Properties

Label 825.4.c.d.199.1
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.d.199.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +9.00000 q^{6} -7.00000i q^{7} -21.0000i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +9.00000 q^{6} -7.00000i q^{7} -21.0000i q^{8} -9.00000 q^{9} +11.0000 q^{11} -3.00000i q^{12} +16.0000i q^{13} -21.0000 q^{14} -71.0000 q^{16} +21.0000i q^{17} +27.0000i q^{18} -125.000 q^{19} +21.0000 q^{21} -33.0000i q^{22} +81.0000i q^{23} +63.0000 q^{24} +48.0000 q^{26} -27.0000i q^{27} +7.00000i q^{28} -186.000 q^{29} -58.0000 q^{31} +45.0000i q^{32} +33.0000i q^{33} +63.0000 q^{34} +9.00000 q^{36} -253.000i q^{37} +375.000i q^{38} -48.0000 q^{39} +63.0000 q^{41} -63.0000i q^{42} +100.000i q^{43} -11.0000 q^{44} +243.000 q^{46} -219.000i q^{47} -213.000i q^{48} +294.000 q^{49} -63.0000 q^{51} -16.0000i q^{52} +192.000i q^{53} -81.0000 q^{54} -147.000 q^{56} -375.000i q^{57} +558.000i q^{58} -249.000 q^{59} -64.0000 q^{61} +174.000i q^{62} +63.0000i q^{63} -433.000 q^{64} +99.0000 q^{66} +272.000i q^{67} -21.0000i q^{68} -243.000 q^{69} -645.000 q^{71} +189.000i q^{72} +112.000i q^{73} -759.000 q^{74} +125.000 q^{76} -77.0000i q^{77} +144.000i q^{78} -509.000 q^{79} +81.0000 q^{81} -189.000i q^{82} +1254.00i q^{83} -21.0000 q^{84} +300.000 q^{86} -558.000i q^{87} -231.000i q^{88} -756.000 q^{89} +112.000 q^{91} -81.0000i q^{92} -174.000i q^{93} -657.000 q^{94} -135.000 q^{96} +839.000i q^{97} -882.000i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 18q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 18q^{6} - 18q^{9} + 22q^{11} - 42q^{14} - 142q^{16} - 250q^{19} + 42q^{21} + 126q^{24} + 96q^{26} - 372q^{29} - 116q^{31} + 126q^{34} + 18q^{36} - 96q^{39} + 126q^{41} - 22q^{44} + 486q^{46} + 588q^{49} - 126q^{51} - 162q^{54} - 294q^{56} - 498q^{59} - 128q^{61} - 866q^{64} + 198q^{66} - 486q^{69} - 1290q^{71} - 1518q^{74} + 250q^{76} - 1018q^{79} + 162q^{81} - 42q^{84} + 600q^{86} - 1512q^{89} + 224q^{91} - 1314q^{94} - 270q^{96} - 198q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.00000i − 1.06066i −0.847791 0.530330i \(-0.822068\pi\)
0.847791 0.530330i \(-0.177932\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) 9.00000 0.612372
\(7\) − 7.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) − 21.0000i − 0.928078i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) − 3.00000i − 0.0721688i
\(13\) 16.0000i 0.341354i 0.985327 + 0.170677i \(0.0545955\pi\)
−0.985327 + 0.170677i \(0.945405\pi\)
\(14\) −21.0000 −0.400892
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 21.0000i 0.299603i 0.988716 + 0.149801i \(0.0478634\pi\)
−0.988716 + 0.149801i \(0.952137\pi\)
\(18\) 27.0000i 0.353553i
\(19\) −125.000 −1.50931 −0.754657 0.656119i \(-0.772197\pi\)
−0.754657 + 0.656119i \(0.772197\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) − 33.0000i − 0.319801i
\(23\) 81.0000i 0.734333i 0.930155 + 0.367167i \(0.119672\pi\)
−0.930155 + 0.367167i \(0.880328\pi\)
\(24\) 63.0000 0.535826
\(25\) 0 0
\(26\) 48.0000 0.362061
\(27\) − 27.0000i − 0.192450i
\(28\) 7.00000i 0.0472456i
\(29\) −186.000 −1.19101 −0.595506 0.803351i \(-0.703048\pi\)
−0.595506 + 0.803351i \(0.703048\pi\)
\(30\) 0 0
\(31\) −58.0000 −0.336036 −0.168018 0.985784i \(-0.553737\pi\)
−0.168018 + 0.985784i \(0.553737\pi\)
\(32\) 45.0000i 0.248592i
\(33\) 33.0000i 0.174078i
\(34\) 63.0000 0.317777
\(35\) 0 0
\(36\) 9.00000 0.0416667
\(37\) − 253.000i − 1.12413i −0.827092 0.562067i \(-0.810006\pi\)
0.827092 0.562067i \(-0.189994\pi\)
\(38\) 375.000i 1.60087i
\(39\) −48.0000 −0.197081
\(40\) 0 0
\(41\) 63.0000 0.239974 0.119987 0.992775i \(-0.461715\pi\)
0.119987 + 0.992775i \(0.461715\pi\)
\(42\) − 63.0000i − 0.231455i
\(43\) 100.000i 0.354648i 0.984153 + 0.177324i \(0.0567440\pi\)
−0.984153 + 0.177324i \(0.943256\pi\)
\(44\) −11.0000 −0.0376889
\(45\) 0 0
\(46\) 243.000 0.778878
\(47\) − 219.000i − 0.679669i −0.940485 0.339834i \(-0.889629\pi\)
0.940485 0.339834i \(-0.110371\pi\)
\(48\) − 213.000i − 0.640498i
\(49\) 294.000 0.857143
\(50\) 0 0
\(51\) −63.0000 −0.172976
\(52\) − 16.0000i − 0.0426692i
\(53\) 192.000i 0.497608i 0.968554 + 0.248804i \(0.0800375\pi\)
−0.968554 + 0.248804i \(0.919962\pi\)
\(54\) −81.0000 −0.204124
\(55\) 0 0
\(56\) −147.000 −0.350780
\(57\) − 375.000i − 0.871403i
\(58\) 558.000i 1.26326i
\(59\) −249.000 −0.549441 −0.274721 0.961524i \(-0.588585\pi\)
−0.274721 + 0.961524i \(0.588585\pi\)
\(60\) 0 0
\(61\) −64.0000 −0.134334 −0.0671669 0.997742i \(-0.521396\pi\)
−0.0671669 + 0.997742i \(0.521396\pi\)
\(62\) 174.000i 0.356420i
\(63\) 63.0000i 0.125988i
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) 99.0000 0.184637
\(67\) 272.000i 0.495971i 0.968764 + 0.247986i \(0.0797686\pi\)
−0.968764 + 0.247986i \(0.920231\pi\)
\(68\) − 21.0000i − 0.0374504i
\(69\) −243.000 −0.423968
\(70\) 0 0
\(71\) −645.000 −1.07813 −0.539066 0.842263i \(-0.681223\pi\)
