Properties

Label 405.4.a.m.1.7
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-5.38503\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.38503 q^{2} +20.9986 q^{4} -5.00000 q^{5} +12.5702 q^{7} +69.9976 q^{8} +O(q^{10})\) \(q+5.38503 q^{2} +20.9986 q^{4} -5.00000 q^{5} +12.5702 q^{7} +69.9976 q^{8} -26.9252 q^{10} +12.8307 q^{11} +59.8645 q^{13} +67.6909 q^{14} +208.951 q^{16} -110.011 q^{17} -12.0872 q^{19} -104.993 q^{20} +69.0936 q^{22} -67.7215 q^{23} +25.0000 q^{25} +322.372 q^{26} +263.956 q^{28} +199.958 q^{29} +76.6286 q^{31} +565.226 q^{32} -592.412 q^{34} -62.8510 q^{35} -22.4815 q^{37} -65.0898 q^{38} -349.988 q^{40} +87.7615 q^{41} +119.410 q^{43} +269.426 q^{44} -364.682 q^{46} +243.155 q^{47} -184.990 q^{49} +134.626 q^{50} +1257.07 q^{52} -293.518 q^{53} -64.1534 q^{55} +879.884 q^{56} +1076.78 q^{58} -581.383 q^{59} -773.693 q^{61} +412.648 q^{62} +1372.15 q^{64} -299.322 q^{65} -231.438 q^{67} -2310.07 q^{68} -338.455 q^{70} -744.342 q^{71} -264.839 q^{73} -121.063 q^{74} -253.813 q^{76} +161.284 q^{77} +559.717 q^{79} -1044.75 q^{80} +472.598 q^{82} -1220.89 q^{83} +550.054 q^{85} +643.027 q^{86} +898.117 q^{88} -255.905 q^{89} +752.509 q^{91} -1422.05 q^{92} +1309.40 q^{94} +60.4359 q^{95} +1049.52 q^{97} -996.177 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8} + 10 q^{10} - 23 q^{11} + 96 q^{13} + 21 q^{14} + 324 q^{16} - 161 q^{17} + 279 q^{19} - 180 q^{20} + 311 q^{22} - 96 q^{23} + 175 q^{25} + 358 q^{26} + 337 q^{28} + 296 q^{29} + 244 q^{31} + 314 q^{32} + 125 q^{34} - 110 q^{35} + 404 q^{37} - 305 q^{38} + 90 q^{40} + 47 q^{41} + 525 q^{43} - 55 q^{44} + 717 q^{46} - 164 q^{47} + 1225 q^{49} - 50 q^{50} + 1682 q^{52} - 506 q^{53} + 115 q^{55} + 981 q^{56} + 1183 q^{58} + 85 q^{59} + 828 q^{61} + 786 q^{62} + 2236 q^{64} - 480 q^{65} + 1093 q^{67} - 2473 q^{68} - 105 q^{70} - 328 q^{71} + 2085 q^{73} + 1316 q^{74} + 2789 q^{76} - 24 q^{77} + 2110 q^{79} - 1620 q^{80} - 62 q^{82} - 1290 q^{83} + 805 q^{85} + 2569 q^{86} + 2271 q^{88} + 3048 q^{89} + 3338 q^{91} - 2763 q^{92} - 517 q^{94} - 1395 q^{95} + 1787 q^{97} - 1279 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.38503 1.90390 0.951948 0.306260i \(-0.0990777\pi\)
0.951948 + 0.306260i \(0.0990777\pi\)
\(3\) 0 0
\(4\) 20.9986 2.62482
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 12.5702 0.678727 0.339363 0.940655i \(-0.389788\pi\)
0.339363 + 0.940655i \(0.389788\pi\)
\(8\) 69.9976 3.09349
\(9\) 0 0
\(10\) −26.9252 −0.851448
\(11\) 12.8307 0.351691 0.175845 0.984418i \(-0.443734\pi\)
0.175845 + 0.984418i \(0.443734\pi\)
\(12\) 0 0
\(13\) 59.8645 1.27719 0.638593 0.769544i \(-0.279517\pi\)
0.638593 + 0.769544i \(0.279517\pi\)
\(14\) 67.6909 1.29223
\(15\) 0 0
\(16\) 208.951 3.26486
\(17\) −110.011 −1.56950 −0.784752 0.619810i \(-0.787210\pi\)
−0.784752 + 0.619810i \(0.787210\pi\)
\(18\) 0 0
\(19\) −12.0872 −0.145947 −0.0729733 0.997334i \(-0.523249\pi\)
−0.0729733 + 0.997334i \(0.523249\pi\)
\(20\) −104.993 −1.17385
\(21\) 0 0
\(22\) 69.0936 0.669582
\(23\) −67.7215 −0.613953 −0.306976 0.951717i \(-0.599317\pi\)
−0.306976 + 0.951717i \(0.599317\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 322.372 2.43163
\(27\) 0 0
\(28\) 263.956 1.78154
\(29\) 199.958 1.28039 0.640194 0.768213i \(-0.278854\pi\)
0.640194 + 0.768213i \(0.278854\pi\)
\(30\) 0 0
\(31\) 76.6286 0.443965 0.221982 0.975051i \(-0.428747\pi\)
0.221982 + 0.975051i \(0.428747\pi\)
\(32\) 565.226 3.12246
\(33\) 0 0
\(34\) −592.412 −2.98817
\(35\) −62.8510 −0.303536
\(36\) 0 0
\(37\) −22.4815 −0.0998900 −0.0499450 0.998752i \(-0.515905\pi\)
−0.0499450 + 0.998752i \(0.515905\pi\)
\(38\) −65.0898 −0.277867
\(39\) 0 0
\(40\) −349.988 −1.38345
\(41\) 87.7615 0.334294 0.167147 0.985932i \(-0.446545\pi\)
0.167147 + 0.985932i \(0.446545\pi\)
\(42\) 0 0
\(43\) 119.410 0.423485 0.211743 0.977325i \(-0.432086\pi\)
0.211743 + 0.977325i \(0.432086\pi\)
\(44\) 269.426 0.923124
\(45\) 0 0
\(46\) −364.682 −1.16890
\(47\) 243.155 0.754635 0.377317 0.926084i \(-0.376847\pi\)
0.377317 + 0.926084i \(0.376847\pi\)
\(48\) 0 0
\(49\) −184.990 −0.539330
\(50\) 134.626 0.380779
\(51\) 0 0
\(52\) 1257.07 3.35238
\(53\) −293.518 −0.760712 −0.380356 0.924840i \(-0.624199\pi\)
−0.380356 + 0.924840i \(0.624199\pi\)
\(54\) 0 0
\(55\) −64.1534 −0.157281
\(56\) 879.884 2.09963
\(57\) 0 0
\(58\) 1076.78 2.43773
\(59\) −581.383 −1.28288 −0.641438 0.767175i \(-0.721662\pi\)
−0.641438 + 0.767175i \(0.721662\pi\)
\(60\) 0 0
\(61\) −773.693 −1.62395 −0.811977 0.583689i \(-0.801609\pi\)
−0.811977 + 0.583689i \(0.801609\pi\)
