Properties

Label 8025.2.a.bb.1.6
Level $8025$
Weight $2$
Character 8025.1
Self dual yes
Analytic conductor $64.080$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8025,2,Mod(1,8025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8025, 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("8025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8025.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: \(64.0799476221\)
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: \( x^{7} - 3x^{6} - 9x^{5} + 24x^{4} + 13x^{3} - 47x^{2} + 19x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 321)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.67795\) of defining polynomial
Character \(\chi\) \(=\) 8025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.39839 q^{2} +1.00000 q^{3} +3.75227 q^{4} +2.39839 q^{6} -2.59863 q^{7} +4.20262 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.39839 q^{2} +1.00000 q^{3} +3.75227 q^{4} +2.39839 q^{6} -2.59863 q^{7} +4.20262 q^{8} +1.00000 q^{9} -2.31906 q^{11} +3.75227 q^{12} +0.510992 q^{13} -6.23251 q^{14} +2.57497 q^{16} +1.70532 q^{17} +2.39839 q^{18} +7.72152 q^{19} -2.59863 q^{21} -5.56201 q^{22} -0.207050 q^{23} +4.20262 q^{24} +1.22556 q^{26} +1.00000 q^{27} -9.75073 q^{28} +4.79678 q^{29} +8.71600 q^{31} -2.22945 q^{32} -2.31906 q^{33} +4.09001 q^{34} +3.75227 q^{36} +5.97891 q^{37} +18.5192 q^{38} +0.510992 q^{39} +6.04876 q^{41} -6.23251 q^{42} -3.98090 q^{43} -8.70174 q^{44} -0.496586 q^{46} -4.96026 q^{47} +2.57497 q^{48} -0.247147 q^{49} +1.70532 q^{51} +1.91738 q^{52} -0.544225 q^{53} +2.39839 q^{54} -10.9210 q^{56} +7.72152 q^{57} +11.5045 q^{58} +11.6915 q^{59} +0.943204 q^{61} +20.9044 q^{62} -2.59863 q^{63} -10.4970 q^{64} -5.56201 q^{66} +0.605901 q^{67} +6.39880 q^{68} -0.207050 q^{69} -1.91240 q^{71} +4.20262 q^{72} +13.0474 q^{73} +14.3397 q^{74} +28.9732 q^{76} +6.02637 q^{77} +1.22556 q^{78} +1.42355 q^{79} +1.00000 q^{81} +14.5073 q^{82} +12.5487 q^{83} -9.75073 q^{84} -9.54775 q^{86} +4.79678 q^{87} -9.74613 q^{88} +13.4753 q^{89} -1.32788 q^{91} -0.776906 q^{92} +8.71600 q^{93} -11.8966 q^{94} -2.22945 q^{96} -12.7949 q^{97} -0.592753 q^{98} -2.31906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{3} + 14 q^{4} - 6 q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{3} + 14 q^{4} - 6 q^{7} - 3 q^{8} + 7 q^{9} + 4 q^{11} + 14 q^{12} - 6 q^{13} + 12 q^{14} + 32 q^{16} + 10 q^{17} + 8 q^{19} - 6 q^{21} - 10 q^{22} - 6 q^{23} - 3 q^{24} + 7 q^{26} + 7 q^{27} - 8 q^{28} + 16 q^{31} - 6 q^{32} + 4 q^{33} - 11 q^{34} + 14 q^{36} - 10 q^{37} + 13 q^{38} - 6 q^{39} - 2 q^{41} + 12 q^{42} - 2 q^{43} + 2 q^{44} - 30 q^{46} - 16 q^{47} + 32 q^{48} + 17 q^{49} + 10 q^{51} + 23 q^{52} + 16 q^{53} + 30 q^{56} + 8 q^{57} + 56 q^{58} + 20 q^{59} + 2 q^{61} + 52 q^{62} - 6 q^{63} + 43 q^{64} - 10 q^{66} - 30 q^{67} + 61 q^{68} - 6 q^{69} + 32 q^{71} - 3 q^{72} + 12 q^{73} - q^{74} - 49 q^{76} + 46 q^{77} + 7 q^{78} + 36 q^{79} + 7 q^{81} - 2 q^{82} + 10 q^{83} - 8 q^{84} - 20 q^{86} + 14 q^{88} - 4 q^{89} + 12 q^{91} - 10 q^{92} + 16 q^{93} - 26 q^{94} - 6 q^{96} - 24 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.39839 1.69592 0.847958 0.530063i \(-0.177832\pi\)
0.847958 + 0.530063i \(0.177832\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.75227 1.87613
\(5\) 0 0
\(6\) 2.39839 0.979138
\(7\) −2.59863 −0.982188 −0.491094 0.871107i \(-0.663403\pi\)
−0.491094 + 0.871107i \(0.663403\pi\)
\(8\) 4.20262 1.48585
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.31906 −0.699224 −0.349612 0.936895i \(-0.613687\pi\)
−0.349612 + 0.936895i \(0.613687\pi\)
\(12\) 3.75227 1.08319
\(13\) 0.510992 0.141724 0.0708618 0.997486i \(-0.477425\pi\)
0.0708618 + 0.997486i \(0.477425\pi\)
\(14\) −6.23251 −1.66571
\(15\) 0 0
\(16\) 2.57497 0.643743
\(17\) 1.70532 0.413600 0.206800 0.978383i \(-0.433695\pi\)
0.206800 + 0.978383i \(0.433695\pi\)
\(18\) 2.39839 0.565306
\(19\) 7.72152 1.77144 0.885719 0.464221i \(-0.153666\pi\)
0.885719 + 0.464221i \(0.153666\pi\)
\(20\) 0 0
\(21\) −2.59863 −0.567067
\(22\) −5.56201 −1.18582
\(23\) −0.207050 −0.0431728 −0.0215864 0.999767i \(-0.506872\pi\)
−0.0215864 + 0.999767i \(0.506872\pi\)
\(24\) 4.20262 0.857855
\(25\) 0 0
\(26\) 1.22556 0.240351
\(27\) 1.00000 0.192450
\(28\) −9.75073 −1.84272
\(29\) 4.79678 0.890739 0.445370 0.895347i \(-0.353072\pi\)
0.445370 + 0.895347i \(0.353072\pi\)
\(30\) 0 0
\(31\) 8.71600 1.56544 0.782720 0.622374i \(-0.213832\pi\)
0.782720 + 0.622374i \(0.213832\pi\)
\(32\) −2.22945 −0.394115
\(33\) −2.31906 −0.403697
\(34\) 4.09001 0.701431
\(35\) 0 0
\(36\) 3.75227 0.625378
\(37\) 5.97891 0.982926 0.491463 0.870898i \(-0.336462\pi\)
0.491463 + 0.870898i \(0.336462\pi\)
\(38\) 18.5192 3.00421
\(39\) 0.510992 0.0818242
\(40\) 0 0
\(41\) 6.04876 0.944657 0.472329 0.881423i \(-0.343414\pi\)
0.472329 + 0.881423i \(0.343414\pi\)