−0.539066 + 0.842263i \(0.681223\pi\)
\(72\) 189.000i 0.309359i
\(73\) 112.000i 0.179570i 0.995961 + 0.0897850i \(0.0286180\pi\)
−0.995961 + 0.0897850i \(0.971382\pi\)
\(74\) −759.000 −1.19232
\(75\) 0 0
\(76\) 125.000 0.188664
\(77\) − 77.0000i − 0.113961i
\(78\) 144.000i 0.209036i
\(79\) −509.000 −0.724898 −0.362449 0.932004i \(-0.618059\pi\)
−0.362449 + 0.932004i \(0.618059\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 189.000i − 0.254531i
\(83\) 1254.00i 1.65837i 0.558977 + 0.829183i \(0.311194\pi\)
−0.558977 + 0.829183i \(0.688806\pi\)
\(84\) −21.0000 −0.0272772
\(85\) 0 0
\(86\) 300.000 0.376161
\(87\) − 558.000i − 0.687631i
\(88\) − 231.000i − 0.279826i
\(89\) −756.000 −0.900403 −0.450201 0.892927i \(-0.648648\pi\)
−0.450201 + 0.892927i \(0.648648\pi\)
\(90\) 0 0
\(91\) 112.000 0.129020
\(92\) − 81.0000i − 0.0917917i
\(93\) − 174.000i − 0.194010i
\(94\) −657.000 −0.720898
\(95\) 0 0
\(96\) −135.000 −0.143525
\(97\) 839.000i 0.878222i 0.898433 + 0.439111i \(0.144707\pi\)
−0.898433 + 0.439111i \(0.855293\pi\)
\(98\) − 882.000i − 0.909137i
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) −1413.00 −1.39207 −0.696033 0.718009i \(-0.745053\pi\)
−0.696033 + 0.718009i \(0.745053\pi\)
\(102\) 189.000i 0.183469i
\(103\) − 1634.00i − 1.56313i −0.623821 0.781567i \(-0.714421\pi\)
0.623821 0.781567i \(-0.285579\pi\)
\(104\) 336.000 0.316803
\(105\) 0 0
\(106\) 576.000 0.527793
\(107\) 726.000i 0.655935i 0.944689 + 0.327968i \(0.106364\pi\)
−0.944689 + 0.327968i \(0.893636\pi\)
\(108\) 27.0000i 0.0240563i
\(109\) −1712.00 −1.50440 −0.752201 0.658934i \(-0.771008\pi\)
−0.752201 + 0.658934i \(0.771008\pi\)
\(110\) 0 0
\(111\) 759.000 0.649019
\(112\) 497.000i 0.419304i
\(113\) 1128.00i 0.939056i 0.882918 + 0.469528i \(0.155576\pi\)
−0.882918 + 0.469528i \(0.844424\pi\)
\(114\) −1125.00 −0.924262
\(115\) 0 0
\(116\) 186.000 0.148876
\(117\) − 144.000i − 0.113785i
\(118\) 747.000i 0.582771i
\(119\) 147.000 0.113239
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 192.000i 0.142482i
\(123\) 189.000i 0.138549i
\(124\) 58.0000 0.0420045
\(125\) 0 0
\(126\) 189.000 0.133631
\(127\) 1127.00i 0.787442i 0.919230 + 0.393721i \(0.128812\pi\)
−0.919230 + 0.393721i \(0.871188\pi\)
\(128\) 1659.00i 1.14560i
\(129\) −300.000 −0.204756
\(130\) 0 0
\(131\) −1122.00 −0.748318 −0.374159 0.927365i \(-0.622068\pi\)
−0.374159 + 0.927365i \(0.622068\pi\)
\(132\) − 33.0000i − 0.0217597i
\(133\) 875.000i 0.570467i
\(134\) 816.000 0.526057
\(135\) 0 0
\(136\) 441.000 0.278055
\(137\) 54.0000i 0.0336754i 0.999858 + 0.0168377i \(0.00535986\pi\)
−0.999858 + 0.0168377i \(0.994640\pi\)
\(138\) 729.000i 0.449686i
\(139\) −1748.00 −1.06664 −0.533322 0.845913i \(-0.679056\pi\)
−0.533322 + 0.845913i \(0.679056\pi\)
\(140\) 0 0
\(141\) 657.000 0.392407
\(142\) 1935.00i 1.14353i
\(143\) 176.000i 0.102922i
\(144\) 639.000 0.369792
\(145\) 0 0
\(146\) 336.000 0.190463
\(147\) 882.000i 0.494872i
\(148\) 253.000i 0.140517i
\(149\) −1797.00 −0.988027 −0.494013 0.869454i \(-0.664471\pi\)
−0.494013 + 0.869454i \(0.664471\pi\)
\(150\) 0 0
\(151\) 1040.00 0.560490 0.280245 0.959928i \(-0.409584\pi\)
0.280245 + 0.959928i \(0.409584\pi\)
\(152\) 2625.00i 1.40076i
\(153\) − 189.000i − 0.0998676i
\(154\) −231.000 −0.120873
\(155\) 0 0
\(156\) 48.0000 0.0246351
\(157\) − 562.000i − 0.285685i −0.989745 0.142842i \(-0.954376\pi\)
0.989745 0.142842i \(-0.0456242\pi\)
\(158\) 1527.00i 0.768871i
\(159\) −576.000 −0.287294
\(160\) 0 0
\(161\) 567.000 0.277552
\(162\) − 243.000i − 0.117851i
\(163\) − 2432.00i − 1.16864i −0.811522 0.584322i \(-0.801361\pi\)
0.811522 0.584322i \(-0.198639\pi\)
\(164\) −63.0000 −0.0299968
\(165\) 0 0
\(166\) 3762.00 1.75896
\(167\) 2340.00i 1.08428i 0.840288 + 0.542140i \(0.182386\pi\)
−0.840288 + 0.542140i \(0.817614\pi\)
\(168\) − 441.000i − 0.202523i
\(169\) 1941.00 0.883477
\(170\) 0 0
\(171\) 1125.00 0.503105
\(172\) − 100.000i − 0.0443310i
\(173\) − 3747.00i − 1.64670i −0.567534 0.823350i \(-0.692102\pi\)
0.567534 0.823350i \(-0.307898\pi\)
\(174\) −1674.00 −0.729343
\(175\) 0 0
\(176\) −781.000 −0.334489
\(177\) − 747.000i − 0.317220i
\(178\) 2268.00i 0.955021i
\(179\) 267.000 0.111489 0.0557445 0.998445i \(-0.482247\pi\)
0.0557445 + 0.998445i \(0.482247\pi\)
\(180\) 0 0
\(181\) 4277.00 1.75639 0.878196 0.478301i \(-0.158747\pi\)
0.878196 + 0.478301i \(0.158747\pi\)
\(182\) − 336.000i − 0.136846i
\(183\) − 192.000i − 0.0775576i
\(184\) 1701.00 0.681518
\(185\) 0 0
\(186\) −522.000 −0.205779
\(187\) 231.000i 0.0903337i
\(188\) 219.000i 0.0849586i
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) −1797.00 −0.680766 −0.340383 0.940287i \(-0.610557\pi\)
−0.340383 + 0.940287i \(0.610557\pi\)
\(192\) − 1299.00i − 0.488267i
\(193\) − 1988.00i − 0.741448i −0.928743 0.370724i \(-0.879110\pi\)
0.928743 0.370724i \(-0.120890\pi\)
\(194\) 2517.00 0.931495
\(195\) 0 0
\(196\) −294.000 −0.107143
