Properties

Label 1815.2.a.w.1.1
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \(x^{4} - x^{3} - 3 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.09529\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.09529 q^{2} +1.00000 q^{3} -0.800331 q^{4} +1.00000 q^{5} -1.09529 q^{6} +0.705037 q^{7} +3.06719 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.09529 q^{2} +1.00000 q^{3} -0.800331 q^{4} +1.00000 q^{5} -1.09529 q^{6} +0.705037 q^{7} +3.06719 q^{8} +1.00000 q^{9} -1.09529 q^{10} -0.800331 q^{12} +4.71333 q^{13} -0.772223 q^{14} +1.00000 q^{15} -1.75881 q^{16} +7.78051 q^{17} -1.09529 q^{18} +1.19967 q^{19} -0.800331 q^{20} +0.705037 q^{21} -6.89318 q^{23} +3.06719 q^{24} +1.00000 q^{25} -5.16248 q^{26} +1.00000 q^{27} -0.564263 q^{28} -1.32741 q^{29} -1.09529 q^{30} -7.68126 q^{31} -4.20796 q^{32} -8.52195 q^{34} +0.705037 q^{35} -0.800331 q^{36} +8.43763 q^{37} -1.31399 q^{38} +4.71333 q^{39} +3.06719 q^{40} +0.232901 q^{41} -0.772223 q^{42} -7.32892 q^{43} +1.00000 q^{45} +7.55006 q^{46} +8.32228 q^{47} -1.75881 q^{48} -6.50292 q^{49} -1.09529 q^{50} +7.78051 q^{51} -3.77222 q^{52} +6.82332 q^{53} -1.09529 q^{54} +2.16248 q^{56} +1.19967 q^{57} +1.45390 q^{58} -3.54011 q^{59} -0.800331 q^{60} +10.8719 q^{61} +8.41324 q^{62} +0.705037 q^{63} +8.12657 q^{64} +4.71333 q^{65} -2.04036 q^{67} -6.22699 q^{68} -6.89318 q^{69} -0.772223 q^{70} +0.670527 q^{71} +3.06719 q^{72} +5.00433 q^{73} -9.24168 q^{74} +1.00000 q^{75} -0.960132 q^{76} -5.16248 q^{78} -2.28027 q^{79} -1.75881 q^{80} +1.00000 q^{81} -0.255095 q^{82} +2.10999 q^{83} -0.564263 q^{84} +7.78051 q^{85} +8.02732 q^{86} -1.32741 q^{87} -3.34722 q^{89} -1.09529 q^{90} +3.32307 q^{91} +5.51683 q^{92} -7.68126 q^{93} -9.11534 q^{94} +1.19967 q^{95} -4.20796 q^{96} +3.32228 q^{97} +7.12261 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{2} + 4q^{3} + q^{4} + 4q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 3q^{2} + 4q^{3} + q^{4} + 4q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 4q^{9} + 3q^{10} + q^{12} + 7q^{13} + 3q^{14} + 4q^{15} - q^{16} + 10q^{17} + 3q^{18} + 9q^{19} + q^{20} + 6q^{21} - 3q^{23} + 3q^{24} + 4q^{25} - 4q^{26} + 4q^{27} - 7q^{28} + 15q^{29} + 3q^{30} - 13q^{31} - 6q^{32} - 3q^{34} + 6q^{35} + q^{36} - 3q^{37} + 15q^{38} + 7q^{39} + 3q^{40} + 22q^{41} + 3q^{42} + 4q^{45} + q^{46} - 2q^{47} - q^{48} - 12q^{49} + 3q^{50} + 10q^{51} - 9q^{52} + 10q^{53} + 3q^{54} - 8q^{56} + 9q^{57} + 39q^{58} - 21q^{59} + q^{60} + 11q^{61} + 10q^{62} + 6q^{63} - 3q^{64} + 7q^{65} + q^{67} + 3q^{68} - 3q^{69} + 3q^{70} - 13q^{71} + 3q^{72} + q^{73} - 11q^{74} + 4q^{75} + 19q^{76} - 4q^{78} - 4q^{79} - q^{80} + 4q^{81} + 25q^{82} + 3q^{83} - 7q^{84} + 10q^{85} + 15q^{87} - 10q^{89} + 3q^{90} + 12q^{91} - 24q^{92} - 13q^{93} - 35q^{94} + 9q^{95} - 6q^{96} - 22q^{97} - 11q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.09529 −0.774490 −0.387245 0.921977i \(-0.626573\pi\)
−0.387245 + 0.921977i \(0.626573\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.800331 −0.400166
\(5\) 1.00000 0.447214
\(6\) −1.09529 −0.447152
\(7\) 0.705037 0.266479 0.133239 0.991084i \(-0.457462\pi\)
0.133239 + 0.991084i \(0.457462\pi\)
\(8\) 3.06719 1.08441
\(9\) 1.00000 0.333333
\(10\) −1.09529 −0.346362
\(11\) 0 0
\(12\) −0.800331 −0.231036
\(13\) 4.71333 1.30724 0.653621 0.756822i \(-0.273249\pi\)
0.653621 + 0.756822i \(0.273249\pi\)
\(14\) −0.772223 −0.206385
\(15\) 1.00000 0.258199
\(16\) −1.75881 −0.439702
\(17\) 7.78051 1.88705 0.943526 0.331299i \(-0.107487\pi\)
0.943526 + 0.331299i \(0.107487\pi\)
\(18\) −1.09529 −0.258163
\(19\) 1.19967 0.275223 0.137611 0.990486i \(-0.456057\pi\)
0.137611 + 0.990486i \(0.456057\pi\)
\(20\) −0.800331 −0.178959
\(21\) 0.705037 0.153852
\(22\) 0 0
\(23\) −6.89318 −1.43733 −0.718664 0.695358i \(-0.755246\pi\)
−0.718664 + 0.695358i \(0.755246\pi\)
\(24\) 3.06719 0.626087
\(25\) 1.00000 0.200000
\(26\) −5.16248 −1.01245
\(27\) 1.00000 0.192450
\(28\) −0.564263 −0.106636
\(29\) −1.32741 −0.246493 −0.123246 0.992376i \(-0.539331\pi\)
−0.123246 + 0.992376i \(0.539331\pi\)
\(30\) −1.09529 −0.199972
\(31\) −7.68126 −1.37960 −0.689798 0.724002i \(-0.742301\pi\)
−0.689798 + 0.724002i \(0.742301\pi\)
\(32\) −4.20796 −0.743869
\(33\) 0 0
\(34\) −8.52195 −1.46150
\(35\) 0.705037 0.119173
\(36\) −0.800331 −0.133389
\(37\) 8.43763 1.38714 0.693569 0.720391i \(-0.256037\pi\)
0.693569 + 0.720391i \(0.256037\pi\)
\(38\) −1.31399 −0.213157
\(39\) 4.71333 0.754737
\(40\) 3.06719 0.484965
\(41\) 0.232901 0.0363730 0.0181865 0.999835i \(-0.494211\pi\)
0.0181865 + 0.999835i \(0.494211\pi\)
\(42\) −0.772223 −0.119157
\(43\) −7.32892 −1.11765 −0.558825 0.829286i \(-0.688748\pi\)
−0.558825 + 0.829286i \(0.688748\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 7.55006 1.11320
\(47\) 8.32228 1.21393 0.606965 0.794729i \(-0.292387\pi\)
0.606965 + 0.794729i \(0.292387\pi\)
\(48\) −1.75881 −0.253862
\(49\) −6.50292 −0.928989
\(50\) −1.09529 −0.154898
\(51\) 7.78051 1.08949
\(52\) −3.77222 −0.523113
\(53\) 6.82332 0.937254 0.468627 0.883396i \(-0.344749\pi\)
0.468627 + 0.883396i \(0.344749\pi\)
\(54\) −1.09529 −0.149051
\(55\) 0 0