\(62\) 412.648 0.845263
\(63\) 0 0
\(64\) 1372.15 2.67998
\(65\) −299.322 −0.571175
\(66\) 0 0
\(67\) −231.438 −0.422010 −0.211005 0.977485i \(-0.567674\pi\)
−0.211005 + 0.977485i \(0.567674\pi\)
\(68\) −2310.07 −4.11966
\(69\) 0 0
\(70\) −338.455 −0.577901
\(71\) −744.342 −1.24418 −0.622092 0.782944i \(-0.713717\pi\)
−0.622092 + 0.782944i \(0.713717\pi\)
\(72\) 0 0
\(73\) −264.839 −0.424616 −0.212308 0.977203i \(-0.568098\pi\)
−0.212308 + 0.977203i \(0.568098\pi\)
\(74\) −121.063 −0.190180
\(75\) 0 0
\(76\) −253.813 −0.383084
\(77\) 161.284 0.238702
\(78\) 0 0
\(79\) 559.717 0.797127 0.398564 0.917141i \(-0.369509\pi\)
0.398564 + 0.917141i \(0.369509\pi\)
\(80\) −1044.75 −1.46009
\(81\) 0 0
\(82\) 472.598 0.636460
\(83\) −1220.89 −1.61458 −0.807288 0.590158i \(-0.799065\pi\)
−0.807288 + 0.590158i \(0.799065\pi\)
\(84\) 0 0
\(85\) 550.054 0.701903
\(86\) 643.027 0.806272
\(87\) 0 0
\(88\) 898.117 1.08795
\(89\) −255.905 −0.304785 −0.152392 0.988320i \(-0.548698\pi\)
−0.152392 + 0.988320i \(0.548698\pi\)
\(90\) 0 0
\(91\) 752.509 0.866861
\(92\) −1422.05 −1.61151
\(93\) 0 0
\(94\) 1309.40 1.43675
\(95\) 60.4359 0.0652693
\(96\) 0 0
\(97\) 1049.52 1.09858 0.549291 0.835631i \(-0.314898\pi\)
0.549291 + 0.835631i \(0.314898\pi\)
\(98\) −996.177 −1.02683
\(99\) 0 0
\(100\) 524.964 0.524964
\(101\) 88.7270 0.0874125 0.0437062 0.999044i \(-0.486083\pi\)
0.0437062 + 0.999044i \(0.486083\pi\)
\(102\) 0 0
\(103\) −1544.07 −1.47710 −0.738552 0.674196i \(-0.764490\pi\)
−0.738552 + 0.674196i \(0.764490\pi\)
\(104\) 4190.37 3.95096
\(105\) 0 0
\(106\) −1580.60 −1.44832
\(107\) −585.772 −0.529240 −0.264620 0.964353i \(-0.585247\pi\)
−0.264620 + 0.964353i \(0.585247\pi\)
\(108\) 0 0
\(109\) 1367.04 1.20127 0.600634 0.799524i \(-0.294915\pi\)
0.600634 + 0.799524i \(0.294915\pi\)
\(110\) −345.468 −0.299446
\(111\) 0 0
\(112\) 2626.55 2.21595
\(113\) 170.429 0.141882 0.0709408 0.997481i \(-0.477400\pi\)
0.0709408 + 0.997481i \(0.477400\pi\)
\(114\) 0 0
\(115\) 338.608 0.274568
\(116\) 4198.83 3.36079
\(117\) 0 0
\(118\) −3130.77 −2.44246
\(119\) −1382.86 −1.06526
\(120\) 0 0
\(121\) −1166.37 −0.876314
\(122\) −4166.36 −3.09184
\(123\) 0 0
\(124\) 1609.09 1.16533
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1809.74 1.26447 0.632237 0.774775i \(-0.282137\pi\)
0.632237 + 0.774775i \(0.282137\pi\)
\(128\) 2867.27 1.97995
\(129\) 0 0
\(130\) −1611.86 −1.08746
\(131\) −1238.29 −0.825879 −0.412940 0.910758i \(-0.635498\pi\)
−0.412940 + 0.910758i \(0.635498\pi\)
\(132\) 0 0
\(133\) −151.938 −0.0990580
\(134\) −1246.30 −0.803462
\(135\) 0 0
\(136\) −7700.50 −4.85524
\(137\) 506.183 0.315665 0.157832 0.987466i \(-0.449549\pi\)
0.157832 + 0.987466i \(0.449549\pi\)
\(138\) 0 0
\(139\) −213.680 −0.130389 −0.0651946 0.997873i \(-0.520767\pi\)
−0.0651946 + 0.997873i \(0.520767\pi\)
\(140\) −1319.78 −0.796727
\(141\) 0 0
\(142\) −4008.30 −2.36880
\(143\) 768.103 0.449175
\(144\) 0 0
\(145\) −999.790 −0.572607
\(146\) −1426.16 −0.808426
\(147\) 0 0
\(148\) −472.078 −0.262193
\(149\) −905.310 −0.497758 −0.248879 0.968535i \(-0.580062\pi\)
−0.248879 + 0.968535i \(0.580062\pi\)
\(150\) 0 0
\(151\) 1358.67 0.732231 0.366116 0.930569i \(-0.380687\pi\)
0.366116 + 0.930569i \(0.380687\pi\)
\(152\) −846.073 −0.451484
\(153\) 0 0
\(154\) 868.521 0.454464
\(155\) −383.143 −0.198547
\(156\) 0 0
\(157\) 1901.34 0.966519 0.483259 0.875477i \(-0.339453\pi\)
0.483259 + 0.875477i \(0.339453\pi\)
\(158\) 3014.09 1.51765
\(159\) 0 0
\(160\) −2826.13 −1.39641
\(161\) −851.273 −0.416706
\(162\) 0 0
\(163\) −2325.15 −1.11730 −0.558649 0.829404i \(-0.688680\pi\)
−0.558649 + 0.829404i \(0.688680\pi\)
\(164\) 1842.86 0.877461
\(165\) 0 0
\(166\) −6574.51 −3.07398
\(167\) 1965.80 0.910888 0.455444 0.890264i \(-0.349480\pi\)
0.455444 + 0.890264i \(0.349480\pi\)
\(168\) 0 0
\(169\) 1386.76 0.631205
\(170\) 2962.06 1.33635
\(171\) 0 0
\(172\) 2507.44 1.11157
\(173\) −2297.90 −1.00986 −0.504932 0.863159i \(-0.668482\pi\)
−0.504932 + 0.863159i \(0.668482\pi\)
\(174\) 0 0
\(175\) 314.255 0.135745
\(176\) 2680.98 1.14822
\(177\) 0 0
\(178\) −1378.05 −0.580278
\(179\) −873.696 −0.364822 −0.182411 0.983222i \(-0.558390\pi\)
−0.182411 + 0.983222i \(0.558390\pi\)
\(180\) 0 0
\(181\) 1494.20 0.613609 0.306805 0.951772i \(-0.400740\pi\)
0.306805 + 0.951772i \(0.400740\pi\)
\(182\) 4052.28 1.65041
\(183\) 0 0
\(184\) −4740.34 −1.89925
\(185\) 112.407 0.0446722
\(186\) 0 0
\(187\) −1411.52 −0.551980
\(188\) 5105.91 1.98078
\(189\) 0 0
\(190\) 325.449 0.124266
\(191\) 5050.26 1.91321 0.956607 0.291383i \(-0.0941153\pi\)
