Properties

Label 825.2.c.f.199.6
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.6
Root \(0.432320 - 0.432320i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.f.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.76156i q^{2} -1.00000i q^{3} -5.62620 q^{4} +2.76156 q^{6} -1.86464i q^{7} -10.0140i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+2.76156i q^{2} -1.00000i q^{3} -5.62620 q^{4} +2.76156 q^{6} -1.86464i q^{7} -10.0140i q^{8} -1.00000 q^{9} +1.00000 q^{11} +5.62620i q^{12} +4.62620i q^{13} +5.14931 q^{14} +16.4017 q^{16} -2.49084i q^{17} -2.76156i q^{18} +5.38776 q^{19} -1.86464 q^{21} +2.76156i q^{22} +7.14931i q^{23} -10.0140 q^{24} -12.7755 q^{26} +1.00000i q^{27} +10.4908i q^{28} +3.52311 q^{29} +8.62620 q^{31} +25.2663i q^{32} -1.00000i q^{33} +6.87859 q^{34} +5.62620 q^{36} +8.87859i q^{37} +14.8786i q^{38} +4.62620 q^{39} -0.761557 q^{41} -5.14931i q^{42} -7.40171i q^{43} -5.62620 q^{44} -19.7432 q^{46} +0.373802i q^{47} -16.4017i q^{48} +3.52311 q^{49} -2.49084 q^{51} -26.0279i q^{52} -5.45856i q^{53} -2.76156 q^{54} -18.6724 q^{56} -5.38776i q^{57} +9.72928i q^{58} -5.14931 q^{59} +4.42003 q^{61} +23.8217i q^{62} +1.86464i q^{63} -36.9711 q^{64} +2.76156 q^{66} -11.9431i q^{67} +14.0140i q^{68} +7.14931 q^{69} +11.6262 q^{71} +10.0140i q^{72} +6.77551i q^{73} -24.5187 q^{74} -30.3126 q^{76} -1.86464i q^{77} +12.7755i q^{78} +6.01395 q^{79} +1.00000 q^{81} -2.10308i q^{82} +14.5693i q^{83} +10.4908 q^{84} +20.4402 q^{86} -3.52311i q^{87} -10.0140i q^{88} -9.04623 q^{89} +8.62620 q^{91} -40.2234i q^{92} -8.62620i q^{93} -1.03228 q^{94} +25.2663 q^{96} -16.3169i q^{97} +9.72928i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46} - 4 q^{49} + 8 q^{51} - 4 q^{54} - 44 q^{56} + 12 q^{59} - 6 q^{61} - 68 q^{64} + 4 q^{66} + 52 q^{71} - 28 q^{74} - 48 q^{76} - 12 q^{79} + 6 q^{81} + 40 q^{84} + 56 q^{86} - 4 q^{89} + 34 q^{91} - 4 q^{94} + 68 q^{96} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.76156i 1.95272i 0.216160 + 0.976358i \(0.430647\pi\)
−0.216160 + 0.976358i \(0.569353\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −5.62620 −2.81310
\(5\) 0 0
\(6\) 2.76156 1.12740
\(7\) − 1.86464i − 0.704768i −0.935855 0.352384i \(-0.885371\pi\)
0.935855 0.352384i \(-0.114629\pi\)
\(8\) − 10.0140i − 3.54047i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 5.62620i 1.62414i
\(13\) 4.62620i 1.28308i 0.767091 + 0.641538i \(0.221703\pi\)
−0.767091 + 0.641538i \(0.778297\pi\)
\(14\) 5.14931 1.37621
\(15\) 0 0
\(16\) 16.4017 4.10043
\(17\) − 2.49084i − 0.604117i −0.953289 0.302059i \(-0.902326\pi\)
0.953289 0.302059i \(-0.0976738\pi\)
\(18\) − 2.76156i − 0.650905i
\(19\) 5.38776 1.23604 0.618018 0.786164i \(-0.287936\pi\)
0.618018 + 0.786164i \(0.287936\pi\)
\(20\) 0 0
\(21\) −1.86464 −0.406898
\(22\) 2.76156i 0.588766i
\(23\) 7.14931i 1.49073i 0.666654 + 0.745367i \(0.267726\pi\)
−0.666654 + 0.745367i \(0.732274\pi\)
\(24\) −10.0140 −2.04409
\(25\) 0 0
\(26\) −12.7755 −2.50548
\(27\) 1.00000i 0.192450i
\(28\) 10.4908i 1.98258i
\(29\) 3.52311 0.654226 0.327113 0.944985i \(-0.393924\pi\)
0.327113 + 0.944985i \(0.393924\pi\)
\(30\) 0 0
\(31\) 8.62620 1.54931 0.774655 0.632384i \(-0.217923\pi\)
0.774655 + 0.632384i \(0.217923\pi\)
\(32\) 25.2663i 4.46650i
\(33\) − 1.00000i − 0.174078i
\(34\) 6.87859 1.17967
\(35\) 0 0
\(36\) 5.62620 0.937700
\(37\) 8.87859i 1.45963i 0.683644 + 0.729816i \(0.260394\pi\)
−0.683644 + 0.729816i \(0.739606\pi\)
\(38\) 14.8786i 2.41363i
\(39\) 4.62620 0.740785
\(40\) 0 0
\(41\) −0.761557 −0.118935 −0.0594676 0.998230i \(-0.518940\pi\)
−0.0594676 + 0.998230i \(0.518940\pi\)
\(42\) − 5.14931i − 0.794556i
\(43\) − 7.40171i − 1.12875i −0.825519 0.564375i \(-0.809117\pi\)
0.825519 0.564375i \(-0.190883\pi\)
\(44\) −5.62620 −0.848181
\(45\) 0 0
\(46\) −19.7432 −2.91098
\(47\) 0.373802i 0.0545246i 0.999628 + 0.0272623i \(0.00867894\pi\)
−0.999628 + 0.0272623i \(0.991321\pi\)
\(48\) − 16.4017i − 2.36738i
\(49\) 3.52311 0.503302
\(50\) 0 0
\(51\) −2.49084 −0.348787
\(52\) − 26.0279i − 3.60942i
\(53\) − 5.45856i − 0.749791i −0.927067 0.374896i \(-0.877679\pi\)
0.927067 0.374896i \(-0.122321\pi\)
\(54\) −2.76156 −0.375800
\(55\) 0 0
\(56\) −18.6724 −2.49521
\(57\) − 5.38776i − 0.713626i
\(58\) 9.72928i 1.27752i
\(59\) −5.14931 −0.670383 −0.335192 0.942150i \(-0.608801\pi\)
−0.335192 + 0.942150i \(0.608801\pi\)
\(60\) 0 0
\(61\) 4.42003 0.565927 0.282963 0.959131i \(-0.408682\pi\)
0.282963 + 0.959131i \(0.408682\pi\)
\(62\) 23.8217i 3.02536i
\(63\) 1.86464i 0.234923i
\(64\) −36.9711 −4.62138
\(65\) 0 0
\(66\) 2.76156 0.339924
\(67\) − 11.9431i − 1.45909i −0.683934 0.729544i \(-0.739732\pi\)
0.683934 0.729544i \(-0.260268\pi\)
\(68\) 14.0140i 1.69944i
\(69\) 7.14931 0.860676
\(70\) 0 0
\(71\) 11.6262 1.37978 0.689888 0.723916i \(-0.257660\pi\)
0.689888 + 0.723916i \(0.257660\pi\)
\(72\) 10.0140i 1.18016i