\(56\) 2.16248 0.288974
\(57\) 1.19967 0.158900
\(58\) 1.45390 0.190906
\(59\) −3.54011 −0.460883 −0.230442 0.973086i \(-0.574017\pi\)
−0.230442 + 0.973086i \(0.574017\pi\)
\(60\) −0.800331 −0.103322
\(61\) 10.8719 1.39200 0.695999 0.718043i \(-0.254962\pi\)
0.695999 + 0.718043i \(0.254962\pi\)
\(62\) 8.41324 1.06848
\(63\) 0.705037 0.0888263
\(64\) 8.12657 1.01582
\(65\) 4.71333 0.584616
\(66\) 0 0
\(67\) −2.04036 −0.249269 −0.124635 0.992203i \(-0.539776\pi\)
−0.124635 + 0.992203i \(0.539776\pi\)
\(68\) −6.22699 −0.755133
\(69\) −6.89318 −0.829841
\(70\) −0.772223 −0.0922983
\(71\) 0.670527 0.0795769 0.0397884 0.999208i \(-0.487332\pi\)
0.0397884 + 0.999208i \(0.487332\pi\)
\(72\) 3.06719 0.361471
\(73\) 5.00433 0.585713 0.292856 0.956156i \(-0.405394\pi\)
0.292856 + 0.956156i \(0.405394\pi\)
\(74\) −9.24168 −1.07432
\(75\) 1.00000 0.115470
\(76\) −0.960132 −0.110135
\(77\) 0 0
\(78\) −5.16248 −0.584536
\(79\) −2.28027 −0.256550 −0.128275 0.991739i \(-0.540944\pi\)
−0.128275 + 0.991739i \(0.540944\pi\)
\(80\) −1.75881 −0.196641
\(81\) 1.00000 0.111111
\(82\) −0.255095 −0.0281706
\(83\) 2.10999 0.231601 0.115801 0.993272i \(-0.463057\pi\)
0.115801 + 0.993272i \(0.463057\pi\)
\(84\) −0.564263 −0.0615662
\(85\) 7.78051 0.843915
\(86\) 8.02732 0.865608
\(87\) −1.32741 −0.142313
\(88\) 0 0
\(89\) −3.34722 −0.354805 −0.177402 0.984138i \(-0.556769\pi\)
−0.177402 + 0.984138i \(0.556769\pi\)
\(90\) −1.09529 −0.115454
\(91\) 3.32307 0.348353
\(92\) 5.51683 0.575169
\(93\) −7.68126 −0.796510
\(94\) −9.11534 −0.940176
\(95\) 1.19967 0.123083
\(96\) −4.20796 −0.429473
\(97\) 3.32228 0.337327 0.168663 0.985674i \(-0.446055\pi\)
0.168663 + 0.985674i \(0.446055\pi\)
\(98\) 7.12261 0.719492
\(99\) 0 0
\(100\) −0.800331 −0.0800331
\(101\) 18.9218 1.88279 0.941394 0.337310i \(-0.109517\pi\)
0.941394 + 0.337310i \(0.109517\pi\)
\(102\) −8.52195 −0.843799
\(103\) −18.0964 −1.78309 −0.891545 0.452932i \(-0.850378\pi\)
−0.891545 + 0.452932i \(0.850378\pi\)
\(104\) 14.4567 1.41759
\(105\) 0.705037 0.0688046
\(106\) −7.47354 −0.725894
\(107\) 6.11945 0.591589 0.295795 0.955252i \(-0.404416\pi\)
0.295795 + 0.955252i \(0.404416\pi\)
\(108\) −0.800331 −0.0770119
\(109\) −9.03128 −0.865039 −0.432520 0.901625i \(-0.642375\pi\)
−0.432520 + 0.901625i \(0.642375\pi\)
\(110\) 0 0
\(111\) 8.43763 0.800864
\(112\) −1.24002 −0.117171
\(113\) 3.91684 0.368466 0.184233 0.982883i \(-0.441020\pi\)
0.184233 + 0.982883i \(0.441020\pi\)
\(114\) −1.31399 −0.123066
\(115\) −6.89318 −0.642792
\(116\) 1.06236 0.0986380
\(117\) 4.71333 0.435747
\(118\) 3.87746 0.356949
\(119\) 5.48555 0.502860
\(120\) 3.06719 0.279994
\(121\) 0 0
\(122\) −11.9079 −1.07809
\(123\) 0.232901 0.0210000
\(124\) 6.14755 0.552067
\(125\) 1.00000 0.0894427
\(126\) −0.772223 −0.0687951
\(127\) −10.8675 −0.964336 −0.482168 0.876079i \(-0.660151\pi\)
−0.482168 + 0.876079i \(0.660151\pi\)
\(128\) −0.485063 −0.0428739
\(129\) −7.32892 −0.645275
\(130\) −5.16248 −0.452779
\(131\) 11.7094 1.02305 0.511526 0.859268i \(-0.329080\pi\)
0.511526 + 0.859268i \(0.329080\pi\)
\(132\) 0 0
\(133\) 0.845811 0.0733411
\(134\) 2.23479 0.193056
\(135\) 1.00000 0.0860663
\(136\) 23.8643 2.04635
\(137\) 20.4302 1.74547 0.872735 0.488194i \(-0.162344\pi\)
0.872735 + 0.488194i \(0.162344\pi\)
\(138\) 7.55006 0.642704
\(139\) 8.17100 0.693055 0.346528 0.938040i \(-0.387361\pi\)
0.346528 + 0.938040i \(0.387361\pi\)
\(140\) −0.564263 −0.0476889
\(141\) 8.32228 0.700862
\(142\) −0.734424 −0.0616315
\(143\) 0 0
\(144\) −1.75881 −0.146567
\(145\) −1.32741 −0.110235
\(146\) −5.48122 −0.453629
\(147\) −6.50292 −0.536352
\(148\) −6.75289 −0.555084
\(149\) 9.60217 0.786641 0.393320 0.919401i \(-0.371326\pi\)
0.393320 + 0.919401i \(0.371326\pi\)
\(150\) −1.09529 −0.0894304
\(151\) −6.90776 −0.562146 −0.281073 0.959686i \(-0.590690\pi\)
−0.281073 + 0.959686i \(0.590690\pi\)
\(152\) 3.67961 0.298456
\(153\) 7.78051 0.629017
\(154\) 0 0
\(155\) −7.68126 −0.616974
\(156\) −3.77222 −0.302020
\(157\) 11.3519 0.905980 0.452990 0.891516i \(-0.350357\pi\)
0.452990 + 0.891516i \(0.350357\pi\)
\(158\) 2.49757 0.198696
\(159\) 6.82332 0.541124
\(160\) −4.20796 −0.332668
\(161\) −4.85995 −0.383018
\(162\) −1.09529 −0.0860544
\(163\) −1.79253 −0.140402 −0.0702008 0.997533i \(-0.522364\pi\)
−0.0702008 + 0.997533i \(0.522364\pi\)
\(164\) −0.186398 −0.0145552
\(165\) 0 0
\(166\) −2.31106 −0.179373
\(167\) −6.02450 −0.466189 −0.233095 0.972454i \(-0.574885\pi\)
−0.233095 + 0.972454i \(0.574885\pi\)
\(168\) 2.16248 0.166839
\(169\) 9.21546 0.708882
\(170\) −8.52195 −0.653604
\(171\) 1.19967 0.0917410
\(172\) 5.86556 0.447245
\(173\) −4.18674 −0.318312 −0.159156 0.987253i \(-0.550877\pi\)
−0.159156 + 0.987253i \(0.550877\pi\)
\(174\) 1.45390 0.110220
\(175\) 0.705037 0.0532958
\(176\) 0 0
\(177\) −3.54011 −0.266091
\(178\) 3.66619 0.274793
\(179\) 4.85166 0.362630 0.181315 0.983425i \(-0.441965\pi\)
0.181315 + 0.983425i \(0.441965\pi\)
\(180\) −0.800331 −0.0596532
\(181\) −23.0877 −1.71610 −0.858049 0.513569i \(-0.828323\pi\)
−0.858049 + 0.513569i \(0.828323\pi\)
\(182\) −3.63974 −0.269795
\(183\) 10.8719 0.803670
\(184\) −21.1427 −1.55866
