Properties

Label 5445.2.a.ca.1.6
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $8$
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] = [5445,2,Mod(1,5445)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5445, 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("5445.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.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: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 3x^{6} + 22x^{5} - 3x^{4} - 32x^{3} + 9x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.48523\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.485227 q^{2} -1.76455 q^{4} -1.00000 q^{5} -1.61947 q^{7} -1.82666 q^{8} +O(q^{10})\) \(q+0.485227 q^{2} -1.76455 q^{4} -1.00000 q^{5} -1.61947 q^{7} -1.82666 q^{8} -0.485227 q^{10} +4.95793 q^{13} -0.785812 q^{14} +2.64276 q^{16} -3.13866 q^{17} +1.09825 q^{19} +1.76455 q^{20} -6.75778 q^{23} +1.00000 q^{25} +2.40572 q^{26} +2.85765 q^{28} +3.55099 q^{29} +8.53688 q^{31} +4.93567 q^{32} -1.52296 q^{34} +1.61947 q^{35} -5.35976 q^{37} +0.532901 q^{38} +1.82666 q^{40} -5.32036 q^{41} +3.41085 q^{43} -3.27906 q^{46} +6.60156 q^{47} -4.37731 q^{49} +0.485227 q^{50} -8.74854 q^{52} +9.71622 q^{53} +2.95823 q^{56} +1.72304 q^{58} +7.18545 q^{59} -6.63497 q^{61} +4.14232 q^{62} -2.89061 q^{64} -4.95793 q^{65} +2.36043 q^{67} +5.53834 q^{68} +0.785812 q^{70} -16.6559 q^{71} +8.49548 q^{73} -2.60070 q^{74} -1.93792 q^{76} -1.99189 q^{79} -2.64276 q^{80} -2.58158 q^{82} +9.54929 q^{83} +3.13866 q^{85} +1.65504 q^{86} -2.65371 q^{89} -8.02924 q^{91} +11.9245 q^{92} +3.20326 q^{94} -1.09825 q^{95} -14.5239 q^{97} -2.12399 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8} + 4 q^{10} + 6 q^{13} - 14 q^{14} + 14 q^{16} - 8 q^{17} - 2 q^{19} - 6 q^{20} - 4 q^{23} + 8 q^{25} + 2 q^{26} + 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} + 20 q^{38} + 12 q^{40} - 22 q^{41} + 14 q^{43} - 2 q^{46} - 10 q^{47} - 4 q^{50} - 10 q^{52} + 18 q^{53} - 34 q^{56} + 12 q^{58} - 2 q^{59} - 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} + 2 q^{71} + 16 q^{73} - 24 q^{74} - 22 q^{76} - 16 q^{79} - 14 q^{80} + 10 q^{82} - 46 q^{83} + 8 q^{85} + 28 q^{86} - 38 q^{89} + 8 q^{91} + 24 q^{92} - 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.485227 0.343107 0.171554 0.985175i \(-0.445121\pi\)
0.171554 + 0.985175i \(0.445121\pi\)
\(3\) 0 0
\(4\) −1.76455 −0.882277
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.61947 −0.612103 −0.306052 0.952015i \(-0.599008\pi\)
−0.306052 + 0.952015i \(0.599008\pi\)
\(8\) −1.82666 −0.645823
\(9\) 0 0
\(10\) −0.485227 −0.153442
\(11\) 0 0
\(12\) 0 0
\(13\) 4.95793 1.37508 0.687542 0.726145i \(-0.258690\pi\)
0.687542 + 0.726145i \(0.258690\pi\)
\(14\) −0.785812 −0.210017
\(15\) 0 0
\(16\) 2.64276 0.660691
\(17\) −3.13866 −0.761238 −0.380619 0.924732i \(-0.624289\pi\)
−0.380619 + 0.924732i \(0.624289\pi\)
\(18\) 0 0
\(19\) 1.09825 0.251956 0.125978 0.992033i \(-0.459793\pi\)
0.125978 + 0.992033i \(0.459793\pi\)
\(20\) 1.76455 0.394566
\(21\) 0 0
\(22\) 0 0
\(23\) −6.75778 −1.40909 −0.704547 0.709657i \(-0.748850\pi\)
−0.704547 + 0.709657i \(0.748850\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.40572 0.471801
\(27\) 0 0
\(28\) 2.85765 0.540045
\(29\) 3.55099 0.659402 0.329701 0.944085i \(-0.393052\pi\)
0.329701 + 0.944085i \(0.393052\pi\)
\(30\) 0 0
\(31\) 8.53688 1.53327 0.766634 0.642084i \(-0.221930\pi\)
0.766634 + 0.642084i \(0.221930\pi\)
\(32\) 4.93567 0.872511
\(33\) 0 0
\(34\) −1.52296 −0.261186
\(35\) 1.61947 0.273741
\(36\) 0 0
\(37\) −5.35976 −0.881139 −0.440570 0.897718i \(-0.645223\pi\)
−0.440570 + 0.897718i \(0.645223\pi\)
\(38\) 0.532901 0.0864479
\(39\) 0 0
\(40\) 1.82666 0.288821
\(41\) −5.32036 −0.830901 −0.415451 0.909616i \(-0.636376\pi\)
−0.415451 + 0.909616i \(0.636376\pi\)
\(42\) 0 0
\(43\) 3.41085 0.520150 0.260075 0.965588i \(-0.416253\pi\)
0.260075 + 0.965588i \(0.416253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.27906 −0.483471
\(47\) 6.60156 0.962937 0.481469 0.876463i \(-0.340104\pi\)
0.481469 + 0.876463i \(0.340104\pi\)
\(48\) 0 0
\(49\) −4.37731 −0.625330
\(50\) 0.485227 0.0686215
\(51\) 0 0
\(52\) −8.74854 −1.21320
\(53\) 9.71622 1.33463 0.667313 0.744778i \(-0.267444\pi\)
0.667313 + 0.744778i \(0.267444\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.95823 0.395310
\(57\) 0 0
\(58\) 1.72304 0.226246
\(59\) 7.18545 0.935466 0.467733 0.883870i \(-0.345071\pi\)
0.467733 + 0.883870i \(0.345071\pi\)
\(60\) 0 0