\(197\) 3327.00i 1.20324i 0.798781 + 0.601622i \(0.205478\pi\)
−0.798781 + 0.601622i \(0.794522\pi\)
\(198\) 297.000i 0.106600i
\(199\) 1780.00 0.634075 0.317037 0.948413i \(-0.397312\pi\)
0.317037 + 0.948413i \(0.397312\pi\)
\(200\) 0 0
\(201\) −816.000 −0.286349
\(202\) 4239.00i 1.47651i
\(203\) 1302.00i 0.450160i
\(204\) 63.0000 0.0216220
\(205\) 0 0
\(206\) −4902.00 −1.65795
\(207\) − 729.000i − 0.244778i
\(208\) − 1136.00i − 0.378690i
\(209\) −1375.00 −0.455075
\(210\) 0 0
\(211\) 2180.00 0.711267 0.355634 0.934625i \(-0.384265\pi\)
0.355634 + 0.934625i \(0.384265\pi\)
\(212\) − 192.000i − 0.0622010i
\(213\) − 1935.00i − 0.622460i
\(214\) 2178.00 0.695724
\(215\) 0 0
\(216\) −567.000 −0.178609
\(217\) 406.000i 0.127010i
\(218\) 5136.00i 1.59566i
\(219\) −336.000 −0.103675
\(220\) 0 0
\(221\) −336.000 −0.102271
\(222\) − 2277.00i − 0.688388i
\(223\) − 3848.00i − 1.15552i −0.816206 0.577760i \(-0.803927\pi\)
0.816206 0.577760i \(-0.196073\pi\)
\(224\) 315.000 0.0939590
\(225\) 0 0
\(226\) 3384.00 0.996019
\(227\) − 1386.00i − 0.405251i −0.979256 0.202626i \(-0.935053\pi\)
0.979256 0.202626i \(-0.0649475\pi\)
\(228\) 375.000i 0.108925i
\(229\) 991.000 0.285970 0.142985 0.989725i \(-0.454330\pi\)
0.142985 + 0.989725i \(0.454330\pi\)
\(230\) 0 0
\(231\) 231.000 0.0657952
\(232\) 3906.00i 1.10535i
\(233\) − 975.000i − 0.274139i −0.990561 0.137069i \(-0.956232\pi\)
0.990561 0.137069i \(-0.0437684\pi\)
\(234\) −432.000 −0.120687
\(235\) 0 0
\(236\) 249.000 0.0686802
\(237\) − 1527.00i − 0.418520i
\(238\) − 441.000i − 0.120108i
\(239\) 1524.00 0.412466 0.206233 0.978503i \(-0.433880\pi\)
0.206233 + 0.978503i \(0.433880\pi\)
\(240\) 0 0
\(241\) −2230.00 −0.596045 −0.298023 0.954559i \(-0.596327\pi\)
−0.298023 + 0.954559i \(0.596327\pi\)
\(242\) − 363.000i − 0.0964237i
\(243\) 243.000i 0.0641500i
\(244\) 64.0000 0.0167917
\(245\) 0 0
\(246\) 567.000 0.146954
\(247\) − 2000.00i − 0.515210i
\(248\) 1218.00i 0.311867i
\(249\) −3762.00 −0.957458
\(250\) 0 0
\(251\) 3864.00 0.971687 0.485844 0.874046i \(-0.338512\pi\)
0.485844 + 0.874046i \(0.338512\pi\)
\(252\) − 63.0000i − 0.0157485i
\(253\) 891.000i 0.221410i
\(254\) 3381.00 0.835208
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) − 4518.00i − 1.09660i −0.836283 0.548298i \(-0.815276\pi\)
0.836283 0.548298i \(-0.184724\pi\)
\(258\) 900.000i 0.217177i
\(259\) −1771.00 −0.424883
\(260\) 0 0
\(261\) 1674.00 0.397004
\(262\) 3366.00i 0.793711i
\(263\) 438.000i 0.102693i 0.998681 + 0.0513465i \(0.0163513\pi\)
−0.998681 + 0.0513465i \(0.983649\pi\)
\(264\) 693.000 0.161558
\(265\) 0 0
\(266\) 2625.00 0.605072
\(267\) − 2268.00i − 0.519848i
\(268\) − 272.000i − 0.0619964i
\(269\) −4902.00 −1.11108 −0.555539 0.831490i \(-0.687488\pi\)
−0.555539 + 0.831490i \(0.687488\pi\)
\(270\) 0 0
\(271\) −2455.00 −0.550298 −0.275149 0.961402i \(-0.588727\pi\)
−0.275149 + 0.961402i \(0.588727\pi\)
\(272\) − 1491.00i − 0.332372i
\(273\) 336.000i 0.0744895i
\(274\) 162.000 0.0357182
\(275\) 0 0
\(276\) 243.000 0.0529959
\(277\) 1868.00i 0.405189i 0.979263 + 0.202594i \(0.0649373\pi\)
−0.979263 + 0.202594i \(0.935063\pi\)
\(278\) 5244.00i 1.13135i
\(279\) 522.000 0.112012
\(280\) 0 0
\(281\) −6093.00 −1.29352 −0.646758 0.762695i \(-0.723876\pi\)
−0.646758 + 0.762695i \(0.723876\pi\)
\(282\) − 1971.00i − 0.416210i
\(283\) − 2543.00i − 0.534154i −0.963675 0.267077i \(-0.913942\pi\)
0.963675 0.267077i \(-0.0860579\pi\)
\(284\) 645.000 0.134767
\(285\) 0 0
\(286\) 528.000 0.109165
\(287\) − 441.000i − 0.0907018i
\(288\) − 405.000i − 0.0828641i
\(289\) 4472.00 0.910238
\(290\) 0 0
\(291\) −2517.00 −0.507042
\(292\) − 112.000i − 0.0224462i
\(293\) 4623.00i 0.921770i 0.887460 + 0.460885i \(0.152468\pi\)
−0.887460 + 0.460885i \(0.847532\pi\)
\(294\) 2646.00 0.524891
\(295\) 0 0
\(296\) −5313.00 −1.04328
\(297\) − 297.000i − 0.0580259i
\(298\) 5391.00i 1.04796i
\(299\) −1296.00 −0.250668
\(300\) 0 0
\(301\) 700.000 0.134044
\(302\) − 3120.00i − 0.594489i
\(303\) − 4239.00i − 0.803710i
\(304\) 8875.00 1.67440
\(305\) 0 0
\(306\) −567.000 −0.105926
\(307\) 644.000i 0.119723i 0.998207 + 0.0598616i \(0.0190659\pi\)
−0.998207 + 0.0598616i \(0.980934\pi\)
\(308\) 77.0000i 0.0142451i
\(309\) 4902.00 0.902476
\(310\) 0 0
\(311\) 2616.00 0.476977 0.238488 0.971145i \(-0.423348\pi\)
0.238488 + 0.971145i \(0.423348\pi\)
\(312\) 1008.00i 0.182906i
\(313\) − 4079.00i − 0.736609i −0.929705 0.368305i \(-0.879938\pi\)
0.929705 0.368305i \(-0.120062\pi\)
\(314\) −1686.00 −0.303014
\(315\) 0 0
\(316\) 509.000 0.0906123
\(317\) 3504.00i 0.620834i 0.950601 + 0.310417i \(0.100469\pi\)
−0.950601 + 0.310417i \(0.899531\pi\)
\(318\) 1728.00i 0.304721i
\(319\) −2046.00 −0.359103
\(320\) 0 0
\(321\) −2178.00 −0.378704
\(322\) − 1701.00i − 0.294388i
\(323\) − 2625.00i − 0.452195i
\(324\) −81.0000 −0.0138889
\(325\) 0 0
\(326\) −7296.00 −1.23953
\(327\) − 5136.00i − 0.868567i
\(328\) − 1323.00i − 0.222715i
\(329\) −1533.00 −0.256891
