Properties

Label 8001.2.a.p.1.13
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $14$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, 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("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} - 875 x^{5} + 1134 x^{4} + 301 x^{3} - 418 x^{2} - 42 x + 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(2.66571\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.66571 q^{2} +5.10602 q^{4} -0.697799 q^{5} -1.00000 q^{7} +8.27977 q^{8} +O(q^{10})\) \(q+2.66571 q^{2} +5.10602 q^{4} -0.697799 q^{5} -1.00000 q^{7} +8.27977 q^{8} -1.86013 q^{10} +1.70540 q^{11} -0.0102354 q^{13} -2.66571 q^{14} +11.8594 q^{16} +3.99137 q^{17} +3.49924 q^{19} -3.56298 q^{20} +4.54610 q^{22} +0.546914 q^{23} -4.51308 q^{25} -0.0272847 q^{26} -5.10602 q^{28} -9.46348 q^{29} +7.67461 q^{31} +15.0543 q^{32} +10.6398 q^{34} +0.697799 q^{35} +7.99554 q^{37} +9.32796 q^{38} -5.77761 q^{40} +11.1674 q^{41} -9.72337 q^{43} +8.70781 q^{44} +1.45792 q^{46} -1.31107 q^{47} +1.00000 q^{49} -12.0306 q^{50} -0.0522622 q^{52} +11.1980 q^{53} -1.19002 q^{55} -8.27977 q^{56} -25.2269 q^{58} -6.94604 q^{59} -5.17872 q^{61} +20.4583 q^{62} +16.4116 q^{64} +0.00714225 q^{65} +15.8318 q^{67} +20.3800 q^{68} +1.86013 q^{70} +1.12772 q^{71} +11.3097 q^{73} +21.3138 q^{74} +17.8672 q^{76} -1.70540 q^{77} -11.4520 q^{79} -8.27549 q^{80} +29.7692 q^{82} +12.8969 q^{83} -2.78517 q^{85} -25.9197 q^{86} +14.1203 q^{88} +6.13283 q^{89} +0.0102354 q^{91} +2.79256 q^{92} -3.49493 q^{94} -2.44176 q^{95} -6.18095 q^{97} +2.66571 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.66571 1.88494 0.942472 0.334286i \(-0.108495\pi\)
0.942472 + 0.334286i \(0.108495\pi\)
\(3\) 0 0
\(4\) 5.10602 2.55301
\(5\) −0.697799 −0.312065 −0.156032 0.987752i \(-0.549870\pi\)
−0.156032 + 0.987752i \(0.549870\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 8.27977 2.92734
\(9\) 0 0
\(10\) −1.86013 −0.588225
\(11\) 1.70540 0.514197 0.257099 0.966385i \(-0.417233\pi\)
0.257099 + 0.966385i \(0.417233\pi\)
\(12\) 0 0
\(13\) −0.0102354 −0.00283879 −0.00141940 0.999999i \(-0.500452\pi\)
−0.00141940 + 0.999999i \(0.500452\pi\)
\(14\) −2.66571 −0.712442
\(15\) 0 0
\(16\) 11.8594 2.96486
\(17\) 3.99137 0.968049 0.484025 0.875054i \(-0.339174\pi\)
0.484025 + 0.875054i \(0.339174\pi\)
\(18\) 0 0
\(19\) 3.49924 0.802780 0.401390 0.915907i \(-0.368527\pi\)
0.401390 + 0.915907i \(0.368527\pi\)
\(20\) −3.56298 −0.796706
\(21\) 0 0
\(22\) 4.54610 0.969233
\(23\) 0.546914 0.114040 0.0570198 0.998373i \(-0.481840\pi\)
0.0570198 + 0.998373i \(0.481840\pi\)
\(24\) 0 0
\(25\) −4.51308 −0.902615
\(26\) −0.0272847 −0.00535096
\(27\) 0 0
\(28\) −5.10602 −0.964948
\(29\) −9.46348 −1.75732 −0.878662 0.477444i \(-0.841564\pi\)
−0.878662 + 0.477444i \(0.841564\pi\)
\(30\) 0 0
\(31\) 7.67461 1.37840 0.689200 0.724571i \(-0.257962\pi\)
0.689200 + 0.724571i \(0.257962\pi\)
\(32\) 15.0543 2.66125
\(33\) 0 0
\(34\) 10.6398 1.82472
\(35\) 0.697799 0.117949
\(36\) 0 0
\(37\) 7.99554 1.31446 0.657229 0.753691i \(-0.271728\pi\)
0.657229 + 0.753691i \(0.271728\pi\)
\(38\) 9.32796 1.51319
\(39\) 0 0
\(40\) −5.77761 −0.913520
\(41\) 11.1674 1.74406 0.872030 0.489453i \(-0.162803\pi\)
0.872030 + 0.489453i \(0.162803\pi\)
\(42\) 0 0
\(43\) −9.72337 −1.48280 −0.741400 0.671064i \(-0.765838\pi\)
−0.741400 + 0.671064i \(0.765838\pi\)
\(44\) 8.70781 1.31275
\(45\) 0 0
\(46\) 1.45792 0.214958
\(47\) −1.31107 −0.191239 −0.0956195 0.995418i \(-0.530483\pi\)
−0.0956195 + 0.995418i \(0.530483\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −12.0306 −1.70138
\(51\) 0 0
\(52\) −0.0522622 −0.00724747
\(53\) 11.1980 1.53816 0.769080 0.639152i \(-0.220715\pi\)
0.769080 + 0.639152i \(0.220715\pi\)
\(54\) 0 0
\(55\) −1.19002 −0.160463
\(56\) −8.27977 −1.10643
\(57\) 0 0
\(58\) −25.2269 −3.31246
\(59\) −6.94604 −0.904297 −0.452149 0.891943i \(-0.649342\pi\)
−0.452149 + 0.891943i \(0.649342\pi\)
\(60\) 0 0
\(61\) −5.17872 −0.663067 −0.331534 0.943443i \(-0.607566\pi\)
−0.331534 + 0.943443i \(0.607566\pi\)
\(62\) 20.4583 2.59821
\(63\) 0 0
\(64\) 16.4116 2.05145
\(65\) 0.00714225 0.000885887 0
\(66\) 0 0
\(67\) 15.8318 1.93416 0.967082 0.254464i \(-0.0818989\pi\)
0.967082 + 0.254464i \(0.0818989\pi\)
\(68\) 20.3800 2.47144
\(69\) 0 0
\(70\) 1.86013 0.222328
\(71\) 1.12772 0.133835 0.0669176 0.997759i \(-0.478684\pi\)
0.0669176 + 0.997759i \(0.478684\pi\)
\(72\) 0 0
\(73\) 11.3097 1.32370 0.661852 0.749634i \(-0.269771\pi\)