\(73\) 6.77551i 0.793014i 0.918032 + 0.396507i \(0.129778\pi\)
−0.918032 + 0.396507i \(0.870222\pi\)
\(74\) −24.5187 −2.85025
\(75\) 0 0
\(76\) −30.3126 −3.47709
\(77\) − 1.86464i − 0.212496i
\(78\) 12.7755i 1.44654i
\(79\) 6.01395 0.676623 0.338311 0.941034i \(-0.390144\pi\)
0.338311 + 0.941034i \(0.390144\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.10308i − 0.232247i
\(83\) 14.5693i 1.59919i 0.600538 + 0.799597i \(0.294953\pi\)
−0.600538 + 0.799597i \(0.705047\pi\)
\(84\) 10.4908 1.14464
\(85\) 0 0
\(86\) 20.4402 2.20413
\(87\) − 3.52311i − 0.377718i
\(88\) − 10.0140i − 1.06749i
\(89\) −9.04623 −0.958898 −0.479449 0.877570i \(-0.659164\pi\)
−0.479449 + 0.877570i \(0.659164\pi\)
\(90\) 0 0
\(91\) 8.62620 0.904271
\(92\) − 40.2234i − 4.19358i
\(93\) − 8.62620i − 0.894495i
\(94\) −1.03228 −0.106471
\(95\) 0 0
\(96\) 25.2663 2.57874
\(97\) − 16.3169i − 1.65673i −0.560185 0.828367i \(-0.689270\pi\)
0.560185 0.828367i \(-0.310730\pi\)
\(98\) 9.72928i 0.982806i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −1.03228 −0.102715 −0.0513576 0.998680i \(-0.516355\pi\)
−0.0513576 + 0.998680i \(0.516355\pi\)
\(102\) − 6.87859i − 0.681082i
\(103\) 3.04623i 0.300154i 0.988674 + 0.150077i \(0.0479521\pi\)
−0.988674 + 0.150077i \(0.952048\pi\)
\(104\) 46.3265 4.54269
\(105\) 0 0
\(106\) 15.0741 1.46413
\(107\) 8.50479i 0.822189i 0.911593 + 0.411095i \(0.134853\pi\)
−0.911593 + 0.411095i \(0.865147\pi\)
\(108\) − 5.62620i − 0.541381i
\(109\) 6.14931 0.588997 0.294499 0.955652i \(-0.404847\pi\)
0.294499 + 0.955652i \(0.404847\pi\)
\(110\) 0 0
\(111\) 8.87859 0.842719
\(112\) − 30.5833i − 2.88985i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 14.8786 1.39351
\(115\) 0 0
\(116\) −19.8217 −1.84040
\(117\) − 4.62620i − 0.427692i
\(118\) − 14.2201i − 1.30907i
\(119\) −4.64452 −0.425762
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 12.2062i 1.10509i
\(123\) 0.761557i 0.0686673i
\(124\) −48.5327 −4.35837
\(125\) 0 0
\(126\) −5.14931 −0.458737
\(127\) 9.26635i 0.822256i 0.911578 + 0.411128i \(0.134865\pi\)
−0.911578 + 0.411128i \(0.865135\pi\)
\(128\) − 51.5650i − 4.55774i
\(129\) −7.40171 −0.651684
\(130\) 0 0
\(131\) 4.06455 0.355121 0.177561 0.984110i \(-0.443179\pi\)
0.177561 + 0.984110i \(0.443179\pi\)
\(132\) 5.62620i 0.489698i
\(133\) − 10.0462i − 0.871119i
\(134\) 32.9817 2.84918
\(135\) 0 0
\(136\) −24.9431 −2.13886
\(137\) − 5.93545i − 0.507100i −0.967322 0.253550i \(-0.918402\pi\)
0.967322 0.253550i \(-0.0815982\pi\)
\(138\) 19.7432i 1.68066i
\(139\) −6.71096 −0.569216 −0.284608 0.958644i \(-0.591863\pi\)
−0.284608 + 0.958644i \(0.591863\pi\)
\(140\) 0 0
\(141\) 0.373802 0.0314798
\(142\) 32.1064i 2.69431i
\(143\) 4.62620i 0.386862i
\(144\) −16.4017 −1.36681
\(145\) 0 0
\(146\) −18.7110 −1.54853
\(147\) − 3.52311i − 0.290582i
\(148\) − 49.9527i − 4.10609i
\(149\) −5.74324 −0.470504 −0.235252 0.971934i \(-0.575592\pi\)
−0.235252 + 0.971934i \(0.575592\pi\)
\(150\) 0 0
\(151\) 10.5616 0.859495 0.429747 0.902949i \(-0.358603\pi\)
0.429747 + 0.902949i \(0.358603\pi\)
\(152\) − 53.9527i − 4.37614i
\(153\) 2.49084i 0.201372i
\(154\) 5.14931 0.414943
\(155\) 0 0
\(156\) −26.0279 −2.08390
\(157\) − 3.10308i − 0.247653i −0.992304 0.123827i \(-0.960483\pi\)
0.992304 0.123827i \(-0.0395166\pi\)
\(158\) 16.6079i 1.32125i
\(159\) −5.45856 −0.432892
\(160\) 0 0
\(161\) 13.3309 1.05062
\(162\) 2.76156i 0.216968i
\(163\) 15.6724i 1.22756i 0.789477 + 0.613780i \(0.210352\pi\)
−0.789477 + 0.613780i \(0.789648\pi\)
\(164\) 4.28467 0.334577
\(165\) 0 0
\(166\) −40.2341 −3.12277
\(167\) 8.98168i 0.695023i 0.937676 + 0.347512i \(0.112973\pi\)
−0.937676 + 0.347512i \(0.887027\pi\)
\(168\) 18.6724i 1.44061i
\(169\) −8.40171 −0.646285
\(170\) 0 0
\(171\) −5.38776 −0.412012
\(172\) 41.6435i 3.17529i
\(173\) − 11.5092i − 0.875025i −0.899212 0.437513i \(-0.855860\pi\)
0.899212 0.437513i \(-0.144140\pi\)
\(174\) 9.72928 0.737575
\(175\) 0 0
\(176\) 16.4017 1.23633
\(177\) 5.14931i 0.387046i
\(178\) − 24.9817i − 1.87246i
\(179\) −10.3738 −0.775374 −0.387687 0.921791i \(-0.626726\pi\)
−0.387687 + 0.921791i \(0.626726\pi\)
\(180\) 0 0
\(181\) −2.66473 −0.198068 −0.0990339 0.995084i \(-0.531575\pi\)
−0.0990339 + 0.995084i \(0.531575\pi\)
\(182\) 23.8217i 1.76578i
\(183\) − 4.42003i − 0.326738i
\(184\) 71.5929 5.27790
\(185\) 0 0
\(186\) 23.8217 1.74669
\(187\) − 2.49084i − 0.182148i
\(188\) − 2.10308i − 0.153383i
\(189\) 1.86464 0.135633
\(190\) 0 0
\(191\) 14.6724 1.06166 0.530830 0.847478i \(-0.321880\pi\)
0.530830 + 0.847478i \(0.321880\pi\)
\(192\) 36.9711i 2.66816i
\(193\) 5.10308i 0.367328i 0.982989 + 0.183664i \(0.0587958\pi\)
−0.982989 + 0.183664i \(0.941204\pi\)
\(194\) 45.0602 3.23513
\(195\) 0 0
\(196\) −19.8217 −1.41584
\(197\) − 3.74324i − 0.266694i −0.991069 0.133347i \(-0.957427\pi\)
0.991069 0.133347i \(-0.0425725\pi\)
\(198\) − 2.76156i − 0.196255i