\(42\) −6.23251 −0.961698
\(43\) −3.98090 −0.607082 −0.303541 0.952818i \(-0.598169\pi\)
−0.303541 + 0.952818i \(0.598169\pi\)
\(44\) −8.70174 −1.31184
\(45\) 0 0
\(46\) −0.496586 −0.0732176
\(47\) −4.96026 −0.723529 −0.361764 0.932270i \(-0.617826\pi\)
−0.361764 + 0.932270i \(0.617826\pi\)
\(48\) 2.57497 0.371665
\(49\) −0.247147 −0.0353067
\(50\) 0 0
\(51\) 1.70532 0.238792
\(52\) 1.91738 0.265892
\(53\) −0.544225 −0.0747551 −0.0373775 0.999301i \(-0.511900\pi\)
−0.0373775 + 0.999301i \(0.511900\pi\)
\(54\) 2.39839 0.326379
\(55\) 0 0
\(56\) −10.9210 −1.45938
\(57\) 7.72152 1.02274
\(58\) 11.5045 1.51062
\(59\) 11.6915 1.52211 0.761055 0.648688i \(-0.224682\pi\)
0.761055 + 0.648688i \(0.224682\pi\)
\(60\) 0 0
\(61\) 0.943204 0.120765 0.0603825 0.998175i \(-0.480768\pi\)
0.0603825 + 0.998175i \(0.480768\pi\)
\(62\) 20.9044 2.65486
\(63\) −2.59863 −0.327396
\(64\) −10.4970 −1.31213
\(65\) 0 0
\(66\) −5.56201 −0.684636
\(67\) 0.605901 0.0740226 0.0370113 0.999315i \(-0.488216\pi\)
0.0370113 + 0.999315i \(0.488216\pi\)
\(68\) 6.39880 0.775969
\(69\) −0.207050 −0.0249259
\(70\) 0 0
\(71\) −1.91240 −0.226960 −0.113480 0.993540i \(-0.536200\pi\)
−0.113480 + 0.993540i \(0.536200\pi\)
\(72\) 4.20262 0.495283
\(73\) 13.0474 1.52709 0.763544 0.645756i \(-0.223458\pi\)
0.763544 + 0.645756i \(0.223458\pi\)
\(74\) 14.3397 1.66696
\(75\) 0 0
\(76\) 28.9732 3.32345
\(77\) 6.02637 0.686769
\(78\) 1.22556 0.138767
\(79\) 1.42355 0.160162 0.0800810 0.996788i \(-0.474482\pi\)
0.0800810 + 0.996788i \(0.474482\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 14.5073 1.60206
\(83\) 12.5487 1.37740 0.688701 0.725046i \(-0.258181\pi\)
0.688701 + 0.725046i \(0.258181\pi\)
\(84\) −9.75073 −1.06389
\(85\) 0 0
\(86\) −9.54775 −1.02956
\(87\) 4.79678 0.514268
\(88\) −9.74613 −1.03894
\(89\) 13.4753 1.42838 0.714188 0.699954i \(-0.246796\pi\)
0.714188 + 0.699954i \(0.246796\pi\)
\(90\) 0 0
\(91\) −1.32788 −0.139199
\(92\) −0.776906 −0.0809980
\(93\) 8.71600 0.903807
\(94\) −11.8966 −1.22704
\(95\) 0 0
\(96\) −2.22945 −0.227542
\(97\) −12.7949 −1.29912 −0.649562 0.760308i \(-0.725048\pi\)
−0.649562 + 0.760308i \(0.725048\pi\)
\(98\) −0.592753 −0.0598771
\(99\) −2.31906 −0.233075
\(100\) 0 0
\(101\) −8.38268 −0.834108 −0.417054 0.908882i \(-0.636937\pi\)
−0.417054 + 0.908882i \(0.636937\pi\)
\(102\) 4.09001 0.404972
\(103\) −15.1868 −1.49640 −0.748201 0.663472i \(-0.769082\pi\)
−0.748201 + 0.663472i \(0.769082\pi\)
\(104\) 2.14750 0.210580
\(105\) 0 0
\(106\) −1.30526 −0.126778
\(107\) −1.00000 −0.0966736
\(108\) 3.75227 0.361062
\(109\) −7.45820 −0.714366 −0.357183 0.934034i \(-0.616263\pi\)
−0.357183 + 0.934034i \(0.616263\pi\)
\(110\) 0 0
\(111\) 5.97891 0.567493
\(112\) −6.69138 −0.632276
\(113\) −9.64461 −0.907289 −0.453644 0.891183i \(-0.649876\pi\)
−0.453644 + 0.891183i \(0.649876\pi\)
\(114\) 18.5192 1.73448
\(115\) 0 0
\(116\) 17.9988 1.67115
\(117\) 0.510992 0.0472412
\(118\) 28.0409 2.58137
\(119\) −4.43148 −0.406233
\(120\) 0 0
\(121\) −5.62195 −0.511086
\(122\) 2.26217 0.204807
\(123\) 6.04876 0.545398
\(124\) 32.7048 2.93697
\(125\) 0 0
\(126\) −6.23251 −0.555236
\(127\) 18.9347 1.68018 0.840090 0.542448i \(-0.182502\pi\)
0.840090 + 0.542448i \(0.182502\pi\)
\(128\) −20.7171 −1.83115
\(129\) −3.98090 −0.350499
\(130\) 0 0
\(131\) 13.7388 1.20037 0.600184 0.799862i \(-0.295094\pi\)
0.600184 + 0.799862i \(0.295094\pi\)
\(132\) −8.70174 −0.757389
\(133\) −20.0653 −1.73989
\(134\) 1.45319 0.125536
\(135\) 0 0
\(136\) 7.16679 0.614547
\(137\) −7.93715 −0.678117 −0.339058 0.940765i \(-0.610108\pi\)
−0.339058 + 0.940765i \(0.610108\pi\)
\(138\) −0.496586 −0.0422722
\(139\) −18.0551 −1.53142 −0.765708 0.643189i \(-0.777611\pi\)
−0.765708 + 0.643189i \(0.777611\pi\)
\(140\) 0 0
\(141\) −4.96026 −0.417730
\(142\) −4.58667 −0.384905
\(143\) −1.18502 −0.0990965
\(144\) 2.57497 0.214581
\(145\) 0 0
\(146\) 31.2928 2.58981
\(147\) −0.247147 −0.0203843
\(148\) 22.4345 1.84410
\(149\) −7.28825 −0.597077 −0.298539 0.954398i \(-0.596499\pi\)
−0.298539 + 0.954398i \(0.596499\pi\)
\(150\) 0 0
\(151\) 11.7419 0.955544 0.477772 0.878484i \(-0.341445\pi\)
0.477772 + 0.878484i \(0.341445\pi\)
\(152\) 32.4506 2.63209
\(153\) 1.70532 0.137867
\(154\) 14.4536 1.16470
\(155\) 0 0
\(156\) 1.91738 0.153513
\(157\) 9.56141 0.763084 0.381542 0.924352i \(-0.375393\pi\)
0.381542 + 0.924352i \(0.375393\pi\)
\(158\) 3.41423 0.271621
\(159\) −0.544225 −0.0431599
\(160\) 0 0
\(161\) 0.538045 0.0424039
\(162\) 2.39839 0.188435
\(163\) −11.2774 −0.883318 −0.441659 0.897183i \(-0.645610\pi\)
−0.441659 + 0.897183i \(0.645610\pi\)
\(164\) 22.6966 1.77230
\(165\) 0 0
\(166\) 30.0967 2.33596
\(167\) 21.3638 1.65318 0.826591 0.562802i \(-0.190277\pi\)
0.826591 + 0.562802i \(0.190277\pi\)
\(168\) −10.9210 −0.842575
\(169\) −12.7389 −0.979914
\(170\) 0 0
\(171\) 7.72152 0.590479
\(172\) −14.9374 −1.13897