\(61\) −6.63497 −0.849521 −0.424760 0.905306i \(-0.639642\pi\)
−0.424760 + 0.905306i \(0.639642\pi\)
\(62\) 4.14232 0.526076
\(63\) 0 0
\(64\) −2.89061 −0.361326
\(65\) −4.95793 −0.614956
\(66\) 0 0
\(67\) 2.36043 0.288373 0.144186 0.989551i \(-0.453944\pi\)
0.144186 + 0.989551i \(0.453944\pi\)
\(68\) 5.53834 0.671623
\(69\) 0 0
\(70\) 0.785812 0.0939225
\(71\) −16.6559 −1.97670 −0.988348 0.152212i \(-0.951360\pi\)
−0.988348 + 0.152212i \(0.951360\pi\)
\(72\) 0 0
\(73\) 8.49548 0.994320 0.497160 0.867659i \(-0.334376\pi\)
0.497160 + 0.867659i \(0.334376\pi\)
\(74\) −2.60070 −0.302325
\(75\) 0 0
\(76\) −1.93792 −0.222295
\(77\) 0 0
\(78\) 0 0
\(79\) −1.99189 −0.224105 −0.112053 0.993702i \(-0.535742\pi\)
−0.112053 + 0.993702i \(0.535742\pi\)
\(80\) −2.64276 −0.295470
\(81\) 0 0
\(82\) −2.58158 −0.285088
\(83\) 9.54929 1.04817 0.524085 0.851666i \(-0.324407\pi\)
0.524085 + 0.851666i \(0.324407\pi\)
\(84\) 0 0
\(85\) 3.13866 0.340436
\(86\) 1.65504 0.178467
\(87\) 0 0
\(88\) 0 0
\(89\) −2.65371 −0.281293 −0.140647 0.990060i \(-0.544918\pi\)
−0.140647 + 0.990060i \(0.544918\pi\)
\(90\) 0 0
\(91\) −8.02924 −0.841693
\(92\) 11.9245 1.24321
\(93\) 0 0
\(94\) 3.20326 0.330391
\(95\) −1.09825 −0.112678
\(96\) 0 0
\(97\) −14.5239 −1.47468 −0.737338 0.675524i \(-0.763918\pi\)
−0.737338 + 0.675524i \(0.763918\pi\)
\(98\) −2.12399 −0.214555
\(99\) 0 0
\(100\) −1.76455 −0.176455
\(101\) −17.6927 −1.76049 −0.880247 0.474517i \(-0.842623\pi\)
−0.880247 + 0.474517i \(0.842623\pi\)
\(102\) 0 0
\(103\) −4.93631 −0.486389 −0.243195 0.969978i \(-0.578195\pi\)
−0.243195 + 0.969978i \(0.578195\pi\)
\(104\) −9.05648 −0.888061
\(105\) 0 0
\(106\) 4.71457 0.457920
\(107\) −12.7475 −1.23235 −0.616173 0.787611i \(-0.711318\pi\)
−0.616173 + 0.787611i \(0.711318\pi\)
\(108\) 0 0
\(109\) 15.5953 1.49376 0.746879 0.664960i \(-0.231551\pi\)
0.746879 + 0.664960i \(0.231551\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.27988 −0.404411
\(113\) 5.63373 0.529977 0.264988 0.964252i \(-0.414632\pi\)
0.264988 + 0.964252i \(0.414632\pi\)
\(114\) 0 0
\(115\) 6.75778 0.630166
\(116\) −6.26592 −0.581776
\(117\) 0 0
\(118\) 3.48658 0.320965
\(119\) 5.08298 0.465956
\(120\) 0 0
\(121\) 0 0
\(122\) −3.21947 −0.291477
\(123\) 0 0
\(124\) −15.0638 −1.35277
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 7.36289 0.653351 0.326676 0.945137i \(-0.394072\pi\)
0.326676 + 0.945137i \(0.394072\pi\)
\(128\) −11.2739 −0.996485
\(129\) 0 0
\(130\) −2.40572 −0.210996
\(131\) 6.74024 0.588897 0.294449 0.955667i \(-0.404864\pi\)
0.294449 + 0.955667i \(0.404864\pi\)
\(132\) 0 0
\(133\) −1.77859 −0.154223
\(134\) 1.14535 0.0989428
\(135\) 0 0
\(136\) 5.73328 0.491625
\(137\) −6.32003 −0.539957 −0.269978 0.962866i \(-0.587017\pi\)
−0.269978 + 0.962866i \(0.587017\pi\)
\(138\) 0 0
\(139\) −14.0838 −1.19457 −0.597285 0.802029i \(-0.703754\pi\)
−0.597285 + 0.802029i \(0.703754\pi\)
\(140\) −2.85765 −0.241515
\(141\) 0 0
\(142\) −8.08191 −0.678219
\(143\) 0 0
\(144\) 0 0
\(145\) −3.55099 −0.294894
\(146\) 4.12224 0.341159
\(147\) 0 0
\(148\) 9.45759 0.777409
\(149\) −17.3475 −1.42116 −0.710582 0.703614i \(-0.751568\pi\)
−0.710582 + 0.703614i \(0.751568\pi\)
\(150\) 0 0
\(151\) −9.01455 −0.733593 −0.366797 0.930301i \(-0.619546\pi\)
−0.366797 + 0.930301i \(0.619546\pi\)
\(152\) −2.00613 −0.162719
\(153\) 0 0
\(154\) 0 0
\(155\) −8.53688 −0.685698
\(156\) 0 0
\(157\) 11.9364 0.952628 0.476314 0.879275i \(-0.341972\pi\)
0.476314 + 0.879275i \(0.341972\pi\)
\(158\) −0.966519 −0.0768921
\(159\) 0 0
\(160\) −4.93567 −0.390199
\(161\) 10.9440 0.862511
\(162\) 0 0
\(163\) −2.32743 −0.182298 −0.0911491 0.995837i \(-0.529054\pi\)
−0.0911491 + 0.995837i \(0.529054\pi\)
\(164\) 9.38807 0.733085
\(165\) 0 0
\(166\) 4.63357 0.359635
\(167\) −9.82442 −0.760237 −0.380118 0.924938i \(-0.624117\pi\)
−0.380118 + 0.924938i \(0.624117\pi\)
\(168\) 0 0
\(169\) 11.5811 0.890854
\(170\) 1.52296 0.116806
\(171\) 0 0
\(172\) −6.01864 −0.458917
\(173\) −23.2400 −1.76690 −0.883452 0.468521i \(-0.844787\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(174\) 0 0
\(175\) −1.61947 −0.122421
\(176\) 0 0
\(177\) 0 0
\(178\) −1.28765 −0.0965137
\(179\) 0.376550 0.0281447 0.0140724 0.999901i \(-0.495520\pi\)
0.0140724 + 0.999901i \(0.495520\pi\)
\(180\) 0 0
\(181\) −7.58158 −0.563534 −0.281767 0.959483i \(-0.590921\pi\)
−0.281767 + 0.959483i \(0.590921\pi\)
\(182\) −3.89600 −0.288791
\(183\) 0 0
\(184\) 12.3442 0.910026
\(185\) 5.35976 0.394057
\(186\) 0 0
\(187\) 0 0
