Properties

Label 1305.2.a.r.1.2
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
Dimension $4$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.13856\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.138564 q^{2} -1.98080 q^{4} +1.00000 q^{5} +5.07830 q^{7} +0.551597 q^{8} +O(q^{10})\) \(q-0.138564 q^{2} -1.98080 q^{4} +1.00000 q^{5} +5.07830 q^{7} +0.551597 q^{8} -0.138564 q^{10} +5.07830 q^{11} -3.67096 q^{13} -0.703671 q^{14} +3.88517 q^{16} +2.60617 q^{17} -6.74926 q^{19} -1.98080 q^{20} -0.703671 q^{22} +3.21234 q^{23} +1.00000 q^{25} +0.508664 q^{26} -10.0591 q^{28} +1.00000 q^{29} -0.787665 q^{31} -1.64154 q^{32} -0.361122 q^{34} +5.07830 q^{35} -5.13021 q^{37} +0.935207 q^{38} +0.551597 q^{40} +8.81469 q^{41} -2.61968 q^{43} -10.0591 q^{44} -0.445115 q^{46} +1.11670 q^{47} +18.7892 q^{49} -0.138564 q^{50} +7.27144 q^{52} -0.619678 q^{53} +5.07830 q^{55} +2.80118 q^{56} -0.138564 q^{58} -12.7763 q^{59} +8.90587 q^{61} +0.109142 q^{62} -7.54288 q^{64} -3.67096 q^{65} -0.524047 q^{67} -5.16230 q^{68} -0.703671 q^{70} -0.195007 q^{71} +1.13021 q^{73} +0.710864 q^{74} +13.3689 q^{76} +25.7892 q^{77} +15.3035 q^{79} +3.88517 q^{80} -1.22140 q^{82} +18.1182 q^{83} +2.60617 q^{85} +0.362994 q^{86} +2.80118 q^{88} +15.5942 q^{89} -18.6423 q^{91} -6.36299 q^{92} -0.154735 q^{94} -6.74926 q^{95} -12.6671 q^{97} -2.60351 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 5 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 5 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8} + 3 q^{10} + 2 q^{11} - 8 q^{13} + 3 q^{14} + 11 q^{16} + 10 q^{17} - 2 q^{19} + 5 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 7 q^{26} - 9 q^{28} + 4 q^{29} - 4 q^{31} + 17 q^{32} - q^{34} + 2 q^{35} - 16 q^{37} + 10 q^{38} + 12 q^{40} + 12 q^{41} + 2 q^{43} - 9 q^{44} - 8 q^{46} + 12 q^{47} + 6 q^{49} + 3 q^{50} - 3 q^{52} + 10 q^{53} + 2 q^{55} + 3 q^{58} - 2 q^{59} - 26 q^{61} - 20 q^{62} + 34 q^{64} - 8 q^{65} + 2 q^{67} - 9 q^{68} + 3 q^{70} + 10 q^{71} - 48 q^{74} + 16 q^{76} + 34 q^{77} + 22 q^{79} + 11 q^{80} + 38 q^{82} + 10 q^{83} + 10 q^{85} + 4 q^{86} + 4 q^{89} - 8 q^{91} - 28 q^{92} + 39 q^{94} - 2 q^{95} - 22 q^{97} - 34 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.138564 −0.0979797 −0.0489899 0.998799i \(-0.515600\pi\)
−0.0489899 + 0.998799i \(0.515600\pi\)
\(3\) 0 0
\(4\) −1.98080 −0.990400
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 5.07830 1.91942 0.959709 0.280995i \(-0.0906645\pi\)
0.959709 + 0.280995i \(0.0906645\pi\)
\(8\) 0.551597 0.195019
\(9\) 0 0
\(10\) −0.138564 −0.0438179
\(11\) 5.07830 1.53117 0.765583 0.643337i \(-0.222451\pi\)
0.765583 + 0.643337i \(0.222451\pi\)
\(12\) 0 0
\(13\) −3.67096 −1.01814 −0.509071 0.860725i \(-0.670011\pi\)
−0.509071 + 0.860725i \(0.670011\pi\)
\(14\) −0.703671 −0.188064
\(15\) 0 0
\(16\) 3.88517 0.971292
\(17\) 2.60617 0.632089 0.316044 0.948744i \(-0.397645\pi\)
0.316044 + 0.948744i \(0.397645\pi\)
\(18\) 0 0
\(19\) −6.74926 −1.54839 −0.774194 0.632949i \(-0.781844\pi\)
−0.774194 + 0.632949i \(0.781844\pi\)
\(20\) −1.98080 −0.442920
\(21\) 0 0
\(22\) −0.703671 −0.150023
\(23\) 3.21234 0.669818 0.334909 0.942250i \(-0.391294\pi\)
0.334909 + 0.942250i \(0.391294\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.508664 0.0997572
\(27\) 0 0
\(28\) −10.0591 −1.90099
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −0.787665 −0.141469 −0.0707344 0.997495i \(-0.522534\pi\)
−0.0707344 + 0.997495i \(0.522534\pi\)
\(32\) −1.64154 −0.290186
\(33\) 0 0
\(34\) −0.361122 −0.0619319
\(35\) 5.07830 0.858390
\(36\) 0 0
\(37\) −5.13021 −0.843402 −0.421701 0.906735i \(-0.638567\pi\)
−0.421701 + 0.906735i \(0.638567\pi\)
\(38\) 0.935207 0.151711
\(39\) 0 0
\(40\) 0.551597 0.0872151
\(41\) 8.81469 1.37662 0.688311 0.725415i \(-0.258352\pi\)
0.688311 + 0.725415i \(0.258352\pi\)
\(42\) 0 0
\(43\) −2.61968 −0.399497 −0.199749 0.979847i \(-0.564013\pi\)
−0.199749 + 0.979847i \(0.564013\pi\)
\(44\) −10.0591 −1.51647
\(45\) 0 0
\(46\) −0.445115 −0.0656286
\(47\) 1.11670 0.162888 0.0814440 0.996678i \(-0.474047\pi\)
0.0814440 + 0.996678i \(0.474047\pi\)
\(48\) 0 0
\(49\) 18.7892 2.68417
\(50\) −0.138564 −0.0195959
\(51\) 0 0
\(52\) 7.27144 1.00837
\(53\) −0.619678 −0.0851194 −0.0425597 0.999094i \(-0.513551\pi\)
−0.0425597 + 0.999094i \(0.513551\pi\)
\(54\) 0 0
\(55\) 5.07830 0.684758
\(56\) 2.80118 0.374323
\(57\) 0 0
\(58\) −0.138564 −0.0181944
\(59\) −12.7763 −1.66333 −0.831665 0.555277i \(-0.812612\pi\)
−0.831665 + 0.555277i \(0.812612\pi\)
\(60\) 0 0
\(61\) 8.90587 1.14028 0.570140 0.821548i \(-0.306889\pi\)