\(199\) 8.08476 0.573114 0.286557 0.958063i \(-0.407489\pi\)
0.286557 + 0.958063i \(0.407489\pi\)
\(200\) 0 0
\(201\) −11.9431 −0.842404
\(202\) − 2.85069i − 0.200574i
\(203\) − 6.56934i − 0.461077i
\(204\) 14.0140 0.981173
\(205\) 0 0
\(206\) −8.41233 −0.586115
\(207\) − 7.14931i − 0.496912i
\(208\) 75.8776i 5.26116i
\(209\) 5.38776 0.372679
\(210\) 0 0
\(211\) −15.9431 −1.09757 −0.548786 0.835963i \(-0.684910\pi\)
−0.548786 + 0.835963i \(0.684910\pi\)
\(212\) 30.7110i 2.10924i
\(213\) − 11.6262i − 0.796614i
\(214\) −23.4865 −1.60550
\(215\) 0 0
\(216\) 10.0140 0.681363
\(217\) − 16.0848i − 1.09190i
\(218\) 16.9817i 1.15014i
\(219\) 6.77551 0.457847
\(220\) 0 0
\(221\) 11.5231 0.775129
\(222\) 24.5187i 1.64559i
\(223\) − 22.1772i − 1.48510i −0.669793 0.742548i \(-0.733617\pi\)
0.669793 0.742548i \(-0.266383\pi\)
\(224\) 47.1127 3.14785
\(225\) 0 0
\(226\) 16.5693 1.10218
\(227\) − 5.93545i − 0.393950i −0.980409 0.196975i \(-0.936888\pi\)
0.980409 0.196975i \(-0.0631117\pi\)
\(228\) 30.3126i 2.00750i
\(229\) 8.31695 0.549599 0.274800 0.961502i \(-0.411388\pi\)
0.274800 + 0.961502i \(0.411388\pi\)
\(230\) 0 0
\(231\) −1.86464 −0.122684
\(232\) − 35.2803i − 2.31627i
\(233\) 28.5187i 1.86833i 0.356848 + 0.934163i \(0.383852\pi\)
−0.356848 + 0.934163i \(0.616148\pi\)
\(234\) 12.7755 0.835161
\(235\) 0 0
\(236\) 28.9711 1.88585
\(237\) − 6.01395i − 0.390648i
\(238\) − 12.8261i − 0.831393i
\(239\) −24.0925 −1.55841 −0.779206 0.626768i \(-0.784377\pi\)
−0.779206 + 0.626768i \(0.784377\pi\)
\(240\) 0 0
\(241\) −5.16763 −0.332877 −0.166438 0.986052i \(-0.553227\pi\)
−0.166438 + 0.986052i \(0.553227\pi\)
\(242\) 2.76156i 0.177520i
\(243\) − 1.00000i − 0.0641500i
\(244\) −24.8680 −1.59201
\(245\) 0 0
\(246\) −2.10308 −0.134088
\(247\) 24.9248i 1.58593i
\(248\) − 86.3823i − 5.48528i
\(249\) 14.5693 0.923295
\(250\) 0 0
\(251\) −9.25240 −0.584006 −0.292003 0.956417i \(-0.594322\pi\)
−0.292003 + 0.956417i \(0.594322\pi\)
\(252\) − 10.4908i − 0.660861i
\(253\) 7.14931i 0.449473i
\(254\) −25.5896 −1.60563
\(255\) 0 0
\(256\) 68.4575 4.27860
\(257\) 25.2158i 1.57292i 0.617644 + 0.786458i \(0.288087\pi\)
−0.617644 + 0.786458i \(0.711913\pi\)
\(258\) − 20.4402i − 1.27255i
\(259\) 16.5554 1.02870
\(260\) 0 0
\(261\) −3.52311 −0.218075
\(262\) 11.2245i 0.693451i
\(263\) 5.45856i 0.336589i 0.985737 + 0.168295i \(0.0538260\pi\)
−0.985737 + 0.168295i \(0.946174\pi\)
\(264\) −10.0140 −0.616316
\(265\) 0 0
\(266\) 27.7432 1.70105
\(267\) 9.04623i 0.553620i
\(268\) 67.1945i 4.10456i
\(269\) −17.0741 −1.04103 −0.520514 0.853853i \(-0.674260\pi\)
−0.520514 + 0.853853i \(0.674260\pi\)
\(270\) 0 0
\(271\) 9.77988 0.594085 0.297043 0.954864i \(-0.404000\pi\)
0.297043 + 0.954864i \(0.404000\pi\)
\(272\) − 40.8540i − 2.47714i
\(273\) − 8.62620i − 0.522081i
\(274\) 16.3911 0.990221
\(275\) 0 0
\(276\) −40.2234 −2.42117
\(277\) − 5.91524i − 0.355412i −0.984084 0.177706i \(-0.943132\pi\)
0.984084 0.177706i \(-0.0568676\pi\)
\(278\) − 18.5327i − 1.11152i
\(279\) −8.62620 −0.516437
\(280\) 0 0
\(281\) −6.76156 −0.403361 −0.201680 0.979451i \(-0.564640\pi\)
−0.201680 + 0.979451i \(0.564640\pi\)
\(282\) 1.03228i 0.0614711i
\(283\) 8.81841i 0.524200i 0.965041 + 0.262100i \(0.0844151\pi\)
−0.965041 + 0.262100i \(0.915585\pi\)
\(284\) −65.4113 −3.88145
\(285\) 0 0
\(286\) −12.7755 −0.755432
\(287\) 1.42003i 0.0838218i
\(288\) − 25.2663i − 1.48883i
\(289\) 10.7957 0.635042
\(290\) 0 0
\(291\) −16.3169 −0.956516
\(292\) − 38.1204i − 2.23083i
\(293\) − 30.4908i − 1.78129i −0.454696 0.890647i \(-0.650252\pi\)
0.454696 0.890647i \(-0.349748\pi\)
\(294\) 9.72928 0.567423
\(295\) 0 0
\(296\) 88.9098 5.16778
\(297\) 1.00000i 0.0580259i
\(298\) − 15.8603i − 0.918761i
\(299\) −33.0741 −1.91273
\(300\) 0 0
\(301\) −13.8015 −0.795507
\(302\) 29.1666i 1.67835i
\(303\) 1.03228i 0.0593027i
\(304\) 88.3684 5.06827
\(305\) 0 0
\(306\) −6.87859 −0.393223
\(307\) − 0.767815i − 0.0438215i −0.999760 0.0219107i \(-0.993025\pi\)
0.999760 0.0219107i \(-0.00697497\pi\)
\(308\) 10.4908i 0.597771i
\(309\) 3.04623 0.173294
\(310\) 0 0
\(311\) −33.8496 −1.91944 −0.959719 0.280963i \(-0.909346\pi\)
−0.959719 + 0.280963i \(0.909346\pi\)
\(312\) − 46.3265i − 2.62272i
\(313\) − 8.11078i − 0.458448i −0.973374 0.229224i \(-0.926381\pi\)
0.973374 0.229224i \(-0.0736189\pi\)
\(314\) 8.56934 0.483596
\(315\) 0 0
\(316\) −33.8357 −1.90341
\(317\) − 8.06455i − 0.452950i −0.974017 0.226475i \(-0.927280\pi\)
0.974017 0.226475i \(-0.0727202\pi\)
\(318\) − 15.0741i − 0.845316i
\(319\) 3.52311 0.197257
\(320\) 0 0
\(321\) 8.50479 0.474691
\(322\) 36.8140i 2.05157i
\(323\) − 13.4200i − 0.746710i
\(324\) −5.62620 −0.312567
\(325\) 0 0
\(326\) −43.2803 −2.39707
\(327\) − 6.14931i − 0.340058i
\(328\) 7.62620i 0.421086i
\(329\) 0.697006 0.0384272
\(330\) 0 0
\(331\) 22.4769 1.23544 0.617721 0.786398i \(-0.288056\pi\)
0.617721 + 0.786398i \(0.288056\pi\)
\(332\) − 81.9700i − 4.49869i
