Properties

Label 5225.2.a.v.1.1
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $15$
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: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 21 x^{13} + 21 x^{12} + 168 x^{11} - 165 x^{10} - 645 x^{9} + 606 x^{8} + 1239 x^{7} + \cdots - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.70405\) 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)
 
\(f(q)\) \(=\) \(q-2.70405 q^{2} +0.205258 q^{3} +5.31188 q^{4} -0.555028 q^{6} -2.55265 q^{7} -8.95548 q^{8} -2.95787 q^{9} +O(q^{10})\) \(q-2.70405 q^{2} +0.205258 q^{3} +5.31188 q^{4} -0.555028 q^{6} -2.55265 q^{7} -8.95548 q^{8} -2.95787 q^{9} -1.00000 q^{11} +1.09031 q^{12} -0.727602 q^{13} +6.90249 q^{14} +13.5923 q^{16} +4.75998 q^{17} +7.99822 q^{18} +1.00000 q^{19} -0.523952 q^{21} +2.70405 q^{22} -5.27242 q^{23} -1.83819 q^{24} +1.96747 q^{26} -1.22290 q^{27} -13.5594 q^{28} +6.31246 q^{29} +2.60117 q^{31} -18.8433 q^{32} -0.205258 q^{33} -12.8712 q^{34} -15.7118 q^{36} -7.54786 q^{37} -2.70405 q^{38} -0.149346 q^{39} +6.59622 q^{41} +1.41679 q^{42} -0.578218 q^{43} -5.31188 q^{44} +14.2569 q^{46} +5.00075 q^{47} +2.78993 q^{48} -0.483971 q^{49} +0.977025 q^{51} -3.86494 q^{52} -5.82375 q^{53} +3.30678 q^{54} +22.8602 q^{56} +0.205258 q^{57} -17.0692 q^{58} +8.71096 q^{59} +10.3898 q^{61} -7.03368 q^{62} +7.55041 q^{63} +23.7686 q^{64} +0.555028 q^{66} -6.24347 q^{67} +25.2845 q^{68} -1.08221 q^{69} +9.50218 q^{71} +26.4892 q^{72} +10.7796 q^{73} +20.4098 q^{74} +5.31188 q^{76} +2.55265 q^{77} +0.403840 q^{78} -13.9182 q^{79} +8.62260 q^{81} -17.8365 q^{82} +8.52531 q^{83} -2.78317 q^{84} +1.56353 q^{86} +1.29568 q^{87} +8.95548 q^{88} -13.9001 q^{89} +1.85732 q^{91} -28.0064 q^{92} +0.533911 q^{93} -13.5223 q^{94} -3.86774 q^{96} +14.6606 q^{97} +1.30868 q^{98} +2.95787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 17 q^{7} - 3 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 17 q^{7} - 3 q^{8} + 15 q^{9} - 15 q^{11} - 9 q^{12} - 3 q^{13} - 11 q^{14} + 13 q^{16} - q^{17} + 10 q^{18} + 15 q^{19} + 16 q^{21} - q^{22} - 18 q^{23} - 27 q^{24} + 15 q^{26} - 31 q^{27} - 34 q^{28} + 11 q^{29} - 2 q^{31} - 10 q^{32} + 4 q^{33} - 13 q^{34} + 8 q^{36} - 27 q^{37} + q^{38} + 4 q^{39} + 6 q^{41} - 2 q^{42} - 38 q^{43} - 13 q^{44} - 9 q^{46} - 14 q^{47} - 32 q^{48} + 28 q^{49} - 32 q^{51} - 16 q^{52} - 11 q^{53} - 11 q^{54} + 2 q^{56} - 4 q^{57} + 6 q^{58} + 13 q^{59} + 8 q^{61} - 7 q^{62} - 49 q^{63} + 9 q^{64} + q^{66} - 31 q^{67} + 26 q^{68} + 39 q^{69} - 3 q^{71} + 18 q^{72} - 18 q^{73} + 7 q^{74} + 13 q^{76} + 17 q^{77} - 18 q^{78} + 10 q^{79} + 31 q^{81} - 58 q^{82} - 16 q^{83} + 112 q^{84} - 63 q^{86} - 67 q^{87} + 3 q^{88} - 7 q^{89} - 50 q^{91} - 98 q^{92} - 26 q^{93} + 22 q^{94} - 37 q^{96} - 24 q^{97} + 46 q^{98} - 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.70405 −1.91205 −0.956026 0.293283i \(-0.905252\pi\)
−0.956026 + 0.293283i \(0.905252\pi\)
\(3\) 0.205258 0.118506 0.0592529 0.998243i \(-0.481128\pi\)
0.0592529 + 0.998243i \(0.481128\pi\)
\(4\) 5.31188 2.65594
\(5\) 0 0
\(6\) −0.555028 −0.226589
\(7\) −2.55265 −0.964812 −0.482406 0.875948i \(-0.660237\pi\)
−0.482406 + 0.875948i \(0.660237\pi\)
\(8\) −8.95548 −3.16624
\(9\) −2.95787 −0.985956
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 1.09031 0.314744
\(13\) −0.727602 −0.201801 −0.100900 0.994897i \(-0.532172\pi\)
−0.100900 + 0.994897i \(0.532172\pi\)
\(14\) 6.90249 1.84477
\(15\) 0 0
\(16\) 13.5923 3.39808
\(17\) 4.75998 1.15447 0.577233 0.816580i \(-0.304133\pi\)
0.577233 + 0.816580i \(0.304133\pi\)
\(18\) 7.99822 1.88520
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.523952 −0.114336
\(22\) 2.70405 0.576505
\(23\) −5.27242 −1.09937 −0.549687 0.835370i \(-0.685253\pi\)
−0.549687 + 0.835370i \(0.685253\pi\)
\(24\) −1.83819 −0.375218
\(25\) 0 0
\(26\) 1.96747 0.385853
\(27\) −1.22290 −0.235347
\(28\) −13.5594 −2.56248
\(29\) 6.31246 1.17219 0.586097 0.810241i \(-0.300664\pi\)
0.586097 + 0.810241i \(0.300664\pi\)
\(30\) 0 0
\(31\) 2.60117 0.467184 0.233592 0.972335i \(-0.424952\pi\)
0.233592 + 0.972335i \(0.424952\pi\)
\(32\) −18.8433 −3.33106
\(33\) −0.205258 −0.0357309
\(34\) −12.8712 −2.20740
\(35\) 0 0
\(36\) −15.7118 −2.61864
\(37\) −7.54786 −1.24086 −0.620430 0.784261i \(-0.713042\pi\)
−0.620430 + 0.784261i \(0.713042\pi\)
\(38\) −2.70405 −0.438655
\(39\) −0.149346 −0.0239145
\(40\) 0 0
\(41\) 6.59622 1.03016 0.515078 0.857143i \(-0.327763\pi\)
0.515078 + 0.857143i \(0.327763\pi\)
\(42\) 1.41679 0.218616
\(43\) −0.578218 −0.0881774 −0.0440887 0.999028i \(-0.514038\pi\)
−0.0440887 + 0.999028i \(0.514038\pi\)
\(44\) −5.31188 −0.800796
\(45\) 0 0
\(46\) 14.2569 2.10206
\(47\) 5.00075 0.729434 0.364717 0.931118i \(-0.381166\pi\)
0.364717 + 0.931118i \(0.381166\pi\)
\(48\) 2.78993 0.402692
