\) 0 0
\(446\) −1.18902 −0.0563018
\(447\) 5.63628 0.266587
\(448\) −22.2314 −1.05033
\(449\) −9.81985 −0.463428 −0.231714 0.972784i \(-0.574433\pi\)
−0.231714 + 0.972784i \(0.574433\pi\)
\(450\) 0 0
\(451\) 2.29509 0.108072
\(452\) −33.5942 −1.58014
\(453\) −28.3573 −1.33234
\(454\) −66.7307 −3.13182
\(455\) 0 0
\(456\) −5.02862 −0.235487
\(457\) 8.44589 0.395082 0.197541 0.980295i \(-0.436704\pi\)
0.197541 + 0.980295i \(0.436704\pi\)
\(458\) −6.38105 −0.298167
\(459\) −3.38211 −0.157863
\(460\) 0 0
\(461\) −25.0196 −1.16528 −0.582639 0.812731i \(-0.697980\pi\)
−0.582639 + 0.812731i \(0.697980\pi\)
\(462\) 6.11000 0.284263
\(463\) 5.25947 0.244428 0.122214 0.992504i \(-0.461001\pi\)
0.122214 + 0.992504i \(0.461001\pi\)
\(464\) 19.7551 0.917108
\(465\) 0 0
\(466\) 47.6630 2.20795
\(467\) −16.4456 −0.761013 −0.380507 0.924778i \(-0.624250\pi\)
−0.380507 + 0.924778i \(0.624250\pi\)
\(468\) −21.5448 −0.995910
\(469\) 0.488671 0.0225647
\(470\) 0 0
\(471\) 15.3234 0.706065
\(472\) 38.3626 1.76578
\(473\) −0.194427 −0.00893975
\(474\) 41.0868 1.88718
\(475\) 0 0
\(476\) 5.06314 0.232069
\(477\) −11.1762 −0.511722
\(478\) 22.4516 1.02691
\(479\) −21.0736 −0.962876 −0.481438 0.876480i \(-0.659885\pi\)
−0.481438 + 0.876480i \(0.659885\pi\)
\(480\) 0 0
\(481\) 32.6281 1.48771
\(482\) −35.2330 −1.60482
\(483\) 14.4388 0.656989
\(484\) 3.77440 0.171564
\(485\) 0 0
\(486\) 34.6996 1.57400
\(487\) 19.2996 0.874547 0.437274 0.899329i \(-0.355944\pi\)
0.437274 + 0.899329i \(0.355944\pi\)
\(488\) −58.5779 −2.65170
\(489\) 23.2628 1.05198
\(490\) 0 0
\(491\) 0.412492 0.0186155 0.00930775 0.999957i \(-0.497037\pi\)
0.00930775 + 0.999957i \(0.497037\pi\)
\(492\) 10.2162 0.460583
\(493\) 4.55693 0.205234
\(494\) −8.52423 −0.383523
\(495\) 0 0
\(496\) −21.2991 −0.956360
\(497\) 0.740698 0.0332248
\(498\) −8.94158 −0.400682
\(499\) 35.3500 1.58248 0.791241 0.611504i \(-0.209435\pi\)
0.791241 + 0.611504i \(0.209435\pi\)
\(500\) 0 0
\(501\) −11.6984 −0.522644
\(502\) 43.2929 1.93226
\(503\) −9.66966 −0.431149 −0.215575 0.976487i \(-0.569162\pi\)
−0.215575 + 0.976487i \(0.569162\pi\)
\(504\) −14.7926 −0.658915
\(505\) 0 0
\(506\) 13.6457 0.606627
\(507\) −0.491139 −0.0218123
\(508\) −56.5442 −2.50874
\(509\) 32.5575 1.44308 0.721542 0.692370i \(-0.243434\pi\)
0.721542 + 0.692370i \(0.243434\pi\)
\(510\) 0 0
\(511\) −19.8071 −0.876215
\(512\) −28.5213 −1.26047
\(513\) 5.43578 0.239996