\(173\) 18.3110 1.39216 0.696079 0.717965i \(-0.254926\pi\)
0.696079 + 0.717965i \(0.254926\pi\)
\(174\) 11.5045 0.872156
\(175\) 0 0
\(176\) −5.97152 −0.450120
\(177\) 11.6915 0.878790
\(178\) 32.3189 2.42241
\(179\) −18.3871 −1.37432 −0.687158 0.726508i \(-0.741142\pi\)
−0.687158 + 0.726508i \(0.741142\pi\)
\(180\) 0 0
\(181\) 8.36497 0.621763 0.310882 0.950449i \(-0.399376\pi\)
0.310882 + 0.950449i \(0.399376\pi\)
\(182\) −3.18476 −0.236070
\(183\) 0.943204 0.0697237
\(184\) −0.870150 −0.0641483
\(185\) 0 0
\(186\) 20.9044 1.53278
\(187\) −3.95474 −0.289199
\(188\) −18.6122 −1.35744
\(189\) −2.59863 −0.189022
\(190\) 0 0
\(191\) 13.4092 0.970254 0.485127 0.874444i \(-0.338773\pi\)
0.485127 + 0.874444i \(0.338773\pi\)
\(192\) −10.4970 −0.757558
\(193\) 10.1874 0.733303 0.366652 0.930358i \(-0.380504\pi\)
0.366652 + 0.930358i \(0.380504\pi\)
\(194\) −30.6871 −2.20321
\(195\) 0 0
\(196\) −0.927360 −0.0662400
\(197\) 9.51148 0.677665 0.338832 0.940847i \(-0.389968\pi\)
0.338832 + 0.940847i \(0.389968\pi\)
\(198\) −5.56201 −0.395275
\(199\) −8.83850 −0.626544 −0.313272 0.949663i \(-0.601425\pi\)
−0.313272 + 0.949663i \(0.601425\pi\)
\(200\) 0 0
\(201\) 0.605901 0.0427370
\(202\) −20.1049 −1.41458
\(203\) −12.4650 −0.874873
\(204\) 6.39880 0.448006
\(205\) 0 0
\(206\) −36.4239 −2.53777
\(207\) −0.207050 −0.0143909
\(208\) 1.31579 0.0912336
\(209\) −17.9067 −1.23863
\(210\) 0 0
\(211\) 2.98608 0.205570 0.102785 0.994704i \(-0.467225\pi\)
0.102785 + 0.994704i \(0.467225\pi\)
\(212\) −2.04208 −0.140250
\(213\) −1.91240 −0.131035
\(214\) −2.39839 −0.163950
\(215\) 0 0
\(216\) 4.20262 0.285952
\(217\) −22.6496 −1.53756
\(218\) −17.8877 −1.21150
\(219\) 13.0474 0.881665
\(220\) 0 0
\(221\) 0.871403 0.0586169
\(222\) 14.3397 0.962420
\(223\) −12.6326 −0.845941 −0.422971 0.906143i \(-0.639013\pi\)
−0.422971 + 0.906143i \(0.639013\pi\)
\(224\) 5.79351 0.387095
\(225\) 0 0
\(226\) −23.1315 −1.53869
\(227\) 29.4478 1.95452 0.977260 0.212046i \(-0.0680125\pi\)
0.977260 + 0.212046i \(0.0680125\pi\)
\(228\) 28.9732 1.91880
\(229\) −25.3001 −1.67187 −0.835937 0.548825i \(-0.815075\pi\)
−0.835937 + 0.548825i \(0.815075\pi\)
\(230\) 0 0
\(231\) 6.02637 0.396506
\(232\) 20.1590 1.32350
\(233\) −8.78721 −0.575669 −0.287835 0.957680i \(-0.592935\pi\)
−0.287835 + 0.957680i \(0.592935\pi\)
\(234\) 1.22556 0.0801172
\(235\) 0 0
\(236\) 43.8698 2.85568
\(237\) 1.42355 0.0924696
\(238\) −10.6284 −0.688937
\(239\) −15.3859 −0.995229 −0.497614 0.867398i \(-0.665791\pi\)
−0.497614 + 0.867398i \(0.665791\pi\)
\(240\) 0 0
\(241\) 2.74583 0.176874 0.0884371 0.996082i \(-0.471813\pi\)
0.0884371 + 0.996082i \(0.471813\pi\)
\(242\) −13.4836 −0.866760
\(243\) 1.00000 0.0641500
\(244\) 3.53915 0.226571
\(245\) 0 0
\(246\) 14.5073 0.924950
\(247\) 3.94563 0.251055
\(248\) 36.6300 2.32601
\(249\) 12.5487 0.795243
\(250\) 0 0
\(251\) −9.51184 −0.600382 −0.300191 0.953879i \(-0.597050\pi\)
−0.300191 + 0.953879i \(0.597050\pi\)
\(252\) −9.75073 −0.614239
\(253\) 0.480161 0.0301875
\(254\) 45.4127 2.84944
\(255\) 0 0
\(256\) −28.6935 −1.79334
\(257\) −9.59376 −0.598442 −0.299221 0.954184i \(-0.596727\pi\)
−0.299221 + 0.954184i \(0.596727\pi\)
\(258\) −9.54775 −0.594417
\(259\) −15.5369 −0.965418
\(260\) 0 0
\(261\) 4.79678 0.296913
\(262\) 32.9511 2.03572
\(263\) −6.68810 −0.412406 −0.206203 0.978509i \(-0.566111\pi\)
−0.206203 + 0.978509i \(0.566111\pi\)
\(264\) −9.74613 −0.599833
\(265\) 0 0
\(266\) −48.1245 −2.95070
\(267\) 13.4753 0.824673
\(268\) 2.27350 0.138876
\(269\) −12.7174 −0.775391 −0.387696 0.921787i \(-0.626729\pi\)
−0.387696 + 0.921787i \(0.626729\pi\)
\(270\) 0 0
\(271\) 14.0805 0.855327 0.427663 0.903938i \(-0.359337\pi\)
0.427663 + 0.903938i \(0.359337\pi\)
\(272\) 4.39114 0.266252
\(273\) −1.32788 −0.0803667
\(274\) −19.0364 −1.15003
\(275\) 0 0
\(276\) −0.776906 −0.0467642
\(277\) 25.7886 1.54949 0.774743 0.632277i \(-0.217879\pi\)
0.774743 + 0.632277i \(0.217879\pi\)
\(278\) −43.3032 −2.59715
\(279\) 8.71600 0.521813
\(280\) 0 0
\(281\) −19.9704 −1.19133 −0.595667 0.803231i \(-0.703112\pi\)
−0.595667 + 0.803231i \(0.703112\pi\)
\(282\) −11.8966 −0.708435
\(283\) −29.5986 −1.75945 −0.879727 0.475480i \(-0.842275\pi\)
−0.879727 + 0.475480i \(0.842275\pi\)
\(284\) −7.17582 −0.425807
\(285\) 0 0
\(286\) −2.84214 −0.168059
\(287\) −15.7185 −0.927831
\(288\) −2.22945 −0.131372
\(289\) −14.0919 −0.828935
\(290\) 0 0
\(291\) −12.7949 −0.750050
\(292\) 48.9575 2.86502
\(293\) 6.23225 0.364092 0.182046 0.983290i \(-0.441728\pi\)
0.182046 + 0.983290i \(0.441728\pi\)
\(294\) −0.592753 −0.0345701
\(295\) 0 0
\(296\) 25.1270 1.46048
\(297\) −2.31906 −0.134566
\(298\) −17.4801 −1.01259
\(299\) −0.105801 −0.00611861
\(300\) 0 0
\(301\) 10.3449 0.596269
\(302\) 28.1617 1.62052
\(303\) −8.38268 −0.481572
\(304\) 19.8827 1.14035
\(305\) 0 0
\(306\) 4.09001 0.233810
\(307\) 27.6374 1.57735 0.788674 0.614812i \(-0.210768\pi\)