\(49\) −0.483971 −0.0691387
\(50\) 0 0
\(51\) 0.977025 0.136811
\(52\) −3.86494 −0.535970
\(53\) −5.82375 −0.799954 −0.399977 0.916525i \(-0.630982\pi\)
−0.399977 + 0.916525i \(0.630982\pi\)
\(54\) 3.30678 0.449996
\(55\) 0 0
\(56\) 22.8602 3.05483
\(57\) 0.205258 0.0271871
\(58\) −17.0692 −2.24129
\(59\) 8.71096 1.13407 0.567035 0.823694i \(-0.308090\pi\)
0.567035 + 0.823694i \(0.308090\pi\)
\(60\) 0 0
\(61\) 10.3898 1.33028 0.665140 0.746719i \(-0.268372\pi\)
0.665140 + 0.746719i \(0.268372\pi\)
\(62\) −7.03368 −0.893279
\(63\) 7.55041 0.951262
\(64\) 23.7686 2.97107
\(65\) 0 0
\(66\) 0.555028 0.0683192
\(67\) −6.24347 −0.762761 −0.381381 0.924418i \(-0.624551\pi\)
−0.381381 + 0.924418i \(0.624551\pi\)
\(68\) 25.2845 3.06619
\(69\) −1.08221 −0.130282
\(70\) 0 0
\(71\) 9.50218 1.12770 0.563851 0.825877i \(-0.309319\pi\)
0.563851 + 0.825877i \(0.309319\pi\)
\(72\) 26.4892 3.12178
\(73\) 10.7796 1.26165 0.630826 0.775924i \(-0.282716\pi\)
0.630826 + 0.775924i \(0.282716\pi\)
\(74\) 20.4098 2.37259
\(75\) 0 0
\(76\) 5.31188 0.609314
\(77\) 2.55265 0.290902
\(78\) 0.403840 0.0457258
\(79\) −13.9182 −1.56591 −0.782957 0.622075i \(-0.786290\pi\)
−0.782957 + 0.622075i \(0.786290\pi\)
\(80\) 0 0
\(81\) 8.62260 0.958066
\(82\) −17.8365 −1.96971
\(83\) 8.52531 0.935774 0.467887 0.883788i \(-0.345015\pi\)
0.467887 + 0.883788i \(0.345015\pi\)
\(84\) −2.78317 −0.303669
\(85\) 0 0
\(86\) 1.56353 0.168600
\(87\) 1.29568 0.138912
\(88\) 8.95548 0.954658
\(89\) −13.9001 −1.47341 −0.736705 0.676214i \(-0.763619\pi\)
−0.736705 + 0.676214i \(0.763619\pi\)
\(90\) 0 0
\(91\) 1.85732 0.194700
\(92\) −28.0064 −2.91987
\(93\) 0.533911 0.0553640
\(94\) −13.5223 −1.39472
\(95\) 0 0
\(96\) −3.86774 −0.394749
\(97\) 14.6606 1.48856 0.744278 0.667870i \(-0.232794\pi\)
0.744278 + 0.667870i \(0.232794\pi\)
\(98\) 1.30868 0.132197
\(99\) 2.95787 0.297277
\(100\) 0 0
\(101\) −6.36247 −0.633089 −0.316545 0.948578i \(-0.602523\pi\)
−0.316545 + 0.948578i \(0.602523\pi\)
\(102\) −2.64192 −0.261589
\(103\) −3.79471 −0.373904 −0.186952 0.982369i \(-0.559861\pi\)
−0.186952 + 0.982369i \(0.559861\pi\)
\(104\) 6.51603 0.638950
\(105\) 0 0
\(106\) 15.7477 1.52955
\(107\) 10.7131 1.03568 0.517839 0.855478i \(-0.326737\pi\)
0.517839 + 0.855478i \(0.326737\pi\)
\(108\) −6.49590 −0.625069
\(109\) −14.7235 −1.41025 −0.705127 0.709081i \(-0.749110\pi\)
−0.705127 + 0.709081i \(0.749110\pi\)
\(110\) 0 0
\(111\) −1.54926 −0.147049
\(112\) −34.6964 −3.27850
\(113\) 12.2567 1.15301 0.576506 0.817093i \(-0.304416\pi\)
0.576506 + 0.817093i \(0.304416\pi\)
\(114\) −0.555028 −0.0519831
\(115\) 0 0
\(116\) 33.5310 3.11328
\(117\) 2.15215 0.198967
\(118\) −23.5549 −2.16840
\(119\) −12.1506 −1.11384
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −28.0946 −2.54356
\(123\) 1.35393 0.122079
\(124\) 13.8171 1.24081
\(125\) 0 0
\(126\) −20.4167 −1.81886
\(127\) −0.503914 −0.0447151 −0.0223576 0.999750i \(-0.507117\pi\)
−0.0223576 + 0.999750i \(0.507117\pi\)
\(128\) −26.5848 −2.34979
\(129\) −0.118684 −0.0104495
\(130\) 0 0
\(131\) −11.6934 −1.02165 −0.510827 0.859684i \(-0.670661\pi\)
−0.510827 + 0.859684i \(0.670661\pi\)
\(132\) −1.09031 −0.0948990
\(133\) −2.55265 −0.221343
\(134\) 16.8826 1.45844
\(135\) 0 0
\(136\) −42.6280 −3.65532
\(137\) −20.5569 −1.75630 −0.878148 0.478389i \(-0.841221\pi\)
−0.878148 + 0.478389i \(0.841221\pi\)
\(138\) 2.92634 0.249107
\(139\) −11.4656 −0.972503 −0.486251 0.873819i \(-0.661636\pi\)
−0.486251 + 0.873819i \(0.661636\pi\)
\(140\) 0 0
\(141\) 1.02644 0.0864422
\(142\) −25.6944 −2.15622
\(143\) 0.727602 0.0608452
\(144\) −40.2043 −3.35036
\(145\) 0 0
\(146\) −29.1485 −2.41234
\(147\) −0.0993389 −0.00819334
\(148\) −40.0933 −3.29565
\(149\) 7.78170 0.637502 0.318751 0.947839i \(-0.396737\pi\)
0.318751 + 0.947839i \(0.396737\pi\)
\(150\) 0 0
\(151\) 1.79747 0.146276 0.0731382 0.997322i \(-0.476699\pi\)
0.0731382 + 0.997322i \(0.476699\pi\)
\(152\) −8.95548 −0.726386
\(153\) −14.0794 −1.13825
\(154\) −6.90249 −0.556219
\(155\) 0 0
\(156\) −0.793310 −0.0635156
\(157\) −10.5222 −0.839767 −0.419883 0.907578i \(-0.637929\pi\)
−0.419883 + 0.907578i \(0.637929\pi\)
\(158\) 37.6354 2.99411
\(159\) −1.19537 −0.0947992
\(160\) 0 0
\(161\) 13.4586 1.06069
\(162\) −23.3159 −1.83187
\(163\) 1.74536 0.136707 0.0683535 0.997661i \(-0.478225\pi\)
0.0683535 + 0.997661i \(0.478225\pi\)
\(164\) 35.0383 2.73603
\(165\) 0 0
\(166\) −23.0528 −1.78925
\(167\) 2.35461 0.182205 0.0911026 0.995842i \(-0.470961\pi\)
0.0911026 + 0.995842i \(0.470961\pi\)
\(168\) 4.69225 0.362015
\(169\) −12.4706 −0.959277
\(170\) 0 0
\(171\) −2.95787 −0.226194
\(172\) −3.07142 −0.234194
\(173\) 13.0946 0.995561 0.497780 0.867303i \(-0.334149\pi\)
0.497780 + 0.867303i \(0.334149\pi\)
\(174\) −3.50359 −0.265606
\(175\) 0 0
\(176\) −13.5923 −1.02456
\(177\) 1.78800 0.134394