\(333\) − 8.87859i − 0.486544i
\(334\) −24.8034 −1.35718
\(335\) 0 0
\(336\) −30.5833 −1.66846
\(337\) − 6.32757i − 0.344685i −0.985037 0.172342i \(-0.944866\pi\)
0.985037 0.172342i \(-0.0551336\pi\)
\(338\) − 23.2018i − 1.26201i
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 8.62620 0.467135
\(342\) − 14.8786i − 0.804542i
\(343\) − 19.6218i − 1.05948i
\(344\) −74.1204 −3.99630
\(345\) 0 0
\(346\) 31.7832 1.70868
\(347\) 0.178261i 0.00956954i 0.999989 + 0.00478477i \(0.00152305\pi\)
−0.999989 + 0.00478477i \(0.998477\pi\)
\(348\) 19.8217i 1.06256i
\(349\) 19.7293 1.05608 0.528042 0.849218i \(-0.322926\pi\)
0.528042 + 0.849218i \(0.322926\pi\)
\(350\) 0 0
\(351\) −4.62620 −0.246928
\(352\) 25.2663i 1.34670i
\(353\) − 19.5231i − 1.03911i −0.854437 0.519555i \(-0.826098\pi\)
0.854437 0.519555i \(-0.173902\pi\)
\(354\) −14.2201 −0.755791
\(355\) 0 0
\(356\) 50.8959 2.69748
\(357\) 4.64452i 0.245814i
\(358\) − 28.6478i − 1.51409i
\(359\) −35.6926 −1.88379 −0.941893 0.335914i \(-0.890955\pi\)
−0.941893 + 0.335914i \(0.890955\pi\)
\(360\) 0 0
\(361\) 10.0279 0.527785
\(362\) − 7.35881i − 0.386770i
\(363\) − 1.00000i − 0.0524864i
\(364\) −48.5327 −2.54380
\(365\) 0 0
\(366\) 12.2062 0.638027
\(367\) − 20.1127i − 1.04987i −0.851141 0.524936i \(-0.824089\pi\)
0.851141 0.524936i \(-0.175911\pi\)
\(368\) 117.261i 6.11265i
\(369\) 0.761557 0.0396451
\(370\) 0 0
\(371\) −10.1783 −0.528429
\(372\) 48.5327i 2.51630i
\(373\) − 2.56165i − 0.132637i −0.997799 0.0663185i \(-0.978875\pi\)
0.997799 0.0663185i \(-0.0211253\pi\)
\(374\) 6.87859 0.355684
\(375\) 0 0
\(376\) 3.74324 0.193043
\(377\) 16.2986i 0.839422i
\(378\) 5.14931i 0.264852i
\(379\) −23.2601 −1.19479 −0.597395 0.801947i \(-0.703798\pi\)
−0.597395 + 0.801947i \(0.703798\pi\)
\(380\) 0 0
\(381\) 9.26635 0.474729
\(382\) 40.5187i 2.07312i
\(383\) 25.7938i 1.31800i 0.752142 + 0.659002i \(0.229021\pi\)
−0.752142 + 0.659002i \(0.770979\pi\)
\(384\) −51.5650 −2.63141
\(385\) 0 0
\(386\) −14.0925 −0.717287
\(387\) 7.40171i 0.376250i
\(388\) 91.8024i 4.66056i
\(389\) −2.33527 −0.118403 −0.0592014 0.998246i \(-0.518855\pi\)
−0.0592014 + 0.998246i \(0.518855\pi\)
\(390\) 0 0
\(391\) 17.8078 0.900578
\(392\) − 35.2803i − 1.78192i
\(393\) − 4.06455i − 0.205029i
\(394\) 10.3372 0.520778
\(395\) 0 0
\(396\) 5.62620 0.282727
\(397\) − 17.1955i − 0.863019i −0.902108 0.431510i \(-0.857981\pi\)
0.902108 0.431510i \(-0.142019\pi\)
\(398\) 22.3265i 1.11913i
\(399\) −10.0462 −0.502941
\(400\) 0 0
\(401\) −26.5693 −1.32681 −0.663405 0.748261i \(-0.730889\pi\)
−0.663405 + 0.748261i \(0.730889\pi\)
\(402\) − 32.9817i − 1.64498i
\(403\) 39.9065i 1.98788i
\(404\) 5.80779 0.288948
\(405\) 0 0
\(406\) 18.1416 0.900353
\(407\) 8.87859i 0.440096i
\(408\) 24.9431i 1.23487i
\(409\) 20.4200 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(410\) 0 0
\(411\) −5.93545 −0.292774
\(412\) − 17.1387i − 0.844362i
\(413\) 9.60162i 0.472465i
\(414\) 19.7432 0.970327
\(415\) 0 0
\(416\) −116.887 −5.73086
\(417\) 6.71096i 0.328637i
\(418\) 14.8786i 0.727736i
\(419\) 13.5896 0.663893 0.331947 0.943298i \(-0.392295\pi\)
0.331947 + 0.943298i \(0.392295\pi\)
\(420\) 0 0
\(421\) −39.6175 −1.93084 −0.965418 0.260705i \(-0.916045\pi\)
−0.965418 + 0.260705i \(0.916045\pi\)
\(422\) − 44.0279i − 2.14324i
\(423\) − 0.373802i − 0.0181749i
\(424\) −54.6618 −2.65461
\(425\) 0 0
\(426\) 32.1064 1.55556
\(427\) − 8.24177i − 0.398847i
\(428\) − 47.8496i − 2.31290i
\(429\) 4.62620 0.223355
\(430\) 0 0
\(431\) −6.56934 −0.316434 −0.158217 0.987404i \(-0.550575\pi\)
−0.158217 + 0.987404i \(0.550575\pi\)
\(432\) 16.4017i 0.789128i
\(433\) − 36.4113i − 1.74982i −0.484290 0.874908i \(-0.660922\pi\)
0.484290 0.874908i \(-0.339078\pi\)
\(434\) 44.4190 2.13218
\(435\) 0 0
\(436\) −34.5972 −1.65691
\(437\) 38.5187i 1.84260i
\(438\) 18.7110i 0.894044i
\(439\) −25.3878 −1.21169 −0.605846 0.795582i \(-0.707165\pi\)
−0.605846 + 0.795582i \(0.707165\pi\)
\(440\) 0 0
\(441\) −3.52311 −0.167767
\(442\) 31.8217i 1.51361i
\(443\) − 20.9065i − 0.993298i −0.867952 0.496649i \(-0.834564\pi\)
0.867952 0.496649i \(-0.165436\pi\)
\(444\) −49.9527 −2.37065
\(445\) 0 0
\(446\) 61.2437 2.89997
\(447\) 5.74324i 0.271646i
\(448\) 68.9377i 3.25700i
\(449\) 34.9450 1.64916 0.824579 0.565747i \(-0.191412\pi\)
0.824579 + 0.565747i \(0.191412\pi\)
\(450\) 0 0
\(451\) −0.761557 −0.0358603
\(452\) 33.7572i 1.58780i
\(453\) − 10.5616i − 0.496229i
\(454\) 16.3911 0.769272
\(455\) 0 0
\(456\) −53.9527 −2.52657
\(457\) − 7.31695i − 0.342272i −0.985247 0.171136i \(-0.945256\pi\)
0.985247 0.171136i \(-0.0547438\pi\)
\(458\) 22.9677i 1.07321i
\(459\) 2.49084 0.116262
\(460\) 0 0
\(461\) −14.0279 −0.653345 −0.326672 0.945138i \(-0.605927\pi\)
−0.326672 + 0.945138i \(0.605927\pi\)
\(462\) − 5.14931i − 0.239568i
\(463\) − 4.33527i − 0.201477i −0.994913 0.100739i \(-0.967879\pi\)
0.994913 0.100739i \(-0.0321206\pi\)
