Properties

Label 405.4.a.m.1.1
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 1.1
Root \(5.31551\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.31551 q^{2} +20.2546 q^{4} -5.00000 q^{5} -13.4337 q^{7} -65.1396 q^{8} +O(q^{10})\) \(q-5.31551 q^{2} +20.2546 q^{4} -5.00000 q^{5} -13.4337 q^{7} -65.1396 q^{8} +26.5775 q^{10} -46.9256 q^{11} -36.1347 q^{13} +71.4071 q^{14} +184.213 q^{16} -54.6071 q^{17} +111.339 q^{19} -101.273 q^{20} +249.434 q^{22} -35.9740 q^{23} +25.0000 q^{25} +192.074 q^{26} -272.095 q^{28} +58.1175 q^{29} -295.667 q^{31} -458.068 q^{32} +290.265 q^{34} +67.1686 q^{35} -53.0417 q^{37} -591.822 q^{38} +325.698 q^{40} +128.359 q^{41} -164.172 q^{43} -950.461 q^{44} +191.220 q^{46} +87.8318 q^{47} -162.535 q^{49} -132.888 q^{50} -731.894 q^{52} -479.247 q^{53} +234.628 q^{55} +875.067 q^{56} -308.924 q^{58} +635.614 q^{59} +48.0257 q^{61} +1571.62 q^{62} +961.163 q^{64} +180.673 q^{65} -28.9183 q^{67} -1106.05 q^{68} -357.035 q^{70} +576.183 q^{71} +835.057 q^{73} +281.944 q^{74} +2255.12 q^{76} +630.386 q^{77} -203.737 q^{79} -921.064 q^{80} -682.294 q^{82} +464.478 q^{83} +273.036 q^{85} +872.656 q^{86} +3056.72 q^{88} +993.782 q^{89} +485.423 q^{91} -728.640 q^{92} -466.871 q^{94} -556.693 q^{95} +881.415 q^{97} +863.956 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.31551 −1.87932 −0.939658 0.342115i \(-0.888857\pi\)
−0.939658 + 0.342115i \(0.888857\pi\)
\(3\) 0 0
\(4\) 20.2546 2.53183
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −13.4337 −0.725353 −0.362676 0.931915i \(-0.618137\pi\)
−0.362676 + 0.931915i \(0.618137\pi\)
\(8\) −65.1396 −2.87879
\(9\) 0 0
\(10\) 26.5775 0.840456
\(11\) −46.9256 −1.28624 −0.643119 0.765766i \(-0.722360\pi\)
−0.643119 + 0.765766i \(0.722360\pi\)
\(12\) 0 0
\(13\) −36.1347 −0.770920 −0.385460 0.922725i \(-0.625957\pi\)
−0.385460 + 0.922725i \(0.625957\pi\)
\(14\) 71.4071 1.36317
\(15\) 0 0
\(16\) 184.213 2.87833
\(17\) −54.6071 −0.779069 −0.389535 0.921012i \(-0.627364\pi\)
−0.389535 + 0.921012i \(0.627364\pi\)
\(18\) 0 0
\(19\) 111.339 1.34436 0.672180 0.740388i \(-0.265358\pi\)
0.672180 + 0.740388i \(0.265358\pi\)
\(20\) −101.273 −1.13227
\(21\) 0 0
\(22\) 249.434 2.41725
\(23\) −35.9740 −0.326135 −0.163067 0.986615i \(-0.552139\pi\)
−0.163067 + 0.986615i \(0.552139\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 192.074 1.44880
\(27\) 0 0
\(28\) −272.095 −1.83647
\(29\) 58.1175 0.372143 0.186072 0.982536i \(-0.440424\pi\)
0.186072 + 0.982536i \(0.440424\pi\)
\(30\) 0 0
\(31\) −295.667 −1.71301 −0.856506 0.516137i \(-0.827369\pi\)
−0.856506 + 0.516137i \(0.827369\pi\)
\(32\) −458.068 −2.53049
\(33\) 0 0
\(34\) 290.265 1.46412
\(35\) 67.1686 0.324388
\(36\) 0 0
\(37\) −53.0417 −0.235676 −0.117838 0.993033i \(-0.537596\pi\)
−0.117838 + 0.993033i \(0.537596\pi\)
\(38\) −591.822 −2.52648
\(39\) 0 0
\(40\) 325.698 1.28743
\(41\) 128.359 0.488935 0.244467 0.969658i \(-0.421387\pi\)
0.244467 + 0.969658i \(0.421387\pi\)
\(42\) 0 0
\(43\) −164.172 −0.582231 −0.291116 0.956688i \(-0.594026\pi\)
−0.291116 + 0.956688i \(0.594026\pi\)
\(44\) −950.461 −3.25653
\(45\) 0 0
\(46\) 191.220 0.612910
\(47\) 87.8318 0.272587 0.136294 0.990668i \(-0.456481\pi\)
0.136294 + 0.990668i \(0.456481\pi\)
\(48\) 0 0
\(49\) −162.535 −0.473863
\(50\) −132.888 −0.375863
\(51\) 0 0
\(52\) −731.894 −1.95184
\(53\) −479.247 −1.24207 −0.621035 0.783783i \(-0.713287\pi\)
−0.621035 + 0.783783i \(0.713287\pi\)
\(54\) 0 0
\(55\) 234.628 0.575223
\(56\) 875.067 2.08814
\(57\) 0 0
\(58\) −308.924 −0.699375
\(59\) 635.614 1.40254 0.701271 0.712895i \(-0.252616\pi\)
0.701271 + 0.712895i \(0.252616\pi\)
\(60\) 0 0