0.788674 + 0.614812i \(0.210768\pi\)
\(308\) 22.6126 1.28847
\(309\) −15.1868 −0.863948
\(310\) 0 0
\(311\) −14.7005 −0.833590 −0.416795 0.909001i \(-0.636847\pi\)
−0.416795 + 0.909001i \(0.636847\pi\)
\(312\) 2.14750 0.121578
\(313\) −12.6308 −0.713933 −0.356966 0.934117i \(-0.616189\pi\)
−0.356966 + 0.934117i \(0.616189\pi\)
\(314\) 22.9320 1.29413
\(315\) 0 0
\(316\) 5.34154 0.300485
\(317\) −3.42979 −0.192636 −0.0963182 0.995351i \(-0.530707\pi\)
−0.0963182 + 0.995351i \(0.530707\pi\)
\(318\) −1.30526 −0.0731955
\(319\) −11.1240 −0.622826
\(320\) 0 0
\(321\) −1.00000 −0.0558146
\(322\) 1.29044 0.0719134
\(323\) 13.1676 0.732667
\(324\) 3.75227 0.208459
\(325\) 0 0
\(326\) −27.0477 −1.49803
\(327\) −7.45820 −0.412439
\(328\) 25.4206 1.40362
\(329\) 12.8899 0.710641
\(330\) 0 0
\(331\) 2.45780 0.135093 0.0675466 0.997716i \(-0.478483\pi\)
0.0675466 + 0.997716i \(0.478483\pi\)
\(332\) 47.0862 2.58419
\(333\) 5.97891 0.327642
\(334\) 51.2388 2.80366
\(335\) 0 0
\(336\) −6.69138 −0.365045
\(337\) −18.2240 −0.992726 −0.496363 0.868115i \(-0.665332\pi\)
−0.496363 + 0.868115i \(0.665332\pi\)
\(338\) −30.5528 −1.66185
\(339\) −9.64461 −0.523823
\(340\) 0 0
\(341\) −20.2130 −1.09459
\(342\) 18.5192 1.00140
\(343\) 18.8326 1.01687
\(344\) −16.7302 −0.902032
\(345\) 0 0
\(346\) 43.9169 2.36099
\(347\) −24.3524 −1.30731 −0.653653 0.756794i \(-0.726764\pi\)
−0.653653 + 0.756794i \(0.726764\pi\)
\(348\) 17.9988 0.964836
\(349\) 33.5583 1.79634 0.898168 0.439653i \(-0.144899\pi\)
0.898168 + 0.439653i \(0.144899\pi\)
\(350\) 0 0
\(351\) 0.510992 0.0272747
\(352\) 5.17024 0.275574
\(353\) −20.2678 −1.07874 −0.539372 0.842068i \(-0.681338\pi\)
−0.539372 + 0.842068i \(0.681338\pi\)
\(354\) 28.0409 1.49035
\(355\) 0 0
\(356\) 50.5628 2.67982
\(357\) −4.43148 −0.234539
\(358\) −44.0994 −2.33072
\(359\) 16.9473 0.894444 0.447222 0.894423i \(-0.352413\pi\)
0.447222 + 0.894423i \(0.352413\pi\)
\(360\) 0 0
\(361\) 40.6219 2.13799
\(362\) 20.0624 1.05446
\(363\) −5.62195 −0.295076
\(364\) −4.98255 −0.261156
\(365\) 0 0
\(366\) 2.26217 0.118246
\(367\) −28.3838 −1.48162 −0.740811 0.671714i \(-0.765558\pi\)
−0.740811 + 0.671714i \(0.765558\pi\)
\(368\) −0.533147 −0.0277922
\(369\) 6.04876 0.314886
\(370\) 0 0
\(371\) 1.41424 0.0734235
\(372\) 32.7048 1.69566
\(373\) −33.0422 −1.71086 −0.855431 0.517917i \(-0.826708\pi\)
−0.855431 + 0.517917i \(0.826708\pi\)
\(374\) −9.48499 −0.490457
\(375\) 0 0
\(376\) −20.8461 −1.07505
\(377\) 2.45111 0.126239
\(378\) −6.23251 −0.320566
\(379\) −25.7958 −1.32504 −0.662519 0.749045i \(-0.730513\pi\)
−0.662519 + 0.749045i \(0.730513\pi\)
\(380\) 0 0
\(381\) 18.9347 0.970052
\(382\) 32.1604 1.64547
\(383\) 23.4961 1.20059 0.600297 0.799777i \(-0.295049\pi\)
0.600297 + 0.799777i \(0.295049\pi\)
\(384\) −20.7171 −1.05721
\(385\) 0 0
\(386\) 24.4333 1.24362
\(387\) −3.98090 −0.202361
\(388\) −48.0099 −2.43733
\(389\) −23.7803 −1.20571 −0.602853 0.797852i \(-0.705970\pi\)
−0.602853 + 0.797852i \(0.705970\pi\)
\(390\) 0 0
\(391\) −0.353085 −0.0178563
\(392\) −1.03866 −0.0524603
\(393\) 13.7388 0.693033
\(394\) 22.8122 1.14926
\(395\) 0 0
\(396\) −8.70174 −0.437279
\(397\) −17.3777 −0.872163 −0.436081 0.899907i \(-0.643634\pi\)
−0.436081 + 0.899907i \(0.643634\pi\)
\(398\) −21.1981 −1.06257
\(399\) −20.0653 −1.00452
\(400\) 0 0
\(401\) −8.79926 −0.439414 −0.219707 0.975566i \(-0.570510\pi\)
−0.219707 + 0.975566i \(0.570510\pi\)
\(402\) 1.45319 0.0724783
\(403\) 4.45381 0.221860
\(404\) −31.4540 −1.56490
\(405\) 0 0
\(406\) −29.8960 −1.48371
\(407\) −13.8655 −0.687285
\(408\) 7.16679 0.354809
\(409\) −0.340432 −0.0168333 −0.00841663 0.999965i \(-0.502679\pi\)
−0.00841663 + 0.999965i \(0.502679\pi\)
\(410\) 0 0
\(411\) −7.93715 −0.391511
\(412\) −56.9850 −2.80745
\(413\) −30.3819 −1.49500
\(414\) −0.496586 −0.0244059
\(415\) 0 0
\(416\) −1.13923 −0.0558554
\(417\) −18.0551 −0.884163
\(418\) −42.9472 −2.10062
\(419\) −1.81254 −0.0885486 −0.0442743 0.999019i \(-0.514098\pi\)
−0.0442743 + 0.999019i \(0.514098\pi\)
\(420\) 0 0
\(421\) −8.91272 −0.434379 −0.217190 0.976129i \(-0.569689\pi\)
−0.217190 + 0.976129i \(0.569689\pi\)
\(422\) 7.16178 0.348630
\(423\) −4.96026 −0.241176
\(424\) −2.28717 −0.111075
\(425\) 0 0
\(426\) −4.58667 −0.222225
\(427\) −2.45103 −0.118614
\(428\) −3.75227 −0.181373
\(429\) −1.18502 −0.0572134
\(430\) 0 0
\(431\) 21.1473 1.01863 0.509315 0.860580i \(-0.329899\pi\)
0.509315 + 0.860580i \(0.329899\pi\)
\(432\) 2.57497 0.123888
\(433\) 34.4959 1.65777 0.828883 0.559422i \(-0.188977\pi\)
0.828883 + 0.559422i \(0.188977\pi\)
\(434\) −54.3226 −2.60757
\(435\) 0 0
\(436\) −27.9851 −1.34025
\(437\) −1.59874 −0.0764780
\(438\) 31.2928 1.49523
\(439\) −13.4740 −0.643080 −0.321540 0.946896i \(-0.604200\pi\)
−0.321540 + 0.946896i \(0.604200\pi\)
\(440\) 0 0
\(441\) −0.247147 −0.0117689
\(442\) 2.08996 0.0994094