\(185\) 8.43763 0.620347
\(186\) 8.41324 0.616889
\(187\) 0 0
\(188\) −6.66058 −0.485773
\(189\) 0.705037 0.0512839
\(190\) −1.31399 −0.0953269
\(191\) 6.22571 0.450476 0.225238 0.974304i \(-0.427684\pi\)
0.225238 + 0.974304i \(0.427684\pi\)
\(192\) 8.12657 0.586485
\(193\) −19.0868 −1.37390 −0.686950 0.726705i \(-0.741051\pi\)
−0.686950 + 0.726705i \(0.741051\pi\)
\(194\) −3.63887 −0.261256
\(195\) 4.71333 0.337528
\(196\) 5.20449 0.371749
\(197\) −6.80056 −0.484520 −0.242260 0.970211i \(-0.577889\pi\)
−0.242260 + 0.970211i \(0.577889\pi\)
\(198\) 0 0
\(199\) 21.2972 1.50972 0.754860 0.655886i \(-0.227705\pi\)
0.754860 + 0.655886i \(0.227705\pi\)
\(200\) 3.06719 0.216883
\(201\) −2.04036 −0.143916
\(202\) −20.7249 −1.45820
\(203\) −0.935870 −0.0656852
\(204\) −6.22699 −0.435976
\(205\) 0.232901 0.0162665
\(206\) 19.8209 1.38099
\(207\) −6.89318 −0.479109
\(208\) −8.28984 −0.574797
\(209\) 0 0
\(210\) −0.772223 −0.0532884
\(211\) 8.89073 0.612063 0.306032 0.952021i \(-0.400999\pi\)
0.306032 + 0.952021i \(0.400999\pi\)
\(212\) −5.46091 −0.375057
\(213\) 0.670527 0.0459437
\(214\) −6.70259 −0.458180
\(215\) −7.32892 −0.499828
\(216\) 3.06719 0.208696
\(217\) −5.41558 −0.367633
\(218\) 9.89190 0.669964
\(219\) 5.00433 0.338162
\(220\) 0 0
\(221\) 36.6721 2.46683
\(222\) −9.24168 −0.620261
\(223\) 5.41720 0.362762 0.181381 0.983413i \(-0.441943\pi\)
0.181381 + 0.983413i \(0.441943\pi\)
\(224\) −2.96677 −0.198226
\(225\) 1.00000 0.0666667
\(226\) −4.29009 −0.285373
\(227\) 8.43842 0.560077 0.280039 0.959989i \(-0.409653\pi\)
0.280039 + 0.959989i \(0.409653\pi\)
\(228\) −0.960132 −0.0635863
\(229\) −11.2053 −0.740466 −0.370233 0.928939i \(-0.620722\pi\)
−0.370233 + 0.928939i \(0.620722\pi\)
\(230\) 7.55006 0.497836
\(231\) 0 0
\(232\) −4.07140 −0.267300
\(233\) −5.83979 −0.382577 −0.191289 0.981534i \(-0.561267\pi\)
−0.191289 + 0.981534i \(0.561267\pi\)
\(234\) −5.16248 −0.337482
\(235\) 8.32228 0.542886
\(236\) 2.83326 0.184430
\(237\) −2.28027 −0.148119
\(238\) −6.00829 −0.389460
\(239\) 16.5261 1.06899 0.534494 0.845173i \(-0.320502\pi\)
0.534494 + 0.845173i \(0.320502\pi\)
\(240\) −1.75881 −0.113531
\(241\) 3.29180 0.212043 0.106022 0.994364i \(-0.466189\pi\)
0.106022 + 0.994364i \(0.466189\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −8.70108 −0.557030
\(245\) −6.50292 −0.415456
\(246\) −0.255095 −0.0162643
\(247\) 5.65443 0.359783
\(248\) −23.5599 −1.49605
\(249\) 2.10999 0.133715
\(250\) −1.09529 −0.0692725
\(251\) −7.39934 −0.467042 −0.233521 0.972352i \(-0.575025\pi\)
−0.233521 + 0.972352i \(0.575025\pi\)
\(252\) −0.564263 −0.0355452
\(253\) 0 0
\(254\) 11.9031 0.746869
\(255\) 7.78051 0.487235
\(256\) −15.7219 −0.982616
\(257\) −18.6991 −1.16642 −0.583210 0.812322i \(-0.698203\pi\)
−0.583210 + 0.812322i \(0.698203\pi\)
\(258\) 8.02732 0.499759
\(259\) 5.94884 0.369643
\(260\) −3.77222 −0.233943
\(261\) −1.32741 −0.0821643
\(262\) −12.8252 −0.792344
\(263\) −3.99020 −0.246046 −0.123023 0.992404i \(-0.539259\pi\)
−0.123023 + 0.992404i \(0.539259\pi\)
\(264\) 0 0
\(265\) 6.82332 0.419153
\(266\) −0.926412 −0.0568020
\(267\) −3.34722 −0.204847
\(268\) 1.63296 0.0997489
\(269\) −12.7150 −0.775246 −0.387623 0.921818i \(-0.626704\pi\)
−0.387623 + 0.921818i \(0.626704\pi\)
\(270\) −1.09529 −0.0666575
\(271\) −23.8280 −1.44745 −0.723724 0.690090i \(-0.757571\pi\)
−0.723724 + 0.690090i \(0.757571\pi\)
\(272\) −13.6844 −0.829740
\(273\) 3.32307 0.201121
\(274\) −22.3771 −1.35185
\(275\) 0 0
\(276\) 5.51683 0.332074
\(277\) −7.41252 −0.445375 −0.222688 0.974890i \(-0.571483\pi\)
−0.222688 + 0.974890i \(0.571483\pi\)
\(278\) −8.94965 −0.536764
\(279\) −7.68126 −0.459865
\(280\) 2.16248 0.129233
\(281\) 29.0815 1.73485 0.867427 0.497564i \(-0.165772\pi\)
0.867427 + 0.497564i \(0.165772\pi\)
\(282\) −9.11534 −0.542811
\(283\) 21.6729 1.28832 0.644160 0.764891i \(-0.277207\pi\)
0.644160 + 0.764891i \(0.277207\pi\)
\(284\) −0.536643 −0.0318439
\(285\) 1.19967 0.0710623
\(286\) 0 0
\(287\) 0.164204 0.00969265
\(288\) −4.20796 −0.247956
\(289\) 43.5364 2.56096
\(290\) 1.45390 0.0853759
\(291\) 3.32228 0.194756
\(292\) −4.00512 −0.234382
\(293\) 8.41220 0.491446 0.245723 0.969340i \(-0.420975\pi\)
0.245723 + 0.969340i \(0.420975\pi\)
\(294\) 7.12261 0.415399
\(295\) −3.54011 −0.206113
\(296\) 25.8798 1.50423
\(297\) 0 0
\(298\) −10.5172 −0.609245
\(299\) −32.4898 −1.87893
\(300\) −0.800331 −0.0462071
\(301\) −5.16716 −0.297830
\(302\) 7.56603 0.435376
\(303\) 18.9218 1.08703
\(304\) −2.10999 −0.121016
\(305\) 10.8719 0.622520
\(306\) −8.52195 −0.487167
\(307\) −23.7431 −1.35509 −0.677545 0.735481i \(-0.736956\pi\)
−0.677545 + 0.735481i \(0.736956\pi\)
\(308\) 0 0
\(309\) −18.0964 −1.02947
\(310\) 8.41324 0.477840
\(311\) 23.0471 1.30688 0.653440 0.756979i \(-0.273325\pi\)
0.653440 + 0.756979i \(0.273325\pi\)
\(312\) 14.4567 0.818447
\(313\) −17.3638 −0.981460 −0.490730 0.871312i \(-0.663270\pi\)
−0.490730 + 0.871312i \(0.663270\pi\)
\(314\) −12.4337 −0.701673
\(315\) 0.705037 0.0397243
\(316\) 1.82497 0.102663
\(317\) −6.15095 −0.345472 −0.172736 0.984968i \(-0.555261\pi\)
−0.172736 + 0.984968i \(0.555261\pi\)
\(318\) −7.47354 −0.419095