0.956607 + 0.291383i \(0.0941153\pi\)
\(192\) 0 0
\(193\) 5161.60 1.92508 0.962539 0.271144i \(-0.0874019\pi\)
0.962539 + 0.271144i \(0.0874019\pi\)
\(194\) 5651.69 2.09158
\(195\) 0 0
\(196\) −3884.52 −1.41564
\(197\) 374.025 0.135270 0.0676350 0.997710i \(-0.478455\pi\)
0.0676350 + 0.997710i \(0.478455\pi\)
\(198\) 0 0
\(199\) 603.342 0.214924 0.107462 0.994209i \(-0.465728\pi\)
0.107462 + 0.994209i \(0.465728\pi\)
\(200\) 1749.94 0.618697
\(201\) 0 0
\(202\) 477.797 0.166424
\(203\) 2513.51 0.869034
\(204\) 0 0
\(205\) −438.807 −0.149501
\(206\) −8314.87 −2.81225
\(207\) 0 0
\(208\) 12508.7 4.16983
\(209\) −155.087 −0.0513281
\(210\) 0 0
\(211\) 2411.81 0.786901 0.393450 0.919346i \(-0.371281\pi\)
0.393450 + 0.919346i \(0.371281\pi\)
\(212\) −6163.45 −1.99673
\(213\) 0 0
\(214\) −3154.40 −1.00762
\(215\) −597.050 −0.189388
\(216\) 0 0
\(217\) 963.237 0.301331
\(218\) 7361.53 2.28709
\(219\) 0 0
\(220\) −1347.13 −0.412834
\(221\) −6585.75 −2.00455
\(222\) 0 0
\(223\) 5481.28 1.64598 0.822990 0.568056i \(-0.192304\pi\)
0.822990 + 0.568056i \(0.192304\pi\)
\(224\) 7105.00 2.11930
\(225\) 0 0
\(226\) 917.766 0.270128
\(227\) −5472.48 −1.60009 −0.800046 0.599938i \(-0.795192\pi\)
−0.800046 + 0.599938i \(0.795192\pi\)
\(228\) 0 0
\(229\) 3275.98 0.945341 0.472670 0.881239i \(-0.343290\pi\)
0.472670 + 0.881239i \(0.343290\pi\)
\(230\) 1823.41 0.522749
\(231\) 0 0
\(232\) 13996.6 3.96086
\(233\) 3446.21 0.968965 0.484483 0.874801i \(-0.339008\pi\)
0.484483 + 0.874801i \(0.339008\pi\)
\(234\) 0 0
\(235\) −1215.78 −0.337483
\(236\) −12208.2 −3.36732
\(237\) 0 0
\(238\) −7446.74 −2.02815
\(239\) −1724.81 −0.466815 −0.233408 0.972379i \(-0.574988\pi\)
−0.233408 + 0.972379i \(0.574988\pi\)
\(240\) 0 0
\(241\) −5575.71 −1.49030 −0.745152 0.666895i \(-0.767623\pi\)
−0.745152 + 0.666895i \(0.767623\pi\)
\(242\) −6280.96 −1.66841
\(243\) 0 0
\(244\) −16246.4 −4.26259
\(245\) 924.951 0.241196
\(246\) 0 0
\(247\) −723.592 −0.186401
\(248\) 5363.82 1.37340
\(249\) 0 0
\(250\) −673.129 −0.170290
\(251\) −1356.38 −0.341090 −0.170545 0.985350i \(-0.554553\pi\)
−0.170545 + 0.985350i \(0.554553\pi\)
\(252\) 0 0
\(253\) −868.913 −0.215921
\(254\) 9745.49 2.40743
\(255\) 0 0
\(256\) 4463.12 1.08963
\(257\) −4087.82 −0.992184 −0.496092 0.868270i \(-0.665232\pi\)
−0.496092 + 0.868270i \(0.665232\pi\)
\(258\) 0 0
\(259\) −282.596 −0.0677980
\(260\) −6285.34 −1.49923
\(261\) 0 0
\(262\) −6668.25 −1.57239
\(263\) −430.260 −0.100878 −0.0504391 0.998727i \(-0.516062\pi\)
−0.0504391 + 0.998727i \(0.516062\pi\)
\(264\) 0 0
\(265\) 1467.59 0.340201
\(266\) −818.192 −0.188596
\(267\) 0 0
\(268\) −4859.86 −1.10770
\(269\) 3467.85 0.786017 0.393008 0.919535i \(-0.371434\pi\)
0.393008 + 0.919535i \(0.371434\pi\)
\(270\) 0 0
\(271\) −55.4415 −0.0124274 −0.00621371 0.999981i \(-0.501978\pi\)
−0.00621371 + 0.999981i \(0.501978\pi\)
\(272\) −22986.9 −5.12420
\(273\) 0 0
\(274\) 2725.81 0.600993
\(275\) 320.767 0.0703381
\(276\) 0 0
\(277\) −1573.54 −0.341317 −0.170658 0.985330i \(-0.554589\pi\)
−0.170658 + 0.985330i \(0.554589\pi\)
\(278\) −1150.67 −0.248247
\(279\) 0 0
\(280\) −4399.42 −0.938984
\(281\) 8146.45 1.72945 0.864727 0.502242i \(-0.167491\pi\)
0.864727 + 0.502242i \(0.167491\pi\)
\(282\) 0 0
\(283\) 3098.31 0.650795 0.325398 0.945577i \(-0.394502\pi\)
0.325398 + 0.945577i \(0.394502\pi\)
\(284\) −15630.1 −3.26576
\(285\) 0 0
\(286\) 4136.26 0.855182
\(287\) 1103.18 0.226894
\(288\) 0 0
\(289\) 7189.39 1.46334
\(290\) −5383.90 −1.09018
\(291\) 0 0
\(292\) −5561.23 −1.11454
\(293\) 122.160 0.0243572 0.0121786 0.999926i \(-0.496123\pi\)
0.0121786 + 0.999926i \(0.496123\pi\)
\(294\) 0 0
\(295\) 2906.92 0.573720
\(296\) −1573.65 −0.309008
\(297\) 0 0
\(298\) −4875.12 −0.947679
\(299\) −4054.11 −0.784132
\(300\) 0 0
\(301\) 1501.01 0.287431
\(302\) 7316.48 1.39409
\(303\) 0 0
\(304\) −2525.62 −0.476495
\(305\) 3868.47 0.726255
\(306\) 0 0
\(307\) −1928.53 −0.358525 −0.179263 0.983801i \(-0.557371\pi\)
−0.179263 + 0.983801i \(0.557371\pi\)
\(308\) 3386.74 0.626549
\(309\) 0 0
\(310\) −2063.24 −0.378013
\(311\) 7939.29 1.44758 0.723788 0.690023i \(-0.242399\pi\)
0.723788 + 0.690023i \(0.242399\pi\)
\(312\) 0 0
\(313\) −6758.89 −1.22056 −0.610279 0.792186i \(-0.708943\pi\)
−0.610279 + 0.792186i \(0.708943\pi\)
\(314\) 10238.8 1.84015
\(315\) 0 0
\(316\) 11753.2 2.09232
\(317\) 8424.23 1.49259 0.746297 0.665614i \(-0.231830\pi\)
0.746297 + 0.665614i \(0.231830\pi\)
\(318\) 0 0
\(319\) 2565.60 0.450301
\(320\) −6860.75 −1.19852
\(321\) 0 0
\(322\) −4584.13 −0.793365
\(323\) 1329.72 0.229064
\(324\) 0 0