\(188\) −11.6488 −0.849578
\(189\) 0 0
\(190\) −0.532901 −0.0386607
\(191\) −2.16401 −0.156583 −0.0782913 0.996931i \(-0.524946\pi\)
−0.0782913 + 0.996931i \(0.524946\pi\)
\(192\) 0 0
\(193\) −18.5881 −1.33800 −0.668999 0.743263i \(-0.733277\pi\)
−0.668999 + 0.743263i \(0.733277\pi\)
\(194\) −7.04738 −0.505972
\(195\) 0 0
\(196\) 7.72400 0.551714
\(197\) 8.70640 0.620306 0.310153 0.950687i \(-0.399620\pi\)
0.310153 + 0.950687i \(0.399620\pi\)
\(198\) 0 0
\(199\) −18.9231 −1.34142 −0.670712 0.741718i \(-0.734011\pi\)
−0.670712 + 0.741718i \(0.734011\pi\)
\(200\) −1.82666 −0.129165
\(201\) 0 0
\(202\) −8.58499 −0.604038
\(203\) −5.75073 −0.403622
\(204\) 0 0
\(205\) 5.32036 0.371590
\(206\) −2.39523 −0.166884
\(207\) 0 0
\(208\) 13.1026 0.908505
\(209\) 0 0
\(210\) 0 0
\(211\) 17.3522 1.19457 0.597287 0.802027i \(-0.296245\pi\)
0.597287 + 0.802027i \(0.296245\pi\)
\(212\) −17.1448 −1.17751
\(213\) 0 0
\(214\) −6.18542 −0.422827
\(215\) −3.41085 −0.232618
\(216\) 0 0
\(217\) −13.8252 −0.938519
\(218\) 7.56726 0.512519
\(219\) 0 0
\(220\) 0 0
\(221\) −15.5613 −1.04677
\(222\) 0 0
\(223\) 7.19082 0.481533 0.240767 0.970583i \(-0.422601\pi\)
0.240767 + 0.970583i \(0.422601\pi\)
\(224\) −7.99318 −0.534067
\(225\) 0 0
\(226\) 2.73364 0.181839
\(227\) −27.3012 −1.81205 −0.906023 0.423229i \(-0.860897\pi\)
−0.906023 + 0.423229i \(0.860897\pi\)
\(228\) 0 0
\(229\) 12.3763 0.817848 0.408924 0.912568i \(-0.365904\pi\)
0.408924 + 0.912568i \(0.365904\pi\)
\(230\) 3.27906 0.216215
\(231\) 0 0
\(232\) −6.48646 −0.425857
\(233\) −14.5447 −0.952857 −0.476429 0.879213i \(-0.658069\pi\)
−0.476429 + 0.879213i \(0.658069\pi\)
\(234\) 0 0
\(235\) −6.60156 −0.430639
\(236\) −12.6791 −0.825341
\(237\) 0 0
\(238\) 2.46640 0.159873
\(239\) −28.9267 −1.87112 −0.935558 0.353173i \(-0.885103\pi\)
−0.935558 + 0.353173i \(0.885103\pi\)
\(240\) 0 0
\(241\) −9.40610 −0.605900 −0.302950 0.953006i \(-0.597972\pi\)
−0.302950 + 0.953006i \(0.597972\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 11.7078 0.749513
\(245\) 4.37731 0.279656
\(246\) 0 0
\(247\) 5.44505 0.346460
\(248\) −15.5940 −0.990220
\(249\) 0 0
\(250\) −0.485227 −0.0306885
\(251\) −0.0325382 −0.00205379 −0.00102690 0.999999i \(-0.500327\pi\)
−0.00102690 + 0.999999i \(0.500327\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 3.57267 0.224170
\(255\) 0 0
\(256\) 0.310793 0.0194246
\(257\) −6.61680 −0.412745 −0.206372 0.978474i \(-0.566166\pi\)
−0.206372 + 0.978474i \(0.566166\pi\)
\(258\) 0 0
\(259\) 8.67999 0.539348
\(260\) 8.74854 0.542562
\(261\) 0 0
\(262\) 3.27055 0.202055
\(263\) −5.19057 −0.320064 −0.160032 0.987112i \(-0.551160\pi\)
−0.160032 + 0.987112i \(0.551160\pi\)
\(264\) 0 0
\(265\) −9.71622 −0.596863
\(266\) −0.863018 −0.0529150
\(267\) 0 0
\(268\) −4.16511 −0.254425
\(269\) 15.8774 0.968060 0.484030 0.875052i \(-0.339173\pi\)
0.484030 + 0.875052i \(0.339173\pi\)
\(270\) 0 0
\(271\) −26.2337 −1.59358 −0.796791 0.604255i \(-0.793471\pi\)
−0.796791 + 0.604255i \(0.793471\pi\)
\(272\) −8.29474 −0.502943
\(273\) 0 0
\(274\) −3.06665 −0.185263
\(275\) 0 0
\(276\) 0 0
\(277\) −6.51114 −0.391216 −0.195608 0.980682i \(-0.562668\pi\)
−0.195608 + 0.980682i \(0.562668\pi\)
\(278\) −6.83383 −0.409866
\(279\) 0 0
\(280\) −2.95823 −0.176788
\(281\) −11.2188 −0.669258 −0.334629 0.942350i \(-0.608611\pi\)
−0.334629 + 0.942350i \(0.608611\pi\)
\(282\) 0 0
\(283\) 8.27653 0.491989 0.245994 0.969271i \(-0.420886\pi\)
0.245994 + 0.969271i \(0.420886\pi\)
\(284\) 29.3903 1.74399
\(285\) 0 0
\(286\) 0 0
\(287\) 8.61619 0.508597
\(288\) 0 0
\(289\) −7.14879 −0.420517
\(290\) −1.72304 −0.101180
\(291\) 0 0
\(292\) −14.9907 −0.877266
\(293\) 0.418459 0.0244466 0.0122233 0.999925i \(-0.496109\pi\)
0.0122233 + 0.999925i \(0.496109\pi\)
\(294\) 0 0
\(295\) −7.18545 −0.418353
\(296\) 9.79048 0.569060
\(297\) 0 0
\(298\) −8.41749 −0.487612
\(299\) −33.5046 −1.93762
\(300\) 0 0
\(301\) −5.52378 −0.318386
\(302\) −4.37410 −0.251701
\(303\) 0 0
\(304\) 2.90241 0.166465
\(305\) 6.63497 0.379917
\(306\) 0 0
\(307\) −32.7941 −1.87166 −0.935828 0.352457i \(-0.885346\pi\)
−0.935828 + 0.352457i \(0.885346\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.14232 −0.235268
\(311\) 0.509602 0.0288969 0.0144484 0.999896i \(-0.495401\pi\)
0.0144484 + 0.999896i \(0.495401\pi\)
\(312\) 0 0
\(313\) 2.83268 0.160113 0.0800563 0.996790i \(-0.474490\pi\)
0.0800563 + 0.996790i \(0.474490\pi\)
\(314\) 5.79186 0.326854
\(315\) 0 0
\(316\) 3.51480 0.197723
\(317\) 26.9913 1.51598 0.757992 0.652264i \(-0.226181\pi\)