\(514\) 40.9873 1.80787
\(515\) 0 0
\(516\) −0.865459 −0.0380997
\(517\) −0.340863 −0.0149912
\(518\) 47.6529 2.09375
\(519\) −12.6734 −0.556299
\(520\) 0 0
\(521\) 13.0671 0.572482 0.286241 0.958158i \(-0.407594\pi\)
0.286241 + 0.958158i \(0.407594\pi\)
\(522\) −28.3200 −1.23953
\(523\) −12.2092 −0.533870 −0.266935 0.963714i \(-0.586011\pi\)
−0.266935 + 0.963714i \(0.586011\pi\)
\(524\) 29.4946 1.28848
\(525\) 0 0
\(526\) −5.31448 −0.231722
\(527\) −4.91309 −0.214018
\(528\) 3.18108 0.138439
\(529\) 9.24686 0.402037
\(530\) 0 0
\(531\) −14.4775 −0.628272
\(532\) −8.13756 −0.352808
\(533\) 8.14144 0.352645
\(534\) 15.4410 0.668198
\(535\) 0 0
\(536\) 0.966445 0.0417441
\(537\) 10.7127 0.462286
\(538\) −65.2179 −2.81174
\(539\) −2.35173 −0.101296
\(540\) 0 0
\(541\) 16.5139 0.709990 0.354995 0.934868i \(-0.384483\pi\)
0.354995 + 0.934868i \(0.384483\pi\)
\(542\) 51.4629 2.21052
\(543\) −7.07666 −0.303689
\(544\) −1.27308 −0.0545831
\(545\) 0 0
\(546\) 21.6742 0.927570
\(547\) −5.86659 −0.250837 −0.125419 0.992104i \(-0.540027\pi\)
−0.125419 + 0.992104i \(0.540027\pi\)
\(548\) −53.3663 −2.27969
\(549\) 22.1065 0.943484
\(550\) 0 0
\(551\) −7.32397 −0.312012
\(552\) 28.5557 1.21541
\(553\) 31.2574 1.32920
\(554\) −27.6127 −1.17315
\(555\) 0 0
\(556\) 47.5396 2.01613
\(557\) 27.3711 1.15975 0.579875 0.814705i \(-0.303101\pi\)
0.579875 + 0.814705i \(0.303101\pi\)
\(558\) 30.5334 1.29258
\(559\) −0.689696 −0.0291710
\(560\) 0 0
\(561\) 0.733784 0.0309804
\(562\) −66.2806 −2.79588
\(563\) −35.1477 −1.48130 −0.740649 0.671892i \(-0.765482\pi\)
−0.740649 + 0.671892i \(0.765482\pi\)
\(564\) −1.51730 −0.0638898
\(565\) 0 0
\(566\) 34.9398 1.46863
\(567\) −3.41350 −0.143354
\(568\) 1.46488 0.0614649
\(569\) −32.5565 −1.36484 −0.682420 0.730960i \(-0.739073\pi\)
−0.682420 + 0.730960i \(0.739073\pi\)
\(570\) 0 0
\(571\) 27.4850 1.15021 0.575106 0.818079i \(-0.304961\pi\)
0.575106 + 0.818079i \(0.304961\pi\)
\(572\) 13.3891 0.559825
\(573\) −14.9533 −0.624684
\(574\) 11.8905 0.496299
\(575\) 0 0
\(576\) 16.5926 0.691357
\(577\) 3.42422 0.142552 0.0712760 0.997457i \(-0.477293\pi\)
0.0712760 + 0.997457i \(0.477293\pi\)
\(578\) −39.9207 −1.66048
\(579\) −17.1267 −0.711760
\(580\) 0 0
\(581\) −6.80243 −0.282212
\(582\) −38.8372 −1.60985
\(583\) 6.94545 0.287651
\(584\) −39.1725 −1.62097
\(585\) 0 0
\(586\) −36.2944 −1.49931
\(587\) 18.7672 0.774604 0.387302 0.921953i \(-0.373407\pi\)