\(325\) 1496.61 0.255437
\(326\) −12521.0 −2.12722
\(327\) 0 0
\(328\) 6143.09 1.03413
\(329\) 3056.51 0.512191
\(330\) 0 0
\(331\) 257.675 0.0427888 0.0213944 0.999771i \(-0.493189\pi\)
0.0213944 + 0.999771i \(0.493189\pi\)
\(332\) −25636.9 −4.23797
\(333\) 0 0
\(334\) 10585.9 1.73424
\(335\) 1157.19 0.188728
\(336\) 0 0
\(337\) −11099.5 −1.79415 −0.897077 0.441875i \(-0.854314\pi\)
−0.897077 + 0.441875i \(0.854314\pi\)
\(338\) 7467.73 1.20175
\(339\) 0 0
\(340\) 11550.3 1.84237
\(341\) 983.198 0.156138
\(342\) 0 0
\(343\) −6636.94 −1.04478
\(344\) 8358.42 1.31005
\(345\) 0 0
\(346\) −12374.3 −1.92267
\(347\) 9886.57 1.52951 0.764753 0.644323i \(-0.222861\pi\)
0.764753 + 0.644323i \(0.222861\pi\)
\(348\) 0 0
\(349\) 6059.20 0.929345 0.464673 0.885483i \(-0.346172\pi\)
0.464673 + 0.885483i \(0.346172\pi\)
\(350\) 1692.27 0.258445
\(351\) 0 0
\(352\) 7252.23 1.09814
\(353\) −9096.69 −1.37158 −0.685790 0.727799i \(-0.740543\pi\)
−0.685790 + 0.727799i \(0.740543\pi\)
\(354\) 0 0
\(355\) 3721.71 0.556416
\(356\) −5373.63 −0.800005
\(357\) 0 0
\(358\) −4704.88 −0.694583
\(359\) −8804.63 −1.29440 −0.647201 0.762319i \(-0.724061\pi\)
−0.647201 + 0.762319i \(0.724061\pi\)
\(360\) 0 0
\(361\) −6712.90 −0.978700
\(362\) 8046.33 1.16825
\(363\) 0 0
\(364\) 15801.6 2.27535
\(365\) 1324.19 0.189894
\(366\) 0 0
\(367\) 8652.36 1.23065 0.615326 0.788272i \(-0.289024\pi\)
0.615326 + 0.788272i \(0.289024\pi\)
\(368\) −14150.5 −2.00447
\(369\) 0 0
\(370\) 605.317 0.0850511
\(371\) −3689.58 −0.516316
\(372\) 0 0
\(373\) −11918.6 −1.65449 −0.827243 0.561844i \(-0.810092\pi\)
−0.827243 + 0.561844i \(0.810092\pi\)
\(374\) −7601.05 −1.05091
\(375\) 0 0
\(376\) 17020.3 2.33445
\(377\) 11970.4 1.63529
\(378\) 0 0
\(379\) 5052.23 0.684738 0.342369 0.939566i \(-0.388771\pi\)
0.342369 + 0.939566i \(0.388771\pi\)
\(380\) 1269.07 0.171320
\(381\) 0 0
\(382\) 27195.8 3.64256
\(383\) −5330.37 −0.711147 −0.355573 0.934648i \(-0.615714\pi\)
−0.355573 + 0.934648i \(0.615714\pi\)
\(384\) 0 0
\(385\) −806.421 −0.106751
\(386\) 27795.4 3.66515
\(387\) 0 0
\(388\) 22038.4 2.88358
\(389\) 3339.52 0.435270 0.217635 0.976030i \(-0.430166\pi\)
0.217635 + 0.976030i \(0.430166\pi\)
\(390\) 0 0
\(391\) 7450.10 0.963601
\(392\) −12948.9 −1.66841
\(393\) 0 0
\(394\) 2014.14 0.257540
\(395\) −2798.58 −0.356486
\(396\) 0 0
\(397\) 9041.65 1.14304 0.571520 0.820588i \(-0.306354\pi\)
0.571520 + 0.820588i \(0.306354\pi\)
\(398\) 3249.02 0.409192
\(399\) 0 0
\(400\) 5223.77 0.652971
\(401\) −1671.96 −0.208213 −0.104107 0.994566i \(-0.533198\pi\)
−0.104107 + 0.994566i \(0.533198\pi\)
\(402\) 0 0
\(403\) 4587.34 0.567026
\(404\) 1863.14 0.229442
\(405\) 0 0
\(406\) 13535.3 1.65455
\(407\) −288.453 −0.0351304
\(408\) 0 0
\(409\) 472.726 0.0571511 0.0285755 0.999592i \(-0.490903\pi\)
0.0285755 + 0.999592i \(0.490903\pi\)
\(410\) −2362.99 −0.284634
\(411\) 0 0
\(412\) −32423.2 −3.87713
\(413\) −7308.11 −0.870722
\(414\) 0 0
\(415\) 6104.43 0.722060
\(416\) 33836.9 3.98796
\(417\) 0 0
\(418\) −835.147 −0.0977233
\(419\) 12539.3 1.46202 0.731008 0.682369i \(-0.239050\pi\)
0.731008 + 0.682369i \(0.239050\pi\)
\(420\) 0 0
\(421\) −6751.76 −0.781616 −0.390808 0.920472i \(-0.627804\pi\)
−0.390808 + 0.920472i \(0.627804\pi\)
\(422\) 12987.7 1.49818
\(423\) 0 0
\(424\) −20545.5 −2.35325
\(425\) −2750.27 −0.313901
\(426\) 0 0
\(427\) −9725.47 −1.10222
\(428\) −12300.4 −1.38916
\(429\) 0 0
\(430\) −3215.13 −0.360576
\(431\) 7535.06 0.842114 0.421057 0.907034i \(-0.361659\pi\)
0.421057 + 0.907034i \(0.361659\pi\)
\(432\) 0 0
\(433\) −5135.13 −0.569928 −0.284964 0.958538i \(-0.591982\pi\)
−0.284964 + 0.958538i \(0.591982\pi\)
\(434\) 5187.06 0.573703
\(435\) 0 0
\(436\) 28705.8 3.15311
\(437\) 818.561 0.0896044
\(438\) 0 0
\(439\) 16973.2 1.84530 0.922650 0.385637i \(-0.126018\pi\)
0.922650 + 0.385637i \(0.126018\pi\)
\(440\) −4490.59 −0.486546
\(441\) 0 0
\(442\) −35464.4 −3.81645
\(443\) 11352.3 1.21753 0.608763 0.793352i \(-0.291666\pi\)
0.608763 + 0.793352i \(0.291666\pi\)
\(444\) 0 0
\(445\) 1279.52 0.136304
\(446\) 29516.9 3.13378
\(447\) 0 0
\(448\) 17248.2 1.81898
\(449\) 11059.8 1.16246 0.581230 0.813740i \(-0.302572\pi\)
0.581230 + 0.813740i \(0.302572\pi\)
\(450\) 0 0
\(451\) 1126.04 0.117568
\(452\) 3578.76 0.372414
\(453\) 0 0
\(454\) −29469.5 −3.04641
\(455\) −3762.54 −0.387672
\(456\) 0 0
\(457\) 356.684 0.0365098 0.0182549 0.999833i \(-0.494189\pi\)
0.0182549 + 0.999833i \(0.494189\pi\)
\(458\) 17641.3 1.79983
\(459\) 0 0
\(460\) 7110.27 0.720691
\(461\) 10319.8 1.04260 0.521300 0.853373i \(-0.325447\pi\)