0.661852 + 0.749634i \(0.269771\pi\)
\(74\) 21.3138 2.47768
\(75\) 0 0
\(76\) 17.8672 2.04951
\(77\) −1.70540 −0.194348
\(78\) 0 0
\(79\) −11.4520 −1.28845 −0.644225 0.764836i \(-0.722820\pi\)
−0.644225 + 0.764836i \(0.722820\pi\)
\(80\) −8.27549 −0.925228
\(81\) 0 0
\(82\) 29.7692 3.28745
\(83\) 12.8969 1.41561 0.707807 0.706406i \(-0.249685\pi\)
0.707807 + 0.706406i \(0.249685\pi\)
\(84\) 0 0
\(85\) −2.78517 −0.302094
\(86\) −25.9197 −2.79499
\(87\) 0 0
\(88\) 14.1203 1.50523
\(89\) 6.13283 0.650079 0.325039 0.945700i \(-0.394622\pi\)
0.325039 + 0.945700i \(0.394622\pi\)
\(90\) 0 0
\(91\) 0.0102354 0.00107296
\(92\) 2.79256 0.291144
\(93\) 0 0
\(94\) −3.49493 −0.360475
\(95\) −2.44176 −0.250519
\(96\) 0 0
\(97\) −6.18095 −0.627581 −0.313790 0.949492i \(-0.601599\pi\)
−0.313790 + 0.949492i \(0.601599\pi\)
\(98\) 2.66571 0.269278
\(99\) 0 0
\(100\) −23.0439 −2.30439
\(101\) −7.01657 −0.698175 −0.349087 0.937090i \(-0.613508\pi\)
−0.349087 + 0.937090i \(0.613508\pi\)
\(102\) 0 0
\(103\) −3.28644 −0.323822 −0.161911 0.986805i \(-0.551766\pi\)
−0.161911 + 0.986805i \(0.551766\pi\)
\(104\) −0.0847468 −0.00831011
\(105\) 0 0
\(106\) 29.8506 2.89935
\(107\) 11.2210 1.08478 0.542389 0.840127i \(-0.317520\pi\)
0.542389 + 0.840127i \(0.317520\pi\)
\(108\) 0 0
\(109\) −0.788467 −0.0755215 −0.0377607 0.999287i \(-0.512022\pi\)
−0.0377607 + 0.999287i \(0.512022\pi\)
\(110\) −3.17226 −0.302464
\(111\) 0 0
\(112\) −11.8594 −1.12061
\(113\) −16.5540 −1.55727 −0.778636 0.627476i \(-0.784088\pi\)
−0.778636 + 0.627476i \(0.784088\pi\)
\(114\) 0 0
\(115\) −0.381636 −0.0355877
\(116\) −48.3208 −4.48647
\(117\) 0 0
\(118\) −18.5161 −1.70455
\(119\) −3.99137 −0.365888
\(120\) 0 0
\(121\) −8.09161 −0.735601
\(122\) −13.8050 −1.24984
\(123\) 0 0
\(124\) 39.1867 3.51907
\(125\) 6.63821 0.593740
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 13.6399 1.20561
\(129\) 0 0
\(130\) 0.0190392 0.00166985
\(131\) −8.91085 −0.778544 −0.389272 0.921123i \(-0.627273\pi\)
−0.389272 + 0.921123i \(0.627273\pi\)
\(132\) 0 0
\(133\) −3.49924 −0.303422
\(134\) 42.2031 3.64579
\(135\) 0 0
\(136\) 33.0476 2.83381
\(137\) 6.18977 0.528827 0.264414 0.964409i \(-0.414822\pi\)
0.264414 + 0.964409i \(0.414822\pi\)
\(138\) 0 0
\(139\) 6.96079 0.590407 0.295203 0.955434i \(-0.404613\pi\)
0.295203 + 0.955434i \(0.404613\pi\)
\(140\) 3.56298 0.301126
\(141\) 0 0
\(142\) 3.00617 0.252272
\(143\) −0.0174555 −0.00145970
\(144\) 0 0
\(145\) 6.60361 0.548400
\(146\) 30.1485 2.49511
\(147\) 0 0
\(148\) 40.8254 3.35583
\(149\) −5.97383 −0.489395 −0.244698 0.969599i \(-0.578689\pi\)
−0.244698 + 0.969599i \(0.578689\pi\)
\(150\) 0 0
\(151\) 8.78433 0.714859 0.357429 0.933940i \(-0.383653\pi\)
0.357429 + 0.933940i \(0.383653\pi\)
\(152\) 28.9729 2.35001
\(153\) 0 0
\(154\) −4.54610 −0.366335
\(155\) −5.35533 −0.430150
\(156\) 0 0
\(157\) 24.8843 1.98598 0.992992 0.118184i \(-0.0377073\pi\)
0.992992 + 0.118184i \(0.0377073\pi\)
\(158\) −30.5277 −2.42866
\(159\) 0 0
\(160\) −10.5049 −0.830483
\(161\) −0.546914 −0.0431029
\(162\) 0 0
\(163\) −4.07138 −0.318895 −0.159448 0.987206i \(-0.550971\pi\)
−0.159448 + 0.987206i \(0.550971\pi\)
\(164\) 57.0212 4.45261
\(165\) 0 0
\(166\) 34.3793 2.66835
\(167\) 7.18537 0.556021 0.278010 0.960578i \(-0.410325\pi\)
0.278010 + 0.960578i \(0.410325\pi\)
\(168\) 0 0
\(169\) −12.9999 −0.999992
\(170\) −7.42447 −0.569431
\(171\) 0 0
\(172\) −49.6477 −3.78561
\(173\) 0.291915 0.0221939 0.0110969 0.999938i \(-0.496468\pi\)
0.0110969 + 0.999938i \(0.496468\pi\)
\(174\) 0 0
\(175\) 4.51308 0.341157
\(176\) 20.2251 1.52452
\(177\) 0 0
\(178\) 16.3484 1.22536
\(179\) −18.3836 −1.37405 −0.687026 0.726633i \(-0.741084\pi\)
−0.687026 + 0.726633i \(0.741084\pi\)
\(180\) 0 0
\(181\) −14.1403 −1.05104 −0.525520 0.850782i \(-0.676129\pi\)
−0.525520 + 0.850782i \(0.676129\pi\)
\(182\) 0.0272847 0.00202247
\(183\) 0 0
\(184\) 4.52832 0.333832
\(185\) −5.57928 −0.410197
\(186\) 0 0
\(187\) 6.80688 0.497768
\(188\) −6.69435 −0.488235
\(189\) 0 0
\(190\) −6.50904 −0.472215
\(191\) −2.42758 −0.175654 −0.0878269 0.996136i \(-0.527992\pi\)
−0.0878269 + 0.996136i \(0.527992\pi\)
\(192\) 0 0
\(193\) −19.3314 −1.39150 −0.695752 0.718282i \(-0.744929\pi\)
−0.695752 + 0.718282i \(0.744929\pi\)
\(194\) −16.4766 −1.18295
\(195\) 0 0
\(196\) 5.10602 0.364716
\(197\) 12.2529 0.872981 0.436491 0.899709i \(-0.356221\pi\)
0.436491 + 0.899709i \(0.356221\pi\)
\(198\) 0 0