\(178\) 37.5866 2.81724
\(179\) −0.853860 −0.0638205 −0.0319103 0.999491i \(-0.510159\pi\)
−0.0319103 + 0.999491i \(0.510159\pi\)
\(180\) 0 0
\(181\) −3.30919 −0.245970 −0.122985 0.992409i \(-0.539247\pi\)
−0.122985 + 0.992409i \(0.539247\pi\)
\(182\) −5.02227 −0.372275
\(183\) 2.13259 0.157646
\(184\) 47.2171 3.48089
\(185\) 0 0
\(186\) −1.44372 −0.105859
\(187\) −4.75998 −0.348084
\(188\) 26.5634 1.93733
\(189\) 3.12164 0.227066
\(190\) 0 0
\(191\) −22.9878 −1.66334 −0.831668 0.555274i \(-0.812614\pi\)
−0.831668 + 0.555274i \(0.812614\pi\)
\(192\) 4.87869 0.352089
\(193\) 3.23091 0.232566 0.116283 0.993216i \(-0.462902\pi\)
0.116283 + 0.993216i \(0.462902\pi\)
\(194\) −39.6429 −2.84619
\(195\) 0 0
\(196\) −2.57079 −0.183628
\(197\) 16.7066 1.19030 0.595149 0.803615i \(-0.297093\pi\)
0.595149 + 0.803615i \(0.297093\pi\)
\(198\) −7.99822 −0.568409
\(199\) −24.2205 −1.71694 −0.858472 0.512861i \(-0.828586\pi\)
−0.858472 + 0.512861i \(0.828586\pi\)
\(200\) 0 0
\(201\) −1.28152 −0.0903916
\(202\) 17.2044 1.21050
\(203\) −16.1135 −1.13095
\(204\) 5.18984 0.363362
\(205\) 0 0
\(206\) 10.2611 0.714923
\(207\) 15.5951 1.08394
\(208\) −9.88980 −0.685734
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 2.74027 0.188648 0.0943239 0.995542i \(-0.469931\pi\)
0.0943239 + 0.995542i \(0.469931\pi\)
\(212\) −30.9351 −2.12463
\(213\) 1.95040 0.133639
\(214\) −28.9688 −1.98027
\(215\) 0 0
\(216\) 10.9517 0.745167
\(217\) −6.63987 −0.450744
\(218\) 39.8130 2.69648
\(219\) 2.21259 0.149513
\(220\) 0 0
\(221\) −3.46338 −0.232972
\(222\) 4.18927 0.281166
\(223\) −15.3500 −1.02791 −0.513957 0.857816i \(-0.671821\pi\)
−0.513957 + 0.857816i \(0.671821\pi\)
\(224\) 48.1004 3.21384
\(225\) 0 0
\(226\) −33.1427 −2.20462
\(227\) −18.5064 −1.22831 −0.614157 0.789184i \(-0.710504\pi\)
−0.614157 + 0.789184i \(0.710504\pi\)
\(228\) 1.09031 0.0722073
\(229\) −4.06791 −0.268815 −0.134408 0.990926i \(-0.542913\pi\)
−0.134408 + 0.990926i \(0.542913\pi\)
\(230\) 0 0
\(231\) 0.523952 0.0344735
\(232\) −56.5311 −3.71145
\(233\) −8.02328 −0.525622 −0.262811 0.964847i \(-0.584650\pi\)
−0.262811 + 0.964847i \(0.584650\pi\)
\(234\) −5.81953 −0.380434
\(235\) 0 0
\(236\) 46.2716 3.01202
\(237\) −2.85681 −0.185570
\(238\) 32.8558 2.12972
\(239\) 4.07616 0.263665 0.131832 0.991272i \(-0.457914\pi\)
0.131832 + 0.991272i \(0.457914\pi\)
\(240\) 0 0
\(241\) 23.0372 1.48396 0.741979 0.670423i \(-0.233888\pi\)
0.741979 + 0.670423i \(0.233888\pi\)
\(242\) −2.70405 −0.173823
\(243\) 5.43856 0.348884
\(244\) 55.1895 3.53314
\(245\) 0 0
\(246\) −3.66108 −0.233422
\(247\) −0.727602 −0.0462962
\(248\) −23.2947 −1.47922
\(249\) 1.74989 0.110895
\(250\) 0 0
\(251\) 11.0036 0.694540 0.347270 0.937765i \(-0.387109\pi\)
0.347270 + 0.937765i \(0.387109\pi\)
\(252\) 40.1069 2.52649
\(253\) 5.27242 0.331474
\(254\) 1.36261 0.0854976
\(255\) 0 0
\(256\) 24.3494 1.52184
\(257\) 18.6522 1.16349 0.581745 0.813371i \(-0.302370\pi\)
0.581745 + 0.813371i \(0.302370\pi\)
\(258\) 0.320927 0.0199800
\(259\) 19.2671 1.19720
\(260\) 0 0
\(261\) −18.6714 −1.15573
\(262\) 31.6194 1.95345
\(263\) 11.6405 0.717783 0.358892 0.933379i \(-0.383155\pi\)
0.358892 + 0.933379i \(0.383155\pi\)
\(264\) 1.83819 0.113133
\(265\) 0 0
\(266\) 6.90249 0.423219
\(267\) −2.85311 −0.174608
\(268\) −33.1646 −2.02585
\(269\) 20.3357 1.23989 0.619945 0.784645i \(-0.287155\pi\)
0.619945 + 0.784645i \(0.287155\pi\)
\(270\) 0 0
\(271\) −14.1613 −0.860239 −0.430120 0.902772i \(-0.641529\pi\)
−0.430120 + 0.902772i \(0.641529\pi\)
\(272\) 64.6992 3.92296
\(273\) 0.381229 0.0230730
\(274\) 55.5869 3.35813
\(275\) 0 0
\(276\) −5.74855 −0.346022
\(277\) 0.252067 0.0151452 0.00757261 0.999971i \(-0.497590\pi\)
0.00757261 + 0.999971i \(0.497590\pi\)
\(278\) 31.0036 1.85948
\(279\) −7.69391 −0.460623
\(280\) 0 0
\(281\) −14.8513 −0.885956 −0.442978 0.896533i \(-0.646078\pi\)
−0.442978 + 0.896533i \(0.646078\pi\)
\(282\) −2.77556 −0.165282
\(283\) −8.11110 −0.482155 −0.241078 0.970506i \(-0.577501\pi\)
−0.241078 + 0.970506i \(0.577501\pi\)
\(284\) 50.4744 2.99511
\(285\) 0 0
\(286\) −1.96747 −0.116339
\(287\) −16.8378 −0.993906
\(288\) 55.7360 3.28428
\(289\) 5.65744 0.332791
\(290\) 0 0
\(291\) 3.00920 0.176402
\(292\) 57.2597 3.35087
\(293\) −6.58970 −0.384974 −0.192487 0.981299i \(-0.561655\pi\)
−0.192487 + 0.981299i \(0.561655\pi\)
\(294\) 0.268617 0.0156661
\(295\) 0 0
\(296\) 67.5948 3.92887
\(297\) 1.22290 0.0709599
\(298\) −21.0421 −1.21894
\(299\) 3.83622 0.221855
\(300\) 0 0
\(301\) 1.47599 0.0850745
\(302\) −4.86046 −0.279688
\(303\) −1.30595 −0.0750248
\(304\) 13.5923 0.779572
\(305\) 0 0
\(306\) 38.0714 2.17640
\(307\) −6.28665 −0.358798 −0.179399 0.983776i \(-0.557415\pi\)
−0.179399 + 0.983776i \(0.557415\pi\)
\(308\) 13.5594 0.772617
\(309\) −0.778894 −0.0443097
\(310\) 0 0