0.757992 + 0.652264i \(0.226181\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.89061 0.161590
\(321\) 0 0
\(322\) 5.31034 0.295934
\(323\) −3.44704 −0.191798
\(324\) 0 0
\(325\) 4.95793 0.275017
\(326\) −1.12933 −0.0625479
\(327\) 0 0
\(328\) 9.71852 0.536615
\(329\) −10.6911 −0.589417
\(330\) 0 0
\(331\) 21.5474 1.18435 0.592175 0.805809i \(-0.298269\pi\)
0.592175 + 0.805809i \(0.298269\pi\)
\(332\) −16.8502 −0.924777
\(333\) 0 0
\(334\) −4.76708 −0.260843
\(335\) −2.36043 −0.128964
\(336\) 0 0
\(337\) 10.9766 0.597932 0.298966 0.954264i \(-0.403358\pi\)
0.298966 + 0.954264i \(0.403358\pi\)
\(338\) 5.61946 0.305658
\(339\) 0 0
\(340\) −5.53834 −0.300359
\(341\) 0 0
\(342\) 0 0
\(343\) 18.4252 0.994870
\(344\) −6.23048 −0.335925
\(345\) 0 0
\(346\) −11.2767 −0.606238
\(347\) −16.7192 −0.897532 −0.448766 0.893649i \(-0.648136\pi\)
−0.448766 + 0.893649i \(0.648136\pi\)
\(348\) 0 0
\(349\) 4.59837 0.246145 0.123072 0.992398i \(-0.460725\pi\)
0.123072 + 0.992398i \(0.460725\pi\)
\(350\) −0.785812 −0.0420034
\(351\) 0 0
\(352\) 0 0
\(353\) 14.4628 0.769775 0.384887 0.922964i \(-0.374240\pi\)
0.384887 + 0.922964i \(0.374240\pi\)
\(354\) 0 0
\(355\) 16.6559 0.884005
\(356\) 4.68262 0.248179
\(357\) 0 0
\(358\) 0.182712 0.00965665
\(359\) 8.56634 0.452114 0.226057 0.974114i \(-0.427416\pi\)
0.226057 + 0.974114i \(0.427416\pi\)
\(360\) 0 0
\(361\) −17.7938 −0.936518
\(362\) −3.67879 −0.193353
\(363\) 0 0
\(364\) 14.1680 0.742607
\(365\) −8.49548 −0.444674
\(366\) 0 0
\(367\) 1.14927 0.0599913 0.0299956 0.999550i \(-0.490451\pi\)
0.0299956 + 0.999550i \(0.490451\pi\)
\(368\) −17.8592 −0.930975
\(369\) 0 0
\(370\) 2.60070 0.135204
\(371\) −15.7352 −0.816929
\(372\) 0 0
\(373\) 32.1348 1.66388 0.831939 0.554867i \(-0.187231\pi\)
0.831939 + 0.554867i \(0.187231\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0588 −0.621887
\(377\) 17.6056 0.906733
\(378\) 0 0
\(379\) 14.7386 0.757073 0.378537 0.925586i \(-0.376427\pi\)
0.378537 + 0.925586i \(0.376427\pi\)
\(380\) 1.93792 0.0994133
\(381\) 0 0
\(382\) −1.05004 −0.0537246
\(383\) −30.0887 −1.53746 −0.768732 0.639572i \(-0.779112\pi\)
−0.768732 + 0.639572i \(0.779112\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9.01943 −0.459077
\(387\) 0 0
\(388\) 25.6282 1.30107
\(389\) 34.6757 1.75813 0.879064 0.476704i \(-0.158169\pi\)
0.879064 + 0.476704i \(0.158169\pi\)
\(390\) 0 0
\(391\) 21.2104 1.07266
\(392\) 7.99587 0.403852
\(393\) 0 0
\(394\) 4.22458 0.212831
\(395\) 1.99189 0.100223
\(396\) 0 0
\(397\) 36.4341 1.82858 0.914288 0.405064i \(-0.132751\pi\)
0.914288 + 0.405064i \(0.132751\pi\)
\(398\) −9.18200 −0.460252
\(399\) 0 0
\(400\) 2.64276 0.132138
\(401\) 21.5435 1.07583 0.537917 0.842998i \(-0.319212\pi\)
0.537917 + 0.842998i \(0.319212\pi\)
\(402\) 0 0
\(403\) 42.3253 2.10837
\(404\) 31.2198 1.55324
\(405\) 0 0
\(406\) −2.79041 −0.138486
\(407\) 0 0
\(408\) 0 0
\(409\) −5.77447 −0.285529 −0.142765 0.989757i \(-0.545599\pi\)
−0.142765 + 0.989757i \(0.545599\pi\)
\(410\) 2.58158 0.127495
\(411\) 0 0
\(412\) 8.71039 0.429130
\(413\) −11.6366 −0.572602
\(414\) 0 0
\(415\) −9.54929 −0.468756
\(416\) 24.4707 1.19978
\(417\) 0 0
\(418\) 0 0
\(419\) −13.4745 −0.658273 −0.329137 0.944282i \(-0.606758\pi\)
−0.329137 + 0.944282i \(0.606758\pi\)
\(420\) 0 0
\(421\) −24.2366 −1.18122 −0.590610 0.806957i \(-0.701113\pi\)
−0.590610 + 0.806957i \(0.701113\pi\)
\(422\) 8.41976 0.409867
\(423\) 0 0
\(424\) −17.7483 −0.861932
\(425\) −3.13866 −0.152248
\(426\) 0 0
\(427\) 10.7452 0.519994
\(428\) 22.4936 1.08727
\(429\) 0 0
\(430\) −1.65504 −0.0798130
\(431\) −0.775904 −0.0373740 −0.0186870 0.999825i \(-0.505949\pi\)
−0.0186870 + 0.999825i \(0.505949\pi\)
\(432\) 0 0
\(433\) 34.5838 1.66199 0.830995 0.556280i \(-0.187772\pi\)
0.830995 + 0.556280i \(0.187772\pi\)
\(434\) −6.70838 −0.322013
\(435\) 0 0
\(436\) −27.5187 −1.31791
\(437\) −7.42173 −0.355030
\(438\) 0 0
\(439\) 7.24100 0.345594 0.172797 0.984957i \(-0.444720\pi\)
0.172797 + 0.984957i \(0.444720\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −7.55076 −0.359153
\(443\) −5.41215 −0.257139 −0.128570 0.991701i \(-0.541039\pi\)
−0.128570 + 0.991701i \(0.541039\pi\)
\(444\) 0 0
\(445\) 2.65371 0.125798
\(446\) 3.48918 0.165218
\(447\) 0 0
\(448\) 4.68126 0.221169
\(449\) 12.8909 0.608359 0.304180 0.952615i \(-0.401618\pi\)
0.304180 + 0.952615i \(0.401618\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −9.94102 −0.467586
\(453\) 0 0
\(454\) −13.2473 −0.621726