0.387302 + 0.921953i \(0.373407\pi\)
\(588\) −10.4684 −0.431708
\(589\) 7.89640 0.325366
\(590\) 0 0
\(591\) 2.16273 0.0889630
\(592\) 24.8098 1.01968
\(593\) −10.8009 −0.443538 −0.221769 0.975099i \(-0.571183\pi\)
−0.221769 + 0.975099i \(0.571183\pi\)
\(594\) −13.0622 −0.535947
\(595\) 0 0
\(596\) 18.0384 0.738883
\(597\) −15.3175 −0.626903
\(598\) 48.4060 1.97947
\(599\) 6.00417 0.245324 0.122662 0.992449i \(-0.460857\pi\)
0.122662 + 0.992449i \(0.460857\pi\)
\(600\) 0 0
\(601\) −34.2605 −1.39751 −0.698757 0.715359i \(-0.746263\pi\)
−0.698757 + 0.715359i \(0.746263\pi\)
\(602\) −1.00729 −0.0410541
\(603\) −0.364723 −0.0148527
\(604\) −90.7550 −3.69277
\(605\) 0 0
\(606\) −13.3328 −0.541608
\(607\) 26.8024 1.08788 0.543938 0.839126i \(-0.316933\pi\)
0.543938 + 0.839126i \(0.316933\pi\)
\(608\) 2.04612 0.0829812
\(609\) 18.6224 0.754616
\(610\) 0 0
\(611\) −1.20915 −0.0489172
\(612\) −3.77892 −0.152754
\(613\) 20.7648 0.838682 0.419341 0.907829i \(-0.362261\pi\)
0.419341 + 0.907829i \(0.362261\pi\)
\(614\) −34.1176 −1.37687
\(615\) 0 0
\(616\) 9.19289 0.370392
\(617\) 1.40456 0.0565455 0.0282728 0.999600i \(-0.490999\pi\)
0.0282728 + 0.999600i \(0.490999\pi\)
\(618\) −37.7863 −1.51999
\(619\) 47.5216 1.91005 0.955027 0.296519i \(-0.0958258\pi\)
0.955027 + 0.296519i \(0.0958258\pi\)
\(620\) 0 0
\(621\) −30.8678 −1.23868
\(622\) 27.4268 1.09971
\(623\) 11.7470 0.470632
\(624\) 11.2844 0.451736
\(625\) 0 0
\(626\) 42.1636 1.68520
\(627\) −1.17935 −0.0470987
\(628\) 49.0412 1.95696
\(629\) 5.72289 0.228187
\(630\) 0 0
\(631\) −25.1060 −0.999454 −0.499727 0.866183i \(-0.666566\pi\)
−0.499727 + 0.866183i \(0.666566\pi\)
\(632\) 61.8177 2.45898
\(633\) 11.2260 0.446193
\(634\) −22.8382 −0.907020
\(635\) 0 0
\(636\) 30.9165 1.22592
\(637\) −8.34237 −0.330537
\(638\) 17.5995 0.696771
\(639\) −0.552826 −0.0218694
\(640\) 0 0
\(641\) 21.3481 0.843199 0.421600 0.906782i \(-0.361469\pi\)
0.421600 + 0.906782i \(0.361469\pi\)
\(642\) 20.0449 0.791108
\(643\) 14.5139 0.572371 0.286185 0.958174i \(-0.407613\pi\)
0.286185 + 0.958174i \(0.407613\pi\)
\(644\) 46.2102 1.82094
\(645\) 0 0
\(646\) −1.49513 −0.0588252
\(647\) 23.8976 0.939511 0.469755 0.882797i \(-0.344342\pi\)
0.469755 + 0.882797i \(0.344342\pi\)
\(648\) −6.75089 −0.265200
\(649\) 8.99709 0.353167
\(650\) 0 0
\(651\) −20.0778 −0.786913
\(652\) 74.4506 2.91571