\(61\) 48.0257 0.100804 0.0504021 0.998729i \(-0.483950\pi\)
0.0504021 + 0.998729i \(0.483950\pi\)
\(62\) 1571.62 3.21929
\(63\) 0 0
\(64\) 961.163 1.87727
\(65\) 180.673 0.344766
\(66\) 0 0
\(67\) −28.9183 −0.0527303 −0.0263652 0.999652i \(-0.508393\pi\)
−0.0263652 + 0.999652i \(0.508393\pi\)
\(68\) −1106.05 −1.97247
\(69\) 0 0
\(70\) −357.035 −0.609627
\(71\) 576.183 0.963103 0.481552 0.876418i \(-0.340073\pi\)
0.481552 + 0.876418i \(0.340073\pi\)
\(72\) 0 0
\(73\) 835.057 1.33885 0.669425 0.742880i \(-0.266541\pi\)
0.669425 + 0.742880i \(0.266541\pi\)
\(74\) 281.944 0.442909
\(75\) 0 0
\(76\) 2255.12 3.40369
\(77\) 630.386 0.932976
\(78\) 0 0
\(79\) −203.737 −0.290155 −0.145078 0.989420i \(-0.546343\pi\)
−0.145078 + 0.989420i \(0.546343\pi\)
\(80\) −921.064 −1.28723
\(81\) 0 0
\(82\) −682.294 −0.918863
\(83\) 464.478 0.614253 0.307127 0.951669i \(-0.400632\pi\)
0.307127 + 0.951669i \(0.400632\pi\)
\(84\) 0 0
\(85\) 273.036 0.348410
\(86\) 872.656 1.09420
\(87\) 0 0
\(88\) 3056.72 3.70281
\(89\) 993.782 1.18360 0.591801 0.806084i \(-0.298417\pi\)
0.591801 + 0.806084i \(0.298417\pi\)
\(90\) 0 0
\(91\) 485.423 0.559189
\(92\) −728.640 −0.825717
\(93\) 0 0
\(94\) −466.871 −0.512277
\(95\) −556.693 −0.601216
\(96\) 0 0
\(97\) 881.415 0.922620 0.461310 0.887239i \(-0.347380\pi\)
0.461310 + 0.887239i \(0.347380\pi\)
\(98\) 863.956 0.890538
\(99\) 0 0
\(100\) 506.366 0.506366
\(101\) 1209.74 1.19182 0.595909 0.803052i \(-0.296792\pi\)
0.595909 + 0.803052i \(0.296792\pi\)
\(102\) 0 0
\(103\) 744.843 0.712539 0.356270 0.934383i \(-0.384048\pi\)
0.356270 + 0.934383i \(0.384048\pi\)
\(104\) 2353.80 2.21931
\(105\) 0 0
\(106\) 2547.44 2.33424
\(107\) −1000.14 −0.903622 −0.451811 0.892114i \(-0.649222\pi\)
−0.451811 + 0.892114i \(0.649222\pi\)
\(108\) 0 0
\(109\) −915.517 −0.804502 −0.402251 0.915530i \(-0.631772\pi\)
−0.402251 + 0.915530i \(0.631772\pi\)
\(110\) −1247.17 −1.08103
\(111\) 0 0
\(112\) −2474.66 −2.08780
\(113\) −1324.90 −1.10298 −0.551488 0.834183i \(-0.685940\pi\)
−0.551488 + 0.834183i \(0.685940\pi\)
\(114\) 0 0
\(115\) 179.870 0.145852
\(116\) 1177.15 0.942202
\(117\) 0 0
\(118\) −3378.61 −2.63582
\(119\) 733.577 0.565100
\(120\) 0 0
\(121\) 871.015 0.654407
\(122\) −255.281 −0.189443
\(123\) 0 0
\(124\) −5988.62 −4.33705
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −993.635 −0.694259 −0.347129 0.937817i \(-0.612843\pi\)
−0.347129 + 0.937817i \(0.612843\pi\)
\(128\) −1444.52 −0.997494
\(129\) 0 0
\(130\) −960.371 −0.647924
\(131\) 1383.99 0.923051 0.461526 0.887127i \(-0.347302\pi\)
0.461526 + 0.887127i \(0.347302\pi\)
\(132\) 0 0
\(133\) −1495.69 −0.975136
\(134\) 153.715 0.0990970
\(135\) 0 0
\(136\) 3557.09 2.24278
\(137\) 1547.14 0.964823 0.482411 0.875945i \(-0.339761\pi\)
0.482411 + 0.875945i \(0.339761\pi\)
\(138\) 0 0
\(139\) −539.505 −0.329210 −0.164605 0.986360i \(-0.552635\pi\)
−0.164605 + 0.986360i \(0.552635\pi\)
\(140\) 1360.48 0.821294
\(141\) 0 0
\(142\) −3062.70 −1.80998
\(143\) 1695.64 0.991586
\(144\) 0 0
\(145\) −290.588 −0.166427
\(146\) −4438.75 −2.51612
\(147\) 0 0
\(148\) −1074.34 −0.596691
\(149\) −1204.67 −0.662351 −0.331176 0.943569i \(-0.607445\pi\)
−0.331176 + 0.943569i \(0.607445\pi\)
\(150\) 0 0
\(151\) 2972.91 1.60220 0.801100 0.598531i \(-0.204249\pi\)
0.801100 + 0.598531i \(0.204249\pi\)
\(152\) −7252.55 −3.87013
\(153\) 0 0
\(154\) −3350.82 −1.75336
\(155\) 1478.33 0.766082
\(156\) 0 0
\(157\) −395.761 −0.201179 −0.100590 0.994928i \(-0.532073\pi\)
−0.100590 + 0.994928i \(0.532073\pi\)
\(158\) 1082.97 0.545293
\(159\) 0 0
\(160\) 2290.34 1.13167
\(161\) 483.265 0.236563
\(162\) 0 0
\(163\) 3861.43 1.85553 0.927764 0.373169i \(-0.121729\pi\)