0.570140 + 0.821548i \(0.306889\pi\)
\(62\) 0.109142 0.0138611
\(63\) 0 0
\(64\) −7.54288 −0.942860
\(65\) −3.67096 −0.455327
\(66\) 0 0
\(67\) −0.524047 −0.0640225 −0.0320112 0.999488i \(-0.510191\pi\)
−0.0320112 + 0.999488i \(0.510191\pi\)
\(68\) −5.16230 −0.626020
\(69\) 0 0
\(70\) −0.703671 −0.0841048
\(71\) −0.195007 −0.0231431 −0.0115716 0.999933i \(-0.503683\pi\)
−0.0115716 + 0.999933i \(0.503683\pi\)
\(72\) 0 0
\(73\) 1.13021 0.132282 0.0661408 0.997810i \(-0.478931\pi\)
0.0661408 + 0.997810i \(0.478931\pi\)
\(74\) 0.710864 0.0826363
\(75\) 0 0
\(76\) 13.3689 1.53352
\(77\) 25.7892 2.93895
\(78\) 0 0
\(79\) 15.3035 1.72178 0.860890 0.508791i \(-0.169907\pi\)
0.860890 + 0.508791i \(0.169907\pi\)
\(80\) 3.88517 0.434375
\(81\) 0 0
\(82\) −1.22140 −0.134881
\(83\) 18.1182 1.98873 0.994366 0.106003i \(-0.0338053\pi\)
0.994366 + 0.106003i \(0.0338053\pi\)
\(84\) 0 0
\(85\) 2.60617 0.282679
\(86\) 0.362994 0.0391426
\(87\) 0 0
\(88\) 2.80118 0.298606
\(89\) 15.5942 1.65298 0.826489 0.562953i \(-0.190335\pi\)
0.826489 + 0.562953i \(0.190335\pi\)
\(90\) 0 0
\(91\) −18.6423 −1.95424
\(92\) −6.36299 −0.663388
\(93\) 0 0
\(94\) −0.154735 −0.0159597
\(95\) −6.74926 −0.692460
\(96\) 0 0
\(97\) −12.6671 −1.28615 −0.643077 0.765802i \(-0.722342\pi\)
−0.643077 + 0.765802i \(0.722342\pi\)
\(98\) −2.60351 −0.262994
\(99\) 0 0
\(100\) −1.98080 −0.198080
\(101\) −7.03990 −0.700497 −0.350248 0.936657i \(-0.613903\pi\)
−0.350248 + 0.936657i \(0.613903\pi\)
\(102\) 0 0
\(103\) 2.65808 0.261908 0.130954 0.991388i \(-0.458196\pi\)
0.130954 + 0.991388i \(0.458196\pi\)
\(104\) −2.02489 −0.198557
\(105\) 0 0
\(106\) 0.0858653 0.00833997
\(107\) −4.81469 −0.465453 −0.232727 0.972542i \(-0.574765\pi\)
−0.232727 + 0.972542i \(0.574765\pi\)
\(108\) 0 0
\(109\) 3.86597 0.370293 0.185146 0.982711i \(-0.440724\pi\)
0.185146 + 0.982711i \(0.440724\pi\)
\(110\) −0.703671 −0.0670924
\(111\) 0 0
\(112\) 19.7301 1.86432
\(113\) 8.73575 0.821791 0.410895 0.911683i \(-0.365216\pi\)
0.410895 + 0.911683i \(0.365216\pi\)
\(114\) 0 0
\(115\) 3.21234 0.299552
\(116\) −1.98080 −0.183913
\(117\) 0 0
\(118\) 1.77034 0.162973
\(119\) 13.2349 1.21324
\(120\) 0 0
\(121\) 14.7892 1.34447
\(122\) −1.23404 −0.111724
\(123\) 0 0
\(124\) 1.56021 0.140111
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 10.7877 0.957250 0.478625 0.878019i \(-0.341135\pi\)
0.478625 + 0.878019i \(0.341135\pi\)
\(128\) 4.32825 0.382567
\(129\) 0 0
\(130\) 0.508664 0.0446128
\(131\) −12.9745 −1.13359 −0.566793 0.823860i \(-0.691816\pi\)
−0.566793 + 0.823860i \(0.691816\pi\)
\(132\) 0 0
\(133\) −34.2748 −2.97200
\(134\) 0.0726141 0.00627291
\(135\) 0 0
\(136\) 1.43755 0.123269
\(137\) 4.08212 0.348759 0.174380 0.984679i \(-0.444208\pi\)
0.174380 + 0.984679i \(0.444208\pi\)
\(138\) 0 0
\(139\) −19.8276 −1.68175 −0.840876 0.541228i \(-0.817960\pi\)
−0.840876 + 0.541228i \(0.817960\pi\)
\(140\) −10.0591 −0.850149
\(141\) 0 0
\(142\) 0.0270211 0.00226756
\(143\) −18.6423 −1.55894
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) −0.156607 −0.0129609
\(147\) 0 0
\(148\) 10.1619 0.835305
\(149\) −10.7763 −0.882828 −0.441414 0.897304i \(-0.645523\pi\)
−0.441414 + 0.897304i \(0.645523\pi\)
\(150\) 0 0
\(151\) −5.49853 −0.447464 −0.223732 0.974651i \(-0.571824\pi\)
−0.223732 + 0.974651i \(0.571824\pi\)
\(152\) −3.72287 −0.301965
\(153\) 0 0
\(154\) −3.57346 −0.287957
\(155\) −0.787665 −0.0632667
\(156\) 0 0
\(157\) 5.77566 0.460948 0.230474 0.973079i \(-0.425972\pi\)
0.230474 + 0.973079i \(0.425972\pi\)
\(158\) −2.12052 −0.168700
\(159\) 0 0
\(160\) −1.64154 −0.129775
\(161\) 16.3132 1.28566
\(162\) 0 0
\(163\) 7.14691 0.559790 0.279895 0.960031i \(-0.409700\pi\)
0.279895 + 0.960031i \(0.409700\pi\)
\(164\) −17.4601 −1.36341
\(165\) 0 0
\(166\) −2.51054 −0.194855
\(167\) −17.1009 −1.32331 −0.661653 0.749810i \(-0.730145\pi\)
−0.661653 + 0.749810i \(0.730145\pi\)
\(168\) 0 0
\(169\) 0.475953 0.0366118
\(170\) −0.361122 −0.0276968
\(171\) 0 0
\(172\) 5.18906 0.395662
\(173\) −10.2862 −0.782045 −0.391022 0.920381i \(-0.627879\pi\)
−0.391022 + 0.920381i \(0.627879\pi\)
\(174\) 0 0
\(175\) 5.07830 0.383884
\(176\) 19.7301 1.48721
\(177\) 0 0
\(178\) −2.16079 −0.161958
\(179\) 12.1182 0.905757 0.452879 0.891572i \(-0.350397\pi\)
0.452879 + 0.891572i \(0.350397\pi\)
\(180\) 0 0
\(181\) −20.9745 −1.55902 −0.779510 0.626389i \(-0.784532\pi\)
−0.779510 + 0.626389i \(0.784532\pi\)
\(182\) 2.58315 0.191476
\(183\) 0 0
\(184\) 1.77191 0.130627
\(185\) −5.13021 −0.377181
\(186\) 0 0
\(187\) 13.2349 0.967833