\(199\) 21.9153 1.55353 0.776767 0.629788i \(-0.216858\pi\)
0.776767 + 0.629788i \(0.216858\pi\)
\(200\) −37.3672 −2.64226
\(201\) 0 0
\(202\) −18.7042 −1.31602
\(203\) 9.46348 0.664206
\(204\) 0 0
\(205\) −7.79262 −0.544260
\(206\) −8.76070 −0.610387
\(207\) 0 0
\(208\) −0.121386 −0.00841661
\(209\) 5.96759 0.412787
\(210\) 0 0
\(211\) −8.68672 −0.598019 −0.299009 0.954250i \(-0.596656\pi\)
−0.299009 + 0.954250i \(0.596656\pi\)
\(212\) 57.1772 3.92694
\(213\) 0 0
\(214\) 29.9121 2.04475
\(215\) 6.78495 0.462730
\(216\) 0 0
\(217\) −7.67461 −0.520986
\(218\) −2.10183 −0.142354
\(219\) 0 0
\(220\) −6.07630 −0.409664
\(221\) −0.0408533 −0.00274809
\(222\) 0 0
\(223\) −11.8361 −0.792602 −0.396301 0.918121i \(-0.629706\pi\)
−0.396301 + 0.918121i \(0.629706\pi\)
\(224\) −15.0543 −1.00586
\(225\) 0 0
\(226\) −44.1283 −2.93537
\(227\) −10.1957 −0.676711 −0.338356 0.941018i \(-0.609871\pi\)
−0.338356 + 0.941018i \(0.609871\pi\)
\(228\) 0 0
\(229\) −24.2567 −1.60293 −0.801465 0.598042i \(-0.795946\pi\)
−0.801465 + 0.598042i \(0.795946\pi\)
\(230\) −1.01733 −0.0670809
\(231\) 0 0
\(232\) −78.3554 −5.14429
\(233\) 23.1487 1.51652 0.758260 0.651953i \(-0.226050\pi\)
0.758260 + 0.651953i \(0.226050\pi\)
\(234\) 0 0
\(235\) 0.914861 0.0596790
\(236\) −35.4666 −2.30868
\(237\) 0 0
\(238\) −10.6398 −0.689679
\(239\) 13.0898 0.846711 0.423355 0.905964i \(-0.360852\pi\)
0.423355 + 0.905964i \(0.360852\pi\)
\(240\) 0 0
\(241\) −28.2424 −1.81925 −0.909627 0.415427i \(-0.863632\pi\)
−0.909627 + 0.415427i \(0.863632\pi\)
\(242\) −21.5699 −1.38657
\(243\) 0 0
\(244\) −26.4427 −1.69282
\(245\) −0.697799 −0.0445807
\(246\) 0 0
\(247\) −0.0358161 −0.00227892
\(248\) 63.5439 4.03504
\(249\) 0 0
\(250\) 17.6956 1.11917
\(251\) 19.8373 1.25212 0.626061 0.779774i \(-0.284666\pi\)
0.626061 + 0.779774i \(0.284666\pi\)
\(252\) 0 0
\(253\) 0.932707 0.0586388
\(254\) −2.66571 −0.167262
\(255\) 0 0
\(256\) 3.53703 0.221064
\(257\) 28.3219 1.76667 0.883335 0.468743i \(-0.155293\pi\)
0.883335 + 0.468743i \(0.155293\pi\)
\(258\) 0 0
\(259\) −7.99554 −0.496819
\(260\) 0.0364685 0.00226168
\(261\) 0 0
\(262\) −23.7538 −1.46751
\(263\) 6.43901 0.397046 0.198523 0.980096i \(-0.436386\pi\)
0.198523 + 0.980096i \(0.436386\pi\)
\(264\) 0 0
\(265\) −7.81393 −0.480006
\(266\) −9.32796 −0.571934
\(267\) 0 0
\(268\) 80.8377 4.93795
\(269\) 27.2248 1.65992 0.829962 0.557819i \(-0.188362\pi\)
0.829962 + 0.557819i \(0.188362\pi\)
\(270\) 0 0
\(271\) 5.70599 0.346614 0.173307 0.984868i \(-0.444555\pi\)
0.173307 + 0.984868i \(0.444555\pi\)
\(272\) 47.3354 2.87013
\(273\) 0 0
\(274\) 16.5001 0.996810
\(275\) −7.69660 −0.464122
\(276\) 0 0
\(277\) 0.494362 0.0297033 0.0148517 0.999890i \(-0.495272\pi\)
0.0148517 + 0.999890i \(0.495272\pi\)
\(278\) 18.5555 1.11288
\(279\) 0 0
\(280\) 5.77761 0.345278
\(281\) −13.8067 −0.823637 −0.411819 0.911266i \(-0.635106\pi\)
−0.411819 + 0.911266i \(0.635106\pi\)
\(282\) 0 0
\(283\) −25.1913 −1.49746 −0.748732 0.662873i \(-0.769337\pi\)
−0.748732 + 0.662873i \(0.769337\pi\)
\(284\) 5.75814 0.341683
\(285\) 0 0
\(286\) −0.0465312 −0.00275145
\(287\) −11.1674 −0.659193
\(288\) 0 0
\(289\) −1.06897 −0.0628804
\(290\) 17.6033 1.03370
\(291\) 0 0
\(292\) 57.7478 3.37943
\(293\) −2.14027 −0.125036 −0.0625178 0.998044i \(-0.519913\pi\)
−0.0625178 + 0.998044i \(0.519913\pi\)
\(294\) 0 0
\(295\) 4.84693 0.282199
\(296\) 66.2012 3.84787
\(297\) 0 0
\(298\) −15.9245 −0.922482
\(299\) −0.00559789 −0.000323734 0
\(300\) 0 0
\(301\) 9.72337 0.560446
\(302\) 23.4165 1.34747
\(303\) 0 0
\(304\) 41.4990 2.38013
\(305\) 3.61370 0.206920
\(306\) 0 0
\(307\) −5.47832 −0.312664 −0.156332 0.987705i \(-0.549967\pi\)
−0.156332 + 0.987705i \(0.549967\pi\)
\(308\) −8.70781 −0.496173
\(309\) 0 0
\(310\) −14.2758 −0.810809
\(311\) −13.3059 −0.754511 −0.377256 0.926109i \(-0.623132\pi\)
−0.377256 + 0.926109i \(0.623132\pi\)
\(312\) 0 0
\(313\) −8.71810 −0.492776 −0.246388 0.969171i \(-0.579244\pi\)
−0.246388 + 0.969171i \(0.579244\pi\)
\(314\) 66.3344 3.74347
\(315\) 0 0
\(316\) −58.4742 −3.28943
\(317\) −14.1949 −0.797264 −0.398632 0.917111i \(-0.630515\pi\)
−0.398632 + 0.917111i \(0.630515\pi\)
\(318\) 0 0
\(319\) −16.1390 −0.903611
\(320\) −11.4520 −0.640185
\(321\) 0 0
\(322\) −1.45792 −0.0812465
\(323\) 13.9667 0.777131
\(324\) 0 0
\(325\) 0.0461932 0.00256234
\(326\) −10.8531 −0.601099
\(327\) 0 0
\(328\) 92.4637 5.10546
\(329\) 1.31107 0.0722815
\(330\) 0 0