\(311\) −31.4799 −1.78506 −0.892530 0.450988i \(-0.851072\pi\)
−0.892530 + 0.450988i \(0.851072\pi\)
\(312\) 1.33747 0.0757192
\(313\) −24.9984 −1.41299 −0.706497 0.707716i \(-0.749726\pi\)
−0.706497 + 0.707716i \(0.749726\pi\)
\(314\) 28.4527 1.60568
\(315\) 0 0
\(316\) −73.9316 −4.15898
\(317\) −1.26448 −0.0710204 −0.0355102 0.999369i \(-0.511306\pi\)
−0.0355102 + 0.999369i \(0.511306\pi\)
\(318\) 3.23234 0.181261
\(319\) −6.31246 −0.353430
\(320\) 0 0
\(321\) 2.19896 0.122734
\(322\) −36.3928 −2.02809
\(323\) 4.75998 0.264853
\(324\) 45.8022 2.54457
\(325\) 0 0
\(326\) −4.71953 −0.261391
\(327\) −3.02212 −0.167123
\(328\) −59.0723 −3.26172
\(329\) −12.7652 −0.703767
\(330\) 0 0
\(331\) −12.8375 −0.705613 −0.352807 0.935696i \(-0.614773\pi\)
−0.352807 + 0.935696i \(0.614773\pi\)
\(332\) 45.2854 2.48536
\(333\) 22.3256 1.22343
\(334\) −6.36698 −0.348386
\(335\) 0 0
\(336\) −7.12172 −0.388522
\(337\) −14.2172 −0.774462 −0.387231 0.921983i \(-0.626568\pi\)
−0.387231 + 0.921983i \(0.626568\pi\)
\(338\) 33.7211 1.83419
\(339\) 2.51578 0.136639
\(340\) 0 0
\(341\) −2.60117 −0.140861
\(342\) 7.99822 0.432494
\(343\) 19.1040 1.03152
\(344\) 5.17822 0.279191
\(345\) 0 0
\(346\) −35.4083 −1.90356
\(347\) 27.0040 1.44965 0.724825 0.688933i \(-0.241920\pi\)
0.724825 + 0.688933i \(0.241920\pi\)
\(348\) 6.88251 0.368941
\(349\) 11.6541 0.623828 0.311914 0.950110i \(-0.399030\pi\)
0.311914 + 0.950110i \(0.399030\pi\)
\(350\) 0 0
\(351\) 0.889786 0.0474932
\(352\) 18.8433 1.00435
\(353\) 12.7049 0.676213 0.338106 0.941108i \(-0.390214\pi\)
0.338106 + 0.941108i \(0.390214\pi\)
\(354\) −4.83483 −0.256968
\(355\) 0 0
\(356\) −73.8358 −3.91329
\(357\) −2.49400 −0.131997
\(358\) 2.30888 0.122028
\(359\) −11.8470 −0.625261 −0.312631 0.949875i \(-0.601210\pi\)
−0.312631 + 0.949875i \(0.601210\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 8.94821 0.470307
\(363\) 0.205258 0.0107733
\(364\) 9.86584 0.517110
\(365\) 0 0
\(366\) −5.76664 −0.301427
\(367\) 0.650593 0.0339607 0.0169803 0.999856i \(-0.494595\pi\)
0.0169803 + 0.999856i \(0.494595\pi\)
\(368\) −71.6643 −3.73576
\(369\) −19.5107 −1.01569
\(370\) 0 0
\(371\) 14.8660 0.771804
\(372\) 2.83607 0.147043
\(373\) −12.3639 −0.640178 −0.320089 0.947388i \(-0.603713\pi\)
−0.320089 + 0.947388i \(0.603713\pi\)
\(374\) 12.8712 0.665555
\(375\) 0 0
\(376\) −44.7841 −2.30957
\(377\) −4.59296 −0.236549
\(378\) −8.44107 −0.434162
\(379\) 37.7295 1.93803 0.969017 0.246994i \(-0.0794428\pi\)
0.969017 + 0.246994i \(0.0794428\pi\)
\(380\) 0 0
\(381\) −0.103432 −0.00529900
\(382\) 62.1600 3.18038
\(383\) −28.8536 −1.47435 −0.737175 0.675702i \(-0.763840\pi\)
−0.737175 + 0.675702i \(0.763840\pi\)
\(384\) −5.45674 −0.278463
\(385\) 0 0
\(386\) −8.73655 −0.444678
\(387\) 1.71029 0.0869390
\(388\) 77.8752 3.95351
\(389\) −30.9376 −1.56860 −0.784298 0.620384i \(-0.786977\pi\)
−0.784298 + 0.620384i \(0.786977\pi\)
\(390\) 0 0
\(391\) −25.0966 −1.26919
\(392\) 4.33419 0.218910
\(393\) −2.40016 −0.121072
\(394\) −45.1756 −2.27591
\(395\) 0 0
\(396\) 15.7118 0.789550
\(397\) −31.9662 −1.60434 −0.802168 0.597099i \(-0.796320\pi\)
−0.802168 + 0.597099i \(0.796320\pi\)
\(398\) 65.4933 3.28288
\(399\) −0.523952 −0.0262304
\(400\) 0 0
\(401\) 31.8777 1.59190 0.795949 0.605364i \(-0.206972\pi\)
0.795949 + 0.605364i \(0.206972\pi\)
\(402\) 3.46530 0.172833
\(403\) −1.89262 −0.0942779
\(404\) −33.7967 −1.68145
\(405\) 0 0
\(406\) 43.5717 2.16243
\(407\) 7.54786 0.374134
\(408\) −8.74974 −0.433176
\(409\) 3.06602 0.151605 0.0758024 0.997123i \(-0.475848\pi\)
0.0758024 + 0.997123i \(0.475848\pi\)
\(410\) 0 0
\(411\) −4.21947 −0.208131
\(412\) −20.1570 −0.993065
\(413\) −22.2360 −1.09416
\(414\) −42.1700 −2.07254
\(415\) 0 0
\(416\) 13.7104 0.672209
\(417\) −2.35342 −0.115247
\(418\) 2.70405 0.132259
\(419\) −31.8947 −1.55816 −0.779080 0.626924i \(-0.784313\pi\)
−0.779080 + 0.626924i \(0.784313\pi\)
\(420\) 0 0
\(421\) 22.8439 1.11334 0.556672 0.830733i \(-0.312078\pi\)
0.556672 + 0.830733i \(0.312078\pi\)
\(422\) −7.40982 −0.360704
\(423\) −14.7916 −0.719190
\(424\) 52.1545 2.53285
\(425\) 0 0
\(426\) −5.27397 −0.255525
\(427\) −26.5216 −1.28347
\(428\) 56.9069 2.75070
\(429\) 0.149346 0.00721051
\(430\) 0 0
\(431\) 20.0366 0.965131 0.482565 0.875860i \(-0.339705\pi\)
0.482565 + 0.875860i \(0.339705\pi\)
\(432\) −16.6220 −0.799729
\(433\) −10.0474 −0.482847 −0.241423 0.970420i \(-0.577614\pi\)
−0.241423 + 0.970420i \(0.577614\pi\)
\(434\) 17.9545 0.861846
\(435\) 0 0
\(436\) −78.2094 −3.74555
\(437\) −5.27242 −0.252214
\(438\) −5.98296 −0.285877
\(439\) −30.8352 −1.47168 −0.735842 0.677153i \(-0.763213\pi\)
−0.735842 + 0.677153i \(0.763213\pi\)
\(440\) 0 0
\(441\) 1.43152 0.0681677
\(442\) 9.36514 0.445454
\(443\) −9.41447 −0.447295 −0.223647 0.974670i \(-0.571796\pi\)
−0.223647 + 0.974670i \(0.571796\pi\)