\(443\) −35.3480 −1.67944 −0.839718 0.543023i \(-0.817279\pi\)
−0.839718 + 0.543023i \(0.817279\pi\)
\(444\) 22.4345 1.06469
\(445\) 0 0
\(446\) −30.2979 −1.43465
\(447\) −7.28825 −0.344723
\(448\) 27.2778 1.28876
\(449\) 9.63784 0.454838 0.227419 0.973797i \(-0.426971\pi\)
0.227419 + 0.973797i \(0.426971\pi\)
\(450\) 0 0
\(451\) −14.0274 −0.660527
\(452\) −36.1892 −1.70219
\(453\) 11.7419 0.551684
\(454\) 70.6273 3.31470
\(455\) 0 0
\(456\) 32.4506 1.51964
\(457\) −11.8024 −0.552091 −0.276045 0.961145i \(-0.589024\pi\)
−0.276045 + 0.961145i \(0.589024\pi\)
\(458\) −60.6794 −2.83536
\(459\) 1.70532 0.0795974
\(460\) 0 0
\(461\) 10.3134 0.480345 0.240172 0.970730i \(-0.422796\pi\)
0.240172 + 0.970730i \(0.422796\pi\)
\(462\) 14.4536 0.672442
\(463\) 5.32448 0.247450 0.123725 0.992317i \(-0.460516\pi\)
0.123725 + 0.992317i \(0.460516\pi\)
\(464\) 12.3516 0.573407
\(465\) 0 0
\(466\) −21.0751 −0.976287
\(467\) −5.00072 −0.231406 −0.115703 0.993284i \(-0.536912\pi\)
−0.115703 + 0.993284i \(0.536912\pi\)
\(468\) 1.91738 0.0886308
\(469\) −1.57451 −0.0727041
\(470\) 0 0
\(471\) 9.56141 0.440567
\(472\) 49.1351 2.26162
\(473\) 9.23196 0.424486
\(474\) 3.41423 0.156821
\(475\) 0 0
\(476\) −16.6281 −0.762147
\(477\) −0.544225 −0.0249184
\(478\) −36.9013 −1.68783
\(479\) −25.5519 −1.16750 −0.583749 0.811934i \(-0.698415\pi\)
−0.583749 + 0.811934i \(0.698415\pi\)
\(480\) 0 0
\(481\) 3.05517 0.139304
\(482\) 6.58556 0.299964
\(483\) 0.538045 0.0244819
\(484\) −21.0951 −0.958866
\(485\) 0 0
\(486\) 2.39839 0.108793
\(487\) 17.7207 0.803002 0.401501 0.915859i \(-0.368489\pi\)
0.401501 + 0.915859i \(0.368489\pi\)
\(488\) 3.96393 0.179438
\(489\) −11.2774 −0.509984
\(490\) 0 0
\(491\) −7.14679 −0.322530 −0.161265 0.986911i \(-0.551557\pi\)
−0.161265 + 0.986911i \(0.551557\pi\)
\(492\) 22.6966 1.02324
\(493\) 8.18002 0.368410
\(494\) 9.46316 0.425768
\(495\) 0 0
\(496\) 22.4435 1.00774
\(497\) 4.96960 0.222917
\(498\) 30.0967 1.34867
\(499\) 15.0773 0.674951 0.337476 0.941334i \(-0.390427\pi\)
0.337476 + 0.941334i \(0.390427\pi\)
\(500\) 0 0
\(501\) 21.3638 0.954466
\(502\) −22.8131 −1.01820
\(503\) −12.2097 −0.544402 −0.272201 0.962240i \(-0.587752\pi\)
−0.272201 + 0.962240i \(0.587752\pi\)
\(504\) −10.9210 −0.486461
\(505\) 0 0
\(506\) 1.15161 0.0511954
\(507\) −12.7389 −0.565754
\(508\) 71.0479 3.15224
\(509\) 29.1949 1.29404 0.647020 0.762473i \(-0.276015\pi\)
0.647020 + 0.762473i \(0.276015\pi\)
\(510\) 0 0
\(511\) −33.9054 −1.49989
\(512\) −27.3840 −1.21021
\(513\) 7.72152 0.340913
\(514\) −23.0096 −1.01491
\(515\) 0 0
\(516\) −14.9374 −0.657583
\(517\) 11.5032 0.505908
\(518\) −37.2636 −1.63727
\(519\) 18.3110 0.803763
\(520\) 0 0
\(521\) −40.9233 −1.79288 −0.896441 0.443163i \(-0.853856\pi\)
−0.896441 + 0.443163i \(0.853856\pi\)
\(522\) 11.5045 0.503540
\(523\) 30.1119 1.31670 0.658350 0.752712i \(-0.271255\pi\)
0.658350 + 0.752712i \(0.271255\pi\)
\(524\) 51.5518 2.25205
\(525\) 0 0
\(526\) −16.0407 −0.699406
\(527\) 14.8635 0.647466
\(528\) −5.97152 −0.259877
\(529\) −22.9571 −0.998136
\(530\) 0 0
\(531\) 11.6915 0.507370
\(532\) −75.2905 −3.26426
\(533\) 3.09087 0.133880
\(534\) 32.3189 1.39858
\(535\) 0 0
\(536\) 2.54637 0.109986
\(537\) −18.3871 −0.793462
\(538\) −30.5012 −1.31500
\(539\) 0.573148 0.0246872
\(540\) 0 0
\(541\) −20.9471 −0.900586 −0.450293 0.892881i \(-0.648680\pi\)
−0.450293 + 0.892881i \(0.648680\pi\)
\(542\) 33.7704 1.45056
\(543\) 8.36497 0.358975
\(544\) −3.80192 −0.163006
\(545\) 0 0
\(546\) −3.18476 −0.136295
\(547\) −14.2139 −0.607744 −0.303872 0.952713i \(-0.598279\pi\)
−0.303872 + 0.952713i \(0.598279\pi\)
\(548\) −29.7823 −1.27224
\(549\) 0.943204 0.0402550
\(550\) 0 0
\(551\) 37.0384 1.57789
\(552\) −0.870150 −0.0370361
\(553\) −3.69928 −0.157309
\(554\) 61.8510 2.62780
\(555\) 0 0
\(556\) −67.7476 −2.87314
\(557\) 43.5974 1.84728 0.923640 0.383261i \(-0.125199\pi\)
0.923640 + 0.383261i \(0.125199\pi\)
\(558\) 20.9044 0.884952
\(559\) −2.03421 −0.0860378
\(560\) 0 0
\(561\) −3.95474 −0.166969
\(562\) −47.8968 −2.02040
\(563\) 14.2699 0.601404 0.300702 0.953718i \(-0.402779\pi\)
0.300702 + 0.953718i \(0.402779\pi\)
\(564\) −18.6122 −0.783716
\(565\) 0 0
\(566\) −70.9889 −2.98389
\(567\) −2.59863 −0.109132
\(568\) −8.03707 −0.337228
\(569\) −24.8762 −1.04287 −0.521433 0.853292i \(-0.674602\pi\)
−0.521433 + 0.853292i \(0.674602\pi\)
\(570\) 0 0
\(571\) 25.4791 1.06627 0.533135 0.846030i \(-0.321014\pi\)
0.533135 + 0.846030i \(0.321014\pi\)
\(572\) −4.44652 −0.185918
\(573\) 13.4092 0.560176
\(574\) −37.6990 −1.57352
\(575\) 0 0
\(576\) −10.4970 −0.437376
\(577\) −12.7043 −0.528888 −0.264444 0.964401i \(-0.585188\pi\)
−0.264444 + 0.964401i \(0.585188\pi\)
\(578\) −33.7978 −1.40580
\(579\) 10.1874 0.423373
\(580\) 0 0
\(581\) −32.6094 −1.35287
\(582\) −30.6871 −1.27202
\(583\) 1.26209 0.0522705
\(584\) 54.8334 2.26902
\(585\) 0 0