0.927764 + 0.373169i \(0.121729\pi\)
\(164\) 2599.86 1.23790
\(165\) 0 0
\(166\) −2468.93 −1.15438
\(167\) 2034.97 0.942939 0.471470 0.881882i \(-0.343724\pi\)
0.471470 + 0.881882i \(0.343724\pi\)
\(168\) 0 0
\(169\) −891.286 −0.405683
\(170\) −1451.32 −0.654773
\(171\) 0 0
\(172\) −3325.24 −1.47411
\(173\) 1546.92 0.679825 0.339913 0.940457i \(-0.389602\pi\)
0.339913 + 0.940457i \(0.389602\pi\)
\(174\) 0 0
\(175\) −335.843 −0.145071
\(176\) −8644.31 −3.70221
\(177\) 0 0
\(178\) −5282.45 −2.22436
\(179\) −823.973 −0.344059 −0.172030 0.985092i \(-0.555033\pi\)
−0.172030 + 0.985092i \(0.555033\pi\)
\(180\) 0 0
\(181\) −4403.55 −1.80836 −0.904180 0.427152i \(-0.859517\pi\)
−0.904180 + 0.427152i \(0.859517\pi\)
\(182\) −2580.27 −1.05089
\(183\) 0 0
\(184\) 2343.33 0.938873
\(185\) 265.209 0.105397
\(186\) 0 0
\(187\) 2562.48 1.00207
\(188\) 1779.00 0.690144
\(189\) 0 0
\(190\) 2959.11 1.12988
\(191\) −3046.93 −1.15428 −0.577141 0.816645i \(-0.695832\pi\)
−0.577141 + 0.816645i \(0.695832\pi\)
\(192\) 0 0
\(193\) 1890.90 0.705235 0.352617 0.935768i \(-0.385292\pi\)
0.352617 + 0.935768i \(0.385292\pi\)
\(194\) −4685.17 −1.73389
\(195\) 0 0
\(196\) −3292.09 −1.19974
\(197\) 3652.00 1.32078 0.660392 0.750921i \(-0.270390\pi\)
0.660392 + 0.750921i \(0.270390\pi\)
\(198\) 0 0
\(199\) 3217.26 1.14606 0.573029 0.819535i \(-0.305768\pi\)
0.573029 + 0.819535i \(0.305768\pi\)
\(200\) −1628.49 −0.575758
\(201\) 0 0
\(202\) −6430.38 −2.23980
\(203\) −780.735 −0.269935
\(204\) 0 0
\(205\) −641.795 −0.218658
\(206\) −3959.22 −1.33909
\(207\) 0 0
\(208\) −6656.47 −2.21896
\(209\) −5224.64 −1.72917
\(210\) 0 0
\(211\) 4947.80 1.61432 0.807159 0.590335i \(-0.201004\pi\)
0.807159 + 0.590335i \(0.201004\pi\)
\(212\) −9706.97 −3.14471
\(213\) 0 0
\(214\) 5316.27 1.69819
\(215\) 820.858 0.260382
\(216\) 0 0
\(217\) 3971.91 1.24254
\(218\) 4866.44 1.51191
\(219\) 0 0
\(220\) 4752.31 1.45637
\(221\) 1973.21 0.600600
\(222\) 0 0
\(223\) 1104.11 0.331555 0.165778 0.986163i \(-0.446987\pi\)
0.165778 + 0.986163i \(0.446987\pi\)
\(224\) 6153.56 1.83550
\(225\) 0 0
\(226\) 7042.53 2.07284
\(227\) −3820.24 −1.11700 −0.558498 0.829506i \(-0.688622\pi\)
−0.558498 + 0.829506i \(0.688622\pi\)
\(228\) 0 0
\(229\) 144.770 0.0417757 0.0208879 0.999782i \(-0.493351\pi\)
0.0208879 + 0.999782i \(0.493351\pi\)
\(230\) −956.101 −0.274102
\(231\) 0 0
\(232\) −3785.75 −1.07132
\(233\) 1286.31 0.361670 0.180835 0.983513i \(-0.442120\pi\)
0.180835 + 0.983513i \(0.442120\pi\)
\(234\) 0 0
\(235\) −439.159 −0.121905
\(236\) 12874.1 3.55099
\(237\) 0 0
\(238\) −3899.34 −1.06200
\(239\) 4219.45 1.14198 0.570991 0.820956i \(-0.306559\pi\)
0.570991 + 0.820956i \(0.306559\pi\)
\(240\) 0 0
\(241\) 2283.13 0.610247 0.305124 0.952313i \(-0.401302\pi\)
0.305124 + 0.952313i \(0.401302\pi\)
\(242\) −4629.89 −1.22984
\(243\) 0 0
\(244\) 972.742 0.255219
\(245\) 812.675 0.211918
\(246\) 0 0
\(247\) −4023.19 −1.03639
\(248\) 19259.6 4.93140
\(249\) 0 0
\(250\) 664.439 0.168091
\(251\) −7922.23 −1.99222 −0.996109 0.0881290i \(-0.971911\pi\)
−0.996109 + 0.0881290i \(0.971911\pi\)
\(252\) 0 0
\(253\) 1688.10 0.419487
\(254\) 5281.67 1.30473
\(255\) 0 0
\(256\) −10.9239 −0.00266698
\(257\) 2262.18 0.549069 0.274535 0.961577i \(-0.411476\pi\)
0.274535 + 0.961577i \(0.411476\pi\)
\(258\) 0 0
\(259\) 712.548 0.170948
\(260\) 3659.47 0.872888
\(261\) 0 0
\(262\) −7356.61 −1.73471
\(263\) 163.470 0.0383269 0.0191635 0.999816i \(-0.493900\pi\)
0.0191635 + 0.999816i \(0.493900\pi\)
\(264\) 0 0
\(265\) 2396.24 0.555470
\(266\) 7950.37 1.83259
\(267\) 0 0
\(268\) −585.729 −0.133504
\(269\) 3304.13 0.748908 0.374454 0.927246i \(-0.377830\pi\)
0.374454 + 0.927246i \(0.377830\pi\)