\(444\) −8.22948 −0.390554
\(445\) 0 0
\(446\) 41.5072 1.96542
\(447\) 1.59726 0.0755477
\(448\) −60.6729 −2.86652
\(449\) 3.24554 0.153166 0.0765832 0.997063i \(-0.475599\pi\)
0.0765832 + 0.997063i \(0.475599\pi\)
\(450\) 0 0
\(451\) −6.59622 −0.310604
\(452\) 65.1060 3.06233
\(453\) 0.368946 0.0173346
\(454\) 50.0423 2.34860
\(455\) 0 0
\(456\) −1.83819 −0.0860809
\(457\) −38.9693 −1.82291 −0.911453 0.411405i \(-0.865038\pi\)
−0.911453 + 0.411405i \(0.865038\pi\)
\(458\) 10.9998 0.513989
\(459\) −5.82099 −0.271700
\(460\) 0 0
\(461\) −1.37595 −0.0640842 −0.0320421 0.999487i \(-0.510201\pi\)
−0.0320421 + 0.999487i \(0.510201\pi\)
\(462\) −1.41679 −0.0659152
\(463\) 12.2145 0.567658 0.283829 0.958875i \(-0.408395\pi\)
0.283829 + 0.958875i \(0.408395\pi\)
\(464\) 85.8009 3.98321
\(465\) 0 0
\(466\) 21.6953 1.00502
\(467\) −7.23601 −0.334843 −0.167421 0.985885i \(-0.553544\pi\)
−0.167421 + 0.985885i \(0.553544\pi\)
\(468\) 11.4320 0.528443
\(469\) 15.9374 0.735921
\(470\) 0 0
\(471\) −2.15978 −0.0995172
\(472\) −78.0109 −3.59074
\(473\) 0.578218 0.0265865
\(474\) 7.72497 0.354819
\(475\) 0 0
\(476\) −64.5424 −2.95830
\(477\) 17.2259 0.788719
\(478\) −11.0221 −0.504141
\(479\) −11.7525 −0.536987 −0.268493 0.963282i \(-0.586526\pi\)
−0.268493 + 0.963282i \(0.586526\pi\)
\(480\) 0 0
\(481\) 5.49184 0.250406
\(482\) −62.2938 −2.83740
\(483\) 2.76250 0.125698
\(484\) 5.31188 0.241449
\(485\) 0 0
\(486\) −14.7061 −0.667084
\(487\) 12.4058 0.562162 0.281081 0.959684i \(-0.409307\pi\)
0.281081 + 0.959684i \(0.409307\pi\)
\(488\) −93.0459 −4.21199
\(489\) 0.358249 0.0162006
\(490\) 0 0
\(491\) −29.3521 −1.32464 −0.662321 0.749220i \(-0.730428\pi\)
−0.662321 + 0.749220i \(0.730428\pi\)
\(492\) 7.19190 0.324236
\(493\) 30.0472 1.35326
\(494\) 1.96747 0.0885208
\(495\) 0 0
\(496\) 35.3559 1.58753
\(497\) −24.2557 −1.08802
\(498\) −4.73178 −0.212036
\(499\) 15.4628 0.692210 0.346105 0.938196i \(-0.387504\pi\)
0.346105 + 0.938196i \(0.387504\pi\)
\(500\) 0 0
\(501\) 0.483303 0.0215924
\(502\) −29.7542 −1.32800
\(503\) 25.3187 1.12890 0.564452 0.825466i \(-0.309088\pi\)
0.564452 + 0.825466i \(0.309088\pi\)
\(504\) −67.6176 −3.01193
\(505\) 0 0
\(506\) −14.2569 −0.633795
\(507\) −2.55969 −0.113680
\(508\) −2.67673 −0.118761
\(509\) −29.5127 −1.30813 −0.654064 0.756439i \(-0.726937\pi\)
−0.654064 + 0.756439i \(0.726937\pi\)
\(510\) 0 0
\(511\) −27.5165 −1.21726
\(512\) −12.6725 −0.560049
\(513\) −1.22290 −0.0539924
\(514\) −50.4364 −2.22465
\(515\) 0 0
\(516\) −0.630434 −0.0277533
\(517\) −5.00075 −0.219933
\(518\) −52.0991 −2.28910
\(519\) 2.68776 0.117980
\(520\) 0 0
\(521\) 25.0642 1.09808 0.549042 0.835795i \(-0.314993\pi\)
0.549042 + 0.835795i \(0.314993\pi\)
\(522\) 50.4884 2.20982
\(523\) −6.49035 −0.283803 −0.141902 0.989881i \(-0.545322\pi\)
−0.141902 + 0.989881i \(0.545322\pi\)
\(524\) −62.1137 −2.71345
\(525\) 0 0
\(526\) −31.4764 −1.37244
\(527\) 12.3815 0.539347
\(528\) −2.78993 −0.121416
\(529\) 4.79838 0.208625
\(530\) 0 0
\(531\) −25.7659 −1.11814
\(532\) −13.5594 −0.587874
\(533\) −4.79942 −0.207886
\(534\) 7.71496 0.333859
\(535\) 0 0
\(536\) 55.9133 2.41509
\(537\) −0.175262 −0.00756310
\(538\) −54.9887 −2.37073
\(539\) 0.483971 0.0208461
\(540\) 0 0
\(541\) 18.9373 0.814178 0.407089 0.913388i \(-0.366544\pi\)
0.407089 + 0.913388i \(0.366544\pi\)
\(542\) 38.2929 1.64482
\(543\) −0.679238 −0.0291489
\(544\) −89.6938 −3.84559
\(545\) 0 0
\(546\) −1.03086 −0.0441168
\(547\) −2.79465 −0.119491 −0.0597454 0.998214i \(-0.519029\pi\)
−0.0597454 + 0.998214i \(0.519029\pi\)
\(548\) −109.196 −4.66462
\(549\) −30.7317 −1.31160
\(550\) 0 0
\(551\) 6.31246 0.268920
\(552\) 9.69168 0.412505
\(553\) 35.5282 1.51081
\(554\) −0.681601 −0.0289584
\(555\) 0 0
\(556\) −60.9041 −2.58291
\(557\) −42.6845 −1.80860 −0.904300 0.426897i \(-0.859607\pi\)
−0.904300 + 0.426897i \(0.859607\pi\)
\(558\) 20.8047 0.880734
\(559\) 0.420713 0.0177942
\(560\) 0 0
\(561\) −0.977025 −0.0412500
\(562\) 40.1587 1.69399
\(563\) −30.2652 −1.27553 −0.637763 0.770233i \(-0.720140\pi\)
−0.637763 + 0.770233i \(0.720140\pi\)
\(564\) 5.45235 0.229585
\(565\) 0 0
\(566\) 21.9328 0.921905
\(567\) −22.0105 −0.924353
\(568\) −85.0966 −3.57057
\(569\) −21.6291 −0.906738 −0.453369 0.891323i \(-0.649778\pi\)
−0.453369 + 0.891323i \(0.649778\pi\)
\(570\) 0 0
\(571\) −30.9060 −1.29337 −0.646687 0.762756i \(-0.723846\pi\)
−0.646687 + 0.762756i \(0.723846\pi\)
\(572\) 3.86494 0.161601
\(573\) −4.71842 −0.197115
\(574\) 45.5303 1.90040
\(575\) 0 0
\(576\) −70.3043 −2.92935
\(577\) 8.58255 0.357296 0.178648 0.983913i \(-0.442828\pi\)
0.178648 + 0.983913i \(0.442828\pi\)
\(578\) −15.2980 −0.636313
\(579\) 0.663171 0.0275604
\(580\) 0 0
\(581\) −21.7621 −0.902846
\(582\) −8.13702 −0.337291
\(583\) 5.82375 0.241195
\(584\) −96.5362 −3.99470
\(585\) 0 0
\(586\) 17.8189 0.736091