\(319\) 0 0
\(320\) 8.12657 0.454289
\(321\) 6.11945 0.341554
\(322\) 5.32307 0.296643
\(323\) 9.33404 0.519360
\(324\) −0.800331 −0.0444628
\(325\) 4.71333 0.261448
\(326\) 1.96335 0.108740
\(327\) −9.03128 −0.499431
\(328\) 0.714351 0.0394434
\(329\) 5.86752 0.323487
\(330\) 0 0
\(331\) 19.4191 1.06737 0.533685 0.845683i \(-0.320807\pi\)
0.533685 + 0.845683i \(0.320807\pi\)
\(332\) −1.68869 −0.0926788
\(333\) 8.43763 0.462379
\(334\) 6.59859 0.361059
\(335\) −2.04036 −0.111477
\(336\) −1.24002 −0.0676489
\(337\) −31.6868 −1.72609 −0.863045 0.505127i \(-0.831446\pi\)
−0.863045 + 0.505127i \(0.831446\pi\)
\(338\) −10.0936 −0.549022
\(339\) 3.91684 0.212734
\(340\) −6.22699 −0.337706
\(341\) 0 0
\(342\) −1.31399 −0.0710524
\(343\) −9.52006 −0.514035
\(344\) −22.4791 −1.21199
\(345\) −6.89318 −0.371116
\(346\) 4.58571 0.246529
\(347\) 27.9434 1.50008 0.750041 0.661392i \(-0.230034\pi\)
0.750041 + 0.661392i \(0.230034\pi\)
\(348\) 1.06236 0.0569487
\(349\) 0.683331 0.0365779 0.0182889 0.999833i \(-0.494178\pi\)
0.0182889 + 0.999833i \(0.494178\pi\)
\(350\) −0.772223 −0.0412771
\(351\) 4.71333 0.251579
\(352\) 0 0
\(353\) 1.55900 0.0829769 0.0414885 0.999139i \(-0.486790\pi\)
0.0414885 + 0.999139i \(0.486790\pi\)
\(354\) 3.87746 0.206085
\(355\) 0.670527 0.0355879
\(356\) 2.67889 0.141981
\(357\) 5.48555 0.290326
\(358\) −5.31399 −0.280853
\(359\) −15.8404 −0.836022 −0.418011 0.908442i \(-0.637273\pi\)
−0.418011 + 0.908442i \(0.637273\pi\)
\(360\) 3.06719 0.161655
\(361\) −17.5608 −0.924252
\(362\) 25.2878 1.32910
\(363\) 0 0
\(364\) −2.65956 −0.139399
\(365\) 5.00433 0.261939
\(366\) −11.9079 −0.622434
\(367\) 15.7361 0.821419 0.410710 0.911766i \(-0.365281\pi\)
0.410710 + 0.911766i \(0.365281\pi\)
\(368\) 12.1238 0.631996
\(369\) 0.232901 0.0121243
\(370\) −9.24168 −0.480452
\(371\) 4.81069 0.249759
\(372\) 6.14755 0.318736
\(373\) 5.27703 0.273234 0.136617 0.990624i \(-0.456377\pi\)
0.136617 + 0.990624i \(0.456377\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 25.5260 1.31640
\(377\) −6.25650 −0.322226
\(378\) −0.772223 −0.0397189
\(379\) 33.5093 1.72125 0.860627 0.509235i \(-0.170072\pi\)
0.860627 + 0.509235i \(0.170072\pi\)
\(380\) −0.960132 −0.0492537
\(381\) −10.8675 −0.556760
\(382\) −6.81898 −0.348889
\(383\) −20.2323 −1.03382 −0.516910 0.856039i \(-0.672918\pi\)
−0.516910 + 0.856039i \(0.672918\pi\)
\(384\) −0.485063 −0.0247532
\(385\) 0 0
\(386\) 20.9057 1.06407
\(387\) −7.32892 −0.372550
\(388\) −2.65892 −0.134986
\(389\) −27.5729 −1.39800 −0.699000 0.715122i \(-0.746371\pi\)
−0.699000 + 0.715122i \(0.746371\pi\)
\(390\) −5.16248 −0.261412
\(391\) −53.6325 −2.71231
\(392\) −19.9457 −1.00741
\(393\) 11.7094 0.590660
\(394\) 7.44862 0.375256
\(395\) −2.28027 −0.114733
\(396\) 0 0
\(397\) 11.7601 0.590222 0.295111 0.955463i \(-0.404643\pi\)
0.295111 + 0.955463i \(0.404643\pi\)
\(398\) −23.3267 −1.16926
\(399\) 0.845811 0.0423435
\(400\) −1.75881 −0.0879404
\(401\) −7.72406 −0.385721 −0.192861 0.981226i \(-0.561777\pi\)
−0.192861 + 0.981226i \(0.561777\pi\)
\(402\) 2.23479 0.111461
\(403\) −36.2043 −1.80347
\(404\) −15.1437 −0.753427
\(405\) 1.00000 0.0496904
\(406\) 1.02505 0.0508725
\(407\) 0 0
\(408\) 23.8643 1.18146
\(409\) −16.7409 −0.827783 −0.413892 0.910326i \(-0.635831\pi\)
−0.413892 + 0.910326i \(0.635831\pi\)
\(410\) −0.255095 −0.0125983
\(411\) 20.4302 1.00775
\(412\) 14.4831 0.713531
\(413\) −2.49591 −0.122816
\(414\) 7.55006 0.371065
\(415\) 2.10999 0.103575
\(416\) −19.8335 −0.972417
\(417\) 8.17100 0.400136
\(418\) 0 0
\(419\) −38.0968 −1.86115 −0.930576 0.366100i \(-0.880693\pi\)
−0.930576 + 0.366100i \(0.880693\pi\)
\(420\) −0.564263 −0.0275332
\(421\) −22.6633 −1.10454 −0.552272 0.833664i \(-0.686239\pi\)
−0.552272 + 0.833664i \(0.686239\pi\)
\(422\) −9.73797 −0.474037
\(423\) 8.32228 0.404643
\(424\) 20.9284 1.01637
\(425\) 7.78051 0.377410
\(426\) −0.734424 −0.0355829
\(427\) 7.66506 0.370938
\(428\) −4.89758 −0.236734
\(429\) 0 0
\(430\) 8.02732 0.387112
\(431\) 33.9766 1.63660 0.818299 0.574793i \(-0.194918\pi\)
0.818299 + 0.574793i \(0.194918\pi\)
\(432\) −1.75881 −0.0846207
\(433\) −36.6753 −1.76250 −0.881251 0.472649i \(-0.843298\pi\)
−0.881251 + 0.472649i \(0.843298\pi\)
\(434\) 5.93165 0.284728
\(435\) −1.32741 −0.0636442
\(436\) 7.22801 0.346159
\(437\) −8.26953 −0.395585
\(438\) −5.48122 −0.261903
\(439\) −1.05012 −0.0501193 −0.0250596 0.999686i \(-0.507978\pi\)
−0.0250596 + 0.999686i \(0.507978\pi\)
\(440\) 0 0
\(441\) −6.50292 −0.309663
\(442\) −40.1667 −1.91054
\(443\) −30.5206 −1.45008 −0.725039 0.688708i \(-0.758178\pi\)
−0.725039 + 0.688708i \(0.758178\pi\)
\(444\) −6.75289 −0.320478
\(445\) −3.34722 −0.158674
\(446\) −5.93342 −0.280956
\(447\) 9.60217 0.454167
\(448\) 5.72953 0.270695
\(449\) −36.5695 −1.72582 −0.862910 0.505357i \(-0.831361\pi\)
−0.862910 + 0.505357i \(0.831361\pi\)
\(450\) −1.09529 −0.0516327
\(451\) 0 0
\(452\) −3.13477 −0.147447
\(453\) −6.90776 −0.324555
\(454\) −9.24255 −0.433774
\(455\) 3.32307 0.155788
\(456\) 3.67961 0.172313
\(457\) 2.38409 0.111523 0.0557615 0.998444i \(-0.482241\pi\)
0.0557615 + 0.998444i \(0.482241\pi\)
\(458\) 12.2731 0.573483