\(270\) 0 0
\(271\) −1954.96 −0.438212 −0.219106 0.975701i \(-0.570314\pi\)
−0.219106 + 0.975701i \(0.570314\pi\)
\(272\) −10059.3 −2.24242
\(273\) 0 0
\(274\) −8223.81 −1.81321
\(275\) −1173.14 −0.257247
\(276\) 0 0
\(277\) −4243.35 −0.920426 −0.460213 0.887809i \(-0.652227\pi\)
−0.460213 + 0.887809i \(0.652227\pi\)
\(278\) 2867.74 0.618689
\(279\) 0 0
\(280\) −4375.33 −0.933844
\(281\) −2622.29 −0.556701 −0.278350 0.960480i \(-0.589788\pi\)
−0.278350 + 0.960480i \(0.589788\pi\)
\(282\) 0 0
\(283\) −7.07471 −0.00148603 −0.000743017 1.00000i \(-0.500237\pi\)
−0.000743017 1.00000i \(0.500237\pi\)
\(284\) 11670.4 2.43841
\(285\) 0 0
\(286\) −9013.20 −1.86350
\(287\) −1724.34 −0.354650
\(288\) 0 0
\(289\) −1931.06 −0.393051
\(290\) 1544.62 0.312770
\(291\) 0 0
\(292\) 16913.8 3.38974
\(293\) −4477.49 −0.892757 −0.446378 0.894844i \(-0.647286\pi\)
−0.446378 + 0.894844i \(0.647286\pi\)
\(294\) 0 0
\(295\) −3178.07 −0.627236
\(296\) 3455.12 0.678461
\(297\) 0 0
\(298\) 6403.43 1.24477
\(299\) 1299.91 0.251424
\(300\) 0 0
\(301\) 2205.44 0.422323
\(302\) −15802.5 −3.01104
\(303\) 0 0
\(304\) 20510.0 3.86951
\(305\) −240.128 −0.0450810
\(306\) 0 0
\(307\) −3889.78 −0.723132 −0.361566 0.932347i \(-0.617758\pi\)
−0.361566 + 0.932347i \(0.617758\pi\)
\(308\) 12768.2 2.36214
\(309\) 0 0
\(310\) −7858.10 −1.43971
\(311\) −7295.66 −1.33022 −0.665111 0.746745i \(-0.731616\pi\)
−0.665111 + 0.746745i \(0.731616\pi\)
\(312\) 0 0
\(313\) −636.072 −0.114866 −0.0574328 0.998349i \(-0.518291\pi\)
−0.0574328 + 0.998349i \(0.518291\pi\)
\(314\) 2103.67 0.378080
\(315\) 0 0
\(316\) −4126.62 −0.734623
\(317\) −6146.36 −1.08900 −0.544501 0.838760i \(-0.683281\pi\)
−0.544501 + 0.838760i \(0.683281\pi\)
\(318\) 0 0
\(319\) −2727.20 −0.478664
\(320\) −4805.82 −0.839542
\(321\) 0 0
\(322\) −2568.80 −0.444576
\(323\) −6079.89 −1.04735
\(324\) 0 0
\(325\) −903.367 −0.154184
\(326\) −20525.5 −3.48712
\(327\) 0 0
\(328\) −8361.25 −1.40754
\(329\) −1179.91 −0.197722
\(330\) 0 0
\(331\) 11734.4 1.94858 0.974291 0.225293i \(-0.0723340\pi\)
0.974291 + 0.225293i \(0.0723340\pi\)
\(332\) 9407.82 1.55518
\(333\) 0 0
\(334\) −10816.9 −1.77208
\(335\) 144.592 0.0235817
\(336\) 0 0
\(337\) 6283.29 1.01565 0.507823 0.861461i \(-0.330450\pi\)
0.507823 + 0.861461i \(0.330450\pi\)
\(338\) 4737.64 0.762407
\(339\) 0 0
\(340\) 5530.24 0.882115
\(341\) 13874.4 2.20334
\(342\) 0 0
\(343\) 6791.22 1.06907
\(344\) 10694.1 1.67612
\(345\) 0 0
\(346\) −8222.64 −1.27761
\(347\) 4800.82 0.742713 0.371357 0.928490i \(-0.378893\pi\)
0.371357 + 0.928490i \(0.378893\pi\)
\(348\) 0 0
\(349\) −3892.46 −0.597016 −0.298508 0.954407i \(-0.596489\pi\)
−0.298508 + 0.954407i \(0.596489\pi\)
\(350\) 1785.18 0.272633
\(351\) 0 0
\(352\) 21495.2 3.25482
\(353\) −2966.27 −0.447249 −0.223624 0.974675i \(-0.571789\pi\)
−0.223624 + 0.974675i \(0.571789\pi\)
\(354\) 0 0
\(355\) −2880.91 −0.430713
\(356\) 20128.7 2.99668
\(357\) 0 0
\(358\) 4379.84 0.646596
\(359\) −9584.91 −1.40911 −0.704557 0.709647i \(-0.748854\pi\)
−0.704557 + 0.709647i \(0.748854\pi\)
\(360\) 0 0
\(361\) 5537.30 0.807304
\(362\) 23407.1 3.39848
\(363\) 0 0
\(364\) 9832.06 1.41577
\(365\) −4175.28 −0.598752
\(366\) 0 0
\(367\) 4537.18 0.645338 0.322669 0.946512i \(-0.395420\pi\)
0.322669 + 0.946512i \(0.395420\pi\)
\(368\) −6626.88 −0.938722
\(369\) 0 0
\(370\) −1409.72 −0.198075
\(371\) 6438.07 0.900938
\(372\) 0 0
\(373\) 13131.3 1.82282 0.911411 0.411497i \(-0.134994\pi\)
0.911411 + 0.411497i \(0.134994\pi\)
\(374\) −13620.9 −1.88320
\(375\) 0 0
\(376\) −5721.33 −0.784721
\(377\) −2100.06 −0.286892
\(378\) 0 0
\(379\) −9451.10 −1.28092 −0.640462 0.767990i \(-0.721257\pi\)