\(188\) −2.21197 −0.161324
\(189\) 0 0
\(190\) 0.935207 0.0678470
\(191\) −26.2094 −1.89645 −0.948223 0.317607i \(-0.897121\pi\)
−0.948223 + 0.317607i \(0.897121\pi\)
\(192\) 0 0
\(193\) −23.7643 −1.71059 −0.855295 0.518141i \(-0.826624\pi\)
−0.855295 + 0.518141i \(0.826624\pi\)
\(194\) 1.75521 0.126017
\(195\) 0 0
\(196\) −37.2176 −2.65840
\(197\) 25.6281 1.82593 0.912964 0.408041i \(-0.133788\pi\)
0.912964 + 0.408041i \(0.133788\pi\)
\(198\) 0 0
\(199\) 7.82757 0.554882 0.277441 0.960743i \(-0.410514\pi\)
0.277441 + 0.960743i \(0.410514\pi\)
\(200\) 0.551597 0.0390038
\(201\) 0 0
\(202\) 0.975479 0.0686345
\(203\) 5.07830 0.356427
\(204\) 0 0
\(205\) 8.81469 0.615644
\(206\) −0.368315 −0.0256617
\(207\) 0 0
\(208\) −14.2623 −0.988913
\(209\) −34.2748 −2.37084
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 1.22746 0.0843022
\(213\) 0 0
\(214\) 0.667143 0.0456050
\(215\) −2.61968 −0.178661
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) −0.535685 −0.0362812
\(219\) 0 0
\(220\) −10.0591 −0.678185
\(221\) −9.56714 −0.643555
\(222\) 0 0
\(223\) 7.86597 0.526744 0.263372 0.964694i \(-0.415165\pi\)
0.263372 + 0.964694i \(0.415165\pi\)
\(224\) −8.33623 −0.556988
\(225\) 0 0
\(226\) −1.21046 −0.0805188
\(227\) 0.0654212 0.00434216 0.00217108 0.999998i \(-0.499309\pi\)
0.00217108 + 0.999998i \(0.499309\pi\)
\(228\) 0 0
\(229\) −3.87041 −0.255764 −0.127882 0.991789i \(-0.540818\pi\)
−0.127882 + 0.991789i \(0.540818\pi\)
\(230\) −0.445115 −0.0293500
\(231\) 0 0
\(232\) 0.551597 0.0362141
\(233\) 8.18363 0.536127 0.268064 0.963401i \(-0.413616\pi\)
0.268064 + 0.963401i \(0.413616\pi\)
\(234\) 0 0
\(235\) 1.11670 0.0728457
\(236\) 25.3073 1.64736
\(237\) 0 0
\(238\) −1.83389 −0.118873
\(239\) 11.2651 0.728680 0.364340 0.931266i \(-0.381295\pi\)
0.364340 + 0.931266i \(0.381295\pi\)
\(240\) 0 0
\(241\) −15.2619 −0.983107 −0.491554 0.870847i \(-0.663571\pi\)
−0.491554 + 0.870847i \(0.663571\pi\)
\(242\) −2.04925 −0.131731
\(243\) 0 0
\(244\) −17.6408 −1.12933
\(245\) 18.7892 1.20040
\(246\) 0 0
\(247\) 24.7763 1.57648
\(248\) −0.434473 −0.0275891
\(249\) 0 0
\(250\) −0.138564 −0.00876357
\(251\) −24.9411 −1.57427 −0.787134 0.616783i \(-0.788436\pi\)
−0.787134 + 0.616783i \(0.788436\pi\)
\(252\) 0 0
\(253\) 16.3132 1.02560
\(254\) −1.49478 −0.0937911
\(255\) 0 0
\(256\) 14.4860 0.905376
\(257\) −1.14691 −0.0715425 −0.0357713 0.999360i \(-0.511389\pi\)
−0.0357713 + 0.999360i \(0.511389\pi\)
\(258\) 0 0
\(259\) −26.0528 −1.61884
\(260\) 7.27144 0.450956
\(261\) 0 0
\(262\) 1.79780 0.111068
\(263\) −0.685099 −0.0422450 −0.0211225 0.999777i \(-0.506724\pi\)
−0.0211225 + 0.999777i \(0.506724\pi\)
\(264\) 0 0
\(265\) −0.619678 −0.0380665
\(266\) 4.74926 0.291196
\(267\) 0 0
\(268\) 1.03803 0.0634079
\(269\) −4.22522 −0.257616 −0.128808 0.991670i \(-0.541115\pi\)
−0.128808 + 0.991670i \(0.541115\pi\)
\(270\) 0 0
\(271\) −0.814686 −0.0494886 −0.0247443 0.999694i \(-0.507877\pi\)
−0.0247443 + 0.999694i \(0.507877\pi\)
\(272\) 10.1254 0.613943
\(273\) 0 0
\(274\) −0.565636 −0.0341713
\(275\) 5.07830 0.306233
\(276\) 0 0
\(277\) 9.85459 0.592105 0.296052 0.955172i \(-0.404330\pi\)
0.296052 + 0.955172i \(0.404330\pi\)
\(278\) 2.74739 0.164778
\(279\) 0 0
\(280\) 2.80118 0.167402
\(281\) −12.2297 −0.729561 −0.364780 0.931094i \(-0.618856\pi\)
−0.364780 + 0.931094i \(0.618856\pi\)
\(282\) 0 0
\(283\) 4.23341 0.251650 0.125825 0.992052i \(-0.459842\pi\)
0.125825 + 0.992052i \(0.459842\pi\)
\(284\) 0.386271 0.0229209
\(285\) 0 0
\(286\) 2.58315 0.152745
\(287\) 44.7637 2.64231
\(288\) 0 0
\(289\) −10.2079 −0.600464
\(290\) −0.138564 −0.00813677
\(291\) 0 0
\(292\) −2.23873 −0.131012
\(293\) 13.3170 0.777989 0.388995 0.921240i \(-0.372822\pi\)
0.388995 + 0.921240i \(0.372822\pi\)
\(294\) 0 0
\(295\) −12.7763 −0.743864
\(296\) −2.82981 −0.164479
\(297\) 0 0
\(298\) 1.49321 0.0864992
\(299\) −11.7924 −0.681970
\(300\) 0 0
\(301\) −13.3035 −0.766802
\(302\) 0.761900 0.0438424
\(303\) 0 0
\(304\) −26.2220 −1.50394
\(305\) 8.90587 0.509949
\(306\) 0 0
\(307\) 14.7379 0.841136 0.420568 0.907261i \(-0.361831\pi\)
0.420568 + 0.907261i \(0.361831\pi\)
\(308\) −51.0832 −2.91073
\(309\) 0 0
\(310\) 0.109142 0.00619886
\(311\) 10.0226 0.568328 0.284164 0.958776i \(-0.408284\pi\)
0.284164 + 0.958776i \(0.408284\pi\)
\(312\) 0 0
\(313\) 4.08800 0.231067 0.115534 0.993304i \(-0.463142\pi\)
0.115534 + 0.993304i \(0.463142\pi\)
\(314\) −0.800300 −0.0451635
\(315\) 0 0
\(316\) −30.3132 −1.70525
\(317\) 12.7628 0.716829 0.358414 0.933563i \(-0.383317\pi\)
0.358414 + 0.933563i \(0.383317\pi\)
\(318\) 0 0