\(455\) 8.02924 0.376416
\(456\) 0 0
\(457\) 1.70545 0.0797777 0.0398888 0.999204i \(-0.487300\pi\)
0.0398888 + 0.999204i \(0.487300\pi\)
\(458\) 6.00531 0.280610
\(459\) 0 0
\(460\) −11.9245 −0.555981
\(461\) −29.9226 −1.39364 −0.696818 0.717248i \(-0.745401\pi\)
−0.696818 + 0.717248i \(0.745401\pi\)
\(462\) 0 0
\(463\) −33.9765 −1.57902 −0.789511 0.613736i \(-0.789666\pi\)
−0.789511 + 0.613736i \(0.789666\pi\)
\(464\) 9.38442 0.435661
\(465\) 0 0
\(466\) −7.05750 −0.326932
\(467\) −0.169922 −0.00786304 −0.00393152 0.999992i \(-0.501251\pi\)
−0.00393152 + 0.999992i \(0.501251\pi\)
\(468\) 0 0
\(469\) −3.82265 −0.176514
\(470\) −3.20326 −0.147755
\(471\) 0 0
\(472\) −13.1254 −0.604146
\(473\) 0 0
\(474\) 0 0
\(475\) 1.09825 0.0503912
\(476\) −8.96920 −0.411103
\(477\) 0 0
\(478\) −14.0360 −0.641994
\(479\) −14.6180 −0.667912 −0.333956 0.942589i \(-0.608384\pi\)
−0.333956 + 0.942589i \(0.608384\pi\)
\(480\) 0 0
\(481\) −26.5733 −1.21164
\(482\) −4.56410 −0.207889
\(483\) 0 0
\(484\) 0 0
\(485\) 14.5239 0.659495
\(486\) 0 0
\(487\) −23.9520 −1.08537 −0.542684 0.839937i \(-0.682592\pi\)
−0.542684 + 0.839937i \(0.682592\pi\)
\(488\) 12.1199 0.548640
\(489\) 0 0
\(490\) 2.12399 0.0959520
\(491\) −31.7288 −1.43190 −0.715950 0.698151i \(-0.754006\pi\)
−0.715950 + 0.698151i \(0.754006\pi\)
\(492\) 0 0
\(493\) −11.1454 −0.501962
\(494\) 2.64209 0.118873
\(495\) 0 0
\(496\) 22.5609 1.01302
\(497\) 26.9738 1.20994
\(498\) 0 0
\(499\) 33.1017 1.48183 0.740917 0.671596i \(-0.234391\pi\)
0.740917 + 0.671596i \(0.234391\pi\)
\(500\) 1.76455 0.0789133
\(501\) 0 0
\(502\) −0.0157884 −0.000704672 0
\(503\) −18.1421 −0.808917 −0.404459 0.914556i \(-0.632540\pi\)
−0.404459 + 0.914556i \(0.632540\pi\)
\(504\) 0 0
\(505\) 17.6927 0.787316
\(506\) 0 0
\(507\) 0 0
\(508\) −12.9922 −0.576437
\(509\) 16.6736 0.739044 0.369522 0.929222i \(-0.379522\pi\)
0.369522 + 0.929222i \(0.379522\pi\)
\(510\) 0 0
\(511\) −13.7582 −0.608627
\(512\) 22.6987 1.00315
\(513\) 0 0
\(514\) −3.21065 −0.141616
\(515\) 4.93631 0.217520
\(516\) 0 0
\(517\) 0 0
\(518\) 4.21176 0.185054
\(519\) 0 0
\(520\) 9.05648 0.397153
\(521\) −20.5225 −0.899108 −0.449554 0.893253i \(-0.648417\pi\)
−0.449554 + 0.893253i \(0.648417\pi\)
\(522\) 0 0
\(523\) 0.404664 0.0176947 0.00884736 0.999961i \(-0.497184\pi\)
0.00884736 + 0.999961i \(0.497184\pi\)
\(524\) −11.8935 −0.519571
\(525\) 0 0
\(526\) −2.51860 −0.109816
\(527\) −26.7944 −1.16718
\(528\) 0 0
\(529\) 22.6676 0.985546
\(530\) −4.71457 −0.204788
\(531\) 0 0
\(532\) 3.13841 0.136067
\(533\) −26.3780 −1.14256
\(534\) 0 0
\(535\) 12.7475 0.551122
\(536\) −4.31171 −0.186238
\(537\) 0 0
\(538\) 7.70412 0.332148
\(539\) 0 0
\(540\) 0 0
\(541\) −17.7037 −0.761142 −0.380571 0.924752i \(-0.624272\pi\)
−0.380571 + 0.924752i \(0.624272\pi\)
\(542\) −12.7293 −0.546769
\(543\) 0 0
\(544\) −15.4914 −0.664188
\(545\) −15.5953 −0.668029
\(546\) 0 0
\(547\) 5.79243 0.247666 0.123833 0.992303i \(-0.460481\pi\)
0.123833 + 0.992303i \(0.460481\pi\)
\(548\) 11.1520 0.476392
\(549\) 0 0
\(550\) 0 0
\(551\) 3.89987 0.166140
\(552\) 0 0
\(553\) 3.22581 0.137175
\(554\) −3.15938 −0.134229
\(555\) 0 0
\(556\) 24.8516 1.05394
\(557\) 41.1689 1.74438 0.872191 0.489166i \(-0.162699\pi\)
0.872191 + 0.489166i \(0.162699\pi\)
\(558\) 0 0
\(559\) 16.9108 0.715250
\(560\) 4.27988 0.180858
\(561\) 0 0
\(562\) −5.44367 −0.229627
\(563\) −12.0373 −0.507314 −0.253657 0.967294i \(-0.581633\pi\)
−0.253657 + 0.967294i \(0.581633\pi\)
\(564\) 0 0
\(565\) −5.63373 −0.237013
\(566\) 4.01600 0.168805
\(567\) 0 0
\(568\) 30.4248 1.27660
\(569\) −3.69378 −0.154851 −0.0774256 0.996998i \(-0.524670\pi\)
−0.0774256 + 0.996998i \(0.524670\pi\)
\(570\) 0 0
\(571\) −5.17733 −0.216665 −0.108332 0.994115i \(-0.534551\pi\)
−0.108332 + 0.994115i \(0.534551\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 4.18081 0.174503
\(575\) −6.75778 −0.281819
\(576\) 0 0
\(577\) 0.984135 0.0409701 0.0204850 0.999790i \(-0.493479\pi\)
0.0204850 + 0.999790i \(0.493479\pi\)
\(578\) −3.46879 −0.144282
\(579\) 0 0
\(580\) 6.26592 0.260178
\(581\) −15.4648 −0.641589
\(582\) 0 0
\(583\) 0 0
\(584\) −15.5184 −0.642155
\(585\) 0 0
\(586\) 0.203048 0.00838782
\(587\) −31.6510 −1.30638 −0.653188 0.757196i \(-0.726569\pi\)
−0.653188 + 0.757196i \(0.726569\pi\)
\(588\) 0 0
\(589\) 9.37563 0.386316
\(590\) −3.48658 −0.143540
\(591\) 0 0
\(592\) −14.1646 −0.582160
\(593\) 38.9330 1.59879 0.799394 0.600808i \(-0.205154\pi\)
0.799394 + 0.600808i \(0.205154\pi\)