−0.640462 + 0.767990i \(0.721257\pi\)
\(380\) −11275.6 −1.52218
\(381\) 0 0
\(382\) 16196.0 2.16926
\(383\) −8597.03 −1.14697 −0.573483 0.819218i \(-0.694408\pi\)
−0.573483 + 0.819218i \(0.694408\pi\)
\(384\) 0 0
\(385\) −3151.93 −0.417240
\(386\) −10051.1 −1.32536
\(387\) 0 0
\(388\) 17852.7 2.33592
\(389\) 4442.08 0.578978 0.289489 0.957181i \(-0.406515\pi\)
0.289489 + 0.957181i \(0.406515\pi\)
\(390\) 0 0
\(391\) 1964.44 0.254082
\(392\) 10587.5 1.36415
\(393\) 0 0
\(394\) −19412.3 −2.48217
\(395\) 1018.69 0.129761
\(396\) 0 0
\(397\) 10445.6 1.32053 0.660265 0.751033i \(-0.270444\pi\)
0.660265 + 0.751033i \(0.270444\pi\)
\(398\) −17101.4 −2.15380
\(399\) 0 0
\(400\) 4605.32 0.575665
\(401\) −12683.4 −1.57950 −0.789751 0.613427i \(-0.789790\pi\)
−0.789751 + 0.613427i \(0.789790\pi\)
\(402\) 0 0
\(403\) 10683.8 1.32059
\(404\) 24502.8 3.01748
\(405\) 0 0
\(406\) 4150.00 0.507293
\(407\) 2489.02 0.303135
\(408\) 0 0
\(409\) −11112.6 −1.34348 −0.671739 0.740788i \(-0.734452\pi\)
−0.671739 + 0.740788i \(0.734452\pi\)
\(410\) 3411.47 0.410928
\(411\) 0 0
\(412\) 15086.5 1.80403
\(413\) −8538.67 −1.01734
\(414\) 0 0
\(415\) −2322.39 −0.274702
\(416\) 16552.2 1.95081
\(417\) 0 0
\(418\) 27771.6 3.24965
\(419\) 2018.52 0.235349 0.117675 0.993052i \(-0.462456\pi\)
0.117675 + 0.993052i \(0.462456\pi\)
\(420\) 0 0
\(421\) −10957.7 −1.26852 −0.634259 0.773121i \(-0.718695\pi\)
−0.634259 + 0.773121i \(0.718695\pi\)
\(422\) −26300.1 −3.03381
\(423\) 0 0
\(424\) 31218.0 3.57565
\(425\) −1365.18 −0.155814
\(426\) 0 0
\(427\) −645.164 −0.0731186
\(428\) −20257.5 −2.28782
\(429\) 0 0
\(430\) −4363.28 −0.489340
\(431\) −2124.75 −0.237460 −0.118730 0.992927i \(-0.537882\pi\)
−0.118730 + 0.992927i \(0.537882\pi\)
\(432\) 0 0
\(433\) −16169.2 −1.79456 −0.897279 0.441463i \(-0.854460\pi\)
−0.897279 + 0.441463i \(0.854460\pi\)
\(434\) −21112.7 −2.33512
\(435\) 0 0
\(436\) −18543.5 −2.03686
\(437\) −4005.30 −0.438443
\(438\) 0 0
\(439\) 692.197 0.0752545 0.0376273 0.999292i \(-0.488020\pi\)
0.0376273 + 0.999292i \(0.488020\pi\)
\(440\) −15283.6 −1.65595
\(441\) 0 0
\(442\) −10488.6 −1.12872
\(443\) −6670.51 −0.715408 −0.357704 0.933835i \(-0.616440\pi\)
−0.357704 + 0.933835i \(0.616440\pi\)
\(444\) 0 0
\(445\) −4968.91 −0.529323
\(446\) −5868.92 −0.623097
\(447\) 0 0
\(448\) −12912.0 −1.36169
\(449\) −9743.18 −1.02407 −0.512037 0.858964i \(-0.671109\pi\)
−0.512037 + 0.858964i \(0.671109\pi\)
\(450\) 0 0
\(451\) −6023.33 −0.628886
\(452\) −26835.4 −2.79255
\(453\) 0 0
\(454\) 20306.5 2.09919
\(455\) −2427.12 −0.250077
\(456\) 0 0
\(457\) 4442.72 0.454752 0.227376 0.973807i \(-0.426985\pi\)
0.227376 + 0.973807i \(0.426985\pi\)
\(458\) −769.524 −0.0785098
\(459\) 0 0
\(460\) 3643.20 0.369272
\(461\) 18939.4 1.91344 0.956720 0.291011i \(-0.0939915\pi\)
0.956720 + 0.291011i \(0.0939915\pi\)
\(462\) 0 0
\(463\) −16792.2 −1.68553 −0.842764 0.538283i \(-0.819073\pi\)
−0.842764 + 0.538283i \(0.819073\pi\)
\(464\) 10706.0 1.07115
\(465\) 0 0
\(466\) −6837.40 −0.679692
\(467\) 1367.93 0.135547 0.0677735 0.997701i \(-0.478410\pi\)
0.0677735 + 0.997701i \(0.478410\pi\)
\(468\) 0 0
\(469\) 388.480 0.0382481
\(470\) 2334.35 0.229097
\(471\) 0 0
\(472\) −41403.6 −4.03762
\(473\) 7703.86 0.748888
\(474\) 0 0
\(475\) 2783.47 0.268872
\(476\) 14858.3 1.43074
\(477\) 0 0
\(478\) −22428.5 −2.14615
\(479\) 202.741 0.0193392 0.00966960 0.999953i \(-0.496922\pi\)
0.00966960 + 0.999953i \(0.496922\pi\)
\(480\) 0 0
\(481\) 1916.65 0.181687
\(482\) −12136.0 −1.14685
\(483\) 0 0
\(484\) 17642.1 1.65685
\(485\) −4407.08 −0.412608
\(486\) 0 0
\(487\) 18961.3 1.76431 0.882153 0.470964i \(-0.156094\pi\)