\(330\) 0 0
\(331\) 4100.00 0.680835 0.340417 0.940274i \(-0.389432\pi\)
0.340417 + 0.940274i \(0.389432\pi\)
\(332\) − 1254.00i − 0.207296i
\(333\) 2277.00i 0.374711i
\(334\) 7020.00 1.15005
\(335\) 0 0
\(336\) −1491.00 −0.242085
\(337\) − 10924.0i − 1.76578i −0.469579 0.882891i \(-0.655594\pi\)
0.469579 0.882891i \(-0.344406\pi\)
\(338\) − 5823.00i − 0.937069i
\(339\) −3384.00 −0.542164
\(340\) 0 0
\(341\) −638.000 −0.101319
\(342\) − 3375.00i − 0.533623i
\(343\) − 4459.00i − 0.701934i
\(344\) 2100.00 0.329141
\(345\) 0 0
\(346\) −11241.0 −1.74659
\(347\) − 3612.00i − 0.558796i −0.960175 0.279398i \(-0.909865\pi\)
0.960175 0.279398i \(-0.0901349\pi\)
\(348\) 558.000i 0.0859538i
\(349\) 406.000 0.0622713 0.0311356 0.999515i \(-0.490088\pi\)
0.0311356 + 0.999515i \(0.490088\pi\)
\(350\) 0 0
\(351\) 432.000 0.0656936
\(352\) 495.000i 0.0749534i
\(353\) 816.000i 0.123035i 0.998106 + 0.0615174i \(0.0195940\pi\)
−0.998106 + 0.0615174i \(0.980406\pi\)
\(354\) −2241.00 −0.336463
\(355\) 0 0
\(356\) 756.000 0.112550
\(357\) 441.000i 0.0653787i
\(358\) − 801.000i − 0.118252i
\(359\) −4818.00 −0.708313 −0.354156 0.935186i \(-0.615232\pi\)
−0.354156 + 0.935186i \(0.615232\pi\)
\(360\) 0 0
\(361\) 8766.00 1.27803
\(362\) − 12831.0i − 1.86293i
\(363\) 363.000i 0.0524864i
\(364\) −112.000 −0.0161275
\(365\) 0 0
\(366\) −576.000 −0.0822623
\(367\) 2306.00i 0.327990i 0.986461 + 0.163995i \(0.0524380\pi\)
−0.986461 + 0.163995i \(0.947562\pi\)
\(368\) − 5751.00i − 0.814651i
\(369\) −567.000 −0.0799914
\(370\) 0 0
\(371\) 1344.00 0.188078
\(372\) 174.000i 0.0242513i
\(373\) − 3134.00i − 0.435047i −0.976055 0.217523i \(-0.930202\pi\)
0.976055 0.217523i \(-0.0697978\pi\)
\(374\) 693.000 0.0958133
\(375\) 0 0
\(376\) −4599.00 −0.630785
\(377\) − 2976.00i − 0.406556i
\(378\) 567.000i 0.0771517i
\(379\) −7202.00 −0.976100 −0.488050 0.872816i \(-0.662292\pi\)
−0.488050 + 0.872816i \(0.662292\pi\)
\(380\) 0 0
\(381\) −3381.00 −0.454630
\(382\) 5391.00i 0.722062i
\(383\) − 11472.0i − 1.53053i −0.643717 0.765263i \(-0.722609\pi\)
0.643717 0.765263i \(-0.277391\pi\)
\(384\) −4977.00 −0.661410
\(385\) 0 0
\(386\) −5964.00 −0.786424
\(387\) − 900.000i − 0.118216i
\(388\) − 839.000i − 0.109778i
\(389\) 3462.00 0.451235 0.225617 0.974216i \(-0.427560\pi\)
0.225617 + 0.974216i \(0.427560\pi\)
\(390\) 0 0
\(391\) −1701.00 −0.220008
\(392\) − 6174.00i − 0.795495i
\(393\) − 3366.00i − 0.432041i
\(394\) 9981.00 1.27623
\(395\) 0 0
\(396\) 99.0000 0.0125630
\(397\) 2486.00i 0.314279i 0.987576 + 0.157140i \(0.0502272\pi\)
−0.987576 + 0.157140i \(0.949773\pi\)
\(398\) − 5340.00i − 0.672538i
\(399\) −2625.00 −0.329359
\(400\) 0 0
\(401\) −9024.00 −1.12378 −0.561892 0.827211i \(-0.689926\pi\)
−0.561892 + 0.827211i \(0.689926\pi\)
\(402\) 2448.00i 0.303719i
\(403\) − 928.000i − 0.114707i
\(404\) 1413.00 0.174008
\(405\) 0 0
\(406\) 3906.00 0.477467
\(407\) − 2783.00i − 0.338939i
\(408\) 1323.00i 0.160535i
\(409\) 14488.0 1.75155 0.875777 0.482716i \(-0.160350\pi\)
0.875777 + 0.482716i \(0.160350\pi\)
\(410\) 0 0
\(411\) −162.000 −0.0194425
\(412\) 1634.00i 0.195392i
\(413\) 1743.00i 0.207669i
\(414\) −2187.00 −0.259626
\(415\) 0 0
\(416\) −720.000 −0.0848579
\(417\) − 5244.00i − 0.615827i
\(418\) 4125.00i 0.482680i
\(419\) 3201.00 0.373220 0.186610 0.982434i \(-0.440250\pi\)
0.186610 + 0.982434i \(0.440250\pi\)
\(420\) 0 0
\(421\) −6721.00 −0.778056 −0.389028 0.921226i \(-0.627189\pi\)
−0.389028 + 0.921226i \(0.627189\pi\)
\(422\) − 6540.00i − 0.754413i
\(423\) 1971.00i 0.226556i
\(424\) 4032.00 0.461819
\(425\) 0 0
\(426\) −5805.00 −0.660219
\(427\) 448.000i 0.0507734i
\(428\) − 726.000i − 0.0819919i
\(429\) −528.000 −0.0594221
\(430\) 0 0
\(431\) −354.000 −0.0395628 −0.0197814 0.999804i \(-0.506297\pi\)
−0.0197814 + 0.999804i \(0.506297\pi\)
\(432\) 1917.00i 0.213499i
\(433\) − 7682.00i − 0.852594i −0.904583 0.426297i \(-0.859818\pi\)
0.904583 0.426297i \(-0.140182\pi\)
\(434\) 1218.00 0.134714
\(435\) 0 0
\(436\) 1712.00 0.188050
\(437\) − 10125.0i − 1.10834i
\(438\) 1008.00i 0.109964i
\(439\) −5465.00 −0.594146 −0.297073 0.954855i \(-0.596011\pi\)
−0.297073 + 0.954855i \(0.596011\pi\)
\(440\) 0 0
\(441\) −2646.00 −0.285714
\(442\) 1008.00i 0.108474i
\(443\) 1551.00i 0.166344i 0.996535 + 0.0831718i \(0.0265050\pi\)
−0.996535 + 0.0831718i \(0.973495\pi\)
\(444\) −759.000 −0.0811274
\(445\) 0 0
\(446\) −11544.0 −1.22561
\(447\) − 5391.00i − 0.570437i
\(448\) 3031.00i 0.319646i
\(449\) 1092.00 0.114777 0.0573883 0.998352i \(-0.481723\pi\)
0.0573883 + 0.998352i \(0.481723\pi\)
\(450\) 0 0
\(451\) 693.000 0.0723550
\(452\) − 1128.00i − 0.117382i
\(453\) 3120.00i 0.323599i
\(454\) −4158.00 −0.429834
\(455\) 0 0
\(456\) −7875.00 −0.808730
\(457\) 10046.0i 1.02830i 0.857701 + 0.514149i \(0.171892\pi\)
−0.857701 + 0.514149i \(0.828108\pi\)
\(458\) − 2973.00i − 0.303317i
\(459\) 567.000 0.0576586
\(460\) 0 0
\(461\) 15510.0 1.56697 0.783485 0.621411i \(-0.213440\pi\)