\(653\) −16.3847 −0.641181 −0.320591 0.947218i \(-0.603881\pi\)
−0.320591 + 0.947218i \(0.603881\pi\)
\(654\) 25.3155 0.989916
\(655\) 0 0
\(656\) 6.19060 0.241702
\(657\) 14.7832 0.576747
\(658\) −1.76595 −0.0688441
\(659\) 3.86889 0.150711 0.0753554 0.997157i \(-0.475991\pi\)
0.0753554 + 0.997157i \(0.475991\pi\)
\(660\) 0 0
\(661\) 4.57508 0.177950 0.0889749 0.996034i \(-0.471641\pi\)
0.0889749 + 0.996034i \(0.471641\pi\)
\(662\) 45.2474 1.75859
\(663\) 2.60297 0.101091
\(664\) −13.4532 −0.522084
\(665\) 0 0
\(666\) −35.5661 −1.37816
\(667\) 41.5901 1.61038
\(668\) −37.4396 −1.44858
\(669\) −0.583550 −0.0225614
\(670\) 0 0
\(671\) −13.7381 −0.530355
\(672\) −5.20259 −0.200694
\(673\) −22.2879 −0.859135 −0.429568 0.903035i \(-0.641334\pi\)
−0.429568 + 0.903035i \(0.641334\pi\)
\(674\) 8.27860 0.318880
\(675\) 0 0
\(676\) −1.57185 −0.0604557
\(677\) 22.4962 0.864600 0.432300 0.901730i \(-0.357702\pi\)
0.432300 + 0.901730i \(0.357702\pi\)
\(678\) −25.2239 −0.968718
\(679\) −29.5459 −1.13387
\(680\) 0 0
\(681\) −32.7502 −1.25499
\(682\) −18.9750 −0.726592
\(683\) −12.8458 −0.491532 −0.245766 0.969329i \(-0.579039\pi\)
−0.245766 + 0.969329i \(0.579039\pi\)
\(684\) 6.07353 0.232227
\(685\) 0 0
\(686\) −48.4497 −1.84982
\(687\) −3.13170 −0.119482
\(688\) −0.524432 −0.0199938
\(689\) 24.6378 0.938625
\(690\) 0 0
\(691\) −24.2131 −0.921110 −0.460555 0.887631i \(-0.652350\pi\)
−0.460555 + 0.887631i \(0.652350\pi\)
\(692\) −40.5600 −1.54186
\(693\) −3.46927 −0.131787
\(694\) −31.8426 −1.20873
\(695\) 0 0
\(696\) 36.8295 1.39602
\(697\) 1.42799 0.0540890
\(698\) 35.7669 1.35380
\(699\) 23.3922 0.884772
\(700\) 0 0
\(701\) −36.6327 −1.38360 −0.691798 0.722091i \(-0.743181\pi\)
−0.691798 + 0.722091i \(0.743181\pi\)
\(702\) −46.3358 −1.74883
\(703\) −9.19792 −0.346906
\(704\) −10.3115 −0.388628
\(705\) 0 0
\(706\) 62.9702 2.36991
\(707\) −10.1431 −0.381471
\(708\) 40.0491 1.50514
\(709\) 37.1353 1.39464 0.697322 0.716758i \(-0.254375\pi\)
0.697322 + 0.716758i \(0.254375\pi\)
\(710\) 0 0
\(711\) −23.3292 −0.874913
\(712\) 23.2320 0.870656
\(713\) −44.8408 −1.67930
\(714\) 3.80161 0.142272
\(715\) 0 0
\(716\) 34.2850 1.28129
\(717\) 11.0188 0.411506
\(718\) −58.5275 −2.18423
\(719\) 5.84694 0.218054 0.109027 0.994039i \(-0.465226\pi\)
0.109027 + 0.994039i \(0.465226\pi\)
\(720\) 0 0
\(721\) −28.7464 −1.07057
\(722\) 2.40300 0.0894304
\(723\) −17.2917 −0.643087
\(724\) −22.6482 −0.841715