\(464\) 57.7851 2.68261
\(465\) 0 0
\(466\) −78.7561 −3.64831
\(467\) 3.70138i 0.171279i 0.996326 + 0.0856396i \(0.0272934\pi\)
−0.996326 + 0.0856396i \(0.972707\pi\)
\(468\) 26.0279i 1.20314i
\(469\) −22.2697 −1.02832
\(470\) 0 0
\(471\) −3.10308 −0.142983
\(472\) 51.5650i 2.37347i
\(473\) − 7.40171i − 0.340331i
\(474\) 16.6079 0.762825
\(475\) 0 0
\(476\) 26.1310 1.19771
\(477\) 5.45856i 0.249930i
\(478\) − 66.5327i − 3.04313i
\(479\) 22.8680 1.04486 0.522432 0.852681i \(-0.325025\pi\)
0.522432 + 0.852681i \(0.325025\pi\)
\(480\) 0 0
\(481\) −41.0741 −1.87282
\(482\) − 14.2707i − 0.650013i
\(483\) − 13.3309i − 0.606577i
\(484\) −5.62620 −0.255736
\(485\) 0 0
\(486\) 2.76156 0.125267
\(487\) − 5.42962i − 0.246039i −0.992404 0.123020i \(-0.960742\pi\)
0.992404 0.123020i \(-0.0392578\pi\)
\(488\) − 44.2620i − 2.00365i
\(489\) 15.6724 0.708732
\(490\) 0 0
\(491\) 21.2803 0.960367 0.480183 0.877168i \(-0.340570\pi\)
0.480183 + 0.877168i \(0.340570\pi\)
\(492\) − 4.28467i − 0.193168i
\(493\) − 8.77551i − 0.395229i
\(494\) −68.8313 −3.09687
\(495\) 0 0
\(496\) 141.484 6.35284
\(497\) − 21.6787i − 0.972422i
\(498\) 40.2341i 1.80293i
\(499\) 16.0077 0.716603 0.358301 0.933606i \(-0.383356\pi\)
0.358301 + 0.933606i \(0.383356\pi\)
\(500\) 0 0
\(501\) 8.98168 0.401272
\(502\) − 25.5510i − 1.14040i
\(503\) − 12.0925i − 0.539176i −0.962976 0.269588i \(-0.913112\pi\)
0.962976 0.269588i \(-0.0868875\pi\)
\(504\) 18.6724 0.831736
\(505\) 0 0
\(506\) −19.7432 −0.877694
\(507\) 8.40171i 0.373133i
\(508\) − 52.1343i − 2.31309i
\(509\) −14.7110 −0.652052 −0.326026 0.945361i \(-0.605710\pi\)
−0.326026 + 0.945361i \(0.605710\pi\)
\(510\) 0 0
\(511\) 12.6339 0.558891
\(512\) 85.9194i 3.79714i
\(513\) 5.38776i 0.237875i
\(514\) −69.6347 −3.07146
\(515\) 0 0
\(516\) 41.6435 1.83325
\(517\) 0.373802i 0.0164398i
\(518\) 45.7187i 2.00876i
\(519\) −11.5092 −0.505196
\(520\) 0 0
\(521\) −17.2803 −0.757064 −0.378532 0.925588i \(-0.623571\pi\)
−0.378532 + 0.925588i \(0.623571\pi\)
\(522\) − 9.72928i − 0.425839i
\(523\) 42.7326i 1.86857i 0.356532 + 0.934283i \(0.383959\pi\)
−0.356532 + 0.934283i \(0.616041\pi\)
\(524\) −22.8680 −0.998992
\(525\) 0 0
\(526\) −15.0741 −0.657264
\(527\) − 21.4865i − 0.935965i
\(528\) − 16.4017i − 0.713793i
\(529\) −28.1127 −1.22229
\(530\) 0 0
\(531\) 5.14931 0.223461
\(532\) 56.5221i 2.45054i
\(533\) − 3.52311i − 0.152603i
\(534\) −24.9817 −1.08106
\(535\) 0 0
\(536\) −119.598 −5.16585
\(537\) 10.3738i 0.447663i
\(538\) − 47.1512i − 2.03283i
\(539\) 3.52311 0.151751
\(540\) 0 0
\(541\) 8.69075 0.373644 0.186822 0.982394i \(-0.440181\pi\)
0.186822 + 0.982394i \(0.440181\pi\)
\(542\) 27.0077i 1.16008i
\(543\) 2.66473i 0.114355i
\(544\) 62.9344 2.69829
\(545\) 0 0
\(546\) 23.8217 1.01948
\(547\) 44.1064i 1.88585i 0.333000 + 0.942927i \(0.391939\pi\)
−0.333000 + 0.942927i \(0.608061\pi\)
\(548\) 33.3940i 1.42652i
\(549\) −4.42003 −0.188642
\(550\) 0 0
\(551\) 18.9817 0.808647
\(552\) − 71.5929i − 3.04720i
\(553\) − 11.2139i − 0.476862i
\(554\) 16.3353 0.694019
\(555\) 0 0
\(556\) 37.7572 1.60126
\(557\) 7.69264i 0.325948i 0.986630 + 0.162974i \(0.0521086\pi\)
−0.986630 + 0.162974i \(0.947891\pi\)
\(558\) − 23.8217i − 1.00845i
\(559\) 34.2418 1.44827
\(560\) 0 0
\(561\) −2.49084 −0.105163
\(562\) − 18.6724i − 0.787649i
\(563\) 9.25240i 0.389942i 0.980809 + 0.194971i \(0.0624613\pi\)
−0.980809 + 0.194971i \(0.937539\pi\)
\(564\) −2.10308 −0.0885558
\(565\) 0 0
\(566\) −24.3525 −1.02361
\(567\) − 1.86464i − 0.0783076i
\(568\) − 116.424i − 4.88505i
\(569\) −4.96772 −0.208258 −0.104129 0.994564i \(-0.533205\pi\)
−0.104129 + 0.994564i \(0.533205\pi\)
\(570\) 0 0
\(571\) 6.02021 0.251938 0.125969 0.992034i \(-0.459796\pi\)
0.125969 + 0.992034i \(0.459796\pi\)
\(572\) − 26.0279i − 1.08828i
\(573\) − 14.6724i − 0.612949i
\(574\) −3.92150 −0.163680
\(575\) 0 0
\(576\) 36.9711 1.54046
\(577\) − 8.25240i − 0.343552i −0.985136 0.171776i \(-0.945050\pi\)
0.985136 0.171776i \(-0.0549505\pi\)
\(578\) 29.8130i 1.24006i
\(579\) 5.10308 0.212077
\(580\) 0 0
\(581\) 27.1666 1.12706
\(582\) − 45.0602i − 1.86780i
\(583\) − 5.45856i − 0.226071i
\(584\) 67.8496 2.80764
\(585\) 0 0
\(586\) 84.2022 3.47836
\(587\) 11.4846i 0.474019i 0.971507 + 0.237010i \(0.0761673\pi\)
−0.971507 + 0.237010i \(0.923833\pi\)
\(588\) 19.8217i 0.817435i
\(589\) 46.4758 1.91500
\(590\) 0 0
\(591\) −3.74324 −0.153976
\(592\) 145.624i 5.98511i
\(593\) − 39.0375i − 1.60308i −0.597943 0.801539i \(-0.704015\pi\)
0.597943 0.801539i \(-0.295985\pi\)
\(594\) −2.76156 −0.113308
\(595\) 0 0
\(596\) 32.3126 1.32357
\(597\) − 8.08476i − 0.330887i
\(598\) − 91.3361i − 3.73501i
\(599\) 30.1589 1.23226 0.616130 0.787645i \(-0.288700\pi\)
0.616130 + 0.787645i \(0.288700\pi\)
\(600\) 0 0
\(601\) 19.1310 0.780369 0.390185 0.920737i \(-0.372411\pi\)
0.390185 + 0.920737i \(0.372411\pi\)
\(602\) − 38.1137i − 1.55340i
\(603\) 11.9431i 0.486362i