0.783485 + 0.621411i \(0.213440\pi\)
\(462\) − 693.000i − 0.0697863i
\(463\) − 6878.00i − 0.690384i −0.938532 0.345192i \(-0.887814\pi\)
0.938532 0.345192i \(-0.112186\pi\)
\(464\) 13206.0 1.32128
\(465\) 0 0
\(466\) −2925.00 −0.290768
\(467\) − 16284.0i − 1.61356i −0.590850 0.806781i \(-0.701208\pi\)
0.590850 0.806781i \(-0.298792\pi\)
\(468\) 144.000i 0.0142231i
\(469\) 1904.00 0.187460
\(470\) 0 0
\(471\) 1686.00 0.164940
\(472\) 5229.00i 0.509924i
\(473\) 1100.00i 0.106930i
\(474\) −4581.00 −0.443908
\(475\) 0 0
\(476\) −147.000 −0.0141549
\(477\) − 1728.00i − 0.165869i
\(478\) − 4572.00i − 0.437486i
\(479\) −6732.00 −0.642156 −0.321078 0.947053i \(-0.604045\pi\)
−0.321078 + 0.947053i \(0.604045\pi\)
\(480\) 0 0
\(481\) 4048.00 0.383727
\(482\) 6690.00i 0.632202i
\(483\) 1701.00i 0.160245i
\(484\) −121.000 −0.0113636
\(485\) 0 0
\(486\) 729.000 0.0680414
\(487\) 17498.0i 1.62815i 0.580758 + 0.814076i \(0.302756\pi\)
−0.580758 + 0.814076i \(0.697244\pi\)
\(488\) 1344.00i 0.124672i
\(489\) 7296.00 0.674717
\(490\) 0 0
\(491\) 2454.00 0.225555 0.112777 0.993620i \(-0.464025\pi\)
0.112777 + 0.993620i \(0.464025\pi\)
\(492\) − 189.000i − 0.0173187i
\(493\) − 3906.00i − 0.356830i
\(494\) −6000.00 −0.546463
\(495\) 0 0
\(496\) 4118.00 0.372790
\(497\) 4515.00i 0.407496i
\(498\) 11286.0i 1.01554i
\(499\) 20716.0 1.85847 0.929234 0.369492i \(-0.120468\pi\)
0.929234 + 0.369492i \(0.120468\pi\)
\(500\) 0 0
\(501\) −7020.00 −0.626009
\(502\) − 11592.0i − 1.03063i
\(503\) 1956.00i 0.173387i 0.996235 + 0.0866936i \(0.0276301\pi\)
−0.996235 + 0.0866936i \(0.972370\pi\)
\(504\) 1323.00 0.116927
\(505\) 0 0
\(506\) 2673.00 0.234841
\(507\) 5823.00i 0.510076i
\(508\) − 1127.00i − 0.0984302i
\(509\) −18240.0 −1.58836 −0.794179 0.607684i \(-0.792099\pi\)
−0.794179 + 0.607684i \(0.792099\pi\)
\(510\) 0 0
\(511\) 784.000 0.0678711
\(512\) 8733.00i 0.753804i
\(513\) 3375.00i 0.290468i
\(514\) −13554.0 −1.16312
\(515\) 0 0
\(516\) 300.000 0.0255945
\(517\) − 2409.00i − 0.204928i
\(518\) 5313.00i 0.450656i
\(519\) 11241.0 0.950723
\(520\) 0 0
\(521\) 20790.0 1.74823 0.874114 0.485721i \(-0.161443\pi\)
0.874114 + 0.485721i \(0.161443\pi\)
\(522\) − 5022.00i − 0.421086i
\(523\) − 11897.0i − 0.994684i −0.867555 0.497342i \(-0.834310\pi\)
0.867555 0.497342i \(-0.165690\pi\)
\(524\) 1122.00 0.0935397
\(525\) 0 0
\(526\) 1314.00 0.108922
\(527\) − 1218.00i − 0.100677i
\(528\) − 2343.00i − 0.193117i
\(529\) 5606.00 0.460754
\(530\) 0 0
\(531\) 2241.00 0.183147
\(532\) − 875.000i − 0.0713084i
\(533\) 1008.00i 0.0819162i
\(534\) −6804.00 −0.551382
\(535\) 0 0
\(536\) 5712.00 0.460300
\(537\) 801.000i 0.0643682i
\(538\) 14706.0i 1.17848i
\(539\) 3234.00 0.258438
\(540\) 0 0
\(541\) 20336.0 1.61611 0.808053 0.589110i \(-0.200522\pi\)
0.808053 + 0.589110i \(0.200522\pi\)
\(542\) 7365.00i 0.583679i
\(543\) 12831.0i 1.01405i
\(544\) −945.000 −0.0744789
\(545\) 0 0
\(546\) 1008.00 0.0790081
\(547\) 16481.0i 1.28826i 0.764917 + 0.644129i \(0.222780\pi\)
−0.764917 + 0.644129i \(0.777220\pi\)
\(548\) − 54.0000i − 0.00420943i
\(549\) 576.000 0.0447779
\(550\) 0 0
\(551\) 23250.0 1.79761
\(552\) 5103.00i 0.393475i
\(553\) 3563.00i 0.273986i
\(554\) 5604.00 0.429767
\(555\) 0 0
\(556\) 1748.00 0.133330
\(557\) − 24618.0i − 1.87271i −0.351058 0.936354i \(-0.614178\pi\)
0.351058 0.936354i \(-0.385822\pi\)
\(558\) − 1566.00i − 0.118807i
\(559\) −1600.00 −0.121060
\(560\) 0 0
\(561\) −693.000 −0.0521542
\(562\) 18279.0i 1.37198i
\(563\) − 6438.00i − 0.481935i −0.970533 0.240967i \(-0.922535\pi\)
0.970533 0.240967i \(-0.0774647\pi\)
\(564\) −657.000 −0.0490509
\(565\) 0 0
\(566\) −7629.00 −0.566556
\(567\) − 567.000i − 0.0419961i
\(568\) 13545.0i 1.00059i
\(569\) 18183.0 1.33967 0.669834 0.742511i \(-0.266365\pi\)
0.669834 + 0.742511i \(0.266365\pi\)
\(570\) 0 0
\(571\) −17656.0 −1.29401 −0.647006 0.762485i \(-0.723979\pi\)
−0.647006 + 0.762485i \(0.723979\pi\)
\(572\) − 176.000i − 0.0128653i
\(573\) − 5391.00i − 0.393041i
\(574\) −1323.00 −0.0962038
\(575\) 0 0
\(576\) 3897.00 0.281901
\(577\) − 17155.0i − 1.23773i −0.785496 0.618867i \(-0.787592\pi\)
0.785496 0.618867i \(-0.212408\pi\)
\(578\) − 13416.0i − 0.965453i
\(579\) 5964.00 0.428075
\(580\) 0 0
\(581\) 8778.00 0.626803
\(582\) 7551.00i 0.537799i
\(583\) 2112.00i 0.150034i
\(584\) 2352.00 0.166655
\(585\) 0 0
\(586\) 13869.0 0.977684
\(587\) 24621.0i 1.73121i 0.500732 + 0.865603i \(0.333064\pi\)
−0.500732 + 0.865603i \(0.666936\pi\)
\(588\) − 882.000i − 0.0618590i
\(589\) 7250.00 0.507183
\(590\) 0 0
\(591\) −9981.00 −0.694693
\(592\) 17963.0i 1.24709i
\(593\) 9066.00i 0.627818i 0.949453 + 0.313909i \(0.101639\pi\)
−0.949453 + 0.313909i \(0.898361\pi\)
\(594\) −891.000 −0.0615457
\(595\) 0 0
\(596\) 1797.00 0.123503
\(597\) 5340.00i 0.366083i
\(598\) 3888.00i 0.265873i
\(599\) −22353.0 −1.52474 −0.762370 0.647142i \(-0.775964\pi\)
−0.762370 + 0.647142i \(0.775964\pi\)