\(725\) 0 0
\(726\) 2.83397 0.105179
\(727\) −52.5801 −1.95009 −0.975044 0.222013i \(-0.928737\pi\)
−0.975044 + 0.222013i \(0.928737\pi\)
\(728\) 32.6102 1.20861
\(729\) 21.7797 0.806657
\(730\) 0 0
\(731\) −0.120971 −0.00447428
\(732\) −61.1531 −2.26028
\(733\) −31.6350 −1.16847 −0.584233 0.811586i \(-0.698605\pi\)
−0.584233 + 0.811586i \(0.698605\pi\)
\(734\) −65.4951 −2.41747
\(735\) 0 0
\(736\) −11.6192 −0.428288
\(737\) 0.226658 0.00834905
\(738\) −8.87455 −0.326677
\(739\) 25.8243 0.949963 0.474982 0.879996i \(-0.342455\pi\)
0.474982 + 0.879996i \(0.342455\pi\)
\(740\) 0 0
\(741\) −4.18354 −0.153686
\(742\) 35.9832 1.32098
\(743\) 22.0227 0.807933 0.403967 0.914774i \(-0.367631\pi\)
0.403967 + 0.914774i \(0.367631\pi\)
\(744\) −39.7080 −1.45577
\(745\) 0 0
\(746\) 69.4826 2.54394
\(747\) 5.07705 0.185760
\(748\) 2.34841 0.0858664
\(749\) 15.2494 0.557201
\(750\) 0 0
\(751\) 25.1467 0.917618 0.458809 0.888535i \(-0.348276\pi\)
0.458809 + 0.888535i \(0.348276\pi\)
\(752\) −0.919419 −0.0335277
\(753\) 21.2474 0.774297
\(754\) 62.4312 2.27361
\(755\) 0 0
\(756\) −44.2340 −1.60877
\(757\) 24.6908 0.897403 0.448702 0.893682i \(-0.351887\pi\)
0.448702 + 0.893682i \(0.351887\pi\)
\(758\) −74.4953 −2.70579
\(759\) 6.69709 0.243089
\(760\) 0 0
\(761\) 22.1274 0.802119 0.401059 0.916052i \(-0.368642\pi\)
0.401059 + 0.916052i \(0.368642\pi\)
\(762\) −42.4556 −1.53800
\(763\) 19.2591 0.697227
\(764\) −47.8568 −1.73140
\(765\) 0 0
\(766\) −56.0555 −2.02537
\(767\) 31.9156 1.15241
\(768\) −34.3026 −1.23779
\(769\) 0.0310222 0.00111869 0.000559344 1.00000i \(-0.499822\pi\)
0.000559344 1.00000i \(0.499822\pi\)
\(770\) 0 0
\(771\) 20.1158 0.724454
\(772\) −54.8125 −1.97274
\(773\) −42.3683 −1.52388 −0.761940 0.647647i \(-0.775753\pi\)
−0.761940 + 0.647647i \(0.775753\pi\)
\(774\) 0.751800 0.0270229
\(775\) 0 0
\(776\) −58.4330 −2.09762
\(777\) 23.3872 0.839010
\(778\) 31.3500 1.12395
\(779\) −2.29509 −0.0822301
\(780\) 0 0
\(781\) 0.343554 0.0122933
\(782\) 8.49030 0.303612
\(783\) −39.8115 −1.42275
\(784\) −6.34338 −0.226549
\(785\) 0 0
\(786\) 22.1458 0.789913
\(787\) 33.4522 1.19244 0.596222 0.802820i \(-0.296668\pi\)
0.596222 + 0.802820i \(0.296668\pi\)
\(788\) 6.92165 0.246573
\(789\) −2.60825 −0.0928562
\(790\) 0 0
\(791\) −19.1894 −0.682298
\(792\) −6.86119 −0.243802
\(793\) −48.7337 −1.73058
\(794\) −46.5739 −1.65285
\(795\) 0 0
\(796\) −49.0223 −1.73755