\(604\) −59.4219 −2.41784
\(605\) 0 0
\(606\) −2.85069 −0.115801
\(607\) − 4.20617i − 0.170723i −0.996350 0.0853615i \(-0.972795\pi\)
0.996350 0.0853615i \(-0.0272045\pi\)
\(608\) 136.129i 5.52076i
\(609\) −6.56934 −0.266203
\(610\) 0 0
\(611\) −1.72928 −0.0699593
\(612\) − 14.0140i − 0.566480i
\(613\) 5.45856i 0.220469i 0.993906 + 0.110235i \(0.0351602\pi\)
−0.993906 + 0.110235i \(0.964840\pi\)
\(614\) 2.12036 0.0855709
\(615\) 0 0
\(616\) −18.6724 −0.752334
\(617\) − 30.1974i − 1.21570i −0.794051 0.607851i \(-0.792032\pi\)
0.794051 0.607851i \(-0.207968\pi\)
\(618\) 8.41233i 0.338394i
\(619\) −23.1955 −0.932308 −0.466154 0.884704i \(-0.654361\pi\)
−0.466154 + 0.884704i \(0.654361\pi\)
\(620\) 0 0
\(621\) −7.14931 −0.286892
\(622\) − 93.4777i − 3.74812i
\(623\) 16.8680i 0.675801i
\(624\) 75.8776 3.03753
\(625\) 0 0
\(626\) 22.3984 0.895219
\(627\) − 5.38776i − 0.215166i
\(628\) 17.4586i 0.696673i
\(629\) 22.1151 0.881789
\(630\) 0 0
\(631\) −3.19554 −0.127212 −0.0636062 0.997975i \(-0.520260\pi\)
−0.0636062 + 0.997975i \(0.520260\pi\)
\(632\) − 60.2234i − 2.39556i
\(633\) 15.9431i 0.633683i
\(634\) 22.2707 0.884483
\(635\) 0 0
\(636\) 30.7110 1.21777
\(637\) 16.2986i 0.645775i
\(638\) 9.72928i 0.385186i
\(639\) −11.6262 −0.459925
\(640\) 0 0
\(641\) 28.1974 1.11373 0.556866 0.830603i \(-0.312004\pi\)
0.556866 + 0.830603i \(0.312004\pi\)
\(642\) 23.4865i 0.926937i
\(643\) 17.9634i 0.708406i 0.935169 + 0.354203i \(0.115248\pi\)
−0.935169 + 0.354203i \(0.884752\pi\)
\(644\) −75.0023 −2.95550
\(645\) 0 0
\(646\) 37.0602 1.45811
\(647\) 40.9990i 1.61184i 0.592028 + 0.805918i \(0.298328\pi\)
−0.592028 + 0.805918i \(0.701672\pi\)
\(648\) − 10.0140i − 0.393385i
\(649\) −5.14931 −0.202128
\(650\) 0 0
\(651\) −16.0848 −0.630412
\(652\) − 88.1762i − 3.45325i
\(653\) 11.6926i 0.457568i 0.973477 + 0.228784i \(0.0734750\pi\)
−0.973477 + 0.228784i \(0.926525\pi\)
\(654\) 16.9817 0.664036
\(655\) 0 0
\(656\) −12.4908 −0.487685
\(657\) − 6.77551i − 0.264338i
\(658\) 1.92482i 0.0750374i
\(659\) 1.72928 0.0673633 0.0336816 0.999433i \(-0.489277\pi\)
0.0336816 + 0.999433i \(0.489277\pi\)
\(660\) 0 0
\(661\) 0.232185 0.00903096 0.00451548 0.999990i \(-0.498563\pi\)
0.00451548 + 0.999990i \(0.498563\pi\)
\(662\) 62.0712i 2.41247i
\(663\) − 11.5231i − 0.447521i
\(664\) 145.897 5.66189
\(665\) 0 0
\(666\) 24.5187 0.950082
\(667\) 25.1878i 0.975277i
\(668\) − 50.5327i − 1.95517i
\(669\) −22.1772 −0.857421
\(670\) 0 0
\(671\) 4.42003 0.170633
\(672\) − 47.1127i − 1.81741i
\(673\) 33.5144i 1.29188i 0.763386 + 0.645942i \(0.223535\pi\)
−0.763386 + 0.645942i \(0.776465\pi\)
\(674\) 17.4740 0.673072
\(675\) 0 0
\(676\) 47.2697 1.81806
\(677\) 13.0183i 0.500335i 0.968203 + 0.250167i \(0.0804857\pi\)
−0.968203 + 0.250167i \(0.919514\pi\)
\(678\) − 16.5693i − 0.636342i
\(679\) −30.4252 −1.16761
\(680\) 0 0
\(681\) −5.93545 −0.227447
\(682\) 23.8217i 0.912182i
\(683\) − 4.54333i − 0.173846i −0.996215 0.0869228i \(-0.972297\pi\)
0.996215 0.0869228i \(-0.0277033\pi\)
\(684\) 30.3126 1.15903
\(685\) 0 0
\(686\) 54.1868 2.06886
\(687\) − 8.31695i − 0.317311i
\(688\) − 121.401i − 4.62836i
\(689\) 25.2524 0.962040
\(690\) 0 0
\(691\) −19.8988 −0.756986 −0.378493 0.925604i \(-0.623558\pi\)
−0.378493 + 0.925604i \(0.623558\pi\)
\(692\) 64.7528i 2.46153i
\(693\) 1.86464i 0.0708319i
\(694\) −0.492277 −0.0186866
\(695\) 0 0
\(696\) −35.2803 −1.33730
\(697\) 1.89692i 0.0718508i
\(698\) 54.4835i 2.06223i
\(699\) 28.5187 1.07868
\(700\) 0 0
\(701\) −15.8444 −0.598436 −0.299218 0.954185i \(-0.596726\pi\)
−0.299218 + 0.954185i \(0.596726\pi\)
\(702\) − 12.7755i − 0.482181i
\(703\) 47.8357i 1.80416i
\(704\) −36.9711 −1.39340
\(705\) 0 0
\(706\) 53.9142 2.02909
\(707\) 1.92482i 0.0723904i
\(708\) − 28.9711i − 1.08880i
\(709\) −8.82174 −0.331307 −0.165654 0.986184i \(-0.552973\pi\)
−0.165654 + 0.986184i \(0.552973\pi\)
\(710\) 0 0
\(711\) −6.01395 −0.225541
\(712\) 90.5885i 3.39495i
\(713\) 61.6714i 2.30961i
\(714\) −12.8261 −0.480005
\(715\) 0 0
\(716\) 58.3651 2.18120
\(717\) 24.0925i 0.899749i
\(718\) − 98.5673i − 3.67850i
\(719\) −36.7668 −1.37117 −0.685585 0.727993i \(-0.740453\pi\)
−0.685585 + 0.727993i \(0.740453\pi\)
\(720\) 0 0
\(721\) 5.68012 0.211539
\(722\) 27.6926i 1.03061i
\(723\) 5.16763i 0.192186i
\(724\) 14.9923 0.557185
\(725\) 0 0
\(726\) 2.76156 0.102491
\(727\) 5.27261i 0.195550i 0.995209 + 0.0977751i \(0.0311726\pi\)
−0.995209 + 0.0977751i \(0.968827\pi\)
\(728\) − 86.3823i − 3.20154i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −18.4365 −0.681897
\(732\) 24.8680i 0.919147i
\(733\) 37.9508i 1.40175i 0.713286 + 0.700873i \(0.247206\pi\)
−0.713286 + 0.700873i \(0.752794\pi\)
\(734\) 55.5423 2.05010
\(735\) 0 0
\(736\) −180.637 −6.65837
\(737\) − 11.9431i − 0.439931i
\(738\) 2.10308i 0.0774156i
\(739\) −37.2943 −1.37189 −0.685946 0.727653i \(-0.740611\pi\)
−0.685946 + 0.727653i \(0.740611\pi\)