\(459\) 7.78051 0.363163
\(460\) 5.51683 0.257223
\(461\) −28.5962 −1.33186 −0.665929 0.746015i \(-0.731965\pi\)
−0.665929 + 0.746015i \(0.731965\pi\)
\(462\) 0 0
\(463\) −30.5806 −1.42120 −0.710600 0.703596i \(-0.751577\pi\)
−0.710600 + 0.703596i \(0.751577\pi\)
\(464\) 2.33465 0.108383
\(465\) −7.68126 −0.356210
\(466\) 6.39628 0.296302
\(467\) −3.74219 −0.173168 −0.0865840 0.996245i \(-0.527595\pi\)
−0.0865840 + 0.996245i \(0.527595\pi\)
\(468\) −3.77222 −0.174371
\(469\) −1.43853 −0.0664250
\(470\) −9.11534 −0.420459
\(471\) 11.3519 0.523068
\(472\) −10.8582 −0.499788
\(473\) 0 0
\(474\) 2.49757 0.114717
\(475\) 1.19967 0.0550446
\(476\) −4.39026 −0.201227
\(477\) 6.82332 0.312418
\(478\) −18.1010 −0.827920
\(479\) 0.944951 0.0431759 0.0215880 0.999767i \(-0.493128\pi\)
0.0215880 + 0.999767i \(0.493128\pi\)
\(480\) −4.20796 −0.192066
\(481\) 39.7693 1.81332
\(482\) −3.60548 −0.164225
\(483\) −4.85995 −0.221135
\(484\) 0 0
\(485\) 3.32228 0.150857
\(486\) −1.09529 −0.0496835
\(487\) −13.3873 −0.606638 −0.303319 0.952889i \(-0.598095\pi\)
−0.303319 + 0.952889i \(0.598095\pi\)
\(488\) 33.3460 1.50950
\(489\) −1.79253 −0.0810609
\(490\) 7.12261 0.321767
\(491\) 2.78887 0.125860 0.0629300 0.998018i \(-0.479955\pi\)
0.0629300 + 0.998018i \(0.479955\pi\)
\(492\) −0.186398 −0.00840347
\(493\) −10.3279 −0.465145
\(494\) −6.19327 −0.278648
\(495\) 0 0
\(496\) 13.5099 0.606611
\(497\) 0.472746 0.0212056
\(498\) −2.31106 −0.103561
\(499\) 17.7790 0.795899 0.397950 0.917407i \(-0.369722\pi\)
0.397950 + 0.917407i \(0.369722\pi\)
\(500\) −0.800331 −0.0357919
\(501\) −6.02450 −0.269155
\(502\) 8.10445 0.361719
\(503\) −25.7838 −1.14964 −0.574822 0.818279i \(-0.694929\pi\)
−0.574822 + 0.818279i \(0.694929\pi\)
\(504\) 2.16248 0.0963245
\(505\) 18.9218 0.842008
\(506\) 0 0
\(507\) 9.21546 0.409273
\(508\) 8.69761 0.385894
\(509\) −28.2301 −1.25128 −0.625639 0.780113i \(-0.715162\pi\)
−0.625639 + 0.780113i \(0.715162\pi\)
\(510\) −8.52195 −0.377358
\(511\) 3.52824 0.156080
\(512\) 18.1902 0.803900
\(513\) 1.19967 0.0529667
\(514\) 20.4810 0.903380
\(515\) −18.0964 −0.797422
\(516\) 5.86556 0.258217
\(517\) 0 0
\(518\) −6.51573 −0.286285
\(519\) −4.18674 −0.183778
\(520\) 14.4567 0.633966
\(521\) 11.6955 0.512388 0.256194 0.966625i \(-0.417531\pi\)
0.256194 + 0.966625i \(0.417531\pi\)
\(522\) 1.45390 0.0636354
\(523\) 19.6871 0.860857 0.430429 0.902625i \(-0.358362\pi\)
0.430429 + 0.902625i \(0.358362\pi\)
\(524\) −9.37137 −0.409390
\(525\) 0.705037 0.0307703
\(526\) 4.37044 0.190560
\(527\) −59.7642 −2.60337
\(528\) 0 0
\(529\) 24.5159 1.06591
\(530\) −7.47354 −0.324630
\(531\) −3.54011 −0.153628
\(532\) −0.676929 −0.0293486
\(533\) 1.09774 0.0475484
\(534\) 3.66619 0.158652
\(535\) 6.11945 0.264567
\(536\) −6.25815 −0.270311
\(537\) 4.85166 0.209364
\(538\) 13.9266 0.600420
\(539\) 0 0
\(540\) −0.800331 −0.0344408
\(541\) 5.45092 0.234353 0.117177 0.993111i \(-0.462616\pi\)
0.117177 + 0.993111i \(0.462616\pi\)
\(542\) 26.0987 1.12103
\(543\) −23.0877 −0.990789
\(544\) −32.7401 −1.40372
\(545\) −9.03128 −0.386857
\(546\) −3.63974 −0.155766
\(547\) −12.7892 −0.546827 −0.273414 0.961897i \(-0.588153\pi\)
−0.273414 + 0.961897i \(0.588153\pi\)
\(548\) −16.3509 −0.698477
\(549\) 10.8719 0.463999
\(550\) 0 0
\(551\) −1.59245 −0.0678405
\(552\) −21.1427 −0.899891
\(553\) −1.60767 −0.0683653
\(554\) 8.11889 0.344939
\(555\) 8.43763 0.358157
\(556\) −6.53951 −0.277337
\(557\) 0.354468 0.0150193 0.00750964 0.999972i \(-0.497610\pi\)
0.00750964 + 0.999972i \(0.497610\pi\)
\(558\) 8.41324 0.356161
\(559\) −34.5436 −1.46104
\(560\) −1.24002 −0.0524006
\(561\) 0 0
\(562\) −31.8528 −1.34363
\(563\) −35.0818 −1.47852 −0.739261 0.673419i \(-0.764825\pi\)
−0.739261 + 0.673419i \(0.764825\pi\)
\(564\) −6.66058 −0.280461
\(565\) 3.91684 0.164783
\(566\) −23.7382 −0.997791
\(567\) 0.705037 0.0296088
\(568\) 2.05663 0.0862943
\(569\) 16.3179 0.684084 0.342042 0.939685i \(-0.388881\pi\)
0.342042 + 0.939685i \(0.388881\pi\)
\(570\) −1.31399 −0.0550370
\(571\) 14.4160 0.603291 0.301645 0.953420i \(-0.402464\pi\)
0.301645 + 0.953420i \(0.402464\pi\)
\(572\) 0 0
\(573\) 6.22571 0.260083
\(574\) −0.179852 −0.00750686
\(575\) −6.89318 −0.287465
\(576\) 8.12657 0.338607
\(577\) 8.90863 0.370871 0.185435 0.982656i \(-0.440630\pi\)
0.185435 + 0.982656i \(0.440630\pi\)
\(578\) −47.6852 −1.98344
\(579\) −19.0868 −0.793221
\(580\) 1.06236 0.0441122
\(581\) 1.48762 0.0617168
\(582\) −3.63887 −0.150836
\(583\) 0 0
\(584\) 15.3492 0.635155
\(585\) 4.71333 0.194872
\(586\) −9.21383 −0.380620
\(587\) −13.8014 −0.569644 −0.284822 0.958580i \(-0.591934\pi\)
−0.284822 + 0.958580i \(0.591934\pi\)
\(588\) 5.20449 0.214630
\(589\) −9.21497 −0.379696
\(590\) 3.87746 0.159633
\(591\) −6.80056 −0.279738
\(592\) −14.8402 −0.609927
\(593\) 16.4676 0.676242 0.338121 0.941103i \(-0.390209\pi\)
0.338121 + 0.941103i \(0.390209\pi\)
\(594\) 0 0
\(595\) 5.48555 0.224886
\(596\) −7.68492 −0.314787
\(597\) 21.2972 0.871638
\(598\) 35.5859 1.45522
\(599\) 37.8599 1.54692 0.773458 0.633848i \(-0.218525\pi\)
0.773458 + 0.633848i \(0.218525\pi\)
\(600\) 3.06719 0.125217