0.521300 + 0.853373i \(0.325447\pi\)
\(462\) 0 0
\(463\) −18168.5 −1.82368 −0.911839 0.410547i \(-0.865338\pi\)
−0.911839 + 0.410547i \(0.865338\pi\)
\(464\) 41781.4 4.18028
\(465\) 0 0
\(466\) 18558.0 1.84481
\(467\) 3817.15 0.378237 0.189118 0.981954i \(-0.439437\pi\)
0.189118 + 0.981954i \(0.439437\pi\)
\(468\) 0 0
\(469\) −2909.22 −0.286429
\(470\) −6546.99 −0.642532
\(471\) 0 0
\(472\) −40695.4 −3.96856
\(473\) 1532.11 0.148936
\(474\) 0 0
\(475\) −302.179 −0.0291893
\(476\) −29038.0 −2.79613
\(477\) 0 0
\(478\) −9288.17 −0.888768
\(479\) 2906.57 0.277254 0.138627 0.990345i \(-0.455731\pi\)
0.138627 + 0.990345i \(0.455731\pi\)
\(480\) 0 0
\(481\) −1345.84 −0.127578
\(482\) −30025.4 −2.83738
\(483\) 0 0
\(484\) −24492.2 −2.30016
\(485\) −5247.59 −0.491301
\(486\) 0 0
\(487\) 10411.1 0.968734 0.484367 0.874865i \(-0.339050\pi\)
0.484367 + 0.874865i \(0.339050\pi\)
\(488\) −54156.7 −5.02368
\(489\) 0 0
\(490\) 4980.89 0.459211
\(491\) 14632.5 1.34492 0.672460 0.740133i \(-0.265238\pi\)
0.672460 + 0.740133i \(0.265238\pi\)
\(492\) 0 0
\(493\) −21997.5 −2.00957
\(494\) −3896.57 −0.354888
\(495\) 0 0
\(496\) 16011.6 1.44948
\(497\) −9356.53 −0.844462
\(498\) 0 0
\(499\) 8166.83 0.732661 0.366330 0.930485i \(-0.380614\pi\)
0.366330 + 0.930485i \(0.380614\pi\)
\(500\) −2624.82 −0.234771
\(501\) 0 0
\(502\) −7304.12 −0.649400
\(503\) −8080.38 −0.716275 −0.358137 0.933669i \(-0.616588\pi\)
−0.358137 + 0.933669i \(0.616588\pi\)
\(504\) 0 0
\(505\) −443.635 −0.0390921
\(506\) −4679.13 −0.411092
\(507\) 0 0
\(508\) 38001.9 3.31902
\(509\) 116.181 0.0101171 0.00505856 0.999987i \(-0.498390\pi\)
0.00505856 + 0.999987i \(0.498390\pi\)
\(510\) 0 0
\(511\) −3329.07 −0.288199
\(512\) 1095.90 0.0945947
\(513\) 0 0
\(514\) −22013.0 −1.88901
\(515\) 7720.35 0.660581
\(516\) 0 0
\(517\) 3119.85 0.265398
\(518\) −1521.79 −0.129080
\(519\) 0 0
\(520\) −20951.9 −1.76692
\(521\) −14479.3 −1.21756 −0.608780 0.793339i \(-0.708341\pi\)
−0.608780 + 0.793339i \(0.708341\pi\)
\(522\) 0 0
\(523\) 6841.05 0.571966 0.285983 0.958235i \(-0.407680\pi\)
0.285983 + 0.958235i \(0.407680\pi\)
\(524\) −26002.4 −2.16778
\(525\) 0 0
\(526\) −2316.96 −0.192062
\(527\) −8429.98 −0.696804
\(528\) 0 0
\(529\) −7580.80 −0.623062
\(530\) 7903.01 0.647707
\(531\) 0 0
\(532\) −3190.48 −0.260009
\(533\) 5253.80 0.426955
\(534\) 0 0
\(535\) 2928.86 0.236684
\(536\) −16200.1 −1.30548
\(537\) 0 0
\(538\) 18674.5 1.49649
\(539\) −2373.55 −0.189677
\(540\) 0 0
\(541\) −12746.0 −1.01293 −0.506465 0.862261i \(-0.669048\pi\)
−0.506465 + 0.862261i \(0.669048\pi\)
\(542\) −298.554 −0.0236605
\(543\) 0 0
\(544\) −62181.0 −4.90071
\(545\) −6835.18 −0.537224
\(546\) 0 0
\(547\) 4980.54 0.389310 0.194655 0.980872i \(-0.437641\pi\)
0.194655 + 0.980872i \(0.437641\pi\)
\(548\) 10629.1 0.828563
\(549\) 0 0
\(550\) 1727.34 0.133916
\(551\) −2416.93 −0.186868
\(552\) 0 0
\(553\) 7035.75 0.541032
\(554\) −8473.55 −0.649831
\(555\) 0 0
\(556\) −4486.97 −0.342248
\(557\) −635.610 −0.0483513 −0.0241756 0.999708i \(-0.507696\pi\)
−0.0241756 + 0.999708i \(0.507696\pi\)
\(558\) 0 0
\(559\) 7148.42 0.540869
\(560\) −13132.8 −0.991001
\(561\) 0 0
\(562\) 43868.9 3.29270
\(563\) −7537.68 −0.564254 −0.282127 0.959377i \(-0.591040\pi\)
−0.282127 + 0.959377i \(0.591040\pi\)
\(564\) 0 0
\(565\) −852.146 −0.0634514
\(566\) 16684.5 1.23905
\(567\) 0 0
\(568\) −52102.2 −3.84887
\(569\) −2258.35 −0.166389 −0.0831943 0.996533i \(-0.526512\pi\)
−0.0831943 + 0.996533i \(0.526512\pi\)
\(570\) 0 0
\(571\) 11377.3 0.833847 0.416923 0.908942i \(-0.363108\pi\)
0.416923 + 0.908942i \(0.363108\pi\)
\(572\) 16129.0 1.17900
\(573\) 0 0
\(574\) 5940.66 0.431983
\(575\) −1693.04 −0.122791
\(576\) 0 0
\(577\) −25027.9 −1.80576 −0.902881 0.429890i \(-0.858553\pi\)
−0.902881 + 0.429890i \(0.858553\pi\)
\(578\) 38715.1 2.78605
\(579\) 0 0
\(580\) −20994.1 −1.50299
\(581\) −15346.8 −1.09586
\(582\) 0 0
\(583\) −3766.03 −0.267535
\(584\) −18538.1 −1.31355
\(585\) 0 0
\(586\) 657.835 0.0463736
\(587\) 148.259 0.0104247 0.00521235 0.999986i \(-0.498341\pi\)
0.00521235 + 0.999986i \(0.498341\pi\)
\(588\) 0 0
\(589\) −926.224 −0.0647952
\(590\) 15653.8 1.09230
\(591\) 0 0
\(592\) −4697.52 −0.326126
\(593\) −27452.6 −1.90109 −0.950544 0.310590i \(-0.899473\pi\)
−0.950544 + 0.310590i \(0.899473\pi\)
\(594\) 0 0
\(595\) 6914.29 0.476401
\(596\) −19010.2 −1.30652
\(597\) 0 0
\(598\) −21831.5 −1.49291
\(599\) −3812.00 −0.260024 −0.130012 0.991512i \(-0.541502\pi\)
−0.130012 + 0.991512i \(0.541502\pi\)
\(600\) 0 0
\(601\) 23168.7 1.57249 0.786247 0.617912i \(-0.212021\pi\)