\(740\) 0 0
\(741\) 24.9248 0.915636
\(742\) − 28.1078i − 1.03187i
\(743\) 22.5693i 0.827989i 0.910279 + 0.413994i \(0.135867\pi\)
−0.910279 + 0.413994i \(0.864133\pi\)
\(744\) −86.3823 −3.16693
\(745\) 0 0
\(746\) 7.07414 0.259002
\(747\) − 14.5693i − 0.533064i
\(748\) 14.0140i 0.512401i
\(749\) 15.8584 0.579453
\(750\) 0 0
\(751\) −9.55102 −0.348522 −0.174261 0.984700i \(-0.555754\pi\)
−0.174261 + 0.984700i \(0.555754\pi\)
\(752\) 6.13099i 0.223574i
\(753\) 9.25240i 0.337176i
\(754\) −45.0096 −1.63915
\(755\) 0 0
\(756\) −10.4908 −0.381548
\(757\) − 30.3188i − 1.10196i −0.834519 0.550978i \(-0.814255\pi\)
0.834519 0.550978i \(-0.185745\pi\)
\(758\) − 64.2341i − 2.33309i
\(759\) 7.14931 0.259504
\(760\) 0 0
\(761\) −1.43066 −0.0518613 −0.0259306 0.999664i \(-0.508255\pi\)
−0.0259306 + 0.999664i \(0.508255\pi\)
\(762\) 25.5896i 0.927012i
\(763\) − 11.4663i − 0.415106i
\(764\) −82.5500 −2.98655
\(765\) 0 0
\(766\) −71.2311 −2.57369
\(767\) − 23.8217i − 0.860153i
\(768\) − 68.4575i − 2.47025i
\(769\) 35.4017 1.27662 0.638309 0.769780i \(-0.279634\pi\)
0.638309 + 0.769780i \(0.279634\pi\)
\(770\) 0 0
\(771\) 25.2158 0.908123
\(772\) − 28.7110i − 1.03333i
\(773\) − 13.4094i − 0.482303i −0.970488 0.241151i \(-0.922475\pi\)
0.970488 0.241151i \(-0.0775250\pi\)
\(774\) −20.4402 −0.734709
\(775\) 0 0
\(776\) −163.397 −5.86562
\(777\) − 16.5554i − 0.593921i
\(778\) − 6.44898i − 0.231207i
\(779\) −4.10308 −0.147008
\(780\) 0 0
\(781\) 11.6262 0.416018
\(782\) 49.1772i 1.75857i
\(783\) 3.52311i 0.125906i
\(784\) 57.7851 2.06375
\(785\) 0 0
\(786\) 11.2245 0.400364
\(787\) 37.3232i 1.33043i 0.746653 + 0.665214i \(0.231660\pi\)
−0.746653 + 0.665214i \(0.768340\pi\)
\(788\) 21.0602i 0.750238i
\(789\) 5.45856 0.194330
\(790\) 0 0
\(791\) −11.1878 −0.397794
\(792\) 10.0140i 0.355830i
\(793\) 20.4479i 0.726128i
\(794\) 47.4865 1.68523
\(795\) 0 0
\(796\) −45.4865 −1.61223
\(797\) 8.05581i 0.285352i 0.989769 + 0.142676i \(0.0455706\pi\)
−0.989769 + 0.142676i \(0.954429\pi\)
\(798\) − 27.7432i − 0.982100i
\(799\) 0.931080 0.0329393
\(800\) 0 0
\(801\) 9.04623 0.319633
\(802\) − 73.3728i − 2.59088i
\(803\) 6.77551i 0.239103i
\(804\) 67.1945 2.36977
\(805\) 0 0
\(806\) −110.204 −3.88177
\(807\) 17.0741i 0.601038i
\(808\) 10.3372i 0.363660i
\(809\) 42.3771 1.48990 0.744950 0.667120i \(-0.232473\pi\)
0.744950 + 0.667120i \(0.232473\pi\)
\(810\) 0 0
\(811\) −21.3878 −0.751026 −0.375513 0.926817i \(-0.622533\pi\)
−0.375513 + 0.926817i \(0.622533\pi\)
\(812\) 36.9604i 1.29706i
\(813\) − 9.77988i − 0.342995i
\(814\) −24.5187 −0.859382
\(815\) 0 0
\(816\) −40.8540 −1.43018
\(817\) − 39.8786i − 1.39518i
\(818\) 56.3911i 1.97167i
\(819\) −8.62620 −0.301424
\(820\) 0 0
\(821\) −47.8776 −1.67094 −0.835469 0.549537i \(-0.814804\pi\)
−0.835469 + 0.549537i \(0.814804\pi\)
\(822\) − 16.3911i − 0.571705i
\(823\) 16.3555i 0.570116i 0.958510 + 0.285058i \(0.0920129\pi\)
−0.958510 + 0.285058i \(0.907987\pi\)
\(824\) 30.5048 1.06268
\(825\) 0 0
\(826\) −26.5154 −0.922589
\(827\) 44.7110i 1.55475i 0.629036 + 0.777376i \(0.283450\pi\)
−0.629036 + 0.777376i \(0.716550\pi\)
\(828\) 40.2234i 1.39786i
\(829\) 38.5972 1.34054 0.670269 0.742118i \(-0.266179\pi\)
0.670269 + 0.742118i \(0.266179\pi\)
\(830\) 0 0
\(831\) −5.91524 −0.205197
\(832\) − 171.035i − 5.92959i
\(833\) − 8.77551i − 0.304053i
\(834\) −18.5327 −0.641735
\(835\) 0 0
\(836\) −30.3126 −1.04838
\(837\) 8.62620i 0.298165i
\(838\) 37.5283i 1.29639i
\(839\) 0.710960 0.0245451 0.0122725 0.999925i \(-0.496093\pi\)
0.0122725 + 0.999925i \(0.496093\pi\)
\(840\) 0 0
\(841\) −16.5877 −0.571988
\(842\) − 109.406i − 3.77038i
\(843\) 6.76156i 0.232880i
\(844\) 89.6993 3.08758
\(845\) 0 0
\(846\) 1.03228 0.0354904
\(847\) − 1.86464i − 0.0640698i
\(848\) − 89.5298i − 3.07446i
\(849\) 8.81841 0.302647
\(850\) 0 0
\(851\) −63.4758 −2.17592
\(852\) 65.4113i 2.24095i
\(853\) 52.7187i 1.80505i 0.430635 + 0.902526i \(0.358290\pi\)
−0.430635 + 0.902526i \(0.641710\pi\)
\(854\) 22.7601 0.778835
\(855\) 0 0
\(856\) 85.1666 2.91093
\(857\) − 16.7124i − 0.570885i −0.958396 0.285442i \(-0.907859\pi\)
0.958396 0.285442i \(-0.0921405\pi\)
\(858\) 12.7755i 0.436149i
\(859\) 41.9142 1.43009 0.715047 0.699076i \(-0.246405\pi\)
0.715047 + 0.699076i \(0.246405\pi\)
\(860\) 0 0
\(861\) 1.42003 0.0483945
\(862\) − 18.1416i − 0.617906i
\(863\) − 41.4219i − 1.41002i −0.709198 0.705009i \(-0.750943\pi\)
0.709198 0.705009i \(-0.249057\pi\)
\(864\) −25.2663 −0.859579
\(865\) 0 0
\(866\) 100.552 3.41689
\(867\) − 10.7957i − 0.366642i
\(868\) 90.4961i 3.07164i
\(869\) 6.01395 0.204009
\(870\) 0 0
\(871\) 55.2514 1.87212
\(872\) − 61.5789i − 2.08533i
\(873\) 16.3169i 0.552245i
\(874\) −106.372 −3.59808
\(875\) 0 0
\(876\) −38.1204 −1.28797
\(877\) 12.0848i 0.408073i 0.978963 + 0.204037i \(0.0654062\pi\)
−0.978963 + 0.204037i \(0.934594\pi\)
\(878\) − 70.1097i − 2.36609i