\(319\) 5.07830 0.284330
\(320\) −7.54288 −0.421660
\(321\) 0 0
\(322\) −2.26043 −0.125969
\(323\) −17.5897 −0.978718
\(324\) 0 0
\(325\) −3.67096 −0.203628
\(326\) −0.990307 −0.0548480
\(327\) 0 0
\(328\) 4.86215 0.268467
\(329\) 5.67096 0.312650
\(330\) 0 0
\(331\) 5.83576 0.320762 0.160381 0.987055i \(-0.448728\pi\)
0.160381 + 0.987055i \(0.448728\pi\)
\(332\) −35.8885 −1.96964
\(333\) 0 0
\(334\) 2.36957 0.129657
\(335\) −0.524047 −0.0286317
\(336\) 0 0
\(337\) 1.95253 0.106361 0.0531807 0.998585i \(-0.483064\pi\)
0.0531807 + 0.998585i \(0.483064\pi\)
\(338\) −0.0659501 −0.00358721
\(339\) 0 0
\(340\) −5.16230 −0.279965
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 59.8690 3.23262
\(344\) −1.44501 −0.0779095
\(345\) 0 0
\(346\) 1.42530 0.0766245
\(347\) −17.3960 −0.933864 −0.466932 0.884293i \(-0.654641\pi\)
−0.466932 + 0.884293i \(0.654641\pi\)
\(348\) 0 0
\(349\) −28.7379 −1.53830 −0.769152 0.639066i \(-0.779321\pi\)
−0.769152 + 0.639066i \(0.779321\pi\)
\(350\) −0.703671 −0.0376128
\(351\) 0 0
\(352\) −8.33623 −0.444323
\(353\) 29.7590 1.58391 0.791955 0.610580i \(-0.209064\pi\)
0.791955 + 0.610580i \(0.209064\pi\)
\(354\) 0 0
\(355\) −0.195007 −0.0103499
\(356\) −30.8889 −1.63711
\(357\) 0 0
\(358\) −1.67915 −0.0887459
\(359\) −7.76659 −0.409905 −0.204953 0.978772i \(-0.565704\pi\)
−0.204953 + 0.978772i \(0.565704\pi\)
\(360\) 0 0
\(361\) 26.5526 1.39750
\(362\) 2.90631 0.152752
\(363\) 0 0
\(364\) 36.9266 1.93548
\(365\) 1.13021 0.0591581
\(366\) 0 0
\(367\) −12.8675 −0.671677 −0.335838 0.941920i \(-0.609020\pi\)
−0.335838 + 0.941920i \(0.609020\pi\)
\(368\) 12.4805 0.650589
\(369\) 0 0
\(370\) 0.710864 0.0369561
\(371\) −3.14691 −0.163380
\(372\) 0 0
\(373\) −13.4639 −0.697133 −0.348566 0.937284i \(-0.613331\pi\)
−0.348566 + 0.937284i \(0.613331\pi\)
\(374\) −1.83389 −0.0948280
\(375\) 0 0
\(376\) 0.615970 0.0317662
\(377\) −3.67096 −0.189064
\(378\) 0 0
\(379\) −9.53318 −0.489687 −0.244843 0.969563i \(-0.578737\pi\)
−0.244843 + 0.969563i \(0.578737\pi\)
\(380\) 13.3689 0.685812
\(381\) 0 0
\(382\) 3.63169 0.185813
\(383\) 20.0836 1.02622 0.513111 0.858322i \(-0.328493\pi\)
0.513111 + 0.858322i \(0.328493\pi\)
\(384\) 0 0
\(385\) 25.7892 1.31434
\(386\) 3.29288 0.167603
\(387\) 0 0
\(388\) 25.0911 1.27381
\(389\) −24.3472 −1.23445 −0.617225 0.786787i \(-0.711743\pi\)
−0.617225 + 0.786787i \(0.711743\pi\)
\(390\) 0 0
\(391\) 8.37188 0.423384
\(392\) 10.3640 0.523463
\(393\) 0 0
\(394\) −3.55114 −0.178904
\(395\) 15.3035 0.770004
\(396\) 0 0
\(397\) −25.9232 −1.30105 −0.650524 0.759486i \(-0.725451\pi\)
−0.650524 + 0.759486i \(0.725451\pi\)
\(398\) −1.08462 −0.0543671
\(399\) 0 0
\(400\) 3.88517 0.194258
\(401\) −31.3576 −1.56592 −0.782961 0.622071i \(-0.786292\pi\)
−0.782961 + 0.622071i \(0.786292\pi\)
\(402\) 0 0
\(403\) 2.89149 0.144035
\(404\) 13.9446 0.693772
\(405\) 0 0
\(406\) −0.703671 −0.0349226
\(407\) −26.0528 −1.29139
\(408\) 0 0
\(409\) 1.67415 0.0827814 0.0413907 0.999143i \(-0.486821\pi\)
0.0413907 + 0.999143i \(0.486821\pi\)
\(410\) −1.22140 −0.0603207
\(411\) 0 0
\(412\) −5.26512 −0.259394
\(413\) −64.8819 −3.19263
\(414\) 0 0
\(415\) 18.1182 0.889388
\(416\) 6.02602 0.295450
\(417\) 0 0
\(418\) 4.74926 0.232294
\(419\) −0.658078 −0.0321492 −0.0160746 0.999871i \(-0.505117\pi\)
−0.0160746 + 0.999871i \(0.505117\pi\)
\(420\) 0 0
\(421\) 5.11989 0.249528 0.124764 0.992186i \(-0.460183\pi\)
0.124764 + 0.992186i \(0.460183\pi\)
\(422\) −0.277129 −0.0134904
\(423\) 0 0
\(424\) −0.341812 −0.0165999
\(425\) 2.60617 0.126418
\(426\) 0 0
\(427\) 45.2267 2.18867
\(428\) 9.53693 0.460985
\(429\) 0 0
\(430\) 0.362994 0.0175051
\(431\) 0.390015 0.0187864 0.00939318 0.999956i \(-0.497010\pi\)
0.00939318 + 0.999956i \(0.497010\pi\)
\(432\) 0 0
\(433\) −29.6989 −1.42724 −0.713618 0.700535i \(-0.752945\pi\)
−0.713618 + 0.700535i \(0.752945\pi\)
\(434\) 0.554257 0.0266052
\(435\) 0 0
\(436\) −7.65771 −0.366738
\(437\) −21.6809 −1.03714
\(438\) 0 0
\(439\) −24.4856 −1.16864 −0.584318 0.811525i \(-0.698638\pi\)
−0.584318 + 0.811525i \(0.698638\pi\)
\(440\) 2.80118 0.133541
\(441\) 0 0
\(442\) 1.32566 0.0630554
\(443\) 11.1091 0.527808 0.263904 0.964549i \(-0.414990\pi\)
0.263904 + 0.964549i \(0.414990\pi\)
\(444\) 0 0
\(445\) 15.5942 0.739234
\(446\) −1.08994 −0.0516103
\(447\) 0 0
\(448\) −38.3050 −1.80974
\(449\) 15.1003 0.712628 0.356314 0.934366i \(-0.384033\pi\)
0.356314 + 0.934366i \(0.384033\pi\)
\(450\) 0 0
\(451\) 44.7637 2.10784
\(452\) −17.3038 −0.813901
\(453\) 0 0
\(454\) −0.00906504 −0.000425443 0
\(455\) −18.6423 −0.873962
\(456\) 0 0