\(331\) −30.6752 −1.68606 −0.843032 0.537864i \(-0.819231\pi\)
−0.843032 + 0.537864i \(0.819231\pi\)
\(332\) 65.8516 3.61408
\(333\) 0 0
\(334\) 19.1541 1.04807
\(335\) −11.0474 −0.603585
\(336\) 0 0
\(337\) 0.340026 0.0185224 0.00926119 0.999957i \(-0.497052\pi\)
0.00926119 + 0.999957i \(0.497052\pi\)
\(338\) −34.6540 −1.88493
\(339\) 0 0
\(340\) −14.2212 −0.771250
\(341\) 13.0883 0.708769
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −80.5072 −4.34066
\(345\) 0 0
\(346\) 0.778161 0.0418342
\(347\) 13.9628 0.749563 0.374782 0.927113i \(-0.377718\pi\)
0.374782 + 0.927113i \(0.377718\pi\)
\(348\) 0 0
\(349\) 14.5697 0.779898 0.389949 0.920836i \(-0.372493\pi\)
0.389949 + 0.920836i \(0.372493\pi\)
\(350\) 12.0306 0.643061
\(351\) 0 0
\(352\) 25.6736 1.36841
\(353\) 9.29100 0.494510 0.247255 0.968950i \(-0.420471\pi\)
0.247255 + 0.968950i \(0.420471\pi\)
\(354\) 0 0
\(355\) −0.786918 −0.0417653
\(356\) 31.3144 1.65966
\(357\) 0 0
\(358\) −49.0053 −2.59001
\(359\) −27.8926 −1.47211 −0.736056 0.676920i \(-0.763314\pi\)
−0.736056 + 0.676920i \(0.763314\pi\)
\(360\) 0 0
\(361\) −6.75535 −0.355544
\(362\) −37.6939 −1.98115
\(363\) 0 0
\(364\) 0.0522622 0.00273929
\(365\) −7.89192 −0.413082
\(366\) 0 0
\(367\) 29.8398 1.55763 0.778813 0.627256i \(-0.215822\pi\)
0.778813 + 0.627256i \(0.215822\pi\)
\(368\) 6.48609 0.338111
\(369\) 0 0
\(370\) −14.8727 −0.773197
\(371\) −11.1980 −0.581370
\(372\) 0 0
\(373\) 37.4048 1.93675 0.968375 0.249500i \(-0.0802664\pi\)
0.968375 + 0.249500i \(0.0802664\pi\)
\(374\) 18.1452 0.938265
\(375\) 0 0
\(376\) −10.8553 −0.559821
\(377\) 0.0968626 0.00498868
\(378\) 0 0
\(379\) −34.0365 −1.74834 −0.874170 0.485620i \(-0.838594\pi\)
−0.874170 + 0.485620i \(0.838594\pi\)
\(380\) −12.4677 −0.639579
\(381\) 0 0
\(382\) −6.47124 −0.331098
\(383\) −1.54300 −0.0788439 −0.0394219 0.999223i \(-0.512552\pi\)
−0.0394219 + 0.999223i \(0.512552\pi\)
\(384\) 0 0
\(385\) 1.19002 0.0606493
\(386\) −51.5319 −2.62291
\(387\) 0 0
\(388\) −31.5601 −1.60222
\(389\) 14.2392 0.721956 0.360978 0.932574i \(-0.382443\pi\)
0.360978 + 0.932574i \(0.382443\pi\)
\(390\) 0 0
\(391\) 2.18294 0.110396
\(392\) 8.27977 0.418191
\(393\) 0 0
\(394\) 32.6626 1.64552
\(395\) 7.99119 0.402080
\(396\) 0 0
\(397\) −4.10450 −0.205999 −0.102999 0.994681i \(-0.532844\pi\)
−0.102999 + 0.994681i \(0.532844\pi\)
\(398\) 58.4199 2.92832
\(399\) 0 0
\(400\) −53.5225 −2.67613
\(401\) 23.1443 1.15577 0.577885 0.816118i \(-0.303878\pi\)
0.577885 + 0.816118i \(0.303878\pi\)
\(402\) 0 0
\(403\) −0.0785527 −0.00391299
\(404\) −35.8268 −1.78245
\(405\) 0 0
\(406\) 25.2269 1.25199
\(407\) 13.6356 0.675891
\(408\) 0 0
\(409\) −22.7189 −1.12338 −0.561689 0.827348i \(-0.689848\pi\)
−0.561689 + 0.827348i \(0.689848\pi\)
\(410\) −20.7729 −1.02590
\(411\) 0 0
\(412\) −16.7806 −0.826723
\(413\) 6.94604 0.341792
\(414\) 0 0
\(415\) −8.99941 −0.441763
\(416\) −0.154087 −0.00755473
\(417\) 0 0
\(418\) 15.9079 0.778080
\(419\) −25.2172 −1.23194 −0.615972 0.787768i \(-0.711236\pi\)
−0.615972 + 0.787768i \(0.711236\pi\)
\(420\) 0 0
\(421\) 2.51746 0.122693 0.0613467 0.998117i \(-0.480460\pi\)
0.0613467 + 0.998117i \(0.480460\pi\)
\(422\) −23.1563 −1.12723
\(423\) 0 0
\(424\) 92.7167 4.50272
\(425\) −18.0134 −0.873776
\(426\) 0 0
\(427\) 5.17872 0.250616
\(428\) 57.2949 2.76945
\(429\) 0 0
\(430\) 18.0867 0.872220
\(431\) 0.00500045 0.000240863 0 0.000120432 1.00000i \(-0.499962\pi\)
0.000120432 1.00000i \(0.499962\pi\)
\(432\) 0 0
\(433\) −10.6977 −0.514098 −0.257049 0.966398i \(-0.582750\pi\)
−0.257049 + 0.966398i \(0.582750\pi\)
\(434\) −20.4583 −0.982029
\(435\) 0 0
\(436\) −4.02593 −0.192807
\(437\) 1.91378 0.0915486
\(438\) 0 0
\(439\) 19.0780 0.910546 0.455273 0.890352i \(-0.349542\pi\)
0.455273 + 0.890352i \(0.349542\pi\)
\(440\) −9.85313 −0.469729
\(441\) 0 0
\(442\) −0.108903 −0.00517999
\(443\) −24.9616 −1.18596 −0.592980 0.805217i \(-0.702049\pi\)
−0.592980 + 0.805217i \(0.702049\pi\)
\(444\) 0 0
\(445\) −4.27948 −0.202867
\(446\) −31.5516 −1.49401
\(447\) 0 0
\(448\) −16.4116 −0.775374
\(449\) 22.7503 1.07365 0.536827 0.843692i \(-0.319623\pi\)
0.536827 + 0.843692i \(0.319623\pi\)
\(450\) 0 0
\(451\) 19.0449 0.896791
\(452\) −84.5253 −3.97573
\(453\) 0 0
\(454\) −27.1788 −1.27556
\(455\) −0.00714225 −0.000334834 0
\(456\) 0 0
\(457\) −27.0127 −1.26360 −0.631801 0.775131i \(-0.717684\pi\)
−0.631801 + 0.775131i \(0.717684\pi\)
\(458\) −64.6615 −3.02143
\(459\) 0 0
\(460\) −1.94864 −0.0908559