0.786247 + 0.617912i \(0.212021\pi\)
\(602\) 8082.97 0.547238
\(603\) 0 0
\(604\) 28530.1 1.92197
\(605\) 5831.87 0.391899
\(606\) 0 0
\(607\) 7352.78 0.491664 0.245832 0.969312i \(-0.420939\pi\)
0.245832 + 0.969312i \(0.420939\pi\)
\(608\) −6831.98 −0.455713
\(609\) 0 0
\(610\) 20831.8 1.38271
\(611\) 14556.4 0.963809
\(612\) 0 0
\(613\) 19902.6 1.31135 0.655676 0.755043i \(-0.272384\pi\)
0.655676 + 0.755043i \(0.272384\pi\)
\(614\) −10385.2 −0.682595
\(615\) 0 0
\(616\) 11289.5 0.738421
\(617\) 25085.2 1.63678 0.818390 0.574664i \(-0.194867\pi\)
0.818390 + 0.574664i \(0.194867\pi\)
\(618\) 0 0
\(619\) −3131.04 −0.203307 −0.101654 0.994820i \(-0.532413\pi\)
−0.101654 + 0.994820i \(0.532413\pi\)
\(620\) −8045.45 −0.521150
\(621\) 0 0
\(622\) 42753.3 2.75603
\(623\) −3216.77 −0.206866
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −36396.8 −2.32382
\(627\) 0 0
\(628\) 39925.4 2.53694
\(629\) 2473.21 0.156778
\(630\) 0 0
\(631\) −22527.2 −1.42123 −0.710614 0.703582i \(-0.751583\pi\)
−0.710614 + 0.703582i \(0.751583\pi\)
\(632\) 39178.8 2.46590
\(633\) 0 0
\(634\) 45364.7 2.84174
\(635\) −9048.69 −0.565490
\(636\) 0 0
\(637\) −11074.3 −0.688825
\(638\) 13815.8 0.857325
\(639\) 0 0
\(640\) −14336.3 −0.885459
\(641\) −6925.76 −0.426757 −0.213378 0.976970i \(-0.568447\pi\)
−0.213378 + 0.976970i \(0.568447\pi\)
\(642\) 0 0
\(643\) −14964.5 −0.917796 −0.458898 0.888489i \(-0.651756\pi\)
−0.458898 + 0.888489i \(0.651756\pi\)
\(644\) −17875.5 −1.09378
\(645\) 0 0
\(646\) 7160.58 0.436114
\(647\) −2371.74 −0.144115 −0.0720577 0.997400i \(-0.522957\pi\)
−0.0720577 + 0.997400i \(0.522957\pi\)
\(648\) 0 0
\(649\) −7459.55 −0.451176
\(650\) 8059.30 0.486326
\(651\) 0 0
\(652\) −48824.7 −2.93270
\(653\) −15971.7 −0.957154 −0.478577 0.878046i \(-0.658847\pi\)
−0.478577 + 0.878046i \(0.658847\pi\)
\(654\) 0 0
\(655\) 6191.47 0.369344
\(656\) 18337.8 1.09142
\(657\) 0 0
\(658\) 16459.4 0.975158
\(659\) −6842.93 −0.404496 −0.202248 0.979334i \(-0.564825\pi\)
−0.202248 + 0.979334i \(0.564825\pi\)
\(660\) 0 0
\(661\) −13391.0 −0.787973 −0.393986 0.919116i \(-0.628904\pi\)
−0.393986 + 0.919116i \(0.628904\pi\)
\(662\) 1387.59 0.0814654
\(663\) 0 0
\(664\) −85459.1 −4.99467
\(665\) 759.691 0.0443001
\(666\) 0 0
\(667\) −13541.5 −0.786098
\(668\) 41279.0 2.39092
\(669\) 0 0
\(670\) 6231.50 0.359319
\(671\) −9927.01 −0.571130
\(672\) 0 0
\(673\) −31902.2 −1.82725 −0.913624 0.406560i \(-0.866728\pi\)
−0.913624 + 0.406560i \(0.866728\pi\)
\(674\) −59771.3 −3.41588
\(675\) 0 0
\(676\) 29119.9 1.65680
\(677\) −11860.8 −0.673332 −0.336666 0.941624i \(-0.609299\pi\)
−0.336666 + 0.941624i \(0.609299\pi\)
\(678\) 0 0
\(679\) 13192.7 0.745637
\(680\) 38502.5 2.17133
\(681\) 0 0
\(682\) 5294.55 0.297271
\(683\) 7639.34 0.427981 0.213991 0.976836i \(-0.431354\pi\)
0.213991 + 0.976836i \(0.431354\pi\)
\(684\) 0 0
\(685\) −2530.91 −0.141170
\(686\) −35740.1 −1.98916
\(687\) 0 0
\(688\) 24950.8 1.38262
\(689\) −17571.3 −0.971571
\(690\) 0 0
\(691\) −15726.0 −0.865767 −0.432883 0.901450i \(-0.642504\pi\)
−0.432883 + 0.901450i \(0.642504\pi\)
\(692\) −48252.6 −2.65071
\(693\) 0 0
\(694\) 53239.5 2.91202
\(695\) 1068.40 0.0583118
\(696\) 0 0
\(697\) −9654.72 −0.524675
\(698\) 32629.0 1.76938
\(699\) 0 0
\(700\) 6598.90 0.356307
\(701\) −6338.17 −0.341497 −0.170748 0.985315i \(-0.554619\pi\)
−0.170748 + 0.985315i \(0.554619\pi\)
\(702\) 0 0
\(703\) 271.737 0.0145786
\(704\) 17605.6 0.942525
\(705\) 0 0
\(706\) −48985.9 −2.61135
\(707\) 1115.32 0.0593292
\(708\) 0 0
\(709\) 34383.6 1.82130 0.910651 0.413176i \(-0.135581\pi\)
0.910651 + 0.413176i \(0.135581\pi\)
\(710\) 20041.5 1.05936
\(711\) 0 0
\(712\) −17912.7 −0.942847
\(713\) −5189.41 −0.272573
\(714\) 0 0
\(715\) −3840.51 −0.200877
\(716\) −18346.4 −0.957592
\(717\) 0 0
\(718\) −47413.2 −2.46441
\(719\) −988.886 −0.0512924 −0.0256462 0.999671i \(-0.508164\pi\)
−0.0256462 + 0.999671i \(0.508164\pi\)
\(720\) 0 0
\(721\) −19409.3 −1.00255
\(722\) −36149.2 −1.86334
\(723\) 0 0
\(724\) 31376.1 1.61061
\(725\) 4998.95 0.256078
\(726\) 0 0
\(727\) −13691.8 −0.698490 −0.349245 0.937032i \(-0.613562\pi\)
−0.349245 + 0.937032i \(0.613562\pi\)
\(728\) 52673.8 2.68162
\(729\) 0 0
\(730\) 7130.82 0.361539
\(731\) −13136.4 −0.664661
\(732\) 0 0
\(733\) −2501.94 −0.126073 −0.0630363 0.998011i \(-0.520078\pi\)
−0.0630363 + 0.998011i \(0.520078\pi\)
\(734\) 46593.2 2.34303
\(735\) 0 0
\(736\) −38277.9 −1.91704
\(737\) −2969.51 −0.148417
\(738\) 0 0
\(739\) 26453.1 1.31677 0.658384 0.752682i \(-0.271240\pi\)
0.658384 + 0.752682i \(0.271240\pi\)
\(740\) 2360.39 0.117256