\(457\) −24.3742 −1.14018 −0.570088 0.821583i \(-0.693091\pi\)
−0.570088 + 0.821583i \(0.693091\pi\)
\(458\) 0.536301 0.0250597
\(459\) 0 0
\(460\) −6.36299 −0.296676
\(461\) 12.9789 0.604489 0.302244 0.953230i \(-0.402264\pi\)
0.302244 + 0.953230i \(0.402264\pi\)
\(462\) 0 0
\(463\) −11.2619 −0.523386 −0.261693 0.965151i \(-0.584281\pi\)
−0.261693 + 0.965151i \(0.584281\pi\)
\(464\) 3.88517 0.180364
\(465\) 0 0
\(466\) −1.13396 −0.0525296
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −2.66127 −0.122886
\(470\) −0.154735 −0.00713740
\(471\) 0 0
\(472\) −7.04736 −0.324381
\(473\) −13.3035 −0.611697
\(474\) 0 0
\(475\) −6.74926 −0.309677
\(476\) −26.2157 −1.20160
\(477\) 0 0
\(478\) −1.56094 −0.0713959
\(479\) 23.8615 1.09026 0.545130 0.838351i \(-0.316480\pi\)
0.545130 + 0.838351i \(0.316480\pi\)
\(480\) 0 0
\(481\) 18.8328 0.858702
\(482\) 2.11476 0.0963246
\(483\) 0 0
\(484\) −29.2944 −1.33156
\(485\) −12.6671 −0.575185
\(486\) 0 0
\(487\) −20.8417 −0.944428 −0.472214 0.881484i \(-0.656545\pi\)
−0.472214 + 0.881484i \(0.656545\pi\)
\(488\) 4.91245 0.222376
\(489\) 0 0
\(490\) −2.60351 −0.117614
\(491\) 19.1021 0.862067 0.431034 0.902336i \(-0.358149\pi\)
0.431034 + 0.902336i \(0.358149\pi\)
\(492\) 0 0
\(493\) 2.60617 0.117376
\(494\) −3.43311 −0.154463
\(495\) 0 0
\(496\) −3.06021 −0.137407
\(497\) −0.990307 −0.0444213
\(498\) 0 0
\(499\) −7.82757 −0.350410 −0.175205 0.984532i \(-0.556059\pi\)
−0.175205 + 0.984532i \(0.556059\pi\)
\(500\) −1.98080 −0.0885841
\(501\) 0 0
\(502\) 3.45594 0.154246
\(503\) −31.5865 −1.40837 −0.704187 0.710015i \(-0.748688\pi\)
−0.704187 + 0.710015i \(0.748688\pi\)
\(504\) 0 0
\(505\) −7.03990 −0.313272
\(506\) −2.26043 −0.100488
\(507\) 0 0
\(508\) −21.3682 −0.948061
\(509\) −19.6167 −0.869497 −0.434748 0.900552i \(-0.643163\pi\)
−0.434748 + 0.900552i \(0.643163\pi\)
\(510\) 0 0
\(511\) 5.73957 0.253904
\(512\) −10.6637 −0.471275
\(513\) 0 0
\(514\) 0.158921 0.00700972
\(515\) 2.65808 0.117129
\(516\) 0 0
\(517\) 5.67096 0.249409
\(518\) 3.60999 0.158614
\(519\) 0 0
\(520\) −2.02489 −0.0887973
\(521\) −24.4057 −1.06923 −0.534616 0.845095i \(-0.679544\pi\)
−0.534616 + 0.845095i \(0.679544\pi\)
\(522\) 0 0
\(523\) −39.9715 −1.74783 −0.873917 0.486076i \(-0.838428\pi\)
−0.873917 + 0.486076i \(0.838428\pi\)
\(524\) 25.6999 1.12270
\(525\) 0 0
\(526\) 0.0949303 0.00413916
\(527\) −2.05279 −0.0894208
\(528\) 0 0
\(529\) −12.6809 −0.551344
\(530\) 0.0858653 0.00372975
\(531\) 0 0
\(532\) 67.8916 2.94347
\(533\) −32.3584 −1.40160
\(534\) 0 0
\(535\) −4.81469 −0.208157
\(536\) −0.289062 −0.0124856
\(537\) 0 0
\(538\) 0.585464 0.0252412
\(539\) 95.4171 4.10991
\(540\) 0 0
\(541\) −8.76064 −0.376649 −0.188325 0.982107i \(-0.560306\pi\)
−0.188325 + 0.982107i \(0.560306\pi\)
\(542\) 0.112886 0.00484888
\(543\) 0 0
\(544\) −4.27813 −0.183423
\(545\) 3.86597 0.165600
\(546\) 0 0
\(547\) 31.3569 1.34072 0.670361 0.742035i \(-0.266139\pi\)
0.670361 + 0.742035i \(0.266139\pi\)
\(548\) −8.08587 −0.345411
\(549\) 0 0
\(550\) −0.703671 −0.0300046
\(551\) −6.74926 −0.287528
\(552\) 0 0
\(553\) 77.7159 3.30482
\(554\) −1.36549 −0.0580143
\(555\) 0 0
\(556\) 39.2744 1.66561
\(557\) −37.6514 −1.59534 −0.797670 0.603094i \(-0.793934\pi\)
−0.797670 + 0.603094i \(0.793934\pi\)
\(558\) 0 0
\(559\) 9.61674 0.406745
\(560\) 19.7301 0.833747
\(561\) 0 0
\(562\) 1.69459 0.0714821
\(563\) 39.0323 1.64501 0.822507 0.568755i \(-0.192575\pi\)
0.822507 + 0.568755i \(0.192575\pi\)
\(564\) 0 0
\(565\) 8.73575 0.367516
\(566\) −0.586599 −0.0246566
\(567\) 0 0
\(568\) −0.107565 −0.00451335
\(569\) 3.03990 0.127439 0.0637197 0.997968i \(-0.479704\pi\)
0.0637197 + 0.997968i \(0.479704\pi\)
\(570\) 0 0
\(571\) 40.1056 1.67837 0.839183 0.543849i \(-0.183034\pi\)
0.839183 + 0.543849i \(0.183034\pi\)
\(572\) 36.9266 1.54398
\(573\) 0 0
\(574\) −6.20264 −0.258893
\(575\) 3.21234 0.133964
\(576\) 0 0
\(577\) −16.1091 −0.670632 −0.335316 0.942106i \(-0.608843\pi\)
−0.335316 + 0.942106i \(0.608843\pi\)
\(578\) 1.41445 0.0588333
\(579\) 0 0
\(580\) −1.98080 −0.0822482
\(581\) 92.0098 3.81721
\(582\) 0 0
\(583\) −3.14691 −0.130332
\(584\) 0.623422 0.0257974
\(585\) 0 0
\(586\) −1.84526 −0.0762272
\(587\) 21.4331 0.884639 0.442320 0.896858i \(-0.354156\pi\)
0.442320 + 0.896858i \(0.354156\pi\)
\(588\) 0 0
\(589\) 5.31616 0.219048
\(590\) 1.77034 0.0728836
\(591\) 0 0
\(592\) −19.9317 −0.819190
\(593\) −23.9886 −0.985095 −0.492547 0.870286i \(-0.663934\pi\)
−0.492547 + 0.870286i \(0.663934\pi\)
\(594\) 0 0
\(595\) 13.2349 0.542578
\(596\) 21.3457 0.874353
\(597\) 0 0