\(797\) 12.2471 0.433816 0.216908 0.976192i \(-0.430403\pi\)
0.216908 + 0.976192i \(0.430403\pi\)
\(798\) −6.11000 −0.216292
\(799\) −0.212083 −0.00750296
\(800\) 0 0
\(801\) −8.76745 −0.309783
\(802\) −41.2627 −1.45704
\(803\) −9.18704 −0.324203
\(804\) 1.00893 0.0355822
\(805\) 0 0
\(806\) −67.3108 −2.37092
\(807\) −32.0078 −1.12673
\(808\) −20.0601 −0.705710
\(809\) −25.7316 −0.904676 −0.452338 0.891846i \(-0.649410\pi\)
−0.452338 + 0.891846i \(0.649410\pi\)
\(810\) 0 0
\(811\) −17.4658 −0.613309 −0.306654 0.951821i \(-0.599210\pi\)
−0.306654 + 0.951821i \(0.599210\pi\)
\(812\) 59.5993 2.09152
\(813\) 25.2570 0.885803
\(814\) 22.1026 0.774696
\(815\) 0 0
\(816\) 1.97925 0.0692877
\(817\) 0.194427 0.00680213
\(818\) 78.1176 2.73132
\(819\) −12.3067 −0.430029
\(820\) 0 0
\(821\) 18.5642 0.647896 0.323948 0.946075i \(-0.394990\pi\)
0.323948 + 0.946075i \(0.394990\pi\)
\(822\) −40.0695 −1.39758
\(823\) 19.1102 0.666139 0.333069 0.942902i \(-0.391916\pi\)
0.333069 + 0.942902i \(0.391916\pi\)
\(824\) −56.8518 −1.98053
\(825\) 0 0
\(826\) 46.6124 1.62185
\(827\) −38.2855 −1.33132 −0.665658 0.746257i \(-0.731849\pi\)
−0.665658 + 0.746257i \(0.731849\pi\)
\(828\) −34.4894 −1.19859
\(829\) 8.64603 0.300289 0.150144 0.988664i \(-0.452026\pi\)
0.150144 + 0.988664i \(0.452026\pi\)
\(830\) 0 0
\(831\) −13.5518 −0.470108
\(832\) −36.5782 −1.26812
\(833\) −1.46323 −0.0506980
\(834\) 35.6946 1.23600
\(835\) 0 0
\(836\) −3.77440 −0.130541
\(837\) 42.9231 1.48364
\(838\) 2.20563 0.0761924
\(839\) 24.7123 0.853165 0.426582 0.904449i \(-0.359717\pi\)
0.426582 + 0.904449i \(0.359717\pi\)
\(840\) 0 0
\(841\) 24.6406 0.849675
\(842\) 34.1382 1.17648
\(843\) −32.5293 −1.12037
\(844\) 35.9279 1.23669
\(845\) 0 0
\(846\) 1.31803 0.0453150
\(847\) 2.15598 0.0740805
\(848\) 18.7341 0.643332
\(849\) 17.1478 0.588512
\(850\) 0 0
\(851\) 52.2316 1.79048
\(852\) 1.52928 0.0523921
\(853\) −7.26768 −0.248841 −0.124420 0.992230i \(-0.539707\pi\)
−0.124420 + 0.992230i \(0.539707\pi\)
\(854\) −71.1749 −2.43556
\(855\) 0 0
\(856\) 30.1588 1.03081
\(857\) 52.0192 1.77694 0.888471 0.458932i \(-0.151768\pi\)
0.888471 + 0.458932i \(0.151768\pi\)
\(858\) 10.0530 0.343205
\(859\) −33.3223 −1.13694 −0.568472 0.822703i \(-0.692465\pi\)
−0.568472 + 0.822703i \(0.692465\pi\)
\(860\) 0 0
\(861\) 5.83563 0.198878
\(862\) −58.5543 −1.99437
\(863\) −12.4776 −0.424743 −0.212371 0.977189i \(-0.568119\pi\)