\(461\) −25.5587 −1.19039 −0.595194 0.803582i \(-0.702925\pi\)
−0.595194 + 0.803582i \(0.702925\pi\)
\(462\) 0 0
\(463\) −37.3000 −1.73348 −0.866740 0.498761i \(-0.833789\pi\)
−0.866740 + 0.498761i \(0.833789\pi\)
\(464\) −112.232 −5.21022
\(465\) 0 0
\(466\) 61.7077 2.85855
\(467\) −8.52627 −0.394549 −0.197274 0.980348i \(-0.563209\pi\)
−0.197274 + 0.980348i \(0.563209\pi\)
\(468\) 0 0
\(469\) −15.8318 −0.731046
\(470\) 2.43876 0.112492
\(471\) 0 0
\(472\) −57.5116 −2.64718
\(473\) −16.5822 −0.762451
\(474\) 0 0
\(475\) −15.7923 −0.724602
\(476\) −20.3800 −0.934117
\(477\) 0 0
\(478\) 34.8937 1.59600
\(479\) 23.6098 1.07876 0.539380 0.842062i \(-0.318659\pi\)
0.539380 + 0.842062i \(0.318659\pi\)
\(480\) 0 0
\(481\) −0.0818376 −0.00373147
\(482\) −75.2861 −3.42919
\(483\) 0 0
\(484\) −41.3160 −1.87800
\(485\) 4.31306 0.195846
\(486\) 0 0
\(487\) −9.23519 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(488\) −42.8786 −1.94102
\(489\) 0 0
\(490\) −1.86013 −0.0840321
\(491\) −1.62814 −0.0734769 −0.0367385 0.999325i \(-0.511697\pi\)
−0.0367385 + 0.999325i \(0.511697\pi\)
\(492\) 0 0
\(493\) −37.7723 −1.70118
\(494\) −0.0954754 −0.00429564
\(495\) 0 0
\(496\) 91.0164 4.08676
\(497\) −1.12772 −0.0505850
\(498\) 0 0
\(499\) −15.9739 −0.715090 −0.357545 0.933896i \(-0.616386\pi\)
−0.357545 + 0.933896i \(0.616386\pi\)
\(500\) 33.8949 1.51582
\(501\) 0 0
\(502\) 52.8807 2.36018
\(503\) 0.578955 0.0258143 0.0129072 0.999917i \(-0.495891\pi\)
0.0129072 + 0.999917i \(0.495891\pi\)
\(504\) 0 0
\(505\) 4.89615 0.217876
\(506\) 2.48633 0.110531
\(507\) 0 0
\(508\) −5.10602 −0.226543
\(509\) 6.21944 0.275672 0.137836 0.990455i \(-0.455985\pi\)
0.137836 + 0.990455i \(0.455985\pi\)
\(510\) 0 0
\(511\) −11.3097 −0.500313
\(512\) −17.8512 −0.788919
\(513\) 0 0
\(514\) 75.4980 3.33007
\(515\) 2.29327 0.101054
\(516\) 0 0
\(517\) −2.23589 −0.0983345
\(518\) −21.3138 −0.936475
\(519\) 0 0
\(520\) 0.0591362 0.00259329
\(521\) −4.12288 −0.180627 −0.0903134 0.995913i \(-0.528787\pi\)
−0.0903134 + 0.995913i \(0.528787\pi\)
\(522\) 0 0
\(523\) −18.3058 −0.800455 −0.400228 0.916416i \(-0.631069\pi\)
−0.400228 + 0.916416i \(0.631069\pi\)
\(524\) −45.4990 −1.98763
\(525\) 0 0
\(526\) 17.1645 0.748409
\(527\) 30.6322 1.33436
\(528\) 0 0
\(529\) −22.7009 −0.986995
\(530\) −20.8297 −0.904784
\(531\) 0 0
\(532\) −17.8672 −0.774641
\(533\) −0.114303 −0.00495102
\(534\) 0 0
\(535\) −7.83002 −0.338521
\(536\) 131.084 5.66196
\(537\) 0 0
\(538\) 72.5735 3.12886
\(539\) 1.70540 0.0734567
\(540\) 0 0
\(541\) −4.80919 −0.206763 −0.103382 0.994642i \(-0.532966\pi\)
−0.103382 + 0.994642i \(0.532966\pi\)
\(542\) 15.2105 0.653349
\(543\) 0 0
\(544\) 60.0873 2.57622
\(545\) 0.550191 0.0235676
\(546\) 0 0
\(547\) −24.1167 −1.03115 −0.515577 0.856843i \(-0.672422\pi\)
−0.515577 + 0.856843i \(0.672422\pi\)
\(548\) 31.6051 1.35010
\(549\) 0 0
\(550\) −20.5169 −0.874844
\(551\) −33.1150 −1.41075
\(552\) 0 0
\(553\) 11.4520 0.486988
\(554\) 1.31783 0.0559891
\(555\) 0 0
\(556\) 35.5420 1.50732
\(557\) −0.973727 −0.0412581 −0.0206291 0.999787i \(-0.506567\pi\)
−0.0206291 + 0.999787i \(0.506567\pi\)
\(558\) 0 0
\(559\) 0.0995226 0.00420936
\(560\) 8.27549 0.349703
\(561\) 0 0
\(562\) −36.8046 −1.55251
\(563\) −36.6265 −1.54362 −0.771811 0.635853i \(-0.780649\pi\)
−0.771811 + 0.635853i \(0.780649\pi\)
\(564\) 0 0
\(565\) 11.5514 0.485970
\(566\) −67.1526 −2.82264
\(567\) 0 0
\(568\) 9.33722 0.391781
\(569\) 31.6724 1.32777 0.663887 0.747833i \(-0.268906\pi\)
0.663887 + 0.747833i \(0.268906\pi\)
\(570\) 0 0
\(571\) −9.40096 −0.393418 −0.196709 0.980462i \(-0.563025\pi\)
−0.196709 + 0.980462i \(0.563025\pi\)
\(572\) −0.0891279 −0.00372663
\(573\) 0 0
\(574\) −29.7692 −1.24254
\(575\) −2.46827 −0.102934
\(576\) 0 0
\(577\) −36.4828 −1.51880 −0.759399 0.650626i \(-0.774507\pi\)
−0.759399 + 0.650626i \(0.774507\pi\)
\(578\) −2.84956 −0.118526
\(579\) 0 0
\(580\) 33.7182 1.40007
\(581\) −12.8969 −0.535052
\(582\) 0 0
\(583\) 19.0970 0.790918
\(584\) 93.6420 3.87493
\(585\) 0 0
\(586\) −5.70533 −0.235685
\(587\) −4.16140 −0.171759 −0.0858797 0.996306i \(-0.527370\pi\)
−0.0858797 + 0.996306i \(0.527370\pi\)
\(588\) 0 0
\(589\) 26.8553 1.10655
\(590\) 12.9205 0.531930
\(591\) 0 0
\(592\) 94.8226 3.89718
\(593\) −27.5376 −1.13084 −0.565418 0.824805i \(-0.691285\pi\)
−0.565418 + 0.824805i \(0.691285\pi\)
\(594\) 0 0
\(595\) 2.78517 0.114181
\(596\) −30.5025 −1.24943
\(597\) 0 0