\(598\) 1.63400 0.0668192
\(599\) 38.1100 1.55713 0.778567 0.627562i \(-0.215947\pi\)
0.778567 + 0.627562i \(0.215947\pi\)
\(600\) 0 0
\(601\) −20.0528 −0.817970 −0.408985 0.912541i \(-0.634117\pi\)
−0.408985 + 0.912541i \(0.634117\pi\)
\(602\) 1.84339 0.0751311
\(603\) 0 0
\(604\) 10.8915 0.443168
\(605\) 14.7892 0.601265
\(606\) 0 0
\(607\) −38.7523 −1.57291 −0.786453 0.617650i \(-0.788085\pi\)
−0.786453 + 0.617650i \(0.788085\pi\)
\(608\) 11.0792 0.449320
\(609\) 0 0
\(610\) −1.23404 −0.0499646
\(611\) −4.09938 −0.165843
\(612\) 0 0
\(613\) 24.6757 0.996640 0.498320 0.866993i \(-0.333950\pi\)
0.498320 + 0.866993i \(0.333950\pi\)
\(614\) −2.04214 −0.0824142
\(615\) 0 0
\(616\) 14.2252 0.573150
\(617\) −10.5249 −0.423717 −0.211859 0.977300i \(-0.567952\pi\)
−0.211859 + 0.977300i \(0.567952\pi\)
\(618\) 0 0
\(619\) 26.5496 1.06712 0.533560 0.845762i \(-0.320854\pi\)
0.533560 + 0.845762i \(0.320854\pi\)
\(620\) 1.56021 0.0626594
\(621\) 0 0
\(622\) −1.38877 −0.0556846
\(623\) 79.1919 3.17276
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −0.566450 −0.0226399
\(627\) 0 0
\(628\) −11.4404 −0.456523
\(629\) −13.3702 −0.533105
\(630\) 0 0
\(631\) −13.2041 −0.525649 −0.262824 0.964844i \(-0.584654\pi\)
−0.262824 + 0.964844i \(0.584654\pi\)
\(632\) 8.44137 0.335780
\(633\) 0 0
\(634\) −1.76846 −0.0702347
\(635\) 10.7877 0.428095
\(636\) 0 0
\(637\) −68.9743 −2.73286
\(638\) −0.703671 −0.0278586
\(639\) 0 0
\(640\) 4.32825 0.171089
\(641\) −9.51435 −0.375794 −0.187897 0.982189i \(-0.560167\pi\)
−0.187897 + 0.982189i \(0.560167\pi\)
\(642\) 0 0
\(643\) 30.1370 1.18849 0.594244 0.804285i \(-0.297451\pi\)
0.594244 + 0.804285i \(0.297451\pi\)
\(644\) −32.3132 −1.27332
\(645\) 0 0
\(646\) 2.43731 0.0958945
\(647\) 42.7535 1.68081 0.840407 0.541955i \(-0.182316\pi\)
0.840407 + 0.541955i \(0.182316\pi\)
\(648\) 0 0
\(649\) −64.8819 −2.54684
\(650\) 0.508664 0.0199514
\(651\) 0 0
\(652\) −14.1566 −0.554416
\(653\) −33.8983 −1.32654 −0.663272 0.748379i \(-0.730833\pi\)
−0.663272 + 0.748379i \(0.730833\pi\)
\(654\) 0 0
\(655\) −12.9745 −0.506955
\(656\) 34.2465 1.33710
\(657\) 0 0
\(658\) −0.785793 −0.0306334
\(659\) −11.7287 −0.456887 −0.228444 0.973557i \(-0.573364\pi\)
−0.228444 + 0.973557i \(0.573364\pi\)
\(660\) 0 0
\(661\) −38.6807 −1.50450 −0.752252 0.658876i \(-0.771032\pi\)
−0.752252 + 0.658876i \(0.771032\pi\)
\(662\) −0.808627 −0.0314282
\(663\) 0 0
\(664\) 9.99394 0.387840
\(665\) −34.2748 −1.32912
\(666\) 0 0
\(667\) 3.21234 0.124382
\(668\) 33.8734 1.31060
\(669\) 0 0
\(670\) 0.0726141 0.00280533
\(671\) 45.2267 1.74596
\(672\) 0 0
\(673\) 30.6410 1.18112 0.590562 0.806992i \(-0.298906\pi\)
0.590562 + 0.806992i \(0.298906\pi\)
\(674\) −0.270552 −0.0104213
\(675\) 0 0
\(676\) −0.942768 −0.0362603
\(677\) 40.8954 1.57174 0.785868 0.618394i \(-0.212216\pi\)
0.785868 + 0.618394i \(0.212216\pi\)
\(678\) 0 0
\(679\) −64.3276 −2.46867
\(680\) 1.43755 0.0551277
\(681\) 0 0
\(682\) 0.554257 0.0212236
\(683\) −21.2317 −0.812409 −0.406205 0.913782i \(-0.633148\pi\)
−0.406205 + 0.913782i \(0.633148\pi\)
\(684\) 0 0
\(685\) 4.08212 0.155970
\(686\) −8.29570 −0.316731
\(687\) 0 0
\(688\) −10.1779 −0.388028
\(689\) 2.27481 0.0866635
\(690\) 0 0
\(691\) −13.9842 −0.531983 −0.265992 0.963975i \(-0.585699\pi\)
−0.265992 + 0.963975i \(0.585699\pi\)
\(692\) 20.3749 0.774537
\(693\) 0 0
\(694\) 2.41046 0.0914998
\(695\) −19.8276 −0.752103
\(696\) 0 0
\(697\) 22.9725 0.870147
\(698\) 3.98204 0.150723
\(699\) 0 0
\(700\) −10.0591 −0.380198
\(701\) −3.16630 −0.119590 −0.0597948 0.998211i \(-0.519045\pi\)
−0.0597948 + 0.998211i \(0.519045\pi\)
\(702\) 0 0
\(703\) 34.6252 1.30591
\(704\) −38.3050 −1.44367
\(705\) 0 0
\(706\) −4.12353 −0.155191
\(707\) −35.7508 −1.34455
\(708\) 0 0
\(709\) 23.3677 0.877592 0.438796 0.898587i \(-0.355405\pi\)
0.438796 + 0.898587i \(0.355405\pi\)
\(710\) 0.0270211 0.00101408
\(711\) 0 0
\(712\) 8.60169 0.322362
\(713\) −2.53024 −0.0947583
\(714\) 0 0
\(715\) −18.6423 −0.697181
\(716\) −24.0037 −0.897062
\(717\) 0 0
\(718\) 1.07617 0.0401624
\(719\) 0.731134 0.0272667 0.0136334 0.999907i \(-0.495660\pi\)
0.0136334 + 0.999907i \(0.495660\pi\)
\(720\) 0 0
\(721\) 13.4985 0.502711
\(722\) −3.67924 −0.136927
\(723\) 0 0
\(724\) 41.5463 1.54405
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −17.7243 −0.657358 −0.328679 0.944442i \(-0.606603\pi\)
−0.328679 + 0.944442i \(0.606603\pi\)
\(728\) −10.2830 −0.381113
\(729\) 0 0
\(730\) −0.156607 −0.00579630
\(731\) −6.82732 −0.252518
\(732\) 0 0
\(733\) −5.67184 −0.209494 −0.104747 0.994499i \(-0.533403\pi\)