\(586\) 14.9474 0.617470
\(587\) 46.5082 1.91960 0.959800 0.280686i \(-0.0905618\pi\)
0.959800 + 0.280686i \(0.0905618\pi\)
\(588\) −0.927360 −0.0382437
\(589\) 67.3008 2.77308
\(590\) 0 0
\(591\) 9.51148 0.391250
\(592\) 15.3955 0.632752
\(593\) 18.6267 0.764908 0.382454 0.923975i \(-0.375079\pi\)
0.382454 + 0.923975i \(0.375079\pi\)
\(594\) −5.56201 −0.228212
\(595\) 0 0
\(596\) −27.3475 −1.12020
\(597\) −8.83850 −0.361736
\(598\) −0.253751 −0.0103767
\(599\) 29.4150 1.20186 0.600931 0.799301i \(-0.294797\pi\)
0.600931 + 0.799301i \(0.294797\pi\)
\(600\) 0 0
\(601\) −7.71448 −0.314680 −0.157340 0.987544i \(-0.550292\pi\)
−0.157340 + 0.987544i \(0.550292\pi\)
\(602\) 24.8110 1.01122
\(603\) 0.605901 0.0246742
\(604\) 44.0588 1.79273
\(605\) 0 0
\(606\) −20.1049 −0.816706
\(607\) −11.5194 −0.467556 −0.233778 0.972290i \(-0.575109\pi\)
−0.233778 + 0.972290i \(0.575109\pi\)
\(608\) −17.2148 −0.698150
\(609\) −12.4650 −0.505108
\(610\) 0 0
\(611\) −2.53465 −0.102541
\(612\) 6.39880 0.258656
\(613\) −4.93728 −0.199415 −0.0997074 0.995017i \(-0.531791\pi\)
−0.0997074 + 0.995017i \(0.531791\pi\)
\(614\) 66.2851 2.67505
\(615\) 0 0
\(616\) 25.3265 1.02044
\(617\) 2.49009 0.100247 0.0501236 0.998743i \(-0.484038\pi\)
0.0501236 + 0.998743i \(0.484038\pi\)
\(618\) −36.4239 −1.46518
\(619\) −34.4243 −1.38363 −0.691815 0.722075i \(-0.743189\pi\)
−0.691815 + 0.722075i \(0.743189\pi\)
\(620\) 0 0
\(621\) −0.207050 −0.00830862
\(622\) −35.2575 −1.41370
\(623\) −35.0172 −1.40293
\(624\) 1.31579 0.0526737
\(625\) 0 0
\(626\) −30.2935 −1.21077
\(627\) −17.9067 −0.715124
\(628\) 35.8770 1.43165
\(629\) 10.1959 0.406538
\(630\) 0 0
\(631\) 45.0793 1.79458 0.897288 0.441445i \(-0.145534\pi\)
0.897288 + 0.441445i \(0.145534\pi\)
\(632\) 5.98264 0.237977
\(633\) 2.98608 0.118686
\(634\) −8.22598 −0.326695
\(635\) 0 0
\(636\) −2.04208 −0.0809736
\(637\) −0.126290 −0.00500379
\(638\) −26.6797 −1.05626
\(639\) −1.91240 −0.0756532
\(640\) 0 0
\(641\) −5.39775 −0.213198 −0.106599 0.994302i \(-0.533996\pi\)
−0.106599 + 0.994302i \(0.533996\pi\)
\(642\) −2.39839 −0.0946568
\(643\) −36.7733 −1.45020 −0.725099 0.688644i \(-0.758206\pi\)
−0.725099 + 0.688644i \(0.758206\pi\)
\(644\) 2.01889 0.0795553
\(645\) 0 0
\(646\) 31.5811 1.24254
\(647\) −40.2852 −1.58378 −0.791888 0.610667i \(-0.790902\pi\)
−0.791888 + 0.610667i \(0.790902\pi\)
\(648\) 4.20262 0.165094
\(649\) −27.1134 −1.06429
\(650\) 0 0
\(651\) −22.6496 −0.887709
\(652\) −42.3160 −1.65722
\(653\) 7.09277 0.277562 0.138781 0.990323i \(-0.455682\pi\)
0.138781 + 0.990323i \(0.455682\pi\)
\(654\) −17.8877 −0.699462
\(655\) 0 0
\(656\) 15.5754 0.608116
\(657\) 13.0474 0.509029
\(658\) 30.9149 1.20519
\(659\) −33.1662 −1.29197 −0.645985 0.763350i \(-0.723553\pi\)
−0.645985 + 0.763350i \(0.723553\pi\)
\(660\) 0 0
\(661\) −6.62018 −0.257495 −0.128748 0.991677i \(-0.541096\pi\)
−0.128748 + 0.991677i \(0.541096\pi\)
\(662\) 5.89477 0.229107
\(663\) 0.871403 0.0338425
\(664\) 52.7375 2.04661
\(665\) 0 0
\(666\) 14.3397 0.555654
\(667\) −0.993171 −0.0384557
\(668\) 80.1628 3.10159
\(669\) −12.6326 −0.488404
\(670\) 0 0
\(671\) −2.18735 −0.0844417
\(672\) 5.79351 0.223489
\(673\) 4.50311 0.173582 0.0867911 0.996227i \(-0.472339\pi\)
0.0867911 + 0.996227i \(0.472339\pi\)
\(674\) −43.7083 −1.68358
\(675\) 0 0
\(676\) −47.7997 −1.83845
\(677\) 12.3244 0.473663 0.236832 0.971551i \(-0.423891\pi\)
0.236832 + 0.971551i \(0.423891\pi\)
\(678\) −23.1315 −0.888361
\(679\) 33.2491 1.27598
\(680\) 0 0
\(681\) 29.4478 1.12844
\(682\) −48.4785 −1.85634
\(683\) 28.0420 1.07300 0.536499 0.843901i \(-0.319747\pi\)
0.536499 + 0.843901i \(0.319747\pi\)
\(684\) 28.9732 1.10782
\(685\) 0 0
\(686\) 45.1679 1.72452
\(687\) −25.3001 −0.965257
\(688\) −10.2507 −0.390805
\(689\) −0.278095 −0.0105946
\(690\) 0 0
\(691\) −25.8054 −0.981684 −0.490842 0.871249i \(-0.663311\pi\)
−0.490842 + 0.871249i \(0.663311\pi\)
\(692\) 68.7077 2.61188
\(693\) 6.02637 0.228923
\(694\) −58.4065 −2.21708
\(695\) 0 0
\(696\) 20.1590 0.764125
\(697\) 10.3150 0.390710
\(698\) 80.4859 3.04643
\(699\) −8.78721 −0.332363
\(700\) 0 0
\(701\) 42.9498 1.62219 0.811095 0.584915i \(-0.198872\pi\)
0.811095 + 0.584915i \(0.198872\pi\)
\(702\) 1.22556 0.0462557
\(703\) 46.1663 1.74119
\(704\) 24.3433 0.917471
\(705\) 0 0
\(706\) −48.6100 −1.82946
\(707\) 21.7834 0.819250
\(708\) 43.8698 1.64873
\(709\) 8.78803 0.330041 0.165021 0.986290i \(-0.447231\pi\)
0.165021 + 0.986290i \(0.447231\pi\)
\(710\) 0 0
\(711\) 1.42355 0.0533873
\(712\) 56.6314 2.12235
\(713\) −1.80465 −0.0675845
\(714\) −10.6284 −0.397758
\(715\) 0 0
\(716\) −68.9933 −2.57840
\(717\) −15.3859 −0.574596
\(718\) 40.6462 1.51690
\(719\) 13.3007 0.496034 0.248017 0.968756i \(-0.420221\pi\)
0.248017 + 0.968756i \(0.420221\pi\)
\(720\) 0 0
\(721\) 39.4648 1.46975
\(722\) 97.4270 3.62586
\(723\) 2.74583 0.102118
\(724\) 31.3876 1.16651