\(598\) −0.0149224 −0.000610221 0
\(599\) −12.0678 −0.493077 −0.246538 0.969133i \(-0.579293\pi\)
−0.246538 + 0.969133i \(0.579293\pi\)
\(600\) 0 0
\(601\) −29.7149 −1.21209 −0.606047 0.795429i \(-0.707246\pi\)
−0.606047 + 0.795429i \(0.707246\pi\)
\(602\) 25.9197 1.05641
\(603\) 0 0
\(604\) 44.8530 1.82504
\(605\) 5.64632 0.229555
\(606\) 0 0
\(607\) 16.3965 0.665514 0.332757 0.943013i \(-0.392021\pi\)
0.332757 + 0.943013i \(0.392021\pi\)
\(608\) 52.6786 2.13640
\(609\) 0 0
\(610\) 9.63309 0.390033
\(611\) 0.0134193 0.000542887 0
\(612\) 0 0
\(613\) 8.61788 0.348073 0.174036 0.984739i \(-0.444319\pi\)
0.174036 + 0.984739i \(0.444319\pi\)
\(614\) −14.6036 −0.589355
\(615\) 0 0
\(616\) −14.1203 −0.568923
\(617\) 12.6133 0.507794 0.253897 0.967231i \(-0.418288\pi\)
0.253897 + 0.967231i \(0.418288\pi\)
\(618\) 0 0
\(619\) 44.2706 1.77938 0.889692 0.456561i \(-0.150919\pi\)
0.889692 + 0.456561i \(0.150919\pi\)
\(620\) −27.3444 −1.09818
\(621\) 0 0
\(622\) −35.4698 −1.42221
\(623\) −6.13283 −0.245707
\(624\) 0 0
\(625\) 17.9333 0.717330
\(626\) −23.2400 −0.928855
\(627\) 0 0
\(628\) 127.060 5.07024
\(629\) 31.9132 1.27246
\(630\) 0 0
\(631\) 25.6008 1.01915 0.509576 0.860425i \(-0.329802\pi\)
0.509576 + 0.860425i \(0.329802\pi\)
\(632\) −94.8198 −3.77173
\(633\) 0 0
\(634\) −37.8395 −1.50280
\(635\) 0.697799 0.0276913
\(636\) 0 0
\(637\) −0.0102354 −0.000405542 0
\(638\) −43.0220 −1.70326
\(639\) 0 0
\(640\) −9.51794 −0.376229
\(641\) −18.3429 −0.724500 −0.362250 0.932081i \(-0.617991\pi\)
−0.362250 + 0.932081i \(0.617991\pi\)
\(642\) 0 0
\(643\) −10.9655 −0.432437 −0.216219 0.976345i \(-0.569372\pi\)
−0.216219 + 0.976345i \(0.569372\pi\)
\(644\) −2.79256 −0.110042
\(645\) 0 0
\(646\) 37.2313 1.46485
\(647\) −34.8051 −1.36833 −0.684164 0.729328i \(-0.739833\pi\)
−0.684164 + 0.729328i \(0.739833\pi\)
\(648\) 0 0
\(649\) −11.8458 −0.464987
\(650\) 0.123138 0.00482986
\(651\) 0 0
\(652\) −20.7886 −0.814143
\(653\) −33.2092 −1.29958 −0.649789 0.760115i \(-0.725143\pi\)
−0.649789 + 0.760115i \(0.725143\pi\)
\(654\) 0 0
\(655\) 6.21797 0.242956
\(656\) 132.439 5.17089
\(657\) 0 0
\(658\) 3.49493 0.136247
\(659\) −5.32002 −0.207239 −0.103619 0.994617i \(-0.533042\pi\)
−0.103619 + 0.994617i \(0.533042\pi\)
\(660\) 0 0
\(661\) −13.8478 −0.538616 −0.269308 0.963054i \(-0.586795\pi\)
−0.269308 + 0.963054i \(0.586795\pi\)
\(662\) −81.7714 −3.17813
\(663\) 0 0
\(664\) 106.783 4.14398
\(665\) 2.44176 0.0946875
\(666\) 0 0
\(667\) −5.17572 −0.200405
\(668\) 36.6887 1.41953
\(669\) 0 0
\(670\) −29.4492 −1.13772
\(671\) −8.83178 −0.340947
\(672\) 0 0
\(673\) 10.0513 0.387451 0.193725 0.981056i \(-0.437943\pi\)
0.193725 + 0.981056i \(0.437943\pi\)
\(674\) 0.906410 0.0349136
\(675\) 0 0
\(676\) −66.3778 −2.55299
\(677\) −23.6492 −0.908912 −0.454456 0.890769i \(-0.650166\pi\)
−0.454456 + 0.890769i \(0.650166\pi\)
\(678\) 0 0
\(679\) 6.18095 0.237203
\(680\) −23.0606 −0.884333
\(681\) 0 0
\(682\) 34.8895 1.33599
\(683\) 37.4785 1.43407 0.717037 0.697035i \(-0.245498\pi\)
0.717037 + 0.697035i \(0.245498\pi\)
\(684\) 0 0
\(685\) −4.31921 −0.165029
\(686\) −2.66571 −0.101777
\(687\) 0 0
\(688\) −115.314 −4.39629
\(689\) −0.114616 −0.00436652
\(690\) 0 0
\(691\) 28.0034 1.06530 0.532650 0.846336i \(-0.321196\pi\)
0.532650 + 0.846336i \(0.321196\pi\)
\(692\) 1.49052 0.0566612
\(693\) 0 0
\(694\) 37.2209 1.41288
\(695\) −4.85723 −0.184245
\(696\) 0 0
\(697\) 44.5734 1.68834
\(698\) 38.8386 1.47006
\(699\) 0 0
\(700\) 23.0439 0.870977
\(701\) 23.3548 0.882099 0.441050 0.897483i \(-0.354606\pi\)
0.441050 + 0.897483i \(0.354606\pi\)
\(702\) 0 0
\(703\) 27.9783 1.05522
\(704\) 27.9883 1.05485
\(705\) 0 0
\(706\) 24.7671 0.932123
\(707\) 7.01657 0.263885
\(708\) 0 0
\(709\) −40.1473 −1.50776 −0.753881 0.657011i \(-0.771821\pi\)
−0.753881 + 0.657011i \(0.771821\pi\)
\(710\) −2.09770 −0.0787252
\(711\) 0 0
\(712\) 50.7784 1.90300
\(713\) 4.19735 0.157192
\(714\) 0 0
\(715\) 0.0121804 0.000455521 0
\(716\) −93.8669 −3.50797
\(717\) 0 0
\(718\) −74.3535 −2.77485
\(719\) −17.7423 −0.661677 −0.330839 0.943687i \(-0.607332\pi\)
−0.330839 + 0.943687i \(0.607332\pi\)
\(720\) 0 0
\(721\) 3.28644 0.122393
\(722\) −18.0078 −0.670181
\(723\) 0 0
\(724\) −72.2006 −2.68332
\(725\) 42.7094 1.58619
\(726\) 0 0
\(727\) −20.4263 −0.757568 −0.378784 0.925485i \(-0.623658\pi\)
−0.378784 + 0.925485i \(0.623658\pi\)
\(728\) 0.0847468 0.00314092
\(729\) 0 0
\(730\) −21.0376 −0.778636
\(731\) −38.8096 −1.43542