0.882153 + 0.470964i \(0.156094\pi\)
\(488\) −3128.37 −0.290194
\(489\) 0 0
\(490\) −4319.78 −0.398261
\(491\) 11712.3 1.07651 0.538257 0.842781i \(-0.319083\pi\)
0.538257 + 0.842781i \(0.319083\pi\)
\(492\) 0 0
\(493\) −3173.63 −0.289925
\(494\) 21385.3 1.94771
\(495\) 0 0
\(496\) −54465.7 −4.93061
\(497\) −7740.28 −0.698590
\(498\) 0 0
\(499\) 8624.87 0.773752 0.386876 0.922132i \(-0.373554\pi\)
0.386876 + 0.922132i \(0.373554\pi\)
\(500\) −2531.83 −0.226454
\(501\) 0 0
\(502\) 42110.7 3.74401
\(503\) 4856.12 0.430465 0.215232 0.976563i \(-0.430949\pi\)
0.215232 + 0.976563i \(0.430949\pi\)
\(504\) 0 0
\(505\) −6048.69 −0.532997
\(506\) −8973.13 −0.788348
\(507\) 0 0
\(508\) −20125.7 −1.75774
\(509\) −10850.3 −0.944858 −0.472429 0.881369i \(-0.656623\pi\)
−0.472429 + 0.881369i \(0.656623\pi\)
\(510\) 0 0
\(511\) −11217.9 −0.971138
\(512\) 11614.3 1.00251
\(513\) 0 0
\(514\) −12024.6 −1.03187
\(515\) −3724.21 −0.318657
\(516\) 0 0
\(517\) −4121.57 −0.350612
\(518\) −3787.55 −0.321266
\(519\) 0 0
\(520\) −11769.0 −0.992508
\(521\) 3559.30 0.299301 0.149651 0.988739i \(-0.452185\pi\)
0.149651 + 0.988739i \(0.452185\pi\)
\(522\) 0 0
\(523\) 22191.7 1.85540 0.927702 0.373321i \(-0.121781\pi\)
0.927702 + 0.373321i \(0.121781\pi\)
\(524\) 28032.2 2.33701
\(525\) 0 0
\(526\) −868.925 −0.0720284
\(527\) 16145.5 1.33455
\(528\) 0 0
\(529\) −10872.9 −0.893636
\(530\) −12737.2 −1.04390
\(531\) 0 0
\(532\) −30294.7 −2.46888
\(533\) −4638.21 −0.376929
\(534\) 0 0
\(535\) 5000.72 0.404112
\(536\) 1883.73 0.151800
\(537\) 0 0
\(538\) −17563.1 −1.40743
\(539\) 7627.06 0.609500
\(540\) 0 0
\(541\) 4257.08 0.338311 0.169155 0.985589i \(-0.445896\pi\)
0.169155 + 0.985589i \(0.445896\pi\)
\(542\) 10391.6 0.823539
\(543\) 0 0
\(544\) 25013.8 1.97143
\(545\) 4577.59 0.359784
\(546\) 0 0
\(547\) −21218.2 −1.65854 −0.829272 0.558845i \(-0.811244\pi\)
−0.829272 + 0.558845i \(0.811244\pi\)
\(548\) 31336.7 2.44277
\(549\) 0 0
\(550\) 6235.84 0.483449
\(551\) 6470.73 0.500294
\(552\) 0 0
\(553\) 2736.95 0.210465
\(554\) 22555.5 1.72977
\(555\) 0 0
\(556\) −10927.5 −0.833503
\(557\) 2506.42 0.190665 0.0953324 0.995445i \(-0.469609\pi\)
0.0953324 + 0.995445i \(0.469609\pi\)
\(558\) 0 0
\(559\) 5932.29 0.448854
\(560\) 12373.3 0.933693
\(561\) 0 0
\(562\) 13938.8 1.04622
\(563\) 17279.6 1.29351 0.646757 0.762696i \(-0.276125\pi\)
0.646757 + 0.762696i \(0.276125\pi\)
\(564\) 0 0
\(565\) 6624.51 0.493266
\(566\) 37.6057 0.00279273
\(567\) 0 0
\(568\) −37532.3 −2.77257
\(569\) −20508.1 −1.51097 −0.755487 0.655164i \(-0.772600\pi\)
−0.755487 + 0.655164i \(0.772600\pi\)
\(570\) 0 0
\(571\) 6825.07 0.500210 0.250105 0.968219i \(-0.419535\pi\)
0.250105 + 0.968219i \(0.419535\pi\)
\(572\) 34344.6 2.51052
\(573\) 0 0
\(574\) 9165.74 0.666500
\(575\) −899.350 −0.0652270
\(576\) 0 0
\(577\) 4249.00 0.306565 0.153283 0.988182i \(-0.451016\pi\)
0.153283 + 0.988182i \(0.451016\pi\)
\(578\) 10264.6 0.738667
\(579\) 0 0
\(580\) −5885.74 −0.421366
\(581\) −6239.66 −0.445551
\(582\) 0 0
\(583\) 22489.0 1.59760
\(584\) −54395.2 −3.85426
\(585\) 0 0
\(586\) 23800.1 1.67777
\(587\) 18688.1 1.31404 0.657018 0.753875i \(-0.271817\pi\)
0.657018 + 0.753875i \(0.271817\pi\)
\(588\) 0 0
\(589\) −32919.2 −2.30290
\(590\) 16893.1 1.17877
\(591\) 0 0
\(592\) −9770.97 −0.678352
\(593\) −12434.5 −0.861084 −0.430542 0.902570i \(-0.641678\pi\)
−0.430542 + 0.902570i \(0.641678\pi\)
\(594\) 0 0
\(595\) −3667.89 −0.252721
\(596\) −24400.1 −1.67696
\(597\) 0 0
\(598\) −6909.68 −0.472505
\(599\) −6043.04 −0.412207 −0.206103 0.978530i \(-0.566078\pi\)
−0.206103 + 0.978530i \(0.566078\pi\)
\(600\) 0 0
\(601\) −631.308 −0.0428479 −0.0214239 0.999770i \(-0.506820\pi\)