\(601\) −1.56441 −0.0638135 −0.0319067 0.999491i \(-0.510158\pi\)
−0.0319067 + 0.999491i \(0.510158\pi\)
\(602\) 5.65956 0.230666
\(603\) −2.04036 −0.0830897
\(604\) 5.52850 0.224951
\(605\) 0 0
\(606\) −20.7249 −0.841892
\(607\) −18.8523 −0.765191 −0.382595 0.923916i \(-0.624970\pi\)
−0.382595 + 0.923916i \(0.624970\pi\)
\(608\) −5.04816 −0.204730
\(609\) −0.935870 −0.0379234
\(610\) −11.9079 −0.482136
\(611\) 39.2256 1.58690
\(612\) −6.22699 −0.251711
\(613\) −32.6700 −1.31953 −0.659765 0.751472i \(-0.729344\pi\)
−0.659765 + 0.751472i \(0.729344\pi\)
\(614\) 26.0057 1.04950
\(615\) 0.232901 0.00939148
\(616\) 0 0
\(617\) −33.6559 −1.35493 −0.677467 0.735553i \(-0.736922\pi\)
−0.677467 + 0.735553i \(0.736922\pi\)
\(618\) 19.8209 0.797312
\(619\) −8.94001 −0.359329 −0.179665 0.983728i \(-0.557501\pi\)
−0.179665 + 0.983728i \(0.557501\pi\)
\(620\) 6.14755 0.246892
\(621\) −6.89318 −0.276614
\(622\) −25.2433 −1.01216
\(623\) −2.35992 −0.0945480
\(624\) −8.28984 −0.331859
\(625\) 1.00000 0.0400000
\(626\) 19.0185 0.760131
\(627\) 0 0
\(628\) −9.08528 −0.362542
\(629\) 65.6491 2.61760
\(630\) −0.772223 −0.0307661
\(631\) −19.2577 −0.766638 −0.383319 0.923616i \(-0.625219\pi\)
−0.383319 + 0.923616i \(0.625219\pi\)
\(632\) −6.99401 −0.278207
\(633\) 8.89073 0.353375
\(634\) 6.73710 0.267565
\(635\) −10.8675 −0.431264
\(636\) −5.46091 −0.216539
\(637\) −30.6504 −1.21441
\(638\) 0 0
\(639\) 0.670527 0.0265256
\(640\) −0.485063 −0.0191738
\(641\) −14.8928 −0.588229 −0.294114 0.955770i \(-0.595025\pi\)
−0.294114 + 0.955770i \(0.595025\pi\)
\(642\) −6.70259 −0.264530
\(643\) 35.3193 1.39286 0.696429 0.717626i \(-0.254771\pi\)
0.696429 + 0.717626i \(0.254771\pi\)
\(644\) 3.88957 0.153270
\(645\) −7.32892 −0.288576
\(646\) −10.2235 −0.402239
\(647\) 31.1428 1.22435 0.612175 0.790722i \(-0.290295\pi\)
0.612175 + 0.790722i \(0.290295\pi\)
\(648\) 3.06719 0.120490
\(649\) 0 0
\(650\) −5.16248 −0.202489
\(651\) −5.41558 −0.212253
\(652\) 1.43462 0.0561839
\(653\) 5.05494 0.197815 0.0989075 0.995097i \(-0.468465\pi\)
0.0989075 + 0.995097i \(0.468465\pi\)
\(654\) 9.89190 0.386804
\(655\) 11.7094 0.457523
\(656\) −0.409628 −0.0159933
\(657\) 5.00433 0.195238
\(658\) −6.42666 −0.250537
\(659\) 14.4486 0.562837 0.281419 0.959585i \(-0.409195\pi\)
0.281419 + 0.959585i \(0.409195\pi\)
\(660\) 0 0
\(661\) 14.8696 0.578361 0.289181 0.957275i \(-0.406617\pi\)
0.289181 + 0.957275i \(0.406617\pi\)
\(662\) −21.2696 −0.826667
\(663\) 36.6721 1.42423
\(664\) 6.47172 0.251152
\(665\) 0.845811 0.0327991
\(666\) −9.24168 −0.358108
\(667\) 9.15004 0.354291
\(668\) 4.82159 0.186553
\(669\) 5.41720 0.209441
\(670\) 2.23479 0.0863375
\(671\) 0 0
\(672\) −2.96677 −0.114446
\(673\) 41.6153 1.60415 0.802075 0.597223i \(-0.203729\pi\)
0.802075 + 0.597223i \(0.203729\pi\)
\(674\) 34.7064 1.33684
\(675\) 1.00000 0.0384900
\(676\) −7.37542 −0.283670
\(677\) −44.2366 −1.70015 −0.850076 0.526660i \(-0.823444\pi\)
−0.850076 + 0.526660i \(0.823444\pi\)
\(678\) −4.29009 −0.164760
\(679\) 2.34233 0.0898904
\(680\) 23.8643 0.915153
\(681\) 8.43842 0.323361
\(682\) 0 0
\(683\) 24.5318 0.938683 0.469342 0.883017i \(-0.344491\pi\)
0.469342 + 0.883017i \(0.344491\pi\)
\(684\) −0.960132 −0.0367116
\(685\) 20.4302 0.780598
\(686\) 10.4273 0.398115
\(687\) −11.2053 −0.427508
\(688\) 12.8902 0.491433
\(689\) 32.1605 1.22522
\(690\) 7.55006 0.287426
\(691\) 23.7670 0.904139 0.452069 0.891983i \(-0.350686\pi\)
0.452069 + 0.891983i \(0.350686\pi\)
\(692\) 3.35078 0.127378
\(693\) 0 0
\(694\) −30.6063 −1.16180
\(695\) 8.17100 0.309944
\(696\) −4.07140 −0.154326
\(697\) 1.81209 0.0686378
\(698\) −0.748449 −0.0283292
\(699\) −5.83979 −0.220881
\(700\) −0.564263 −0.0213271
\(701\) 32.6939 1.23483 0.617416 0.786637i \(-0.288180\pi\)
0.617416 + 0.786637i \(0.288180\pi\)
\(702\) −5.16248 −0.194845
\(703\) 10.1224 0.381772
\(704\) 0 0
\(705\) 8.32228 0.313435
\(706\) −1.70756 −0.0642648
\(707\) 13.3406 0.501723
\(708\) 2.83326 0.106480
\(709\) 23.8264 0.894821 0.447410 0.894329i \(-0.352346\pi\)
0.447410 + 0.894329i \(0.352346\pi\)
\(710\) −0.734424 −0.0275624
\(711\) −2.28027 −0.0855168
\(712\) −10.2666 −0.384755
\(713\) 52.9483 1.98293
\(714\) −6.00829 −0.224855
\(715\) 0 0
\(716\) −3.88293 −0.145112
\(717\) 16.5261 0.617180
\(718\) 17.3499 0.647491
\(719\) −2.84680 −0.106168 −0.0530838 0.998590i \(-0.516905\pi\)
−0.0530838 + 0.998590i \(0.516905\pi\)
\(720\) −1.75881 −0.0655469
\(721\) −12.7586 −0.475156
\(722\) 19.2342 0.715824
\(723\) 3.29180 0.122423
\(724\) 18.4778 0.686723
\(725\) −1.32741 −0.0492986
\(726\) 0 0
\(727\) −13.5192 −0.501399 −0.250700 0.968065i \(-0.580661\pi\)
−0.250700 + 0.968065i \(0.580661\pi\)
\(728\) 10.1925 0.377758
\(729\) 1.00000 0.0370370
\(730\) −5.48122 −0.202869
\(731\) −57.0227 −2.10906
\(732\) −8.70108 −0.321601
\(733\) −32.8352 −1.21280 −0.606399 0.795161i \(-0.707386\pi\)
−0.606399 + 0.795161i \(0.707386\pi\)
\(734\) −17.2357 −0.636181
\(735\) −6.50292 −0.239864
\(736\) 29.0062 1.06918
\(737\) 0 0
\(738\) −0.255095 −0.00939018
\(739\) −50.8927 −1.87212 −0.936058 0.351844i \(-0.885555\pi\)
−0.936058 + 0.351844i \(0.885555\pi\)
\(740\) −6.75289 −0.248241
\(741\) 5.65443 0.207721