−0.212371 + 0.977189i \(0.568119\pi\)
\(864\) 11.1223 0.378387
\(865\) 0 0
\(866\) 82.1720 2.79232
\(867\) −19.5924 −0.665392
\(868\) −64.2574 −2.18104
\(869\) 14.4980 0.491809
\(870\) 0 0
\(871\) 0.804030 0.0272435
\(872\) 38.0888 1.28985
\(873\) 22.0518 0.746342
\(874\) −13.6457 −0.461574
\(875\) 0 0
\(876\) −40.8946 −1.38170
\(877\) −1.06682 −0.0360238 −0.0180119 0.999838i \(-0.505734\pi\)
−0.0180119 + 0.999838i \(0.505734\pi\)
\(878\) −34.3347 −1.15874
\(879\) −17.8126 −0.600805
\(880\) 0 0
\(881\) −34.9901 −1.17885 −0.589423 0.807825i \(-0.700645\pi\)
−0.589423 + 0.807825i \(0.700645\pi\)
\(882\) 9.09357 0.306196
\(883\) 48.3044 1.62557 0.812786 0.582562i \(-0.197950\pi\)
0.812786 + 0.582562i \(0.197950\pi\)
\(884\) 8.33059 0.280188
\(885\) 0 0
\(886\) −63.5754 −2.13586
\(887\) −8.74597 −0.293661 −0.146830 0.989162i \(-0.546907\pi\)
−0.146830 + 0.989162i \(0.546907\pi\)
\(888\) 46.2528 1.55214
\(889\) −32.2987 −1.08326
\(890\) 0 0
\(891\) −1.58327 −0.0530415
\(892\) −1.86760 −0.0625320
\(893\) 0.340863 0.0114066
\(894\) 13.5440 0.452978
\(895\) 0 0
\(896\) −44.5992 −1.48995
\(897\) 23.7568 0.793216
\(898\) −23.5971 −0.787445
\(899\) −57.8330 −1.92884
\(900\) 0 0
\(901\) 4.32141 0.143967
\(902\) 5.51510 0.183633
\(903\) −0.494360 −0.0164513
\(904\) −37.9510 −1.26223
\(905\) 0 0
\(906\) −68.1425 −2.26388
\(907\) 25.8242 0.857479 0.428739 0.903428i \(-0.358958\pi\)
0.428739 + 0.903428i \(0.358958\pi\)
\(908\) −104.814 −3.47838
\(909\) 7.57040 0.251094
\(910\) 0 0
\(911\) −11.3062 −0.374591 −0.187296 0.982304i \(-0.559972\pi\)
−0.187296 + 0.982304i \(0.559972\pi\)
\(912\) −3.18108 −0.105336
\(913\) −3.15514 −0.104420
\(914\) 20.2955 0.671315
\(915\) 0 0
\(916\) −10.0228 −0.331161
\(917\) 16.8477 0.556360
\(918\) −8.12720 −0.268238
\(919\) −1.21432 −0.0400567 −0.0200284 0.999799i \(-0.506376\pi\)
−0.0200284 + 0.999799i \(0.506376\pi\)
\(920\) 0 0
\(921\) −16.7443 −0.551743
\(922\) −60.1220 −1.98001
\(923\) 1.21870 0.0401140
\(924\) 9.59702 0.315719
\(925\) 0 0
\(926\) 12.6385 0.415327
\(927\) 21.4551 0.704679
\(928\) −14.9857 −0.491931
\(929\) −45.7356 −1.50054 −0.750269 0.661133i \(-0.770076\pi\)
−0.750269 + 0.661133i \(0.770076\pi\)
\(930\) 0 0
\(931\) 2.35173 0.0770749
\(932\) 74.8646 2.45227
\(933\) 13.4606 0.440679
\(934\) −39.5188 −1.29310
\(935\) 0 0
\(936\) −24.3389 −0.795542
\(937\) 30.7781 1.00548 0.502739 0.864438i \(-0.332326\pi\)