\(600\) 0 0
\(601\) −6304.00 −0.427863 −0.213931 0.976849i \(-0.568627\pi\)
−0.213931 + 0.976849i \(0.568627\pi\)
\(602\) − 2100.00i − 0.142175i
\(603\) − 2448.00i − 0.165324i
\(604\) −1040.00 −0.0700613
\(605\) 0 0
\(606\) −12717.0 −0.852463
\(607\) 23672.0i 1.58289i 0.611238 + 0.791447i \(0.290672\pi\)
−0.611238 + 0.791447i \(0.709328\pi\)
\(608\) − 5625.00i − 0.375204i
\(609\) −3906.00 −0.259900
\(610\) 0 0
\(611\) 3504.00 0.232008
\(612\) 189.000i 0.0124835i
\(613\) 3028.00i 0.199510i 0.995012 + 0.0997551i \(0.0318059\pi\)
−0.995012 + 0.0997551i \(0.968194\pi\)
\(614\) 1932.00 0.126986
\(615\) 0 0
\(616\) −1617.00 −0.105764
\(617\) − 1170.00i − 0.0763410i −0.999271 0.0381705i \(-0.987847\pi\)
0.999271 0.0381705i \(-0.0121530\pi\)
\(618\) − 14706.0i − 0.957220i
\(619\) −9626.00 −0.625043 −0.312521 0.949911i \(-0.601174\pi\)
−0.312521 + 0.949911i \(0.601174\pi\)
\(620\) 0 0
\(621\) 2187.00 0.141323
\(622\) − 7848.00i − 0.505910i
\(623\) 5292.00i 0.340320i
\(624\) 3408.00 0.218637
\(625\) 0 0
\(626\) −12237.0 −0.781292
\(627\) − 4125.00i − 0.262738i
\(628\) 562.000i 0.0357106i
\(629\) 5313.00 0.336794
\(630\) 0 0
\(631\) −5794.00 −0.365540 −0.182770 0.983156i \(-0.558506\pi\)
−0.182770 + 0.983156i \(0.558506\pi\)
\(632\) 10689.0i 0.672762i
\(633\) 6540.00i 0.410650i
\(634\) 10512.0 0.658493
\(635\) 0 0
\(636\) 576.000 0.0359118
\(637\) 4704.00i 0.292589i
\(638\) 6138.00i 0.380887i
\(639\) 5805.00 0.359378
\(640\) 0 0
\(641\) 28308.0 1.74430 0.872152 0.489235i \(-0.162724\pi\)
0.872152 + 0.489235i \(0.162724\pi\)
\(642\) 6534.00i 0.401677i
\(643\) − 2270.00i − 0.139222i −0.997574 0.0696112i \(-0.977824\pi\)
0.997574 0.0696112i \(-0.0221759\pi\)
\(644\) −567.000 −0.0346940
\(645\) 0 0
\(646\) −7875.00 −0.479625
\(647\) 23361.0i 1.41950i 0.704454 + 0.709749i \(0.251192\pi\)
−0.704454 + 0.709749i \(0.748808\pi\)
\(648\) − 1701.00i − 0.103120i
\(649\) −2739.00 −0.165663
\(650\) 0 0
\(651\) −1218.00 −0.0733290
\(652\) 2432.00i 0.146080i
\(653\) − 12294.0i − 0.736756i −0.929676 0.368378i \(-0.879913\pi\)
0.929676 0.368378i \(-0.120087\pi\)
\(654\) −15408.0 −0.921255
\(655\) 0 0
\(656\) −4473.00 −0.266222
\(657\) − 1008.00i − 0.0598567i
\(658\) 4599.00i 0.272474i
\(659\) −31896.0 −1.88542 −0.942710 0.333613i \(-0.891732\pi\)
−0.942710 + 0.333613i \(0.891732\pi\)
\(660\) 0 0
\(661\) 13469.0 0.792562 0.396281 0.918129i \(-0.370301\pi\)
0.396281 + 0.918129i \(0.370301\pi\)
\(662\) − 12300.0i − 0.722135i
\(663\) − 1008.00i − 0.0590460i
\(664\) 26334.0 1.53909
\(665\) 0 0
\(666\) 6831.00 0.397441
\(667\) − 15066.0i − 0.874599i
\(668\) − 2340.00i − 0.135535i
\(669\) 11544.0 0.667140
\(670\) 0 0
\(671\) −704.000 −0.0405032
\(672\) 945.000i 0.0542473i
\(673\) − 13712.0i − 0.785377i −0.919672 0.392689i \(-0.871545\pi\)
0.919672 0.392689i \(-0.128455\pi\)
\(674\) −32772.0 −1.87289
\(675\) 0 0
\(676\) −1941.00 −0.110435
\(677\) − 3714.00i − 0.210843i −0.994428 0.105421i \(-0.966381\pi\)
0.994428 0.105421i \(-0.0336192\pi\)
\(678\) 10152.0i 0.575052i
\(679\) 5873.00 0.331937
\(680\) 0 0
\(681\) 4158.00 0.233972
\(682\) 1914.00i 0.107465i
\(683\) 13065.0i 0.731945i 0.930626 + 0.365972i \(0.119264\pi\)
−0.930626 + 0.365972i \(0.880736\pi\)
\(684\) −1125.00 −0.0628881
\(685\) 0 0
\(686\) −13377.0 −0.744513
\(687\) 2973.00i 0.165105i
\(688\) − 7100.00i − 0.393437i
\(689\) −3072.00 −0.169860
\(690\) 0 0
\(691\) 12512.0 0.688826 0.344413 0.938818i \(-0.388078\pi\)
0.344413 + 0.938818i \(0.388078\pi\)
\(692\) 3747.00i 0.205838i
\(693\) 693.000i 0.0379869i
\(694\) −10836.0 −0.592693
\(695\) 0 0
\(696\) −11718.0 −0.638175
\(697\) 1323.00i 0.0718970i
\(698\) − 1218.00i − 0.0660487i
\(699\) 2925.00 0.158274
\(700\) 0 0
\(701\) 33405.0 1.79984 0.899921 0.436053i \(-0.143624\pi\)
0.899921 + 0.436053i \(0.143624\pi\)
\(702\) − 1296.00i − 0.0696786i
\(703\) 31625.0i 1.69667i
\(704\) −4763.00 −0.254989
\(705\) 0 0
\(706\) 2448.00 0.130498
\(707\) 9891.00i 0.526152i
\(708\) 747.000i 0.0396525i
\(709\) −22217.0 −1.17684 −0.588418 0.808557i \(-0.700249\pi\)
−0.588418 + 0.808557i \(0.700249\pi\)
\(710\) 0 0
\(711\) 4581.00 0.241633
\(712\) 15876.0i 0.835644i
\(713\) − 4698.00i − 0.246762i
\(714\) 1323.00 0.0693446
\(715\) 0 0
\(716\) −267.000 −0.0139361
\(717\) 4572.00i 0.238137i
\(718\) 14454.0i 0.751279i
\(719\) −12336.0 −0.639854 −0.319927 0.947442i \(-0.603658\pi\)
−0.319927 + 0.947442i \(0.603658\pi\)
\(720\) 0 0
\(721\) −11438.0 −0.590809
\(722\) − 26298.0i − 1.35555i
\(723\) − 6690.00i − 0.344127i
\(724\) −4277.00 −0.219549
\(725\) 0 0
\(726\) 1089.00 0.0556702
\(727\) − 10720.0i − 0.546881i −0.961889 0.273441i \(-0.911838\pi\)
0.961889 0.273441i \(-0.0881617\pi\)
\(728\) − 2352.00i − 0.119740i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −2100.00 −0.106253
\(732\) 192.000i 0.00969471i
\(733\) − 10820.0i − 0.545219i −0.962125 0.272610i \(-0.912113\pi\)
0.962125 0.272610i \(-0.0878868\pi\)
\(734\) 6918.00 0.347886