\(741\) 0 0
\(742\) −19868.5 −0.983012
\(743\) −20882.8 −1.03111 −0.515555 0.856856i \(-0.672414\pi\)
−0.515555 + 0.856856i \(0.672414\pi\)
\(744\) 0 0
\(745\) 4526.55 0.222604
\(746\) −64182.2 −3.14997
\(747\) 0 0
\(748\) −29639.8 −1.44885
\(749\) −7363.27 −0.359210
\(750\) 0 0
\(751\) −17695.3 −0.859801 −0.429901 0.902876i \(-0.641451\pi\)
−0.429901 + 0.902876i \(0.641451\pi\)
\(752\) 50807.5 2.46377
\(753\) 0 0
\(754\) 64460.9 3.11343
\(755\) −6793.35 −0.327464
\(756\) 0 0
\(757\) 7755.61 0.372368 0.186184 0.982515i \(-0.440388\pi\)
0.186184 + 0.982515i \(0.440388\pi\)
\(758\) 27206.4 1.30367
\(759\) 0 0
\(760\) 4230.37 0.201910
\(761\) 17054.3 0.812376 0.406188 0.913790i \(-0.366858\pi\)
0.406188 + 0.913790i \(0.366858\pi\)
\(762\) 0 0
\(763\) 17183.9 0.815333
\(764\) 106048. 5.02184
\(765\) 0 0
\(766\) −28704.2 −1.35395
\(767\) −34804.2 −1.63847
\(768\) 0 0
\(769\) −10048.3 −0.471199 −0.235599 0.971850i \(-0.575705\pi\)
−0.235599 + 0.971850i \(0.575705\pi\)
\(770\) −4342.60 −0.203242
\(771\) 0 0
\(772\) 108386. 5.05298
\(773\) 7293.24 0.339352 0.169676 0.985500i \(-0.445728\pi\)
0.169676 + 0.985500i \(0.445728\pi\)
\(774\) 0 0
\(775\) 1915.72 0.0887930
\(776\) 73463.8 3.39845
\(777\) 0 0
\(778\) 17983.4 0.828710
\(779\) −1060.79 −0.0487891
\(780\) 0 0
\(781\) −9550.42 −0.437568
\(782\) 40119.0 1.83460
\(783\) 0 0
\(784\) −38653.8 −1.76083
\(785\) −9506.70 −0.432240
\(786\) 0 0
\(787\) 28456.3 1.28889 0.644445 0.764651i \(-0.277089\pi\)
0.644445 + 0.764651i \(0.277089\pi\)
\(788\) 7853.99 0.355059
\(789\) 0 0
\(790\) −15070.5 −0.678713
\(791\) 2142.33 0.0962989
\(792\) 0 0
\(793\) −46316.7 −2.07409
\(794\) 48689.5 2.17623
\(795\) 0 0
\(796\) 12669.3 0.564136
\(797\) 4081.46 0.181396 0.0906981 0.995878i \(-0.471090\pi\)
0.0906981 + 0.995878i \(0.471090\pi\)
\(798\) 0 0
\(799\) −26749.7 −1.18440
\(800\) 14130.6 0.624492
\(801\) 0 0
\(802\) −9003.54 −0.396416
\(803\) −3398.06 −0.149334
\(804\) 0 0
\(805\) 4256.36 0.186357
\(806\) 24702.9 1.07956
\(807\) 0 0
\(808\) 6210.67 0.270409
\(809\) 14209.7 0.617536 0.308768 0.951137i \(-0.400083\pi\)
0.308768 + 0.951137i \(0.400083\pi\)
\(810\) 0 0
\(811\) 4901.79 0.212238 0.106119 0.994353i \(-0.466158\pi\)
0.106119 + 0.994353i \(0.466158\pi\)
\(812\) 52780.1 2.28106
\(813\) 0 0
\(814\) −1553.33 −0.0668846
\(815\) 11625.7 0.499671
\(816\) 0 0
\(817\) −1443.33 −0.0618063
\(818\) 2545.64 0.108810
\(819\) 0 0
\(820\) −9214.32 −0.392412
\(821\) −14519.6 −0.617222 −0.308611 0.951188i \(-0.599864\pi\)
−0.308611 + 0.951188i \(0.599864\pi\)
\(822\) 0 0
\(823\) −38406.4 −1.62669 −0.813344 0.581783i \(-0.802355\pi\)
−0.813344 + 0.581783i \(0.802355\pi\)
\(824\) −108081. −4.56940
\(825\) 0 0
\(826\) −39354.4 −1.65776
\(827\) −10446.4 −0.439248 −0.219624 0.975585i \(-0.570483\pi\)
−0.219624 + 0.975585i \(0.570483\pi\)
\(828\) 0 0
\(829\) −6474.00 −0.271232 −0.135616 0.990761i \(-0.543301\pi\)
−0.135616 + 0.990761i \(0.543301\pi\)
\(830\) 32872.6 1.37473
\(831\) 0 0
\(832\) 82143.1 3.42284
\(833\) 20350.9 0.846480
\(834\) 0 0
\(835\) −9829.01 −0.407362
\(836\) −3256.60 −0.134727
\(837\) 0 0
\(838\) 67524.5 2.78353
\(839\) 15839.4 0.651773 0.325886 0.945409i \(-0.394337\pi\)
0.325886 + 0.945409i \(0.394337\pi\)
\(840\) 0 0
\(841\) 15594.2 0.639394
\(842\) −36358.4 −1.48812
\(843\) 0 0
\(844\) 50644.6 2.06547
\(845\) −6933.79 −0.282284
\(846\) 0 0
\(847\) −14661.5 −0.594778
\(848\) −61330.8 −2.48362
\(849\) 0 0
\(850\) −14810.3 −0.597634
\(851\) 1522.48 0.0613277
\(852\) 0 0
\(853\) −18840.3 −0.756249 −0.378125 0.925755i \(-0.623431\pi\)
−0.378125 + 0.925755i \(0.623431\pi\)
\(854\) −52372.0 −2.09852
\(855\) 0 0
\(856\) −41002.6 −1.63720
\(857\) 9056.21 0.360973 0.180487 0.983577i \(-0.442233\pi\)
0.180487 + 0.983577i \(0.442233\pi\)
\(858\) 0 0
\(859\) 22055.4 0.876044 0.438022 0.898964i \(-0.355679\pi\)
0.438022 + 0.898964i \(0.355679\pi\)
\(860\) −12537.2 −0.497110
\(861\) 0 0
\(862\) 40576.5 1.60330
\(863\) 13105.6 0.516941 0.258471 0.966019i \(-0.416781\pi\)
0.258471 + 0.966019i \(0.416781\pi\)
\(864\) 0 0
\(865\) 11489.5 0.451624
\(866\) −27652.8 −1.08508
\(867\) 0 0
\(868\) 20226.6 0.790939
\(869\) 7181.55 0.280342
\(870\) 0 0
\(871\) −13854.9 −0.538985
\(872\) 95689.2 3.71611
\(873\) 0 0
\(874\) 4407.98 0.170597
\(875\) −1571.27 −0.0607072
\(876\) 0 0
\(877\) −36722.2 −1.41393 −0.706967 0.707246i \(-0.749937\pi\)
−0.706967 + 0.707246i \(0.749937\pi\)
\(878\) 91401.2 3.51326
\(879\) 0 0
\(880\) −13404.9 −0.513499
\(881\) 36054.4 1.37878 0.689390 0.724390i \(-0.257879\pi\)
0.689390 + 0.724390i \(0.257879\pi\)
\(882\) 0 0