\(725\) 0 0
\(726\) −13.4836 −0.500424
\(727\) 16.7795 0.622317 0.311159 0.950358i \(-0.399283\pi\)
0.311159 + 0.950358i \(0.399283\pi\)
\(728\) −5.58055 −0.206829
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.78870 −0.251089
\(732\) 3.53915 0.130811
\(733\) −14.8741 −0.549389 −0.274694 0.961532i \(-0.588577\pi\)
−0.274694 + 0.961532i \(0.588577\pi\)
\(734\) −68.0753 −2.51271
\(735\) 0 0
\(736\) 0.461607 0.0170151
\(737\) −1.40512 −0.0517583
\(738\) 14.5073 0.534020
\(739\) 0.303603 0.0111682 0.00558412 0.999984i \(-0.498223\pi\)
0.00558412 + 0.999984i \(0.498223\pi\)
\(740\) 0 0
\(741\) 3.94563 0.144946
\(742\) 3.39189 0.124520
\(743\) −46.5691 −1.70846 −0.854228 0.519899i \(-0.825969\pi\)
−0.854228 + 0.519899i \(0.825969\pi\)
\(744\) 36.6300 1.34292
\(745\) 0 0
\(746\) −79.2481 −2.90148
\(747\) 12.5487 0.459134
\(748\) −14.8392 −0.542576
\(749\) 2.59863 0.0949517
\(750\) 0 0
\(751\) −26.3917 −0.963047 −0.481523 0.876433i \(-0.659916\pi\)
−0.481523 + 0.876433i \(0.659916\pi\)
\(752\) −12.7725 −0.465766
\(753\) −9.51184 −0.346631
\(754\) 5.87872 0.214090
\(755\) 0 0
\(756\) −9.75073 −0.354631
\(757\) 13.9095 0.505551 0.252775 0.967525i \(-0.418657\pi\)
0.252775 + 0.967525i \(0.418657\pi\)
\(758\) −61.8682 −2.24716
\(759\) 0.480161 0.0174287
\(760\) 0 0
\(761\) 12.1860 0.441742 0.220871 0.975303i \(-0.429110\pi\)
0.220871 + 0.975303i \(0.429110\pi\)
\(762\) 45.4127 1.64513
\(763\) 19.3811 0.701641
\(764\) 50.3148 1.82033
\(765\) 0 0
\(766\) 56.3528 2.03611
\(767\) 5.97428 0.215719
\(768\) −28.6935 −1.03539
\(769\) −18.1664 −0.655097 −0.327548 0.944834i \(-0.606222\pi\)
−0.327548 + 0.944834i \(0.606222\pi\)
\(770\) 0 0
\(771\) −9.59376 −0.345511
\(772\) 38.2257 1.37577
\(773\) 9.81626 0.353066 0.176533 0.984295i \(-0.443512\pi\)
0.176533 + 0.984295i \(0.443512\pi\)
\(774\) −9.54775 −0.343187
\(775\) 0 0
\(776\) −53.7720 −1.93030
\(777\) −15.5369 −0.557385
\(778\) −57.0343 −2.04478
\(779\) 46.7056 1.67340
\(780\) 0 0
\(781\) 4.43497 0.158696
\(782\) −0.846836 −0.0302828
\(783\) 4.79678 0.171423
\(784\) −0.636395 −0.0227284
\(785\) 0 0
\(786\) 32.9511 1.17533
\(787\) 29.7235 1.05953 0.529765 0.848145i \(-0.322280\pi\)
0.529765 + 0.848145i \(0.322280\pi\)
\(788\) 35.6896 1.27139
\(789\) −6.68810 −0.238103
\(790\) 0 0
\(791\) 25.0627 0.891128
\(792\) −9.74613 −0.346314
\(793\) 0.481970 0.0171152
\(794\) −41.6785 −1.47912
\(795\) 0 0
\(796\) −33.1644 −1.17548
\(797\) −32.5535 −1.15310 −0.576552 0.817061i \(-0.695602\pi\)
−0.576552 + 0.817061i \(0.695602\pi\)
\(798\) −48.1245 −1.70359
\(799\) −8.45882 −0.299252
\(800\) 0 0
\(801\) 13.4753 0.476125
\(802\) −21.1040 −0.745210
\(803\) −30.2578 −1.06778
\(804\) 2.27350 0.0801802
\(805\) 0 0
\(806\) 10.6820 0.376256
\(807\) −12.7174 −0.447672
\(808\) −35.2292 −1.23936
\(809\) −21.7347 −0.764153 −0.382077 0.924131i \(-0.624791\pi\)
−0.382077 + 0.924131i \(0.624791\pi\)
\(810\) 0 0
\(811\) −6.98018 −0.245107 −0.122554 0.992462i \(-0.539108\pi\)
−0.122554 + 0.992462i \(0.539108\pi\)
\(812\) −46.7721 −1.64138
\(813\) 14.0805 0.493823
\(814\) −33.2548 −1.16558
\(815\) 0 0
\(816\) 4.39114 0.153721
\(817\) −30.7386 −1.07541
\(818\) −0.816487 −0.0285478
\(819\) −1.32788 −0.0463998
\(820\) 0 0
\(821\) −24.0515 −0.839403 −0.419702 0.907662i \(-0.637865\pi\)
−0.419702 + 0.907662i \(0.637865\pi\)
\(822\) −19.0364 −0.663970
\(823\) 4.37135 0.152376 0.0761879 0.997093i \(-0.475725\pi\)
0.0761879 + 0.997093i \(0.475725\pi\)
\(824\) −63.8244 −2.22343
\(825\) 0 0
\(826\) −72.8677 −2.53539
\(827\) 35.9008 1.24839 0.624197 0.781267i \(-0.285426\pi\)
0.624197 + 0.781267i \(0.285426\pi\)
\(828\) −0.776906 −0.0269993
\(829\) 51.9960 1.80590 0.902948 0.429749i \(-0.141398\pi\)
0.902948 + 0.429749i \(0.141398\pi\)
\(830\) 0 0
\(831\) 25.7886 0.894596
\(832\) −5.36390 −0.185960
\(833\) −0.421463 −0.0146028
\(834\) −43.3032 −1.49947
\(835\) 0 0
\(836\) −67.1907 −2.32384
\(837\) 8.71600 0.301269
\(838\) −4.34719 −0.150171
\(839\) 43.5905 1.50491 0.752455 0.658644i \(-0.228869\pi\)
0.752455 + 0.658644i \(0.228869\pi\)
\(840\) 0 0
\(841\) −5.99093 −0.206584
\(842\) −21.3762 −0.736671
\(843\) −19.9704 −0.687817
\(844\) 11.2046 0.385677
\(845\) 0 0
\(846\) −11.8966 −0.409015
\(847\) 14.6093 0.501983
\(848\) −1.40136 −0.0481230
\(849\) −29.5986 −1.01582
\(850\) 0 0
\(851\) −1.23793 −0.0424357
\(852\) −7.17582 −0.245840
\(853\) −26.4502 −0.905638 −0.452819 0.891603i \(-0.649582\pi\)
−0.452819 + 0.891603i \(0.649582\pi\)
\(854\) −5.87853 −0.201159
\(855\) 0 0
\(856\) −4.20262 −0.143642
\(857\) −44.3271 −1.51419 −0.757093 0.653307i \(-0.773381\pi\)
−0.757093 + 0.653307i \(0.773381\pi\)
\(858\) −2.84214 −0.0970291
\(859\) −38.2891 −1.30641 −0.653204 0.757182i \(-0.726576\pi\)
−0.653204 + 0.757182i \(0.726576\pi\)
\(860\) 0 0
\(861\) −15.7185 −0.535684
\(862\) 50.7195 1.72751
\(863\) 1.76848 0.0601997 0.0300998 0.999547i \(-0.490417\pi\)