\(735\) 0 0
\(736\) −3645.00 −0.182550
\(737\) 2992.00i 0.149541i
\(738\) 1701.00i 0.0848437i
\(739\) −10127.0 −0.504097 −0.252049 0.967715i \(-0.581104\pi\)
−0.252049 + 0.967715i \(0.581104\pi\)
\(740\) 0 0
\(741\) 6000.00 0.297457
\(742\) − 4032.00i − 0.199487i
\(743\) 6000.00i 0.296257i 0.988968 + 0.148128i \(0.0473249\pi\)
−0.988968 + 0.148128i \(0.952675\pi\)
\(744\) −3654.00 −0.180057
\(745\) 0 0
\(746\) −9402.00 −0.461437
\(747\) − 11286.0i − 0.552789i
\(748\) − 231.000i − 0.0112917i
\(749\) 5082.00 0.247920
\(750\) 0 0
\(751\) 26132.0 1.26973 0.634867 0.772621i \(-0.281055\pi\)
0.634867 + 0.772621i \(0.281055\pi\)
\(752\) 15549.0i 0.754008i
\(753\) 11592.0i 0.561004i
\(754\) −8928.00 −0.431218
\(755\) 0 0
\(756\) 189.000 0.00909241
\(757\) 25850.0i 1.24113i 0.784156 + 0.620564i \(0.213096\pi\)
−0.784156 + 0.620564i \(0.786904\pi\)
\(758\) 21606.0i 1.03531i
\(759\) −2673.00 −0.127831
\(760\) 0 0
\(761\) −3654.00 −0.174057 −0.0870285 0.996206i \(-0.527737\pi\)
−0.0870285 + 0.996206i \(0.527737\pi\)
\(762\) 10143.0i 0.482208i
\(763\) 11984.0i 0.568611i
\(764\) 1797.00 0.0850958
\(765\) 0 0
\(766\) −34416.0 −1.62337
\(767\) − 3984.00i − 0.187554i
\(768\) 4539.00i 0.213264i
\(769\) −6248.00 −0.292989 −0.146495 0.989211i \(-0.546799\pi\)
−0.146495 + 0.989211i \(0.546799\pi\)
\(770\) 0 0
\(771\) 13554.0 0.633120
\(772\) 1988.00i 0.0926809i
\(773\) 23952.0i 1.11448i 0.830351 + 0.557240i \(0.188140\pi\)
−0.830351 + 0.557240i \(0.811860\pi\)
\(774\) −2700.00 −0.125387
\(775\) 0 0
\(776\) 17619.0 0.815058
\(777\) − 5313.00i − 0.245306i
\(778\) − 10386.0i − 0.478607i
\(779\) −7875.00 −0.362197
\(780\) 0 0
\(781\) −7095.00 −0.325069
\(782\) 5103.00i 0.233354i
\(783\) 5022.00i 0.229210i
\(784\) −20874.0 −0.950893
\(785\) 0 0
\(786\) −10098.0 −0.458249
\(787\) − 39631.0i − 1.79504i −0.440979 0.897518i \(-0.645369\pi\)
0.440979 0.897518i \(-0.354631\pi\)
\(788\) − 3327.00i − 0.150405i
\(789\) −1314.00 −0.0592898
\(790\) 0 0
\(791\) 7896.00 0.354930
\(792\) 2079.00i 0.0932753i
\(793\) − 1024.00i − 0.0458554i
\(794\) 7458.00 0.333343
\(795\) 0 0
\(796\) −1780.00 −0.0792593
\(797\) − 7530.00i − 0.334663i −0.985901 0.167331i \(-0.946485\pi\)
0.985901 0.167331i \(-0.0535150\pi\)
\(798\) 7875.00i 0.349338i
\(799\) 4599.00 0.203631
\(800\) 0 0
\(801\) 6804.00 0.300134
\(802\) 27072.0i 1.19195i
\(803\) 1232.00i 0.0541424i
\(804\) 816.000 0.0357937
\(805\) 0 0
\(806\) −2784.00 −0.121665
\(807\) − 14706.0i − 0.641482i
\(808\) 29673.0i 1.29195i
\(809\) 7239.00 0.314598 0.157299 0.987551i \(-0.449721\pi\)
0.157299 + 0.987551i \(0.449721\pi\)
\(810\) 0 0
\(811\) −35611.0 −1.54189 −0.770944 0.636903i \(-0.780215\pi\)
−0.770944 + 0.636903i \(0.780215\pi\)
\(812\) − 1302.00i − 0.0562700i
\(813\) − 7365.00i − 0.317714i
\(814\) −8349.00 −0.359499
\(815\) 0 0
\(816\) 4473.00 0.191895
\(817\) − 12500.0i − 0.535275i
\(818\) − 43464.0i − 1.85780i
\(819\) −1008.00 −0.0430066
\(820\) 0 0
\(821\) −42054.0 −1.78769 −0.893846 0.448375i \(-0.852003\pi\)
−0.893846 + 0.448375i \(0.852003\pi\)
\(822\) 486.000i 0.0206219i
\(823\) − 10172.0i − 0.430831i −0.976523 0.215415i \(-0.930889\pi\)
0.976523 0.215415i \(-0.0691105\pi\)
\(824\) −34314.0 −1.45071
\(825\) 0 0
\(826\) 5229.00 0.220267
\(827\) 4560.00i 0.191737i 0.995394 + 0.0958686i \(0.0305629\pi\)
−0.995394 + 0.0958686i \(0.969437\pi\)
\(828\) 729.000i 0.0305972i
\(829\) −13202.0 −0.553105 −0.276553 0.960999i \(-0.589192\pi\)
−0.276553 + 0.960999i \(0.589192\pi\)
\(830\) 0 0
\(831\) −5604.00 −0.233936
\(832\) − 6928.00i − 0.288684i
\(833\) 6174.00i 0.256802i
\(834\) −15732.0 −0.653183
\(835\) 0 0
\(836\) 1375.00 0.0568844
\(837\) 1566.00i 0.0646701i
\(838\) − 9603.00i − 0.395859i
\(839\) −6216.00 −0.255781 −0.127890 0.991788i \(-0.540821\pi\)
−0.127890 + 0.991788i \(0.540821\pi\)
\(840\) 0 0
\(841\) 10207.0 0.418508
\(842\) 20163.0i 0.825253i
\(843\) − 18279.0i − 0.746812i
\(844\) −2180.00 −0.0889084
\(845\) 0 0
\(846\) 5913.00 0.240299
\(847\) − 847.000i − 0.0343604i
\(848\) − 13632.0i − 0.552034i
\(849\) 7629.00 0.308394
\(850\) 0 0
\(851\) 20493.0 0.825489
\(852\) 1935.00i 0.0778075i
\(853\) − 22718.0i − 0.911899i −0.890006 0.455949i \(-0.849300\pi\)
0.890006 0.455949i \(-0.150700\pi\)
\(854\) 1344.00 0.0538533
\(855\) 0 0
\(856\) 15246.0 0.608759
\(857\) − 27435.0i − 1.09354i −0.837284 0.546769i \(-0.815858\pi\)
0.837284 0.546769i \(-0.184142\pi\)
\(858\) 1584.00i 0.0630267i
\(859\) −37556.0 −1.49173 −0.745864 0.666098i \(-0.767963\pi\)
−0.745864 + 0.666098i \(0.767963\pi\)
\(860\) 0 0
\(861\) 1323.00 0.0523667
\(862\) 1062.00i 0.0419627i
\(863\) 14976.0i 0.590717i 0.955386 + 0.295359i \(0.0954391\pi\)
−0.955386 + 0.295359i \(0.904561\pi\)
\(864\) 1215.00 0.0478416
\(865\) 0 0
\(866\) −23046.0 −0.904313
\(867\) 13416.0i 0.525526i
\(868\) − 406.000i − 0.0158762i
\(869\) −5599.00 −0.218565
\(870\) 0 0
\(871\) −4352.00 −0.169302
\(872\) 35952.0i 1.39620i
\(873\) − 7551.00i − 0.292741i