\(587\) 21.1713 0.873834 0.436917 0.899502i \(-0.356070\pi\)
0.436917 + 0.899502i \(0.356070\pi\)
\(588\) −0.527677 −0.0217610
\(589\) 2.60117 0.107179
\(590\) 0 0
\(591\) 3.42917 0.141057
\(592\) −102.593 −4.21654
\(593\) 8.54489 0.350897 0.175448 0.984489i \(-0.443863\pi\)
0.175448 + 0.984489i \(0.443863\pi\)
\(594\) −3.30678 −0.135679
\(595\) 0 0
\(596\) 41.3355 1.69317
\(597\) −4.97145 −0.203468
\(598\) −10.3733 −0.424197
\(599\) −5.41629 −0.221304 −0.110652 0.993859i \(-0.535294\pi\)
−0.110652 + 0.993859i \(0.535294\pi\)
\(600\) 0 0
\(601\) −10.2791 −0.419295 −0.209648 0.977777i \(-0.567232\pi\)
−0.209648 + 0.977777i \(0.567232\pi\)
\(602\) −3.99114 −0.162667
\(603\) 18.4674 0.752049
\(604\) 9.54797 0.388501
\(605\) 0 0
\(606\) 3.53135 0.143451
\(607\) −27.4399 −1.11375 −0.556877 0.830595i \(-0.688000\pi\)
−0.556877 + 0.830595i \(0.688000\pi\)
\(608\) −18.8433 −0.764197
\(609\) −3.30743 −0.134024
\(610\) 0 0
\(611\) −3.63856 −0.147200
\(612\) −74.7881 −3.02313
\(613\) −35.0872 −1.41716 −0.708579 0.705632i \(-0.750663\pi\)
−0.708579 + 0.705632i \(0.750663\pi\)
\(614\) 16.9994 0.686040
\(615\) 0 0
\(616\) −22.8602 −0.921065
\(617\) 16.2427 0.653908 0.326954 0.945040i \(-0.393978\pi\)
0.326954 + 0.945040i \(0.393978\pi\)
\(618\) 2.10617 0.0847225
\(619\) −44.7185 −1.79739 −0.898693 0.438578i \(-0.855482\pi\)
−0.898693 + 0.438578i \(0.855482\pi\)
\(620\) 0 0
\(621\) 6.44764 0.258735
\(622\) 85.1231 3.41313
\(623\) 35.4822 1.42156
\(624\) −2.02996 −0.0812635
\(625\) 0 0
\(626\) 67.5970 2.70172
\(627\) −0.205258 −0.00819722
\(628\) −55.8929 −2.23037
\(629\) −35.9277 −1.43253
\(630\) 0 0
\(631\) 26.6254 1.05994 0.529969 0.848017i \(-0.322203\pi\)
0.529969 + 0.848017i \(0.322203\pi\)
\(632\) 124.644 4.95807
\(633\) 0.562462 0.0223559
\(634\) 3.41922 0.135795
\(635\) 0 0
\(636\) −6.34967 −0.251781
\(637\) 0.352138 0.0139522
\(638\) 17.0692 0.675776
\(639\) −28.1062 −1.11186
\(640\) 0 0
\(641\) 18.1107 0.715330 0.357665 0.933850i \(-0.383573\pi\)
0.357665 + 0.933850i \(0.383573\pi\)
\(642\) −5.94609 −0.234673
\(643\) 10.3842 0.409514 0.204757 0.978813i \(-0.434360\pi\)
0.204757 + 0.978813i \(0.434360\pi\)
\(644\) 71.4907 2.81713
\(645\) 0 0
\(646\) −12.8712 −0.506412
\(647\) −5.54001 −0.217800 −0.108900 0.994053i \(-0.534733\pi\)
−0.108900 + 0.994053i \(0.534733\pi\)
\(648\) −77.2195 −3.03347
\(649\) −8.71096 −0.341935
\(650\) 0 0
\(651\) −1.36289 −0.0534158
\(652\) 9.27113 0.363085
\(653\) 22.9691 0.898849 0.449425 0.893318i \(-0.351629\pi\)
0.449425 + 0.893318i \(0.351629\pi\)
\(654\) 8.17195 0.319548
\(655\) 0 0
\(656\) 89.6578 3.50055
\(657\) −31.8845 −1.24393
\(658\) 34.5176 1.34564
\(659\) 47.5854 1.85366 0.926831 0.375478i \(-0.122522\pi\)
0.926831 + 0.375478i \(0.122522\pi\)
\(660\) 0 0
\(661\) 36.5786 1.42274 0.711372 0.702816i \(-0.248074\pi\)
0.711372 + 0.702816i \(0.248074\pi\)
\(662\) 34.7132 1.34917
\(663\) −0.710886 −0.0276085
\(664\) −76.3483 −2.96289
\(665\) 0 0
\(666\) −60.3695 −2.33927
\(667\) −33.2819 −1.28868
\(668\) 12.5074 0.483926
\(669\) −3.15072 −0.121814
\(670\) 0 0
\(671\) −10.3898 −0.401094
\(672\) 9.87299 0.380859
\(673\) 49.8445 1.92136 0.960682 0.277651i \(-0.0895558\pi\)
0.960682 + 0.277651i \(0.0895558\pi\)
\(674\) 38.4441 1.48081
\(675\) 0 0
\(676\) −66.2423 −2.54778
\(677\) −13.7408 −0.528103 −0.264052 0.964509i \(-0.585059\pi\)
−0.264052 + 0.964509i \(0.585059\pi\)
\(678\) −6.80280 −0.261260
\(679\) −37.4233 −1.43618
\(680\) 0 0
\(681\) −3.79859 −0.145562
\(682\) 7.03368 0.269334
\(683\) −47.6403 −1.82291 −0.911453 0.411405i \(-0.865038\pi\)
−0.911453 + 0.411405i \(0.865038\pi\)
\(684\) −15.7118 −0.600757
\(685\) 0 0
\(686\) −51.6581 −1.97231
\(687\) −0.834972 −0.0318562
\(688\) −7.85931 −0.299633
\(689\) 4.23737 0.161431
\(690\) 0 0
\(691\) 5.77892 0.219840 0.109920 0.993940i \(-0.464940\pi\)
0.109920 + 0.993940i \(0.464940\pi\)
\(692\) 69.5567 2.64415
\(693\) −7.55041 −0.286816
\(694\) −73.0201 −2.77181
\(695\) 0 0
\(696\) −11.6035 −0.439828
\(697\) 31.3979 1.18928
\(698\) −31.5132 −1.19279
\(699\) −1.64684 −0.0622893
\(700\) 0 0
\(701\) 14.7926 0.558709 0.279355 0.960188i \(-0.409879\pi\)
0.279355 + 0.960188i \(0.409879\pi\)
\(702\) −2.40602 −0.0908095
\(703\) −7.54786 −0.284673
\(704\) −23.7686 −0.895812
\(705\) 0 0
\(706\) −34.3546 −1.29295
\(707\) 16.2412 0.610812
\(708\) 9.49762 0.356942
\(709\) −42.7910 −1.60705 −0.803524 0.595272i \(-0.797044\pi\)
−0.803524 + 0.595272i \(0.797044\pi\)
\(710\) 0 0
\(711\) 41.1681 1.54392
\(712\) 124.482 4.66517
\(713\) −13.7144 −0.513610
\(714\) 6.74391 0.252385
\(715\) 0 0
\(716\) −4.53560 −0.169503
\(717\) 0.836665 0.0312458
\(718\) 32.0349 1.19553
\(719\) 6.08276 0.226849 0.113424 0.993547i \(-0.463818\pi\)
0.113424 + 0.993547i \(0.463818\pi\)
\(720\) 0 0
\(721\) 9.68656 0.360746
\(722\) −2.70405 −0.100634