−0.104747 + 0.994499i \(0.533403\pi\)
\(734\) 1.78297 0.0658107
\(735\) 0 0
\(736\) −5.27317 −0.194372
\(737\) −2.66127 −0.0980291
\(738\) 0 0
\(739\) 8.72350 0.320899 0.160450 0.987044i \(-0.448706\pi\)
0.160450 + 0.987044i \(0.448706\pi\)
\(740\) 10.1619 0.373560
\(741\) 0 0
\(742\) 0.436050 0.0160079
\(743\) 0.413474 0.0151689 0.00758445 0.999971i \(-0.497586\pi\)
0.00758445 + 0.999971i \(0.497586\pi\)
\(744\) 0 0
\(745\) −10.7763 −0.394813
\(746\) 1.86561 0.0683049
\(747\) 0 0
\(748\) −26.2157 −0.958541
\(749\) −24.4504 −0.893399
\(750\) 0 0
\(751\) 9.71381 0.354462 0.177231 0.984169i \(-0.443286\pi\)
0.177231 + 0.984169i \(0.443286\pi\)
\(752\) 4.33858 0.158212
\(753\) 0 0
\(754\) 0.508664 0.0185244
\(755\) −5.49853 −0.200112
\(756\) 0 0
\(757\) 22.3877 0.813695 0.406847 0.913496i \(-0.366628\pi\)
0.406847 + 0.913496i \(0.366628\pi\)
\(758\) 1.32096 0.0479794
\(759\) 0 0
\(760\) −3.72287 −0.135043
\(761\) −44.0836 −1.59803 −0.799014 0.601313i \(-0.794645\pi\)
−0.799014 + 0.601313i \(0.794645\pi\)
\(762\) 0 0
\(763\) 19.6326 0.710746
\(764\) 51.9156 1.87824
\(765\) 0 0
\(766\) −2.78286 −0.100549
\(767\) 46.9012 1.69351
\(768\) 0 0
\(769\) −14.2761 −0.514808 −0.257404 0.966304i \(-0.582867\pi\)
−0.257404 + 0.966304i \(0.582867\pi\)
\(770\) −3.57346 −0.128778
\(771\) 0 0
\(772\) 47.0723 1.69417
\(773\) 25.5807 0.920072 0.460036 0.887900i \(-0.347836\pi\)
0.460036 + 0.887900i \(0.347836\pi\)
\(774\) 0 0
\(775\) −0.787665 −0.0282937
\(776\) −6.98715 −0.250824
\(777\) 0 0
\(778\) 3.37365 0.120951
\(779\) −59.4926 −2.13155
\(780\) 0 0
\(781\) −0.990307 −0.0354360
\(782\) −1.16004 −0.0414831
\(783\) 0 0
\(784\) 72.9991 2.60711
\(785\) 5.77566 0.206142
\(786\) 0 0
\(787\) −8.73362 −0.311320 −0.155660 0.987811i \(-0.549750\pi\)
−0.155660 + 0.987811i \(0.549750\pi\)
\(788\) −50.7642 −1.80840
\(789\) 0 0
\(790\) −2.12052 −0.0754448
\(791\) 44.3628 1.57736
\(792\) 0 0
\(793\) −32.6931 −1.16097
\(794\) 3.59203 0.127476
\(795\) 0 0
\(796\) −15.5048 −0.549555
\(797\) −8.43874 −0.298915 −0.149458 0.988768i \(-0.547753\pi\)
−0.149458 + 0.988768i \(0.547753\pi\)
\(798\) 0 0
\(799\) 2.91032 0.102960
\(800\) −1.64154 −0.0580372
\(801\) 0 0
\(802\) 4.34504 0.153429
\(803\) 5.73957 0.202545
\(804\) 0 0
\(805\) 16.3132 0.574965
\(806\) −0.400657 −0.0141125
\(807\) 0 0
\(808\) −3.88319 −0.136610
\(809\) 21.7508 0.764716 0.382358 0.924014i \(-0.375112\pi\)
0.382358 + 0.924014i \(0.375112\pi\)
\(810\) 0 0
\(811\) −3.24629 −0.113993 −0.0569963 0.998374i \(-0.518152\pi\)
−0.0569963 + 0.998374i \(0.518152\pi\)
\(812\) −10.0591 −0.353005
\(813\) 0 0
\(814\) 3.60999 0.126530
\(815\) 7.14691 0.250345
\(816\) 0 0
\(817\) 17.6809 0.618576
\(818\) −0.231977 −0.00811090
\(819\) 0 0
\(820\) −17.4601 −0.609734
\(821\) −33.9232 −1.18393 −0.591964 0.805964i \(-0.701647\pi\)
−0.591964 + 0.805964i \(0.701647\pi\)
\(822\) 0 0
\(823\) −36.4648 −1.27108 −0.635542 0.772066i \(-0.719223\pi\)
−0.635542 + 0.772066i \(0.719223\pi\)
\(824\) 1.46619 0.0510770
\(825\) 0 0
\(826\) 8.99031 0.312813
\(827\) 15.9772 0.555583 0.277792 0.960641i \(-0.410398\pi\)
0.277792 + 0.960641i \(0.410398\pi\)
\(828\) 0 0
\(829\) 5.00595 0.173864 0.0869319 0.996214i \(-0.472294\pi\)
0.0869319 + 0.996214i \(0.472294\pi\)
\(830\) −2.51054 −0.0871420
\(831\) 0 0
\(832\) 27.6896 0.959964
\(833\) 48.9677 1.69663
\(834\) 0 0
\(835\) −17.1009 −0.591800
\(836\) 67.8916 2.34808
\(837\) 0 0
\(838\) 0.0911861 0.00314997
\(839\) −31.4713 −1.08651 −0.543255 0.839567i \(-0.682808\pi\)
−0.543255 + 0.839567i \(0.682808\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −0.709434 −0.0244487
\(843\) 0 0
\(844\) −3.96160 −0.136364
\(845\) 0.475953 0.0163733
\(846\) 0 0
\(847\) 75.1039 2.58060
\(848\) −2.40755 −0.0826758
\(849\) 0 0
\(850\) −0.361122 −0.0123864
\(851\) −16.4800 −0.564926
\(852\) 0 0
\(853\) −36.0197 −1.23329 −0.616646 0.787241i \(-0.711509\pi\)
−0.616646 + 0.787241i \(0.711509\pi\)
\(854\) −6.26681 −0.214446
\(855\) 0 0
\(856\) −2.65576 −0.0907722
\(857\) 27.4217 0.936708 0.468354 0.883541i \(-0.344847\pi\)
0.468354 + 0.883541i \(0.344847\pi\)
\(858\) 0 0
\(859\) 14.0642 0.479863 0.239932 0.970790i \(-0.422875\pi\)
0.239932 + 0.970790i \(0.422875\pi\)
\(860\) 5.18906 0.176945
\(861\) 0 0
\(862\) −0.0540421 −0.00184068
\(863\) −19.7540 −0.672433 −0.336216 0.941785i \(-0.609147\pi\)
−0.336216 + 0.941785i \(0.609147\pi\)
\(864\) 0 0
\(865\) −10.2862 −0.349741
\(866\) 4.11520 0.139840
\(867\) 0 0
\(868\) 7.92320 0.268931
\(869\) 77.7159 2.63633
\(870\) 0 0
\(871\) 1.92375 0.0651839
\(872\) 2.13246 0.0722140
\(873\) 0 0