\(594\) 0 0
\(595\) −5.08298 −0.208382
\(596\) 30.6107 1.25386
\(597\) 0 0
\(598\) −16.2573 −0.664812
\(599\) −1.69868 −0.0694063 −0.0347031 0.999398i \(-0.511049\pi\)
−0.0347031 + 0.999398i \(0.511049\pi\)
\(600\) 0 0
\(601\) −41.0817 −1.67576 −0.837879 0.545856i \(-0.816205\pi\)
−0.837879 + 0.545856i \(0.816205\pi\)
\(602\) −2.68029 −0.109240
\(603\) 0 0
\(604\) 15.9067 0.647233
\(605\) 0 0
\(606\) 0 0
\(607\) −21.8428 −0.886573 −0.443287 0.896380i \(-0.646188\pi\)
−0.443287 + 0.896380i \(0.646188\pi\)
\(608\) 5.42060 0.219834
\(609\) 0 0
\(610\) 3.21947 0.130352
\(611\) 32.7301 1.32412
\(612\) 0 0
\(613\) 45.7809 1.84907 0.924536 0.381096i \(-0.124453\pi\)
0.924536 + 0.381096i \(0.124453\pi\)
\(614\) −15.9126 −0.642179
\(615\) 0 0
\(616\) 0 0
\(617\) −37.0763 −1.49263 −0.746317 0.665590i \(-0.768180\pi\)
−0.746317 + 0.665590i \(0.768180\pi\)
\(618\) 0 0
\(619\) 16.2548 0.653334 0.326667 0.945139i \(-0.394074\pi\)
0.326667 + 0.945139i \(0.394074\pi\)
\(620\) 15.0638 0.604976
\(621\) 0 0
\(622\) 0.247273 0.00991474
\(623\) 4.29762 0.172180
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 1.37449 0.0549358
\(627\) 0 0
\(628\) −21.0624 −0.840482
\(629\) 16.8225 0.670756
\(630\) 0 0
\(631\) −9.08331 −0.361601 −0.180800 0.983520i \(-0.557869\pi\)
−0.180800 + 0.983520i \(0.557869\pi\)
\(632\) 3.63851 0.144732
\(633\) 0 0
\(634\) 13.0969 0.520145
\(635\) −7.36289 −0.292187
\(636\) 0 0
\(637\) −21.7024 −0.859880
\(638\) 0 0
\(639\) 0 0
\(640\) 11.2739 0.445641
\(641\) −7.45471 −0.294443 −0.147222 0.989104i \(-0.547033\pi\)
−0.147222 + 0.989104i \(0.547033\pi\)
\(642\) 0 0
\(643\) 44.1591 1.74147 0.870733 0.491756i \(-0.163645\pi\)
0.870733 + 0.491756i \(0.163645\pi\)
\(644\) −19.3114 −0.760974
\(645\) 0 0
\(646\) −1.67260 −0.0658074
\(647\) −11.4388 −0.449704 −0.224852 0.974393i \(-0.572190\pi\)
−0.224852 + 0.974393i \(0.572190\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 2.40572 0.0943602
\(651\) 0 0
\(652\) 4.10687 0.160838
\(653\) 1.97887 0.0774393 0.0387196 0.999250i \(-0.487672\pi\)
0.0387196 + 0.999250i \(0.487672\pi\)
\(654\) 0 0
\(655\) −6.74024 −0.263363
\(656\) −14.0605 −0.548969
\(657\) 0 0
\(658\) −5.18759 −0.202233
\(659\) −18.3062 −0.713107 −0.356553 0.934275i \(-0.616048\pi\)
−0.356553 + 0.934275i \(0.616048\pi\)
\(660\) 0 0
\(661\) −1.16189 −0.0451922 −0.0225961 0.999745i \(-0.507193\pi\)
−0.0225961 + 0.999745i \(0.507193\pi\)
\(662\) 10.4554 0.406359
\(663\) 0 0
\(664\) −17.4433 −0.676933
\(665\) 1.77859 0.0689706
\(666\) 0 0
\(667\) −23.9968 −0.929160
\(668\) 17.3357 0.670740
\(669\) 0 0
\(670\) −1.14535 −0.0442485
\(671\) 0 0
\(672\) 0 0
\(673\) 1.31368 0.0506385 0.0253192 0.999679i \(-0.491940\pi\)
0.0253192 + 0.999679i \(0.491940\pi\)
\(674\) 5.32613 0.205155
\(675\) 0 0
\(676\) −20.4355 −0.785980
\(677\) 19.2668 0.740484 0.370242 0.928935i \(-0.379275\pi\)
0.370242 + 0.928935i \(0.379275\pi\)
\(678\) 0 0
\(679\) 23.5210 0.902654
\(680\) −5.73328 −0.219861
\(681\) 0 0
\(682\) 0 0
\(683\) 7.15293 0.273699 0.136850 0.990592i \(-0.456302\pi\)
0.136850 + 0.990592i \(0.456302\pi\)
\(684\) 0 0
\(685\) 6.32003 0.241476
\(686\) 8.94043 0.341347
\(687\) 0 0
\(688\) 9.01407 0.343658
\(689\) 48.1724 1.83522
\(690\) 0 0
\(691\) 20.8320 0.792485 0.396242 0.918146i \(-0.370314\pi\)
0.396242 + 0.918146i \(0.370314\pi\)
\(692\) 41.0083 1.55890
\(693\) 0 0
\(694\) −8.11259 −0.307950
\(695\) 14.0838 0.534228
\(696\) 0 0
\(697\) 16.6988 0.632513
\(698\) 2.23125 0.0844541
\(699\) 0 0
\(700\) 2.85765 0.108009
\(701\) −18.7296 −0.707408 −0.353704 0.935357i \(-0.615078\pi\)
−0.353704 + 0.935357i \(0.615078\pi\)
\(702\) 0 0
\(703\) −5.88636 −0.222008
\(704\) 0 0
\(705\) 0 0
\(706\) 7.01772 0.264115
\(707\) 28.6529 1.07760
\(708\) 0 0
\(709\) −18.0328 −0.677236 −0.338618 0.940924i \(-0.609959\pi\)
−0.338618 + 0.940924i \(0.609959\pi\)
\(710\) 8.08191 0.303309
\(711\) 0 0
\(712\) 4.84744 0.181666
\(713\) −57.6903 −2.16052
\(714\) 0 0
\(715\) 0 0
\(716\) −0.664444 −0.0248314
\(717\) 0 0
\(718\) 4.15662 0.155124
\(719\) −39.4673 −1.47188 −0.735942 0.677045i \(-0.763260\pi\)
−0.735942 + 0.677045i \(0.763260\pi\)
\(720\) 0 0
\(721\) 7.99423 0.297721
\(722\) −8.63406 −0.321326
\(723\) 0 0
\(724\) 13.3781 0.497193
\(725\) 3.55099 0.131880
\(726\) 0 0
\(727\) 12.4002 0.459899 0.229950 0.973203i \(-0.426144\pi\)
0.229950 + 0.973203i \(0.426144\pi\)
\(728\) 14.6667 0.543585
\(729\) 0 0
\(730\) −4.12224 −0.152571
\(731\) −10.7055 −0.395958
\(732\) 0 0
\(733\) 10.9465 0.404320 0.202160 0.979353i \(-0.435204\pi\)