\(723\) 4.72858 0.175858
\(724\) −17.5780 −0.653282
\(725\) 0 0
\(726\) −0.555028 −0.0205990
\(727\) 12.5655 0.466028 0.233014 0.972473i \(-0.425141\pi\)
0.233014 + 0.972473i \(0.425141\pi\)
\(728\) −16.6332 −0.616466
\(729\) −24.7515 −0.916722
\(730\) 0 0
\(731\) −2.75231 −0.101798
\(732\) 11.3281 0.418698
\(733\) −29.2151 −1.07909 −0.539543 0.841958i \(-0.681403\pi\)
−0.539543 + 0.841958i \(0.681403\pi\)
\(734\) −1.75923 −0.0649345
\(735\) 0 0
\(736\) 99.3497 3.66208
\(737\) 6.24347 0.229981
\(738\) 52.7580 1.94205
\(739\) 40.8838 1.50393 0.751967 0.659201i \(-0.229105\pi\)
0.751967 + 0.659201i \(0.229105\pi\)
\(740\) 0 0
\(741\) −0.149346 −0.00548637
\(742\) −40.1984 −1.47573
\(743\) 46.3327 1.69978 0.849890 0.526960i \(-0.176668\pi\)
0.849890 + 0.526960i \(0.176668\pi\)
\(744\) −4.78143 −0.175296
\(745\) 0 0
\(746\) 33.4326 1.22405
\(747\) −25.2167 −0.922632
\(748\) −25.2845 −0.924491
\(749\) −27.3469 −0.999234
\(750\) 0 0
\(751\) 2.97997 0.108741 0.0543704 0.998521i \(-0.482685\pi\)
0.0543704 + 0.998521i \(0.482685\pi\)
\(752\) 67.9717 2.47867
\(753\) 2.25857 0.0823070
\(754\) 12.4196 0.452295
\(755\) 0 0
\(756\) 16.5818 0.603073
\(757\) 14.0931 0.512224 0.256112 0.966647i \(-0.417558\pi\)
0.256112 + 0.966647i \(0.417558\pi\)
\(758\) −102.022 −3.70562
\(759\) 1.08221 0.0392816
\(760\) 0 0
\(761\) −36.8510 −1.33585 −0.667923 0.744230i \(-0.732817\pi\)
−0.667923 + 0.744230i \(0.732817\pi\)
\(762\) 0.279686 0.0101320
\(763\) 37.5839 1.36063
\(764\) −122.108 −4.41772
\(765\) 0 0
\(766\) 78.0215 2.81903
\(767\) −6.33812 −0.228856
\(768\) 4.99792 0.180347
\(769\) −36.0236 −1.29904 −0.649522 0.760343i \(-0.725031\pi\)
−0.649522 + 0.760343i \(0.725031\pi\)
\(770\) 0 0
\(771\) 3.82851 0.137880
\(772\) 17.1622 0.617682
\(773\) 33.1377 1.19188 0.595941 0.803028i \(-0.296779\pi\)
0.595941 + 0.803028i \(0.296779\pi\)
\(774\) −4.62471 −0.166232
\(775\) 0 0
\(776\) −131.292 −4.71313
\(777\) 3.95472 0.141875
\(778\) 83.6567 2.99924
\(779\) 6.59622 0.236334
\(780\) 0 0
\(781\) −9.50218 −0.340015
\(782\) 67.8625 2.42676
\(783\) −7.71951 −0.275873
\(784\) −6.57828 −0.234939
\(785\) 0 0
\(786\) 6.49014 0.231496
\(787\) −30.1960 −1.07637 −0.538185 0.842827i \(-0.680890\pi\)
−0.538185 + 0.842827i \(0.680890\pi\)
\(788\) 88.7436 3.16136
\(789\) 2.38930 0.0850615
\(790\) 0 0
\(791\) −31.2870 −1.11244
\(792\) −26.4892 −0.941251
\(793\) −7.55966 −0.268451
\(794\) 86.4380 3.06757
\(795\) 0 0
\(796\) −128.656 −4.56010
\(797\) 36.0088 1.27550 0.637749 0.770245i \(-0.279866\pi\)
0.637749 + 0.770245i \(0.279866\pi\)
\(798\) 1.41679 0.0501539
\(799\) 23.8035 0.842107
\(800\) 0 0
\(801\) 41.1147 1.45272
\(802\) −86.1989 −3.04379
\(803\) −10.7796 −0.380403
\(804\) −6.80729 −0.240075
\(805\) 0 0
\(806\) 5.11773 0.180264
\(807\) 4.17407 0.146934
\(808\) 56.9790 2.00451
\(809\) −33.9639 −1.19411 −0.597054 0.802201i \(-0.703662\pi\)
−0.597054 + 0.802201i \(0.703662\pi\)
\(810\) 0 0
\(811\) 6.84873 0.240491 0.120246 0.992744i \(-0.461632\pi\)
0.120246 + 0.992744i \(0.461632\pi\)
\(812\) −85.5930 −3.00373
\(813\) −2.90673 −0.101943
\(814\) −20.4098 −0.715363
\(815\) 0 0
\(816\) 13.2800 0.464894
\(817\) −0.578218 −0.0202293
\(818\) −8.29066 −0.289876
\(819\) −5.49370 −0.191965
\(820\) 0 0
\(821\) 14.9183 0.520652 0.260326 0.965521i \(-0.416170\pi\)
0.260326 + 0.965521i \(0.416170\pi\)
\(822\) 11.4097 0.397958
\(823\) −50.2012 −1.74990 −0.874952 0.484209i \(-0.839107\pi\)
−0.874952 + 0.484209i \(0.839107\pi\)
\(824\) 33.9834 1.18387
\(825\) 0 0
\(826\) 60.1273 2.09210
\(827\) 43.3575 1.50769 0.753844 0.657053i \(-0.228197\pi\)
0.753844 + 0.657053i \(0.228197\pi\)
\(828\) 82.8394 2.87887
\(829\) −31.1594 −1.08221 −0.541106 0.840954i \(-0.681994\pi\)
−0.541106 + 0.840954i \(0.681994\pi\)
\(830\) 0 0
\(831\) 0.0517387 0.00179480
\(832\) −17.2941 −0.599564
\(833\) −2.30369 −0.0798182
\(834\) 6.36375 0.220359
\(835\) 0 0
\(836\) −5.31188 −0.183715
\(837\) −3.18097 −0.109950
\(838\) 86.2449 2.97928
\(839\) −38.4626 −1.32788 −0.663938 0.747788i \(-0.731116\pi\)
−0.663938 + 0.747788i \(0.731116\pi\)
\(840\) 0 0
\(841\) 10.8471 0.374038
\(842\) −61.7710 −2.12877
\(843\) −3.04835 −0.104991
\(844\) 14.5560 0.501037
\(845\) 0 0
\(846\) 39.9971 1.37513
\(847\) −2.55265 −0.0877101
\(848\) −79.1582 −2.71830
\(849\) −1.66487 −0.0571382
\(850\) 0 0
\(851\) 39.7955 1.36417
\(852\) 10.3603 0.354938
\(853\) 29.7541 1.01876 0.509380 0.860542i \(-0.329875\pi\)
0.509380 + 0.860542i \(0.329875\pi\)
\(854\) 71.7157 2.45406
\(855\) 0 0
\(856\) −95.9413 −3.27921
\(857\) 14.0757 0.480818 0.240409 0.970672i \(-0.422718\pi\)
0.240409 + 0.970672i \(0.422718\pi\)
\(858\) −0.403840 −0.0137869
\(859\) 21.8419 0.745235 0.372618 0.927985i \(-0.378460\pi\)
0.372618 + 0.927985i \(0.378460\pi\)
\(860\) 0 0
\(861\) −3.45610 −0.117784
\(862\) −54.1800 −1.84538