0.502739 + 0.864438i \(0.332326\pi\)
\(938\) 1.17428 0.0383414
\(939\) 20.6931 0.675295
\(940\) 0 0
\(941\) 31.6570 1.03199 0.515995 0.856592i \(-0.327422\pi\)
0.515995 + 0.856592i \(0.327422\pi\)
\(942\) 36.8221 1.19973
\(943\) 13.0330 0.424412
\(944\) 24.2680 0.789858
\(945\) 0 0
\(946\) −0.467207 −0.0151902
\(947\) −50.8102 −1.65111 −0.825555 0.564321i \(-0.809138\pi\)
−0.825555 + 0.564321i \(0.809138\pi\)
\(948\) 64.5353 2.09601
\(949\) −32.5895 −1.05790
\(950\) 0 0
\(951\) −11.2086 −0.363463
\(952\) 5.71976 0.185378
\(953\) −26.6043 −0.861799 −0.430899 0.902400i \(-0.641804\pi\)
−0.430899 + 0.902400i \(0.641804\pi\)
\(954\) −26.8563 −0.869506
\(955\) 0 0
\(956\) 35.2649 1.14055
\(957\) 8.63752 0.279211
\(958\) −50.6398 −1.63610
\(959\) −30.4834 −0.984361
\(960\) 0 0
\(961\) 31.3532 1.01139
\(962\) 78.4052 2.52789
\(963\) −11.3815 −0.366764
\(964\) −55.3407 −1.78241
\(965\) 0 0
\(966\) 34.6965 1.11634
\(967\) 24.6428 0.792458 0.396229 0.918152i \(-0.370319\pi\)
0.396229 + 0.918152i \(0.370319\pi\)
\(968\) 4.26389 0.137047
\(969\) −0.733784 −0.0235725
\(970\) 0 0
\(971\) 9.87139 0.316788 0.158394 0.987376i \(-0.449368\pi\)
0.158394 + 0.987376i \(0.449368\pi\)
\(972\) 54.5028 1.74818
\(973\) 27.1552 0.870555
\(974\) 46.3769 1.48601
\(975\) 0 0
\(976\) −37.0562 −1.18614
\(977\) −15.0169 −0.480434 −0.240217 0.970719i \(-0.577219\pi\)
−0.240217 + 0.970719i \(0.577219\pi\)
\(978\) 55.9005 1.78750
\(979\) 5.44854 0.174136
\(980\) 0 0
\(981\) −14.3742 −0.458933
\(982\) 0.991218 0.0316310
\(983\) 7.52621 0.240049 0.120024 0.992771i \(-0.461703\pi\)
0.120024 + 0.992771i \(0.461703\pi\)
\(984\) 11.5411 0.367918
\(985\) 0 0
\(986\) 10.9503 0.348729
\(987\) −0.866699 −0.0275873
\(988\) −13.3891 −0.425963
\(989\) −1.10408 −0.0351076
\(990\) 0 0
\(991\) 49.1175 1.56027 0.780135 0.625611i \(-0.215150\pi\)
0.780135 + 0.625611i \(0.215150\pi\)
\(992\) 16.1570 0.512985
\(993\) 22.2066 0.704706
\(994\) 1.77990 0.0564549
\(995\) 0 0
\(996\) −14.0446 −0.445020
\(997\) −26.7842 −0.848263 −0.424132 0.905601i \(-0.639421\pi\)
−0.424132 + 0.905601i \(0.639421\pi\)
\(998\) 84.9460 2.68892
\(999\) −49.9979 −1.58186
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.m.1.7 7
5.4 even 2 1045.2.a.h.1.1 7
15.14 odd 2 9405.2.a.bd.1.7 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.h.1.1 7 5.4 even 2
5225.2.a.m.1.7 7 1.1 even 1 trivial
9405.2.a.bd.1.7 7 15.14 odd 2