\(874\) −30375.0 −1.17557
\(875\) 0 0
\(876\) 336.000 0.0129593
\(877\) − 30718.0i − 1.18275i −0.806396 0.591376i \(-0.798585\pi\)
0.806396 0.591376i \(-0.201415\pi\)
\(878\) 16395.0i 0.630187i
\(879\) −13869.0 −0.532184
\(880\) 0 0
\(881\) 2916.00 0.111513 0.0557563 0.998444i \(-0.482243\pi\)
0.0557563 + 0.998444i \(0.482243\pi\)
\(882\) 7938.00i 0.303046i
\(883\) 39670.0i 1.51189i 0.654633 + 0.755947i \(0.272823\pi\)
−0.654633 + 0.755947i \(0.727177\pi\)
\(884\) 336.000 0.0127838
\(885\) 0 0
\(886\) 4653.00 0.176434
\(887\) − 23724.0i − 0.898054i −0.893518 0.449027i \(-0.851771\pi\)
0.893518 0.449027i \(-0.148229\pi\)
\(888\) − 15939.0i − 0.602340i
\(889\) 7889.00 0.297625
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 3848.00i 0.144440i
\(893\) 27375.0i 1.02583i
\(894\) −16173.0 −0.605040
\(895\) 0 0
\(896\) 11613.0 0.432995
\(897\) − 3888.00i − 0.144723i
\(898\) − 3276.00i − 0.121739i
\(899\) 10788.0 0.400222
\(900\) 0 0
\(901\) −4032.00 −0.149085
\(902\) − 2079.00i − 0.0767440i
\(903\) 2100.00i 0.0773905i
\(904\) 23688.0 0.871517
\(905\) 0 0
\(906\) 9360.00 0.343229
\(907\) − 42166.0i − 1.54366i −0.635829 0.771830i \(-0.719342\pi\)
0.635829 0.771830i \(-0.280658\pi\)
\(908\) 1386.00i 0.0506564i
\(909\) 12717.0 0.464022
\(910\) 0 0
\(911\) 32139.0 1.16884 0.584420 0.811452i \(-0.301322\pi\)
0.584420 + 0.811452i \(0.301322\pi\)
\(912\) 26625.0i 0.966713i
\(913\) 13794.0i 0.500016i
\(914\) 30138.0 1.09067
\(915\) 0 0
\(916\) −991.000 −0.0357462
\(917\) 7854.00i 0.282837i
\(918\) − 1701.00i − 0.0611562i
\(919\) −44525.0 −1.59820 −0.799099 0.601199i \(-0.794690\pi\)
−0.799099 + 0.601199i \(0.794690\pi\)
\(920\) 0 0
\(921\) −1932.00 −0.0691222
\(922\) − 46530.0i − 1.66202i
\(923\) − 10320.0i − 0.368025i
\(924\) −231.000 −0.00822440
\(925\) 0 0
\(926\) −20634.0 −0.732263
\(927\) 14706.0i 0.521045i
\(928\) − 8370.00i − 0.296076i
\(929\) 5964.00 0.210627 0.105314 0.994439i \(-0.466415\pi\)
0.105314 + 0.994439i \(0.466415\pi\)
\(930\) 0 0
\(931\) −36750.0 −1.29370
\(932\) 975.000i 0.0342674i
\(933\) 7848.00i 0.275383i
\(934\) −48852.0 −1.71144
\(935\) 0 0
\(936\) −3024.00 −0.105601
\(937\) 6662.00i 0.232271i 0.993233 + 0.116136i \(0.0370507\pi\)
−0.993233 + 0.116136i \(0.962949\pi\)
\(938\) − 5712.00i − 0.198831i
\(939\) 12237.0 0.425282
\(940\) 0 0
\(941\) −42129.0 −1.45948 −0.729738 0.683727i \(-0.760358\pi\)
−0.729738 + 0.683727i \(0.760358\pi\)
\(942\) − 5058.00i − 0.174945i
\(943\) 5103.00i 0.176221i
\(944\) 17679.0 0.609536
\(945\) 0 0
\(946\) 3300.00 0.113417
\(947\) 23049.0i 0.790910i 0.918485 + 0.395455i \(0.129413\pi\)
−0.918485 + 0.395455i \(0.870587\pi\)
\(948\) 1527.00i 0.0523150i
\(949\) −1792.00 −0.0612969
\(950\) 0 0
\(951\) −10512.0 −0.358438
\(952\) − 3087.00i − 0.105095i
\(953\) − 10221.0i − 0.347419i −0.984797 0.173710i \(-0.944425\pi\)
0.984797 0.173710i \(-0.0555754\pi\)
\(954\) −5184.00 −0.175931
\(955\) 0 0
\(956\) −1524.00 −0.0515582
\(957\) − 6138.00i − 0.207328i
\(958\) 20196.0i 0.681110i
\(959\) 378.000 0.0127281
\(960\) 0 0
\(961\) −26427.0 −0.887080
\(962\) − 12144.0i − 0.407004i
\(963\) − 6534.00i − 0.218645i
\(964\) 2230.00 0.0745057
\(965\) 0 0
\(966\) 5103.00 0.169965
\(967\) − 1072.00i − 0.0356496i −0.999841 0.0178248i \(-0.994326\pi\)
0.999841 0.0178248i \(-0.00567412\pi\)
\(968\) − 2541.00i − 0.0843707i
\(969\) 7875.00 0.261075
\(970\) 0 0
\(971\) −50337.0 −1.66364 −0.831818 0.555048i \(-0.812700\pi\)
−0.831818 + 0.555048i \(0.812700\pi\)
\(972\) − 243.000i − 0.00801875i
\(973\) 12236.0i 0.403153i
\(974\) 52494.0 1.72692
\(975\) 0 0
\(976\) 4544.00 0.149027
\(977\) 49638.0i 1.62545i 0.582651 + 0.812723i \(0.302016\pi\)
−0.582651 + 0.812723i \(0.697984\pi\)
\(978\) − 21888.0i − 0.715645i
\(979\) −8316.00 −0.271482
\(980\) 0 0
\(981\) 15408.0 0.501467
\(982\) − 7362.00i − 0.239237i
\(983\) 1143.00i 0.0370865i 0.999828 + 0.0185433i \(0.00590284\pi\)
−0.999828 + 0.0185433i \(0.994097\pi\)
\(984\) 3969.00 0.128584
\(985\) 0 0
\(986\) −11718.0 −0.378476
\(987\) − 4599.00i − 0.148316i
\(988\) 2000.00i 0.0644013i
\(989\) −8100.00 −0.260430
\(990\) 0 0
\(991\) 35060.0 1.12383 0.561916 0.827194i \(-0.310064\pi\)
0.561916 + 0.827194i \(0.310064\pi\)
\(992\) − 2610.00i − 0.0835359i
\(993\) 12300.0i 0.393080i
\(994\) 13545.0 0.432215
\(995\) 0 0
\(996\) 3762.00 0.119682
\(997\) − 55582.0i − 1.76560i −0.469752 0.882798i \(-0.655657\pi\)
0.469752 0.882798i \(-0.344343\pi\)
\(998\) − 62148.0i − 1.97120i
\(999\) −6831.00 −0.216340
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 825.4.c.d.199.1 2
5.2 odd 4 825.4.a.g.1.1 yes 1
5.3 odd 4 825.4.a.c.1.1 1
5.4 even 2 inner 825.4.c.d.199.2 2
15.2 even 4 2475.4.a.d.1.1 1
15.8 even 4 2475.4.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.c.1.1 1 5.3 odd 4
825.4.a.g.1.1 yes 1 5.2 odd 4
825.4.c.d.199.1 2 1.1 even 1 trivial
825.4.c.d.199.2 2 5.4 even 2 inner
2475.4.a.d.1.1 1 15.2 even 4
2475.4.a.i.1.1 1 15.8 even 4