\(742\) −5.26912 −0.193435
\(743\) 15.4855 0.568107 0.284054 0.958808i \(-0.408321\pi\)
0.284054 + 0.958808i \(0.408321\pi\)
\(744\) −23.5599 −0.863746
\(745\) 9.60217 0.351796
\(746\) −5.77990 −0.211617
\(747\) 2.10999 0.0772004
\(748\) 0 0
\(749\) 4.31444 0.157646
\(750\) −1.09529 −0.0399945
\(751\) 11.0575 0.403494 0.201747 0.979438i \(-0.435338\pi\)
0.201747 + 0.979438i \(0.435338\pi\)
\(752\) −14.6373 −0.533767
\(753\) −7.39934 −0.269647
\(754\) 6.85270 0.249561
\(755\) −6.90776 −0.251399
\(756\) −0.564263 −0.0205221
\(757\) 9.00282 0.327213 0.163607 0.986526i \(-0.447687\pi\)
0.163607 + 0.986526i \(0.447687\pi\)
\(758\) −36.7025 −1.33309
\(759\) 0 0
\(760\) 3.67961 0.133473
\(761\) −8.29644 −0.300746 −0.150373 0.988629i \(-0.548047\pi\)
−0.150373 + 0.988629i \(0.548047\pi\)
\(762\) 11.9031 0.431205
\(763\) −6.36738 −0.230515
\(764\) −4.98263 −0.180265
\(765\) 7.78051 0.281305
\(766\) 22.1603 0.800684
\(767\) −16.6857 −0.602486
\(768\) −15.7219 −0.567313
\(769\) −2.89088 −0.104248 −0.0521239 0.998641i \(-0.516599\pi\)
−0.0521239 + 0.998641i \(0.516599\pi\)
\(770\) 0 0
\(771\) −18.6991 −0.673432
\(772\) 15.2758 0.549787
\(773\) −27.8477 −1.00161 −0.500807 0.865559i \(-0.666963\pi\)
−0.500807 + 0.865559i \(0.666963\pi\)
\(774\) 8.02732 0.288536
\(775\) −7.68126 −0.275919
\(776\) 10.1901 0.365802
\(777\) 5.94884 0.213413
\(778\) 30.2004 1.08274
\(779\) 0.279404 0.0100107
\(780\) −3.77222 −0.135067
\(781\) 0 0
\(782\) 58.7433 2.10066
\(783\) −1.32741 −0.0474376
\(784\) 11.4374 0.408478
\(785\) 11.3519 0.405167
\(786\) −12.8252 −0.457460
\(787\) 24.8356 0.885292 0.442646 0.896696i \(-0.354040\pi\)
0.442646 + 0.896696i \(0.354040\pi\)
\(788\) 5.44270 0.193888
\(789\) −3.99020 −0.142055
\(790\) 2.49757 0.0888594
\(791\) 2.76152 0.0981883
\(792\) 0 0
\(793\) 51.2426 1.81968
\(794\) −12.8808 −0.457121
\(795\) 6.82332 0.241998
\(796\) −17.0448 −0.604138
\(797\) −2.81107 −0.0995731 −0.0497866 0.998760i \(-0.515854\pi\)
−0.0497866 + 0.998760i \(0.515854\pi\)
\(798\) −0.926412 −0.0327946
\(799\) 64.7516 2.29075
\(800\) −4.20796 −0.148774
\(801\) −3.34722 −0.118268
\(802\) 8.46012 0.298737
\(803\) 0 0
\(804\) 1.63296 0.0575901
\(805\) −4.85995 −0.171291
\(806\) 39.6544 1.39677
\(807\) −12.7150 −0.447589
\(808\) 58.0366 2.04172
\(809\) −11.6766 −0.410526 −0.205263 0.978707i \(-0.565805\pi\)
−0.205263 + 0.978707i \(0.565805\pi\)
\(810\) −1.09529 −0.0384847
\(811\) −24.0690 −0.845175 −0.422588 0.906322i \(-0.638878\pi\)
−0.422588 + 0.906322i \(0.638878\pi\)
\(812\) 0.749006 0.0262849
\(813\) −23.8280 −0.835684
\(814\) 0 0
\(815\) −1.79253 −0.0627895
\(816\) −13.6844 −0.479051
\(817\) −8.79227 −0.307603
\(818\) 18.3362 0.641110
\(819\) 3.32307 0.116118
\(820\) −0.186398 −0.00650930
\(821\) −4.85561 −0.169462 −0.0847310 0.996404i \(-0.527003\pi\)
−0.0847310 + 0.996404i \(0.527003\pi\)
\(822\) −22.3771 −0.780490
\(823\) 26.0601 0.908397 0.454198 0.890901i \(-0.349926\pi\)
0.454198 + 0.890901i \(0.349926\pi\)
\(824\) −55.5050 −1.93361
\(825\) 0 0
\(826\) 2.73376 0.0951195
\(827\) 16.2843 0.566260 0.283130 0.959081i \(-0.408627\pi\)
0.283130 + 0.959081i \(0.408627\pi\)
\(828\) 5.51683 0.191723
\(829\) −10.0141 −0.347805 −0.173903 0.984763i \(-0.555638\pi\)
−0.173903 + 0.984763i \(0.555638\pi\)
\(830\) −2.31106 −0.0802179
\(831\) −7.41252 −0.257137
\(832\) 38.3032 1.32792
\(833\) −50.5961 −1.75305
\(834\) −8.94965 −0.309901
\(835\) −6.02450 −0.208486
\(836\) 0 0
\(837\) −7.68126 −0.265503
\(838\) 41.7272 1.44144
\(839\) 11.7532 0.405766 0.202883 0.979203i \(-0.434969\pi\)
0.202883 + 0.979203i \(0.434969\pi\)
\(840\) 2.16248 0.0746126
\(841\) −27.2380 −0.939241
\(842\) 24.8230 0.855458
\(843\) 29.0815 1.00162
\(844\) −7.11553 −0.244927
\(845\) 9.21546 0.317021
\(846\) −9.11534 −0.313392
\(847\) 0 0
\(848\) −12.0009 −0.412113
\(849\) 21.6729 0.743812
\(850\) −8.52195 −0.292300
\(851\) −58.1621 −1.99377
\(852\) −0.536643 −0.0183851
\(853\) 45.6353 1.56252 0.781262 0.624203i \(-0.214576\pi\)
0.781262 + 0.624203i \(0.214576\pi\)
\(854\) −8.39549 −0.287288
\(855\) 1.19967 0.0410278
\(856\) 18.7695 0.641527
\(857\) 8.41558 0.287471 0.143735 0.989616i \(-0.454089\pi\)
0.143735 + 0.989616i \(0.454089\pi\)
\(858\) 0 0
\(859\) −45.3009 −1.54565 −0.772823 0.634622i \(-0.781156\pi\)
−0.772823 + 0.634622i \(0.781156\pi\)
\(860\) 5.86556 0.200014
\(861\) 0.164204 0.00559606
\(862\) −37.2144 −1.26753
\(863\) −12.0590 −0.410493 −0.205247 0.978710i \(-0.565800\pi\)
−0.205247 + 0.978710i \(0.565800\pi\)
\(864\) −4.20796 −0.143158
\(865\) −4.18674 −0.142354
\(866\) 40.1702 1.36504
\(867\) 43.5364 1.47857
\(868\) 4.33425 0.147114
\(869\) 0 0
\(870\) 1.45390 0.0492918
\(871\) −9.61687 −0.325855
\(872\) −27.7006 −0.938061
\(873\) 3.32228 0.112442
\(874\) 9.05757 0.306377
\(875\) 0.705037 0.0238346
\(876\) −4.00512 −0.135321
\(877\) 21.1439 0.713979 0.356989 0.934108i \(-0.383803\pi\)
0.356989 + 0.934108i \(0.383803\pi\)
\(878\) 1.15019 0.0388169
\(879\) 8.41220 0.283736
\(880\) 0 0
\(881\) −13.1669 −0.443605 −0.221803 0.975092i \(-0.571194\pi\)
−0.221803 + 0.975092i \(0.571194\pi\)
\(882\) 7.12261 0.239831
\(883\) −34.7697 −1.17009 −0.585047 0.810999i \(-0.698924\pi\)