−0.0214239 + 0.999770i \(0.506820\pi\)
\(602\) −11723.0 −0.793679
\(603\) 0 0
\(604\) 60215.2 4.05649
\(605\) −4355.08 −0.292660
\(606\) 0 0
\(607\) −9454.87 −0.632226 −0.316113 0.948722i \(-0.602378\pi\)
−0.316113 + 0.948722i \(0.602378\pi\)
\(608\) −51000.7 −3.40190
\(609\) 0 0
\(610\) 1276.40 0.0847215
\(611\) −3173.77 −0.210143
\(612\) 0 0
\(613\) −2682.66 −0.176757 −0.0883783 0.996087i \(-0.528168\pi\)
−0.0883783 + 0.996087i \(0.528168\pi\)
\(614\) 20676.2 1.35899
\(615\) 0 0
\(616\) −41063.1 −2.68584
\(617\) −2338.53 −0.152586 −0.0762930 0.997085i \(-0.524308\pi\)
−0.0762930 + 0.997085i \(0.524308\pi\)
\(618\) 0 0
\(619\) −11930.2 −0.774663 −0.387331 0.921941i \(-0.626603\pi\)
−0.387331 + 0.921941i \(0.626603\pi\)
\(620\) 29943.1 1.93959
\(621\) 0 0
\(622\) 38780.1 2.49991
\(623\) −13350.2 −0.858530
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 3381.05 0.215869
\(627\) 0 0
\(628\) −8015.99 −0.509352
\(629\) 2896.46 0.183608
\(630\) 0 0
\(631\) 4106.81 0.259096 0.129548 0.991573i \(-0.458647\pi\)
0.129548 + 0.991573i \(0.458647\pi\)
\(632\) 13271.4 0.835295
\(633\) 0 0
\(634\) 32671.0 2.04658
\(635\) 4968.17 0.310482
\(636\) 0 0
\(637\) 5873.15 0.365310
\(638\) 14496.5 0.899562
\(639\) 0 0
\(640\) 7222.62 0.446093
\(641\) 14925.3 0.919679 0.459839 0.888002i \(-0.347907\pi\)
0.459839 + 0.888002i \(0.347907\pi\)
\(642\) 0 0
\(643\) 19005.7 1.16565 0.582823 0.812599i \(-0.301948\pi\)
0.582823 + 0.812599i \(0.301948\pi\)
\(644\) 9788.35 0.598936
\(645\) 0 0
\(646\) 32317.7 1.96830
\(647\) 9252.81 0.562234 0.281117 0.959673i \(-0.409295\pi\)
0.281117 + 0.959673i \(0.409295\pi\)
\(648\) 0 0
\(649\) −29826.6 −1.80400
\(650\) 4801.85 0.289760
\(651\) 0 0
\(652\) 78211.9 4.69788
\(653\) 22950.7 1.37539 0.687695 0.726000i \(-0.258623\pi\)
0.687695 + 0.726000i \(0.258623\pi\)
\(654\) 0 0
\(655\) −6919.95 −0.412801
\(656\) 23645.4 1.40731
\(657\) 0 0
\(658\) 6271.81 0.371582
\(659\) 6517.08 0.385234 0.192617 0.981274i \(-0.438302\pi\)
0.192617 + 0.981274i \(0.438302\pi\)
\(660\) 0 0
\(661\) −5196.32 −0.305769 −0.152885 0.988244i \(-0.548856\pi\)
−0.152885 + 0.988244i \(0.548856\pi\)
\(662\) −62374.3 −3.66200
\(663\) 0 0
\(664\) −30255.9 −1.76831
\(665\) 7478.47 0.436094
\(666\) 0 0
\(667\) −2090.72 −0.121369
\(668\) 41217.6 2.38736
\(669\) 0 0
\(670\) −768.577 −0.0443175
\(671\) −2253.64 −0.129658
\(672\) 0 0
\(673\) 28190.0 1.61463 0.807313 0.590123i \(-0.200921\pi\)
0.807313 + 0.590123i \(0.200921\pi\)
\(674\) −33398.9 −1.90872
\(675\) 0 0
\(676\) −18052.7 −1.02712
\(677\) 8676.88 0.492584 0.246292 0.969196i \(-0.420788\pi\)
0.246292 + 0.969196i \(0.420788\pi\)
\(678\) 0 0
\(679\) −11840.7 −0.669225
\(680\) −17785.4 −1.00300
\(681\) 0 0
\(682\) −73749.3 −4.14077
\(683\) 20624.1 1.15543 0.577714 0.816239i \(-0.303945\pi\)
0.577714 + 0.816239i \(0.303945\pi\)
\(684\) 0 0
\(685\) −7735.68 −0.431482
\(686\) −36098.8 −2.00912
\(687\) 0 0
\(688\) −30242.5 −1.67585
\(689\) 17317.4 0.957535
\(690\) 0 0
\(691\) 6986.06 0.384605 0.192303 0.981336i \(-0.438405\pi\)
0.192303 + 0.981336i \(0.438405\pi\)
\(692\) 31332.2 1.72120
\(693\) 0 0
\(694\) −25518.8 −1.39579
\(695\) 2697.52 0.147227
\(696\) 0 0
\(697\) −7009.32 −0.380914
\(698\) 20690.4 1.12198
\(699\) 0 0
\(700\) −6802.38 −0.367294
\(701\) 7848.55 0.422875 0.211438 0.977391i \(-0.432185\pi\)
0.211438 + 0.977391i \(0.432185\pi\)
\(702\) 0 0
\(703\) −5905.60 −0.316833
\(704\) −45103.2 −2.41462
\(705\) 0 0
\(706\) 15767.3 0.840522
\(707\) −16251.3 −0.864488
\(708\) 0 0
\(709\) −25424.6 −1.34674 −0.673371 0.739305i \(-0.735154\pi\)
−0.673371 + 0.739305i \(0.735154\pi\)
\(710\) 15313.5 0.809445
\(711\) 0 0
\(712\) −64734.5 −3.40734