\(863\) −37.9643 −1.29232 −0.646161 0.763201i \(-0.723626\pi\)
−0.646161 + 0.763201i \(0.723626\pi\)
\(864\) 23.0435 0.783955
\(865\) 0 0
\(866\) 27.1686 0.923228
\(867\) 1.16124 0.0394377
\(868\) −35.2702 −1.19715
\(869\) 13.9182 0.472141
\(870\) 0 0
\(871\) 4.54276 0.153926
\(872\) 131.856 4.46521
\(873\) −43.3640 −1.46765
\(874\) 14.2569 0.482246
\(875\) 0 0
\(876\) 11.7530 0.397098
\(877\) −8.73734 −0.295039 −0.147520 0.989059i \(-0.547129\pi\)
−0.147520 + 0.989059i \(0.547129\pi\)
\(878\) 83.3799 2.81394
\(879\) −1.35259 −0.0456217
\(880\) 0 0
\(881\) −47.7668 −1.60930 −0.804652 0.593747i \(-0.797648\pi\)
−0.804652 + 0.593747i \(0.797648\pi\)
\(882\) −3.87091 −0.130340
\(883\) −35.9428 −1.20957 −0.604786 0.796388i \(-0.706741\pi\)
−0.604786 + 0.796388i \(0.706741\pi\)
\(884\) −18.3970 −0.618759
\(885\) 0 0
\(886\) 25.4572 0.855251
\(887\) −29.7501 −0.998910 −0.499455 0.866340i \(-0.666466\pi\)
−0.499455 + 0.866340i \(0.666466\pi\)
\(888\) 13.8744 0.465594
\(889\) 1.28632 0.0431417
\(890\) 0 0
\(891\) −8.62260 −0.288868
\(892\) −81.5375 −2.73008
\(893\) 5.00075 0.167344
\(894\) −4.31906 −0.144451
\(895\) 0 0
\(896\) 67.8617 2.26710
\(897\) 0.787416 0.0262911
\(898\) −8.77609 −0.292862
\(899\) 16.4198 0.547630
\(900\) 0 0
\(901\) −27.7210 −0.923519
\(902\) 17.8365 0.593890
\(903\) 0.302959 0.0100818
\(904\) −109.765 −3.65071
\(905\) 0 0
\(906\) −0.997649 −0.0331447
\(907\) −10.1512 −0.337066 −0.168533 0.985696i \(-0.553903\pi\)
−0.168533 + 0.985696i \(0.553903\pi\)
\(908\) −98.3039 −3.26233
\(909\) 18.8193 0.624198
\(910\) 0 0
\(911\) 33.0830 1.09609 0.548044 0.836449i \(-0.315372\pi\)
0.548044 + 0.836449i \(0.315372\pi\)
\(912\) 2.78993 0.0923839
\(913\) −8.52531 −0.282146
\(914\) 105.375 3.48549
\(915\) 0 0
\(916\) −21.6083 −0.713957
\(917\) 29.8491 0.985703
\(918\) 15.7402 0.519505
\(919\) 17.3847 0.573468 0.286734 0.958010i \(-0.407430\pi\)
0.286734 + 0.958010i \(0.407430\pi\)
\(920\) 0 0
\(921\) −1.29039 −0.0425196
\(922\) 3.72063 0.122532
\(923\) −6.91381 −0.227571
\(924\) 2.78317 0.0915596
\(925\) 0 0
\(926\) −33.0287 −1.08539
\(927\) 11.2242 0.368653
\(928\) −118.947 −3.90464
\(929\) −0.843879 −0.0276868 −0.0138434 0.999904i \(-0.504407\pi\)
−0.0138434 + 0.999904i \(0.504407\pi\)
\(930\) 0 0
\(931\) −0.483971 −0.0158615
\(932\) −42.6187 −1.39602
\(933\) −6.46150 −0.211540
\(934\) 19.5665 0.640237
\(935\) 0 0
\(936\) −19.2736 −0.629976
\(937\) −28.9038 −0.944246 −0.472123 0.881533i \(-0.656512\pi\)
−0.472123 + 0.881533i \(0.656512\pi\)
\(938\) −43.0955 −1.40712
\(939\) −5.13113 −0.167448
\(940\) 0 0
\(941\) −25.8992 −0.844289 −0.422144 0.906529i \(-0.638722\pi\)
−0.422144 + 0.906529i \(0.638722\pi\)
\(942\) 5.84014 0.190282
\(943\) −34.7780 −1.13253
\(944\) 118.402 3.85366
\(945\) 0 0
\(946\) −1.56353 −0.0508347
\(947\) −13.4248 −0.436249 −0.218124 0.975921i \(-0.569994\pi\)
−0.218124 + 0.975921i \(0.569994\pi\)
\(948\) −15.1751 −0.492863
\(949\) −7.84324 −0.254602
\(950\) 0 0
\(951\) −0.259545 −0.00841633
\(952\) 108.814 3.52669
\(953\) −22.9222 −0.742524 −0.371262 0.928528i \(-0.621075\pi\)
−0.371262 + 0.928528i \(0.621075\pi\)
\(954\) −46.5796 −1.50807
\(955\) 0 0
\(956\) 21.6521 0.700278
\(957\) −1.29568 −0.0418835
\(958\) 31.7794 1.02675
\(959\) 52.4747 1.69449
\(960\) 0 0
\(961\) −24.2339 −0.781740
\(962\) −14.8502 −0.478790
\(963\) −31.6881 −1.02113
\(964\) 122.371 3.94130
\(965\) 0 0
\(966\) −7.46992 −0.240341
\(967\) −28.1998 −0.906845 −0.453423 0.891296i \(-0.649797\pi\)
−0.453423 + 0.891296i \(0.649797\pi\)
\(968\) −8.95548 −0.287840
\(969\) 0.977025 0.0313866
\(970\) 0 0
\(971\) −33.9967 −1.09101 −0.545503 0.838109i \(-0.683661\pi\)
−0.545503 + 0.838109i \(0.683661\pi\)
\(972\) 28.8890 0.926615
\(973\) 29.2678 0.938282
\(974\) −33.5460 −1.07488
\(975\) 0 0
\(976\) 141.222 4.52039
\(977\) −20.5638 −0.657894 −0.328947 0.944348i \(-0.606694\pi\)
−0.328947 + 0.944348i \(0.606694\pi\)
\(978\) −0.968722 −0.0309763
\(979\) 13.9001 0.444250
\(980\) 0 0
\(981\) 43.5501 1.39045
\(982\) 79.3695 2.53278
\(983\) 7.53887 0.240452 0.120226 0.992747i \(-0.461638\pi\)
0.120226 + 0.992747i \(0.461638\pi\)
\(984\) −12.1251 −0.386533
\(985\) 0 0
\(986\) −81.2491 −2.58750
\(987\) −2.62015 −0.0834005
\(988\) −3.86494 −0.122960
\(989\) 3.04860 0.0969400
\(990\) 0 0
\(991\) 48.8053 1.55035 0.775176 0.631746i \(-0.217661\pi\)
0.775176 + 0.631746i \(0.217661\pi\)
\(992\) −49.0146 −1.55621
\(993\) −2.63500 −0.0836193
\(994\) 65.5887 2.08035
\(995\) 0 0
\(996\) 9.29520 0.294530
\(997\) 26.9298 0.852875 0.426438 0.904517i \(-0.359768\pi\)
0.426438 + 0.904517i \(0.359768\pi\)
\(998\) −41.8122 −1.32354
\(999\) 9.23029 0.292033
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.v.1.1 yes 15
5.4 even 2 5225.2.a.u.1.15 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.u.1.15 15 5.4 even 2
5225.2.a.v.1.1 yes 15 1.1 even 1 trivial