−0.585047 + 0.810999i \(0.698924\pi\)
\(884\) −29.3498 −0.987142
\(885\) −3.54011 −0.119000
\(886\) 33.4290 1.12307
\(887\) 20.9565 0.703651 0.351826 0.936066i \(-0.385561\pi\)
0.351826 + 0.936066i \(0.385561\pi\)
\(888\) 25.8798 0.868468
\(889\) −7.66200 −0.256975
\(890\) 3.66619 0.122891
\(891\) 0 0
\(892\) −4.33555 −0.145165
\(893\) 9.98398 0.334101
\(894\) −10.5172 −0.351748
\(895\) 4.85166 0.162173
\(896\) −0.341987 −0.0114250
\(897\) −32.4898 −1.08480
\(898\) 40.0543 1.33663
\(899\) 10.1961 0.340061
\(900\) −0.800331 −0.0266777
\(901\) 53.0889 1.76865
\(902\) 0 0
\(903\) −5.16716 −0.171952
\(904\) 12.0137 0.399569
\(905\) −23.0877 −0.767462
\(906\) 7.56603 0.251365
\(907\) 26.2408 0.871312 0.435656 0.900113i \(-0.356516\pi\)
0.435656 + 0.900113i \(0.356516\pi\)
\(908\) −6.75353 −0.224124
\(909\) 18.9218 0.627596
\(910\) −3.63974 −0.120656
\(911\) −39.6599 −1.31399 −0.656995 0.753895i \(-0.728173\pi\)
−0.656995 + 0.753895i \(0.728173\pi\)
\(912\) −2.10999 −0.0698687
\(913\) 0 0
\(914\) −2.61128 −0.0863734
\(915\) 10.8719 0.359412
\(916\) 8.96793 0.296309
\(917\) 8.25554 0.272622
\(918\) −8.52195 −0.281266
\(919\) −14.9758 −0.494005 −0.247003 0.969015i \(-0.579446\pi\)
−0.247003 + 0.969015i \(0.579446\pi\)
\(920\) −21.1427 −0.697053
\(921\) −23.7431 −0.782361
\(922\) 31.3213 1.03151
\(923\) 3.16041 0.104026
\(924\) 0 0
\(925\) 8.43763 0.277427
\(926\) 33.4947 1.10071
\(927\) −18.0964 −0.594364
\(928\) 5.58567 0.183359
\(929\) −49.6461 −1.62884 −0.814418 0.580279i \(-0.802943\pi\)
−0.814418 + 0.580279i \(0.802943\pi\)
\(930\) 8.41324 0.275881
\(931\) −7.80135 −0.255679
\(932\) 4.67376 0.153094
\(933\) 23.0471 0.754527
\(934\) 4.09880 0.134117
\(935\) 0 0
\(936\) 14.4567 0.472530
\(937\) 54.2914 1.77362 0.886811 0.462132i \(-0.152915\pi\)
0.886811 + 0.462132i \(0.152915\pi\)
\(938\) 1.57561 0.0514455
\(939\) −17.3638 −0.566646
\(940\) −6.66058 −0.217244
\(941\) −6.66066 −0.217131 −0.108566 0.994089i \(-0.534626\pi\)
−0.108566 + 0.994089i \(0.534626\pi\)
\(942\) −12.4337 −0.405111
\(943\) −1.60543 −0.0522800
\(944\) 6.22638 0.202651
\(945\) 0.705037 0.0229349
\(946\) 0 0
\(947\) −42.6250 −1.38513 −0.692563 0.721358i \(-0.743518\pi\)
−0.692563 + 0.721358i \(0.743518\pi\)
\(948\) 1.82497 0.0592723
\(949\) 23.5871 0.765669
\(950\) −1.31399 −0.0426315
\(951\) −6.15095 −0.199458
\(952\) 16.8252 0.545308
\(953\) 16.7539 0.542713 0.271357 0.962479i \(-0.412528\pi\)
0.271357 + 0.962479i \(0.412528\pi\)
\(954\) −7.47354 −0.241965
\(955\) 6.22571 0.201459
\(956\) −13.2264 −0.427772
\(957\) 0 0
\(958\) −1.03500 −0.0334393
\(959\) 14.4040 0.465131
\(960\) 8.12657 0.262284
\(961\) 28.0018 0.903284
\(962\) −43.5591 −1.40440
\(963\) 6.11945 0.197196
\(964\) −2.63453 −0.0848524
\(965\) −19.0868 −0.614427
\(966\) 5.32307 0.171267
\(967\) 50.1233 1.61186 0.805928 0.592014i \(-0.201667\pi\)
0.805928 + 0.592014i \(0.201667\pi\)
\(968\) 0 0
\(969\) 9.33404 0.299853
\(970\) −3.63887 −0.116837
\(971\) 40.1230 1.28761 0.643804 0.765190i \(-0.277355\pi\)
0.643804 + 0.765190i \(0.277355\pi\)
\(972\) −0.800331 −0.0256706
\(973\) 5.76086 0.184685
\(974\) 14.6631 0.469835
\(975\) 4.71333 0.150947
\(976\) −19.1215 −0.612064
\(977\) −24.1682 −0.773209 −0.386604 0.922246i \(-0.626352\pi\)
−0.386604 + 0.922246i \(0.626352\pi\)
\(978\) 1.96335 0.0627809
\(979\) 0 0
\(980\) 5.20449 0.166251
\(981\) −9.03128 −0.288346
\(982\) −3.05464 −0.0974774
\(983\) 23.3021 0.743220 0.371610 0.928389i \(-0.378806\pi\)
0.371610 + 0.928389i \(0.378806\pi\)
\(984\) 0.714351 0.0227727
\(985\) −6.80056 −0.216684
\(986\) 11.3121 0.360250
\(987\) 5.86752 0.186765
\(988\) −4.52542 −0.143973
\(989\) 50.5195 1.60643
\(990\) 0 0
\(991\) −54.0689 −1.71756 −0.858778 0.512348i \(-0.828776\pi\)
−0.858778 + 0.512348i \(0.828776\pi\)
\(992\) 32.3224 1.02624
\(993\) 19.4191 0.616246
\(994\) −0.517796 −0.0164235
\(995\) 21.2972 0.675168
\(996\) −1.68869 −0.0535081
\(997\) −50.7109 −1.60603 −0.803015 0.595959i \(-0.796772\pi\)
−0.803015 + 0.595959i \(0.796772\pi\)
\(998\) −19.4733 −0.616416
\(999\) 8.43763 0.266955
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 1815.2.a.w.1.1 4
3.2 odd 2 5445.2.a.bf.1.4 4
5.4 even 2 9075.2.a.cm.1.4 4
11.7 odd 10 165.2.m.d.16.2 8
11.8 odd 10 165.2.m.d.31.2 yes 8
11.10 odd 2 1815.2.a.p.1.4 4
33.8 even 10 495.2.n.a.361.1 8
33.29 even 10 495.2.n.a.181.1 8
33.32 even 2 5445.2.a.bt.1.1 4
55.7 even 20 825.2.bx.f.49.2 16
55.8 even 20 825.2.bx.f.724.2 16
55.18 even 20 825.2.bx.f.49.3 16
55.19 odd 10 825.2.n.g.526.1 8
55.29 odd 10 825.2.n.g.676.1 8
55.52 even 20 825.2.bx.f.724.3 16
55.54 odd 2 9075.2.a.di.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.d.16.2 8 11.7 odd 10
165.2.m.d.31.2 yes 8 11.8 odd 10
495.2.n.a.181.1 8 33.29 even 10
495.2.n.a.361.1 8 33.8 even 10
825.2.n.g.526.1 8 55.19 odd 10
825.2.n.g.676.1 8 55.29 odd 10
825.2.bx.f.49.2 16 55.7 even 20
825.2.bx.f.49.3 16 55.18 even 20
825.2.bx.f.724.2 16 55.8 even 20
825.2.bx.f.724.3 16 55.52 even 20
1815.2.a.p.1.4 4 11.10 odd 2
1815.2.a.w.1.1 4 1.1 even 1 trivial
5445.2.a.bf.1.4 4 3.2 odd 2
5445.2.a.bt.1.1 4 33.32 even 2
9075.2.a.cm.1.4 4 5.4 even 2
9075.2.a.di.1.1 4 55.54 odd 2