\(732\) 0 0
\(733\) −30.8007 −1.13765 −0.568825 0.822458i \(-0.692602\pi\)
−0.568825 + 0.822458i \(0.692602\pi\)
\(734\) 79.5444 2.93604
\(735\) 0 0
\(736\) 8.23341 0.303488
\(737\) 26.9996 0.994542
\(738\) 0 0
\(739\) −44.8782 −1.65087 −0.825435 0.564497i \(-0.809070\pi\)
−0.825435 + 0.564497i \(0.809070\pi\)
\(740\) −28.4879 −1.04724
\(741\) 0 0
\(742\) −29.8506 −1.09585
\(743\) 14.8611 0.545200 0.272600 0.962127i \(-0.412116\pi\)
0.272600 + 0.962127i \(0.412116\pi\)
\(744\) 0 0
\(745\) 4.16853 0.152723
\(746\) 99.7106 3.65066
\(747\) 0 0
\(748\) 34.7561 1.27081
\(749\) −11.2210 −0.410008
\(750\) 0 0
\(751\) −5.90762 −0.215572 −0.107786 0.994174i \(-0.534376\pi\)
−0.107786 + 0.994174i \(0.534376\pi\)
\(752\) −15.5485 −0.566996
\(753\) 0 0
\(754\) 0.258208 0.00940338
\(755\) −6.12969 −0.223082
\(756\) 0 0
\(757\) −51.5497 −1.87360 −0.936802 0.349860i \(-0.886229\pi\)
−0.936802 + 0.349860i \(0.886229\pi\)
\(758\) −90.7316 −3.29552
\(759\) 0 0
\(760\) −20.2172 −0.733356
\(761\) 8.05458 0.291978 0.145989 0.989286i \(-0.453364\pi\)
0.145989 + 0.989286i \(0.453364\pi\)
\(762\) 0 0
\(763\) 0.788467 0.0285444
\(764\) −12.3953 −0.448446
\(765\) 0 0
\(766\) −4.11321 −0.148616
\(767\) 0.0710955 0.00256711
\(768\) 0 0
\(769\) −10.9759 −0.395800 −0.197900 0.980222i \(-0.563412\pi\)
−0.197900 + 0.980222i \(0.563412\pi\)
\(770\) 3.17226 0.114320
\(771\) 0 0
\(772\) −98.7065 −3.55252
\(773\) 32.9115 1.18374 0.591871 0.806032i \(-0.298389\pi\)
0.591871 + 0.806032i \(0.298389\pi\)
\(774\) 0 0
\(775\) −34.6361 −1.24416
\(776\) −51.1769 −1.83714
\(777\) 0 0
\(778\) 37.9576 1.36085
\(779\) 39.0775 1.40010
\(780\) 0 0
\(781\) 1.92321 0.0688177
\(782\) 5.81908 0.208090
\(783\) 0 0
\(784\) 11.8594 0.423551
\(785\) −17.3642 −0.619756
\(786\) 0 0
\(787\) −36.9996 −1.31889 −0.659446 0.751752i \(-0.729209\pi\)
−0.659446 + 0.751752i \(0.729209\pi\)
\(788\) 62.5635 2.22873
\(789\) 0 0
\(790\) 21.3022 0.757898
\(791\) 16.5540 0.588593
\(792\) 0 0
\(793\) 0.0530063 0.00188231
\(794\) −10.9414 −0.388296
\(795\) 0 0
\(796\) 111.900 3.96619
\(797\) 0.129949 0.00460303 0.00230151 0.999997i \(-0.499267\pi\)
0.00230151 + 0.999997i \(0.499267\pi\)
\(798\) 0 0
\(799\) −5.23296 −0.185129
\(800\) −67.9412 −2.40208
\(801\) 0 0
\(802\) 61.6960 2.17856
\(803\) 19.2876 0.680645
\(804\) 0 0
\(805\) 0.381636 0.0134509
\(806\) −0.209399 −0.00737576
\(807\) 0 0
\(808\) −58.0956 −2.04379
\(809\) 29.4117 1.03406 0.517031 0.855967i \(-0.327037\pi\)
0.517031 + 0.855967i \(0.327037\pi\)
\(810\) 0 0
\(811\) −22.5962 −0.793458 −0.396729 0.917936i \(-0.629855\pi\)
−0.396729 + 0.917936i \(0.629855\pi\)
\(812\) 48.3208 1.69573
\(813\) 0 0
\(814\) 36.3486 1.27402
\(815\) 2.84100 0.0995160
\(816\) 0 0
\(817\) −34.0244 −1.19036
\(818\) −60.5621 −2.11751
\(819\) 0 0
\(820\) −39.7893 −1.38950
\(821\) 12.7636 0.445452 0.222726 0.974881i \(-0.428504\pi\)
0.222726 + 0.974881i \(0.428504\pi\)
\(822\) 0 0
\(823\) 31.8621 1.11064 0.555321 0.831636i \(-0.312595\pi\)
0.555321 + 0.831636i \(0.312595\pi\)
\(824\) −27.2109 −0.947938
\(825\) 0 0
\(826\) 18.5161 0.644259
\(827\) 17.0949 0.594447 0.297223 0.954808i \(-0.403939\pi\)
0.297223 + 0.954808i \(0.403939\pi\)
\(828\) 0 0
\(829\) 51.7797 1.79838 0.899191 0.437556i \(-0.144156\pi\)
0.899191 + 0.437556i \(0.144156\pi\)
\(830\) −23.9898 −0.832699
\(831\) 0 0
\(832\) −0.167979 −0.00582363
\(833\) 3.99137 0.138293
\(834\) 0 0
\(835\) −5.01394 −0.173515
\(836\) 30.4707 1.05385
\(837\) 0 0
\(838\) −67.2219 −2.32214
\(839\) −7.47622 −0.258108 −0.129054 0.991638i \(-0.541194\pi\)
−0.129054 + 0.991638i \(0.541194\pi\)
\(840\) 0 0
\(841\) 60.5575 2.08819
\(842\) 6.71082 0.231270
\(843\) 0 0
\(844\) −44.3546 −1.52675
\(845\) 9.07131 0.312062
\(846\) 0 0
\(847\) 8.09161 0.278031
\(848\) 132.802 4.56043
\(849\) 0 0
\(850\) −48.0184 −1.64702
\(851\) 4.37288 0.149900
\(852\) 0 0
\(853\) −33.2445 −1.13827 −0.569134 0.822244i \(-0.692722\pi\)
−0.569134 + 0.822244i \(0.692722\pi\)
\(854\) 13.8050 0.472397
\(855\) 0 0
\(856\) 92.9076 3.17551
\(857\) −1.97724 −0.0675412 −0.0337706 0.999430i \(-0.510752\pi\)
−0.0337706 + 0.999430i \(0.510752\pi\)
\(858\) 0 0
\(859\) 35.6972 1.21797 0.608986 0.793181i \(-0.291577\pi\)
0.608986 + 0.793181i \(0.291577\pi\)
\(860\) 34.6441 1.18135
\(861\) 0 0
\(862\) 0.0133298 0.000454014 0
\(863\) −6.28198 −0.213841 −0.106920 0.994268i \(-0.534099\pi\)
−0.106920 + 0.994268i \(0.534099\pi\)
\(864\) 0 0
\(865\) −0.203698 −0.00692593