\(879\) −30.4908 −1.02843
\(880\) 0 0
\(881\) −31.8130 −1.07181 −0.535904 0.844279i \(-0.680029\pi\)
−0.535904 + 0.844279i \(0.680029\pi\)
\(882\) − 9.72928i − 0.327602i
\(883\) − 20.9894i − 0.706349i −0.935558 0.353174i \(-0.885102\pi\)
0.935558 0.353174i \(-0.114898\pi\)
\(884\) −64.8313 −2.18051
\(885\) 0 0
\(886\) 57.7345 1.93963
\(887\) − 49.2032i − 1.65208i −0.563609 0.826042i \(-0.690588\pi\)
0.563609 0.826042i \(-0.309412\pi\)
\(888\) − 88.9098i − 2.98362i
\(889\) 17.2784 0.579499
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 124.773i 4.17772i
\(893\) 2.01395i 0.0673944i
\(894\) −15.8603 −0.530447
\(895\) 0 0
\(896\) −96.1502 −3.21215
\(897\) 33.0741i 1.10431i
\(898\) 96.5027i 3.22034i
\(899\) 30.3911 1.01360
\(900\) 0 0
\(901\) −13.5964 −0.452962
\(902\) − 2.10308i − 0.0700250i
\(903\) 13.8015i 0.459286i
\(904\) −60.0837 −1.99835
\(905\) 0 0
\(906\) 29.1666 0.968995
\(907\) 24.8526i 0.825216i 0.910909 + 0.412608i \(0.135382\pi\)
−0.910909 + 0.412608i \(0.864618\pi\)
\(908\) 33.3940i 1.10822i
\(909\) 1.03228 0.0342384
\(910\) 0 0
\(911\) 45.4758 1.50668 0.753341 0.657630i \(-0.228441\pi\)
0.753341 + 0.657630i \(0.228441\pi\)
\(912\) − 88.3684i − 2.92617i
\(913\) 14.5693i 0.482175i
\(914\) 20.2062 0.668361
\(915\) 0 0
\(916\) −46.7928 −1.54608
\(917\) − 7.57893i − 0.250278i
\(918\) 6.87859i 0.227027i
\(919\) 15.7355 0.519068 0.259534 0.965734i \(-0.416431\pi\)
0.259534 + 0.965734i \(0.416431\pi\)
\(920\) 0 0
\(921\) −0.767815 −0.0253004
\(922\) − 38.7389i − 1.27580i
\(923\) 53.7851i 1.77036i
\(924\) 10.4908 0.345123
\(925\) 0 0
\(926\) 11.9721 0.393427
\(927\) − 3.04623i − 0.100051i
\(928\) 89.0162i 2.92210i
\(929\) 2.06455 0.0677357 0.0338679 0.999426i \(-0.489217\pi\)
0.0338679 + 0.999426i \(0.489217\pi\)
\(930\) 0 0
\(931\) 18.9817 0.622099
\(932\) − 160.452i − 5.25578i
\(933\) 33.8496i 1.10819i
\(934\) −10.2216 −0.334460
\(935\) 0 0
\(936\) −46.3265 −1.51423
\(937\) − 40.1493i − 1.31162i −0.754926 0.655810i \(-0.772327\pi\)
0.754926 0.655810i \(-0.227673\pi\)
\(938\) − 61.4990i − 2.00801i
\(939\) −8.11078 −0.264685
\(940\) 0 0
\(941\) 26.4050 0.860780 0.430390 0.902643i \(-0.358376\pi\)
0.430390 + 0.902643i \(0.358376\pi\)
\(942\) − 8.56934i − 0.279204i
\(943\) − 5.44461i − 0.177301i
\(944\) −84.4575 −2.74886
\(945\) 0 0
\(946\) 20.4402 0.664570
\(947\) − 8.67243i − 0.281816i −0.990023 0.140908i \(-0.954998\pi\)
0.990023 0.140908i \(-0.0450022\pi\)
\(948\) 33.8357i 1.09893i
\(949\) −31.3449 −1.01750
\(950\) 0 0
\(951\) −8.06455 −0.261511
\(952\) 46.5100i 1.50740i
\(953\) − 26.8540i − 0.869887i −0.900458 0.434943i \(-0.856768\pi\)
0.900458 0.434943i \(-0.143232\pi\)
\(954\) −15.0741 −0.488043
\(955\) 0 0
\(956\) 135.549 4.38397
\(957\) − 3.52311i − 0.113886i
\(958\) 63.1512i 2.04032i
\(959\) −11.0675 −0.357388
\(960\) 0 0
\(961\) 43.4113 1.40036
\(962\) − 113.429i − 3.65708i
\(963\) − 8.50479i − 0.274063i
\(964\) 29.0741 0.936415
\(965\) 0 0
\(966\) 36.8140 1.18447
\(967\) 55.8496i 1.79600i 0.439992 + 0.898002i \(0.354981\pi\)
−0.439992 + 0.898002i \(0.645019\pi\)
\(968\) − 10.0140i − 0.321861i
\(969\) −13.4200 −0.431113
\(970\) 0 0
\(971\) 30.0664 0.964878 0.482439 0.875930i \(-0.339751\pi\)
0.482439 + 0.875930i \(0.339751\pi\)
\(972\) 5.62620i 0.180460i
\(973\) 12.5135i 0.401165i
\(974\) 14.9942 0.480445
\(975\) 0 0
\(976\) 72.4961 2.32054
\(977\) − 15.7014i − 0.502331i −0.967944 0.251166i \(-0.919186\pi\)
0.967944 0.251166i \(-0.0808139\pi\)
\(978\) 43.2803i 1.38395i
\(979\) −9.04623 −0.289119
\(980\) 0 0
\(981\) −6.14931 −0.196332
\(982\) 58.7668i 1.87532i
\(983\) − 51.9894i − 1.65820i −0.559098 0.829102i \(-0.688852\pi\)
0.559098 0.829102i \(-0.311148\pi\)
\(984\) 7.62620 0.243114
\(985\) 0 0
\(986\) 24.2341 0.771770
\(987\) − 0.697006i − 0.0221860i
\(988\) − 140.232i − 4.46137i
\(989\) 52.9171 1.68267
\(990\) 0 0
\(991\) 27.6445 0.878157 0.439079 0.898449i \(-0.355305\pi\)
0.439079 + 0.898449i \(0.355305\pi\)
\(992\) 217.953i 6.92000i
\(993\) − 22.4769i − 0.713282i
\(994\) 59.8669 1.89886
\(995\) 0 0
\(996\) −81.9700 −2.59732
\(997\) − 40.5048i − 1.28280i −0.767207 0.641400i \(-0.778354\pi\)
0.767207 0.641400i \(-0.221646\pi\)
\(998\) 44.2062i 1.39932i
\(999\) −8.87859 −0.280906
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 825.2.c.f.199.6 6
3.2 odd 2 2475.2.c.q.199.1 6
5.2 odd 4 825.2.a.i.1.1 3
5.3 odd 4 825.2.a.m.1.3 yes 3
5.4 even 2 inner 825.2.c.f.199.1 6
15.2 even 4 2475.2.a.bd.1.3 3
15.8 even 4 2475.2.a.z.1.1 3
15.14 odd 2 2475.2.c.q.199.6 6
55.32 even 4 9075.2.a.cj.1.3 3
55.43 even 4 9075.2.a.cd.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.1 3 5.2 odd 4
825.2.a.m.1.3 yes 3 5.3 odd 4
825.2.c.f.199.1 6 5.4 even 2 inner
825.2.c.f.199.6 6 1.1 even 1 trivial
2475.2.a.z.1.1 3 15.8 even 4
2475.2.a.bd.1.3 3 15.2 even 4
2475.2.c.q.199.1 6 3.2 odd 2
2475.2.c.q.199.6 6 15.14 odd 2
9075.2.a.cd.1.1 3 55.43 even 4
9075.2.a.cj.1.3 3 55.32 even 4