0.202160 + 0.979353i \(0.435204\pi\)
\(734\) 0.557656 0.0205834
\(735\) 0 0
\(736\) −33.3541 −1.22945
\(737\) 0 0
\(738\) 0 0
\(739\) 14.7047 0.540920 0.270460 0.962731i \(-0.412824\pi\)
0.270460 + 0.962731i \(0.412824\pi\)
\(740\) −9.45759 −0.347668
\(741\) 0 0
\(742\) −7.63512 −0.280294
\(743\) 28.8578 1.05869 0.529345 0.848407i \(-0.322438\pi\)
0.529345 + 0.848407i \(0.322438\pi\)
\(744\) 0 0
\(745\) 17.3475 0.635564
\(746\) 15.5927 0.570889
\(747\) 0 0
\(748\) 0 0
\(749\) 20.6442 0.754323
\(750\) 0 0
\(751\) −25.7450 −0.939447 −0.469723 0.882814i \(-0.655646\pi\)
−0.469723 + 0.882814i \(0.655646\pi\)
\(752\) 17.4464 0.636204
\(753\) 0 0
\(754\) 8.54270 0.311107
\(755\) 9.01455 0.328073
\(756\) 0 0
\(757\) 39.6545 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(758\) 7.15159 0.259757
\(759\) 0 0
\(760\) 2.00613 0.0727701
\(761\) 31.1718 1.12998 0.564988 0.825099i \(-0.308881\pi\)
0.564988 + 0.825099i \(0.308881\pi\)
\(762\) 0 0
\(763\) −25.2562 −0.914334
\(764\) 3.81852 0.138149
\(765\) 0 0
\(766\) −14.5999 −0.527515
\(767\) 35.6250 1.28634
\(768\) 0 0
\(769\) −38.5273 −1.38933 −0.694666 0.719333i \(-0.744448\pi\)
−0.694666 + 0.719333i \(0.744448\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 32.7997 1.18049
\(773\) 42.8510 1.54124 0.770621 0.637293i \(-0.219946\pi\)
0.770621 + 0.637293i \(0.219946\pi\)
\(774\) 0 0
\(775\) 8.53688 0.306654
\(776\) 26.5302 0.952380
\(777\) 0 0
\(778\) 16.8256 0.603226
\(779\) −5.84309 −0.209350
\(780\) 0 0
\(781\) 0 0
\(782\) 10.2919 0.368036
\(783\) 0 0
\(784\) −11.5682 −0.413149
\(785\) −11.9364 −0.426028
\(786\) 0 0
\(787\) 27.0485 0.964176 0.482088 0.876123i \(-0.339879\pi\)
0.482088 + 0.876123i \(0.339879\pi\)
\(788\) −15.3629 −0.547282
\(789\) 0 0
\(790\) 0.966519 0.0343872
\(791\) −9.12367 −0.324400
\(792\) 0 0
\(793\) −32.8957 −1.16816
\(794\) 17.6788 0.627398
\(795\) 0 0
\(796\) 33.3909 1.18351
\(797\) −10.3959 −0.368241 −0.184121 0.982904i \(-0.558944\pi\)
−0.184121 + 0.982904i \(0.558944\pi\)
\(798\) 0 0
\(799\) −20.7201 −0.733024
\(800\) 4.93567 0.174502
\(801\) 0 0
\(802\) 10.4535 0.369126
\(803\) 0 0
\(804\) 0 0
\(805\) −10.9440 −0.385727
\(806\) 20.5374 0.723398
\(807\) 0 0
\(808\) 32.3187 1.13697
\(809\) 39.9399 1.40421 0.702106 0.712073i \(-0.252243\pi\)
0.702106 + 0.712073i \(0.252243\pi\)
\(810\) 0 0
\(811\) −15.5894 −0.547417 −0.273709 0.961813i \(-0.588250\pi\)
−0.273709 + 0.961813i \(0.588250\pi\)
\(812\) 10.1475 0.356107
\(813\) 0 0
\(814\) 0 0
\(815\) 2.32743 0.0815263
\(816\) 0 0
\(817\) 3.74597 0.131055
\(818\) −2.80193 −0.0979671
\(819\) 0 0
\(820\) −9.38807 −0.327846
\(821\) −2.46398 −0.0859934 −0.0429967 0.999075i \(-0.513690\pi\)
−0.0429967 + 0.999075i \(0.513690\pi\)
\(822\) 0 0
\(823\) −55.1708 −1.92313 −0.961566 0.274573i \(-0.911464\pi\)
−0.961566 + 0.274573i \(0.911464\pi\)
\(824\) 9.01698 0.314122
\(825\) 0 0
\(826\) −5.64642 −0.196464
\(827\) −30.7906 −1.07069 −0.535346 0.844633i \(-0.679819\pi\)
−0.535346 + 0.844633i \(0.679819\pi\)
\(828\) 0 0
\(829\) −25.8676 −0.898418 −0.449209 0.893427i \(-0.648294\pi\)
−0.449209 + 0.893427i \(0.648294\pi\)
\(830\) −4.63357 −0.160834
\(831\) 0 0
\(832\) −14.3314 −0.496853
\(833\) 13.7389 0.476025
\(834\) 0 0
\(835\) 9.82442 0.339988
\(836\) 0 0
\(837\) 0 0
\(838\) −6.53820 −0.225858
\(839\) −52.4369 −1.81032 −0.905161 0.425070i \(-0.860250\pi\)
−0.905161 + 0.425070i \(0.860250\pi\)
\(840\) 0 0
\(841\) −16.3905 −0.565189
\(842\) −11.7603 −0.405285
\(843\) 0 0
\(844\) −30.6189 −1.05395
\(845\) −11.5811 −0.398402
\(846\) 0 0
\(847\) 0 0
\(848\) 25.6777 0.881775
\(849\) 0 0
\(850\) −1.52296 −0.0522373
\(851\) 36.2201 1.24161
\(852\) 0 0
\(853\) −29.8759 −1.02293 −0.511465 0.859304i \(-0.670897\pi\)
−0.511465 + 0.859304i \(0.670897\pi\)
\(854\) 5.21384 0.178414
\(855\) 0 0
\(856\) 23.2854 0.795877
\(857\) −50.9131 −1.73916 −0.869579 0.493794i \(-0.835610\pi\)
−0.869579 + 0.493794i \(0.835610\pi\)
\(858\) 0 0
\(859\) 30.2338 1.03156 0.515782 0.856720i \(-0.327501\pi\)
0.515782 + 0.856720i \(0.327501\pi\)
\(860\) 6.01864 0.205234
\(861\) 0 0
\(862\) −0.376490 −0.0128233
\(863\) 18.8078 0.640225 0.320112 0.947380i \(-0.396279\pi\)
0.320112 + 0.947380i \(0.396279\pi\)
\(864\) 0 0
\(865\) 23.2400 0.790184
\(866\) 16.7810 0.570241
\(867\) 0 0
\(868\) 24.3954 0.828034
\(869\) 0 0
\(870\) 0 0
\(871\) 11.7029 0.396536
\(872\) −28.4874 −0.964703
\(873\) 0 0
\(874\) −3.60122 −0.121813
\(875\) 1.61947 0.0547482
\(876\) 0 0
\(877\) −45.2915 −1.52939 −0.764693 0.644395i \(-0.777109\pi\)