\(866\) −28.5170 −0.969046
\(867\) 0 0
\(868\) −39.1867 −1.33008
\(869\) −19.5302 −0.662517
\(870\) 0 0
\(871\) −0.162045 −0.00549069
\(872\) −6.52832 −0.221077
\(873\) 0 0
\(874\) 5.10159 0.172564
\(875\) −6.63821 −0.224412
\(876\) 0 0
\(877\) −33.5476 −1.13282 −0.566411 0.824123i \(-0.691668\pi\)
−0.566411 + 0.824123i \(0.691668\pi\)
\(878\) 50.8566 1.71633
\(879\) 0 0
\(880\) −14.1130 −0.475750
\(881\) −18.4440 −0.621393 −0.310696 0.950509i \(-0.600562\pi\)
−0.310696 + 0.950509i \(0.600562\pi\)
\(882\) 0 0
\(883\) −23.3811 −0.786836 −0.393418 0.919360i \(-0.628707\pi\)
−0.393418 + 0.919360i \(0.628707\pi\)
\(884\) −0.208598 −0.00701591
\(885\) 0 0
\(886\) −66.5403 −2.23547
\(887\) 10.3669 0.348086 0.174043 0.984738i \(-0.444317\pi\)
0.174043 + 0.984738i \(0.444317\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −11.4079 −0.382393
\(891\) 0 0
\(892\) −60.4353 −2.02352
\(893\) −4.58774 −0.153523
\(894\) 0 0
\(895\) 12.8280 0.428793
\(896\) −13.6399 −0.455679
\(897\) 0 0
\(898\) 60.6458 2.02378
\(899\) −72.6285 −2.42230
\(900\) 0 0
\(901\) 44.6953 1.48902
\(902\) 50.7683 1.69040
\(903\) 0 0
\(904\) −137.063 −4.55866
\(905\) 9.86707 0.327992
\(906\) 0 0
\(907\) 56.7034 1.88281 0.941403 0.337285i \(-0.109509\pi\)
0.941403 + 0.337285i \(0.109509\pi\)
\(908\) −52.0594 −1.72765
\(909\) 0 0
\(910\) −0.0190392 −0.000631143 0
\(911\) 6.97305 0.231028 0.115514 0.993306i \(-0.463149\pi\)
0.115514 + 0.993306i \(0.463149\pi\)
\(912\) 0 0
\(913\) 21.9943 0.727904
\(914\) −72.0082 −2.38182
\(915\) 0 0
\(916\) −123.855 −4.09230
\(917\) 8.91085 0.294262
\(918\) 0 0
\(919\) 44.1712 1.45707 0.728537 0.685007i \(-0.240201\pi\)
0.728537 + 0.685007i \(0.240201\pi\)
\(920\) −3.15986 −0.104177
\(921\) 0 0
\(922\) −68.1322 −2.24382
\(923\) −0.0115426 −0.000379930 0
\(924\) 0 0
\(925\) −36.0845 −1.18645
\(926\) −99.4312 −3.26751
\(927\) 0 0
\(928\) −142.466 −4.67668
\(929\) 22.7442 0.746215 0.373107 0.927788i \(-0.378292\pi\)
0.373107 + 0.927788i \(0.378292\pi\)
\(930\) 0 0
\(931\) 3.49924 0.114683
\(932\) 118.198 3.87169
\(933\) 0 0
\(934\) −22.7286 −0.743702
\(935\) −4.74983 −0.155336
\(936\) 0 0
\(937\) −4.43632 −0.144928 −0.0724641 0.997371i \(-0.523086\pi\)
−0.0724641 + 0.997371i \(0.523086\pi\)
\(938\) −42.2031 −1.37798
\(939\) 0 0
\(940\) 4.67130 0.152361
\(941\) 21.2474 0.692644 0.346322 0.938116i \(-0.387430\pi\)
0.346322 + 0.938116i \(0.387430\pi\)
\(942\) 0 0
\(943\) 6.10763 0.198892
\(944\) −82.3761 −2.68111
\(945\) 0 0
\(946\) −44.2034 −1.43718
\(947\) 27.7982 0.903320 0.451660 0.892190i \(-0.350832\pi\)
0.451660 + 0.892190i \(0.350832\pi\)
\(948\) 0 0
\(949\) −0.115760 −0.00375772
\(950\) −42.0978 −1.36583
\(951\) 0 0
\(952\) −33.0476 −1.07108
\(953\) −8.88868 −0.287933 −0.143966 0.989583i \(-0.545986\pi\)
−0.143966 + 0.989583i \(0.545986\pi\)
\(954\) 0 0
\(955\) 1.69396 0.0548154
\(956\) 66.8370 2.16166
\(957\) 0 0
\(958\) 62.9370 2.03340
\(959\) −6.18977 −0.199878
\(960\) 0 0
\(961\) 27.8996 0.899986
\(962\) −0.218156 −0.00703362
\(963\) 0 0
\(964\) −144.206 −4.64457
\(965\) 13.4894 0.434240
\(966\) 0 0
\(967\) 14.7376 0.473930 0.236965 0.971518i \(-0.423847\pi\)
0.236965 + 0.971518i \(0.423847\pi\)
\(968\) −66.9967 −2.15335
\(969\) 0 0
\(970\) 11.4974 0.369159
\(971\) −16.9135 −0.542780 −0.271390 0.962469i \(-0.587483\pi\)
−0.271390 + 0.962469i \(0.587483\pi\)
\(972\) 0 0
\(973\) −6.96079 −0.223153
\(974\) −24.6184 −0.788824
\(975\) 0 0
\(976\) −61.4167 −1.96590
\(977\) −7.07125 −0.226229 −0.113115 0.993582i \(-0.536083\pi\)
−0.113115 + 0.993582i \(0.536083\pi\)
\(978\) 0 0
\(979\) 10.4589 0.334269
\(980\) −3.56298 −0.113815
\(981\) 0 0
\(982\) −4.34015 −0.138500
\(983\) 0.986817 0.0314746 0.0157373 0.999876i \(-0.494990\pi\)
0.0157373 + 0.999876i \(0.494990\pi\)
\(984\) 0 0
\(985\) −8.55004 −0.272427
\(986\) −100.690 −3.20662
\(987\) 0 0
\(988\) −0.182878 −0.00581812
\(989\) −5.31785 −0.169098
\(990\) 0 0
\(991\) −18.6954 −0.593877 −0.296939 0.954897i \(-0.595966\pi\)
−0.296939 + 0.954897i \(0.595966\pi\)
\(992\) 115.536 3.66827
\(993\) 0 0
\(994\) −3.00617 −0.0953498
\(995\) −15.2925 −0.484804
\(996\) 0 0
\(997\) 60.3561 1.91150 0.955748 0.294185i \(-0.0950483\pi\)
0.955748 + 0.294185i \(0.0950483\pi\)
\(998\) −42.5818 −1.34790
\(999\) 0 0
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 8001.2.a.p.1.13 14
3.2 odd 2 2667.2.a.m.1.2 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.m.1.2 14 3.2 odd 2
8001.2.a.p.1.13 14 1.1 even 1 trivial