\(874\) 3.00420 0.101619
\(875\) 5.07830 0.171678
\(876\) 0 0
\(877\) 42.2030 1.42509 0.712547 0.701624i \(-0.247541\pi\)
0.712547 + 0.701624i \(0.247541\pi\)
\(878\) 3.39284 0.114503
\(879\) 0 0
\(880\) 19.7301 0.665100
\(881\) 33.0927 1.11492 0.557461 0.830203i \(-0.311776\pi\)
0.557461 + 0.830203i \(0.311776\pi\)
\(882\) 0 0
\(883\) −26.2634 −0.883835 −0.441917 0.897056i \(-0.645702\pi\)
−0.441917 + 0.897056i \(0.645702\pi\)
\(884\) 18.9506 0.637377
\(885\) 0 0
\(886\) −1.53932 −0.0517145
\(887\) −14.0716 −0.472479 −0.236239 0.971695i \(-0.575915\pi\)
−0.236239 + 0.971695i \(0.575915\pi\)
\(888\) 0 0
\(889\) 54.7830 1.83736
\(890\) −2.16079 −0.0724300
\(891\) 0 0
\(892\) −15.5809 −0.521687
\(893\) −7.53693 −0.252214
\(894\) 0 0
\(895\) 12.1182 0.405067
\(896\) 21.9802 0.734306
\(897\) 0 0
\(898\) −2.09237 −0.0698231
\(899\) −0.787665 −0.0262701
\(900\) 0 0
\(901\) −1.61499 −0.0538030
\(902\) −6.20264 −0.206525
\(903\) 0 0
\(904\) 4.81861 0.160265
\(905\) −20.9745 −0.697215
\(906\) 0 0
\(907\) 11.9810 0.397822 0.198911 0.980018i \(-0.436260\pi\)
0.198911 + 0.980018i \(0.436260\pi\)
\(908\) −0.129586 −0.00430047
\(909\) 0 0
\(910\) 2.58315 0.0856306
\(911\) −27.5300 −0.912109 −0.456055 0.889952i \(-0.650738\pi\)
−0.456055 + 0.889952i \(0.650738\pi\)
\(912\) 0 0
\(913\) 92.0098 3.04508
\(914\) 3.37739 0.111714
\(915\) 0 0
\(916\) 7.66652 0.253309
\(917\) −65.8884 −2.17583
\(918\) 0 0
\(919\) −17.7250 −0.584694 −0.292347 0.956312i \(-0.594436\pi\)
−0.292347 + 0.956312i \(0.594436\pi\)
\(920\) 1.77191 0.0584183
\(921\) 0 0
\(922\) −1.79842 −0.0592277
\(923\) 0.715865 0.0235630
\(924\) 0 0
\(925\) −5.13021 −0.168680
\(926\) 1.56050 0.0512813
\(927\) 0 0
\(928\) −1.64154 −0.0538861
\(929\) −36.9971 −1.21383 −0.606917 0.794765i \(-0.707594\pi\)
−0.606917 + 0.794765i \(0.707594\pi\)
\(930\) 0 0
\(931\) −126.813 −4.15613
\(932\) −16.2101 −0.530980
\(933\) 0 0
\(934\) −1.10851 −0.0362717
\(935\) 13.2349 0.432828
\(936\) 0 0
\(937\) −36.0446 −1.17753 −0.588763 0.808306i \(-0.700385\pi\)
−0.588763 + 0.808306i \(0.700385\pi\)
\(938\) 0.368757 0.0120403
\(939\) 0 0
\(940\) −2.21197 −0.0721464
\(941\) 19.6256 0.639777 0.319889 0.947455i \(-0.396355\pi\)
0.319889 + 0.947455i \(0.396355\pi\)
\(942\) 0 0
\(943\) 28.3157 0.922087
\(944\) −49.6380 −1.61558
\(945\) 0 0
\(946\) 1.84339 0.0599339
\(947\) −18.9555 −0.615970 −0.307985 0.951391i \(-0.599655\pi\)
−0.307985 + 0.951391i \(0.599655\pi\)
\(948\) 0 0
\(949\) −4.14897 −0.134681
\(950\) 0.935207 0.0303421
\(951\) 0 0
\(952\) 7.30033 0.236605
\(953\) 30.2761 0.980738 0.490369 0.871515i \(-0.336862\pi\)
0.490369 + 0.871515i \(0.336862\pi\)
\(954\) 0 0
\(955\) −26.2094 −0.848116
\(956\) −22.3140 −0.721685
\(957\) 0 0
\(958\) −3.30635 −0.106823
\(959\) 20.7303 0.669415
\(960\) 0 0
\(961\) −30.3796 −0.979987
\(962\) −2.60956 −0.0841354
\(963\) 0 0
\(964\) 30.2308 0.973670
\(965\) −23.7643 −0.764999
\(966\) 0 0
\(967\) −20.4812 −0.658631 −0.329316 0.944220i \(-0.606818\pi\)
−0.329316 + 0.944220i \(0.606818\pi\)
\(968\) 8.15766 0.262197
\(969\) 0 0
\(970\) 1.75521 0.0563565
\(971\) −44.1566 −1.41705 −0.708526 0.705684i \(-0.750640\pi\)
−0.708526 + 0.705684i \(0.750640\pi\)
\(972\) 0 0
\(973\) −100.690 −3.22799
\(974\) 2.88792 0.0925348
\(975\) 0 0
\(976\) 34.6008 1.10754
\(977\) −0.492580 −0.0157590 −0.00787952 0.999969i \(-0.502508\pi\)
−0.00787952 + 0.999969i \(0.502508\pi\)
\(978\) 0 0
\(979\) 79.1919 2.53098
\(980\) −37.2176 −1.18887
\(981\) 0 0
\(982\) −2.64687 −0.0844651
\(983\) −27.5513 −0.878750 −0.439375 0.898304i \(-0.644800\pi\)
−0.439375 + 0.898304i \(0.644800\pi\)
\(984\) 0 0
\(985\) 25.6281 0.816580
\(986\) −0.361122 −0.0115005
\(987\) 0 0
\(988\) −49.0769 −1.56134
\(989\) −8.41529 −0.267590
\(990\) 0 0
\(991\) 23.0927 0.733563 0.366782 0.930307i \(-0.380460\pi\)
0.366782 + 0.930307i \(0.380460\pi\)
\(992\) 1.29298 0.0410522
\(993\) 0 0
\(994\) 0.137221 0.00435239
\(995\) 7.82757 0.248151
\(996\) 0 0
\(997\) 7.79593 0.246900 0.123450 0.992351i \(-0.460604\pi\)
0.123450 + 0.992351i \(0.460604\pi\)
\(998\) 1.08462 0.0343331
\(999\) 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 1305.2.a.r.1.2 4
3.2 odd 2 435.2.a.j.1.3 4
5.4 even 2 6525.2.a.bi.1.3 4
12.11 even 2 6960.2.a.co.1.1 4
15.2 even 4 2175.2.c.n.349.5 8
15.8 even 4 2175.2.c.n.349.4 8
15.14 odd 2 2175.2.a.v.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.3 4 3.2 odd 2
1305.2.a.r.1.2 4 1.1 even 1 trivial
2175.2.a.v.1.2 4 15.14 odd 2
2175.2.c.n.349.4 8 15.8 even 4
2175.2.c.n.349.5 8 15.2 even 4
6525.2.a.bi.1.3 4 5.4 even 2
6960.2.a.co.1.1 4 12.11 even 2