−0.764693 + 0.644395i \(0.777109\pi\)
\(878\) 3.51353 0.118576
\(879\) 0 0
\(880\) 0 0
\(881\) 24.6233 0.829581 0.414791 0.909917i \(-0.363855\pi\)
0.414791 + 0.909917i \(0.363855\pi\)
\(882\) 0 0
\(883\) 55.9629 1.88330 0.941651 0.336591i \(-0.109274\pi\)
0.941651 + 0.336591i \(0.109274\pi\)
\(884\) 27.4587 0.923537
\(885\) 0 0
\(886\) −2.62612 −0.0882263
\(887\) −23.5352 −0.790234 −0.395117 0.918631i \(-0.629296\pi\)
−0.395117 + 0.918631i \(0.629296\pi\)
\(888\) 0 0
\(889\) −11.9240 −0.399918
\(890\) 1.28765 0.0431623
\(891\) 0 0
\(892\) −12.6886 −0.424846
\(893\) 7.25017 0.242618
\(894\) 0 0
\(895\) −0.376550 −0.0125867
\(896\) 18.2578 0.609951
\(897\) 0 0
\(898\) 6.25501 0.208732
\(899\) 30.3144 1.01104
\(900\) 0 0
\(901\) −30.4959 −1.01597
\(902\) 0 0
\(903\) 0 0
\(904\) −10.2909 −0.342271
\(905\) 7.58158 0.252020
\(906\) 0 0
\(907\) −45.6222 −1.51486 −0.757430 0.652916i \(-0.773545\pi\)
−0.757430 + 0.652916i \(0.773545\pi\)
\(908\) 48.1745 1.59873
\(909\) 0 0
\(910\) 3.89600 0.129151
\(911\) −46.8905 −1.55355 −0.776776 0.629778i \(-0.783146\pi\)
−0.776776 + 0.629778i \(0.783146\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.827531 0.0273723
\(915\) 0 0
\(916\) −21.8386 −0.721569
\(917\) −10.9156 −0.360466
\(918\) 0 0
\(919\) −33.8040 −1.11509 −0.557545 0.830146i \(-0.688257\pi\)
−0.557545 + 0.830146i \(0.688257\pi\)
\(920\) −12.3442 −0.406976
\(921\) 0 0
\(922\) −14.5193 −0.478167
\(923\) −82.5790 −2.71812
\(924\) 0 0
\(925\) −5.35976 −0.176228
\(926\) −16.4863 −0.541774
\(927\) 0 0
\(928\) 17.5265 0.575336
\(929\) 15.5132 0.508971 0.254485 0.967077i \(-0.418094\pi\)
0.254485 + 0.967077i \(0.418094\pi\)
\(930\) 0 0
\(931\) −4.80738 −0.157555
\(932\) 25.6650 0.840684
\(933\) 0 0
\(934\) −0.0824506 −0.00269787
\(935\) 0 0
\(936\) 0 0
\(937\) −4.34747 −0.142026 −0.0710128 0.997475i \(-0.522623\pi\)
−0.0710128 + 0.997475i \(0.522623\pi\)
\(938\) −1.85486 −0.0605632
\(939\) 0 0
\(940\) 11.6488 0.379943
\(941\) −18.7131 −0.610030 −0.305015 0.952348i \(-0.598662\pi\)
−0.305015 + 0.952348i \(0.598662\pi\)
\(942\) 0 0
\(943\) 35.9538 1.17082
\(944\) 18.9894 0.618054
\(945\) 0 0
\(946\) 0 0
\(947\) 18.6269 0.605294 0.302647 0.953103i \(-0.402130\pi\)
0.302647 + 0.953103i \(0.402130\pi\)
\(948\) 0 0
\(949\) 42.1200 1.36727
\(950\) 0.532901 0.0172896
\(951\) 0 0
\(952\) −9.28490 −0.300925
\(953\) 8.74307 0.283216 0.141608 0.989923i \(-0.454773\pi\)
0.141608 + 0.989923i \(0.454773\pi\)
\(954\) 0 0
\(955\) 2.16401 0.0700259
\(956\) 51.0428 1.65084
\(957\) 0 0
\(958\) −7.09304 −0.229166
\(959\) 10.2351 0.330509
\(960\) 0 0
\(961\) 41.8783 1.35091
\(962\) −12.8941 −0.415722
\(963\) 0 0
\(964\) 16.5976 0.534572
\(965\) 18.5881 0.598371
\(966\) 0 0
\(967\) 47.6843 1.53342 0.766712 0.641992i \(-0.221892\pi\)
0.766712 + 0.641992i \(0.221892\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 7.04738 0.226278
\(971\) 32.8229 1.05334 0.526669 0.850070i \(-0.323441\pi\)
0.526669 + 0.850070i \(0.323441\pi\)
\(972\) 0 0
\(973\) 22.8083 0.731200
\(974\) −11.6222 −0.372398
\(975\) 0 0
\(976\) −17.5346 −0.561270
\(977\) −46.1269 −1.47573 −0.737864 0.674949i \(-0.764166\pi\)
−0.737864 + 0.674949i \(0.764166\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −7.72400 −0.246734
\(981\) 0 0
\(982\) −15.3957 −0.491296
\(983\) −16.1706 −0.515761 −0.257881 0.966177i \(-0.583024\pi\)
−0.257881 + 0.966177i \(0.583024\pi\)
\(984\) 0 0
\(985\) −8.70640 −0.277409
\(986\) −5.40803 −0.172227
\(987\) 0 0
\(988\) −9.60809 −0.305674
\(989\) −23.0498 −0.732940
\(990\) 0 0
\(991\) 60.7308 1.92918 0.964589 0.263758i \(-0.0849622\pi\)
0.964589 + 0.263758i \(0.0849622\pi\)
\(992\) 42.1352 1.33779
\(993\) 0 0
\(994\) 13.0884 0.415140
\(995\) 18.9231 0.599903
\(996\) 0 0
\(997\) −46.9915 −1.48824 −0.744118 0.668048i \(-0.767130\pi\)
−0.744118 + 0.668048i \(0.767130\pi\)
\(998\) 16.0618 0.508428
\(999\) 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 5445.2.a.ca.1.6 8
3.2 odd 2 5445.2.a.cd.1.3 8
11.5 even 5 495.2.n.h.91.2 yes 16
11.9 even 5 495.2.n.h.136.2 yes 16
11.10 odd 2 5445.2.a.cc.1.3 8
33.5 odd 10 495.2.n.g.91.3 16
33.20 odd 10 495.2.n.g.136.3 yes 16
33.32 even 2 5445.2.a.cb.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.n.g.91.3 16 33.5 odd 10
495.2.n.g.136.3 yes 16 33.20 odd 10
495.2.n.h.91.2 yes 16 11.5 even 5
495.2.n.h.136.2 yes 16 11.9 even 5
5445.2.a.ca.1.6 8 1.1 even 1 trivial
5445.2.a.cb.1.6 8 33.32 even 2
5445.2.a.cc.1.3 8 11.10 odd 2
5445.2.a.cd.1.3 8 3.2 odd 2