0.0300998 + 0.999547i \(0.490417\pi\)
\(864\) −2.22945 −0.0758475
\(865\) 0 0
\(866\) 82.7345 2.81143
\(867\) −14.0919 −0.478586
\(868\) −84.9874 −2.88466
\(869\) −3.30130 −0.111989
\(870\) 0 0
\(871\) 0.309610 0.0104907
\(872\) −31.3439 −1.06144
\(873\) −12.7949 −0.433042
\(874\) −3.83440 −0.129700
\(875\) 0 0
\(876\) 48.9575 1.65412
\(877\) −37.2241 −1.25697 −0.628484 0.777822i \(-0.716324\pi\)
−0.628484 + 0.777822i \(0.716324\pi\)
\(878\) −32.3159 −1.09061
\(879\) 6.23225 0.210209
\(880\) 0 0
\(881\) 34.4231 1.15974 0.579872 0.814708i \(-0.303103\pi\)
0.579872 + 0.814708i \(0.303103\pi\)
\(882\) −0.592753 −0.0199590
\(883\) −12.4548 −0.419139 −0.209569 0.977794i \(-0.567206\pi\)
−0.209569 + 0.977794i \(0.567206\pi\)
\(884\) 3.26974 0.109973
\(885\) 0 0
\(886\) −84.7783 −2.84818
\(887\) 48.0530 1.61346 0.806731 0.590918i \(-0.201235\pi\)
0.806731 + 0.590918i \(0.201235\pi\)
\(888\) 25.1270 0.843209
\(889\) −49.2041 −1.65025
\(890\) 0 0
\(891\) −2.31906 −0.0776915
\(892\) −47.4009 −1.58710
\(893\) −38.3008 −1.28169
\(894\) −17.4801 −0.584621
\(895\) 0 0
\(896\) 53.8359 1.79853
\(897\) −0.105801 −0.00353258
\(898\) 23.1153 0.771367
\(899\) 41.8087 1.39440
\(900\) 0 0
\(901\) −0.928076 −0.0309187
\(902\) −33.6433 −1.12020
\(903\) 10.3449 0.344256
\(904\) −40.5326 −1.34809
\(905\) 0 0
\(906\) 28.1617 0.935609
\(907\) −28.7360 −0.954164 −0.477082 0.878859i \(-0.658306\pi\)
−0.477082 + 0.878859i \(0.658306\pi\)
\(908\) 110.496 3.66694
\(909\) −8.38268 −0.278036
\(910\) 0 0
\(911\) 13.1645 0.436160 0.218080 0.975931i \(-0.430021\pi\)
0.218080 + 0.975931i \(0.430021\pi\)
\(912\) 19.8827 0.658382
\(913\) −29.1013 −0.963111
\(914\) −28.3066 −0.936300
\(915\) 0 0
\(916\) −94.9326 −3.13666
\(917\) −35.7021 −1.17899
\(918\) 4.09001 0.134991
\(919\) 25.2878 0.834167 0.417084 0.908868i \(-0.363052\pi\)
0.417084 + 0.908868i \(0.363052\pi\)
\(920\) 0 0
\(921\) 27.6374 0.910682
\(922\) 24.7356 0.814625
\(923\) −0.977219 −0.0321656
\(924\) 22.6126 0.743899
\(925\) 0 0
\(926\) 12.7702 0.419654
\(927\) −15.1868 −0.498801
\(928\) −10.6942 −0.351054
\(929\) 22.5813 0.740869 0.370435 0.928859i \(-0.379209\pi\)
0.370435 + 0.928859i \(0.379209\pi\)
\(930\) 0 0
\(931\) −1.90835 −0.0625436
\(932\) −32.9719 −1.08003
\(933\) −14.7005 −0.481273
\(934\) −11.9937 −0.392445
\(935\) 0 0
\(936\) 2.14750 0.0701933
\(937\) −0.720844 −0.0235489 −0.0117745 0.999931i \(-0.503748\pi\)
−0.0117745 + 0.999931i \(0.503748\pi\)
\(938\) −3.77628 −0.123300
\(939\) −12.6308 −0.412189
\(940\) 0 0
\(941\) 10.4512 0.340699 0.170350 0.985384i \(-0.445510\pi\)
0.170350 + 0.985384i \(0.445510\pi\)
\(942\) 22.9320 0.747164
\(943\) −1.25239 −0.0407835
\(944\) 30.1054 0.979847
\(945\) 0 0
\(946\) 22.1418 0.719893
\(947\) 51.3114 1.66740 0.833698 0.552220i \(-0.186219\pi\)
0.833698 + 0.552220i \(0.186219\pi\)
\(948\) 5.34154 0.173485
\(949\) 6.66714 0.216424
\(950\) 0 0
\(951\) −3.42979 −0.111219
\(952\) −18.6238 −0.603601
\(953\) −41.2436 −1.33601 −0.668006 0.744156i \(-0.732852\pi\)
−0.668006 + 0.744156i \(0.732852\pi\)
\(954\) −1.30526 −0.0422595
\(955\) 0 0
\(956\) −57.7319 −1.86718
\(957\) −11.1240 −0.359589
\(958\) −61.2835 −1.97998
\(959\) 20.6257 0.666038
\(960\) 0 0
\(961\) 44.9687 1.45060
\(962\) 7.32749 0.236248
\(963\) −1.00000 −0.0322245
\(964\) 10.3031 0.331840
\(965\) 0 0
\(966\) 1.29044 0.0415192
\(967\) −27.7604 −0.892716 −0.446358 0.894855i \(-0.647279\pi\)
−0.446358 + 0.894855i \(0.647279\pi\)
\(968\) −23.6269 −0.759397
\(969\) 13.1676 0.423006
\(970\) 0 0
\(971\) 52.2741 1.67756 0.838778 0.544474i \(-0.183271\pi\)
0.838778 + 0.544474i \(0.183271\pi\)
\(972\) 3.75227 0.120354
\(973\) 46.9185 1.50414
\(974\) 42.5011 1.36182
\(975\) 0 0
\(976\) 2.42872 0.0777416
\(977\) −45.2329 −1.44713 −0.723565 0.690256i \(-0.757498\pi\)
−0.723565 + 0.690256i \(0.757498\pi\)
\(978\) −27.0477 −0.864890
\(979\) −31.2500 −0.998754
\(980\) 0 0
\(981\) −7.45820 −0.238122
\(982\) −17.1408 −0.546984
\(983\) −14.3566 −0.457906 −0.228953 0.973437i \(-0.573530\pi\)
−0.228953 + 0.973437i \(0.573530\pi\)
\(984\) 25.4206 0.810379
\(985\) 0 0
\(986\) 19.6189 0.624792
\(987\) 12.8899 0.410289
\(988\) 14.8051 0.471012
\(989\) 0.824245 0.0262095
\(990\) 0 0
\(991\) 30.4840 0.968357 0.484179 0.874969i \(-0.339118\pi\)
0.484179 + 0.874969i \(0.339118\pi\)
\(992\) −19.4319 −0.616963
\(993\) 2.45780 0.0779960
\(994\) 11.9190 0.378049
\(995\) 0 0
\(996\) 47.0862 1.49198
\(997\) −7.85984 −0.248924 −0.124462 0.992224i \(-0.539720\pi\)
−0.124462 + 0.992224i \(0.539720\pi\)
\(998\) 36.1611 1.14466
\(999\) 5.97891 0.189164
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 8025.2.a.bb.1.6 7
5.4 even 2 321.2.a.d.1.2 7
15.14 odd 2 963.2.a.e.1.6 7
20.19 odd 2 5136.2.a.bi.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
321.2.a.d.1.2 7 5.4 even 2
963.2.a.e.1.6 7 15.14 odd 2
5136.2.a.bi.1.6 7 20.19 odd 2
8025.2.a.bb.1.6 7 1.1 even 1 trivial