\(883\) 13524.6 0.515446 0.257723 0.966219i \(-0.417028\pi\)
0.257723 + 0.966219i \(0.417028\pi\)
\(884\) −138291. −5.26158
\(885\) 0 0
\(886\) 61132.5 2.31804
\(887\) −39730.2 −1.50396 −0.751978 0.659188i \(-0.770900\pi\)
−0.751978 + 0.659188i \(0.770900\pi\)
\(888\) 0 0
\(889\) 22748.8 0.858233
\(890\) 6890.27 0.259508
\(891\) 0 0
\(892\) 115099. 4.32040
\(893\) −2939.06 −0.110136
\(894\) 0 0
\(895\) 4368.48 0.163153
\(896\) 36042.1 1.34384
\(897\) 0 0
\(898\) 59557.3 2.21320
\(899\) 15322.5 0.568447
\(900\) 0 0
\(901\) 32290.1 1.19394
\(902\) 6063.76 0.223837
\(903\) 0 0
\(904\) 11929.6 0.438909
\(905\) −7471.02 −0.274414
\(906\) 0 0
\(907\) 23022.4 0.842829 0.421415 0.906868i \(-0.361534\pi\)
0.421415 + 0.906868i \(0.361534\pi\)
\(908\) −114914. −4.19995
\(909\) 0 0
\(910\) −20261.4 −0.738087
\(911\) 31880.2 1.15943 0.579714 0.814820i \(-0.303164\pi\)
0.579714 + 0.814820i \(0.303164\pi\)
\(912\) 0 0
\(913\) −15664.8 −0.567831
\(914\) 1920.75 0.0695108
\(915\) 0 0
\(916\) 68790.9 2.48135
\(917\) −15565.6 −0.560547
\(918\) 0 0
\(919\) 38459.6 1.38049 0.690243 0.723578i \(-0.257504\pi\)
0.690243 + 0.723578i \(0.257504\pi\)
\(920\) 23701.7 0.849372
\(921\) 0 0
\(922\) 55572.2 1.98500
\(923\) −44559.7 −1.58906
\(924\) 0 0
\(925\) −562.037 −0.0199780
\(926\) −97838.1 −3.47209
\(927\) 0 0
\(928\) 113021. 3.99796
\(929\) −37201.0 −1.31381 −0.656903 0.753975i \(-0.728134\pi\)
−0.656903 + 0.753975i \(0.728134\pi\)
\(930\) 0 0
\(931\) 2236.01 0.0787134
\(932\) 72365.5 2.54336
\(933\) 0 0
\(934\) 20555.5 0.720123
\(935\) 7057.58 0.246853
\(936\) 0 0
\(937\) −23593.7 −0.822595 −0.411298 0.911501i \(-0.634924\pi\)
−0.411298 + 0.911501i \(0.634924\pi\)
\(938\) −15666.2 −0.545332
\(939\) 0 0
\(940\) −25529.5 −0.885832
\(941\) 36571.7 1.26695 0.633477 0.773762i \(-0.281627\pi\)
0.633477 + 0.773762i \(0.281627\pi\)
\(942\) 0 0
\(943\) −5943.34 −0.205241
\(944\) −121481. −4.18841
\(945\) 0 0
\(946\) 8250.48 0.283558
\(947\) 32911.7 1.12934 0.564671 0.825316i \(-0.309003\pi\)
0.564671 + 0.825316i \(0.309003\pi\)
\(948\) 0 0
\(949\) −15854.4 −0.542314
\(950\) −1627.24 −0.0555735
\(951\) 0 0
\(952\) −96796.8 −3.29538
\(953\) −21564.5 −0.732993 −0.366497 0.930419i \(-0.619443\pi\)
−0.366497 + 0.930419i \(0.619443\pi\)
\(954\) 0 0
\(955\) −25251.3 −0.855615
\(956\) −36218.6 −1.22531
\(957\) 0 0
\(958\) 15652.0 0.527862
\(959\) 6362.82 0.214250
\(960\) 0 0
\(961\) −23919.1 −0.802895
\(962\) −7247.40 −0.242895
\(963\) 0 0
\(964\) −117082. −3.91178
\(965\) −25808.0 −0.860921
\(966\) 0 0
\(967\) 29763.7 0.989801 0.494901 0.868950i \(-0.335204\pi\)
0.494901 + 0.868950i \(0.335204\pi\)
\(968\) −81643.3 −2.71086
\(969\) 0 0
\(970\) −28258.4 −0.935385
\(971\) 29812.3 0.985297 0.492648 0.870228i \(-0.336029\pi\)
0.492648 + 0.870228i \(0.336029\pi\)
\(972\) 0 0
\(973\) −2686.00 −0.0884986
\(974\) 56064.2 1.84437
\(975\) 0 0
\(976\) −161664. −5.30198
\(977\) −5759.73 −0.188608 −0.0943041 0.995543i \(-0.530063\pi\)
−0.0943041 + 0.995543i \(0.530063\pi\)
\(978\) 0 0
\(979\) −3283.43 −0.107190
\(980\) 19422.6 0.633095
\(981\) 0 0
\(982\) 78796.5 2.56059
\(983\) −17651.4 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(984\) 0 0
\(985\) −1870.13 −0.0604946
\(986\) −118457. −3.82602
\(987\) 0 0
\(988\) −15194.4 −0.489269
\(989\) −8086.63 −0.260000
\(990\) 0 0
\(991\) −5186.45 −0.166249 −0.0831246 0.996539i \(-0.526490\pi\)
−0.0831246 + 0.996539i \(0.526490\pi\)
\(992\) 43312.5 1.38626
\(993\) 0 0
\(994\) −50385.2 −1.60777
\(995\) −3016.71 −0.0961167
\(996\) 0 0
\(997\) −16110.2 −0.511752 −0.255876 0.966710i \(-0.582364\pi\)
−0.255876 + 0.966710i \(0.582364\pi\)
\(998\) 43978.6 1.39491
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 405.4.a.m.1.7 7
3.2 odd 2 405.4.a.n.1.1 7
5.4 even 2 2025.4.a.bb.1.1 7
9.2 odd 6 135.4.e.c.91.7 14
9.4 even 3 45.4.e.c.16.1 14
9.5 odd 6 135.4.e.c.46.7 14
9.7 even 3 45.4.e.c.31.1 yes 14
15.14 odd 2 2025.4.a.ba.1.7 7
45.4 even 6 225.4.e.d.151.7 14
45.7 odd 12 225.4.k.d.49.14 28
45.13 odd 12 225.4.k.d.124.14 28
45.22 odd 12 225.4.k.d.124.1 28
45.34 even 6 225.4.e.d.76.7 14
45.43 odd 12 225.4.k.d.49.1 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.e.c.16.1 14 9.4 even 3
45.4.e.c.31.1 yes 14 9.7 even 3
135.4.e.c.46.7 14 9.5 odd 6
135.4.e.c.91.7 14 9.2 odd 6
225.4.e.d.76.7 14 45.34 even 6
225.4.e.d.151.7 14 45.4 even 6
225.4.k.d.49.1 28 45.43 odd 12
225.4.k.d.49.14 28 45.7 odd 12
225.4.k.d.124.1 28 45.22 odd 12
225.4.k.d.124.14 28 45.13 odd 12
405.4.a.m.1.7 7 1.1 even 1 trivial
405.4.a.n.1.1 7 3.2 odd 2
2025.4.a.ba.1.7 7 15.14 odd 2
2025.4.a.bb.1.1 7 5.4 even 2