\(713\) 10636.3 0.558673
\(714\) 0 0
\(715\) −8478.21 −0.443451
\(716\) −16689.3 −0.871099
\(717\) 0 0
\(718\) 50948.7 2.64817
\(719\) 16707.2 0.866584 0.433292 0.901254i \(-0.357352\pi\)
0.433292 + 0.901254i \(0.357352\pi\)
\(720\) 0 0
\(721\) −10006.0 −0.516843
\(722\) −29433.6 −1.51718
\(723\) 0 0
\(724\) −89192.2 −4.57846
\(725\) 1452.94 0.0744286
\(726\) 0 0
\(727\) 1436.48 0.0732821 0.0366410 0.999328i \(-0.488334\pi\)
0.0366410 + 0.999328i \(0.488334\pi\)
\(728\) −31620.3 −1.60979
\(729\) 0 0
\(730\) 22193.8 1.12524
\(731\) 8964.95 0.453599
\(732\) 0 0
\(733\) 11540.9 0.581545 0.290772 0.956792i \(-0.406088\pi\)
0.290772 + 0.956792i \(0.406088\pi\)
\(734\) −24117.4 −1.21279
\(735\) 0 0
\(736\) 16478.6 0.825282
\(737\) 1357.01 0.0678237
\(738\) 0 0
\(739\) 4127.87 0.205475 0.102737 0.994709i \(-0.467240\pi\)
0.102737 + 0.994709i \(0.467240\pi\)
\(740\) 5371.70 0.266848
\(741\) 0 0
\(742\) −34221.6 −1.69315
\(743\) −27497.0 −1.35769 −0.678846 0.734281i \(-0.737520\pi\)
−0.678846 + 0.734281i \(0.737520\pi\)
\(744\) 0 0
\(745\) 6023.35 0.296213
\(746\) −69799.5 −3.42566
\(747\) 0 0
\(748\) 51902.0 2.53706
\(749\) 13435.7 0.655445
\(750\) 0 0
\(751\) 34394.1 1.67118 0.835592 0.549351i \(-0.185125\pi\)
0.835592 + 0.549351i \(0.185125\pi\)
\(752\) 16179.8 0.784594
\(753\) 0 0
\(754\) 11162.9 0.539161
\(755\) −14864.6 −0.716526
\(756\) 0 0
\(757\) −30459.8 −1.46246 −0.731229 0.682132i \(-0.761053\pi\)
−0.731229 + 0.682132i \(0.761053\pi\)
\(758\) 50237.4 2.40726
\(759\) 0 0
\(760\) 36262.8 1.73077
\(761\) 447.168 0.0213007 0.0106504 0.999943i \(-0.496610\pi\)
0.0106504 + 0.999943i \(0.496610\pi\)
\(762\) 0 0
\(763\) 12298.8 0.583548
\(764\) −61714.3 −2.92244
\(765\) 0 0
\(766\) 45697.6 2.15551
\(767\) −22967.7 −1.08125
\(768\) 0 0
\(769\) −8057.17 −0.377827 −0.188914 0.981994i \(-0.560497\pi\)
−0.188914 + 0.981994i \(0.560497\pi\)
\(770\) 16754.1 0.784125
\(771\) 0 0
\(772\) 38299.6 1.78553
\(773\) −19600.4 −0.912001 −0.456000 0.889980i \(-0.650718\pi\)
−0.456000 + 0.889980i \(0.650718\pi\)
\(774\) 0 0
\(775\) −7391.67 −0.342602
\(776\) −57415.0 −2.65603
\(777\) 0 0
\(778\) −23611.9 −1.08808
\(779\) 14291.3 0.657304
\(780\) 0 0
\(781\) −27037.7 −1.23878
\(782\) −10442.0 −0.477500
\(783\) 0 0
\(784\) −29941.0 −1.36393
\(785\) 1978.80 0.0899702
\(786\) 0 0
\(787\) 17936.4 0.812404 0.406202 0.913783i \(-0.366853\pi\)
0.406202 + 0.913783i \(0.366853\pi\)
\(788\) 73970.0 3.34400
\(789\) 0 0
\(790\) −5414.84 −0.243862
\(791\) 17798.4 0.800047
\(792\) 0 0
\(793\) −1735.39 −0.0777119
\(794\) −55523.7 −2.48169
\(795\) 0 0
\(796\) 65164.4 2.90162
\(797\) −17942.6 −0.797440 −0.398720 0.917073i \(-0.630546\pi\)
−0.398720 + 0.917073i \(0.630546\pi\)
\(798\) 0 0
\(799\) −4796.25 −0.212364
\(800\) −11451.7 −0.506099
\(801\) 0 0
\(802\) 67418.9 2.96838
\(803\) −39185.6 −1.72208
\(804\) 0 0
\(805\) −2416.33 −0.105794
\(806\) −56790.0 −2.48181
\(807\) 0 0
\(808\) −78801.9 −3.43099
\(809\) 29094.2 1.26440 0.632200 0.774806i \(-0.282152\pi\)
0.632200 + 0.774806i \(0.282152\pi\)
\(810\) 0 0
\(811\) 1438.31 0.0622760 0.0311380 0.999515i \(-0.490087\pi\)
0.0311380 + 0.999515i \(0.490087\pi\)
\(812\) −15813.5 −0.683429
\(813\) 0 0
\(814\) −13230.4 −0.569687
\(815\) −19307.2 −0.829817
\(816\) 0 0
\(817\) −18278.7 −0.782729
\(818\) 59069.1 2.52482
\(819\) 0 0
\(820\) −12999.3 −0.553605
\(821\) −242.830 −0.0103226 −0.00516128 0.999987i \(-0.501643\pi\)
−0.00516128 + 0.999987i \(0.501643\pi\)
\(822\) 0 0
\(823\) 18171.0 0.769625 0.384812 0.922995i \(-0.374266\pi\)
0.384812 + 0.922995i \(0.374266\pi\)
\(824\) −48518.7 −2.05125
\(825\) 0 0
\(826\) 45387.4 1.91190
\(827\) 12181.7 0.512213 0.256107 0.966649i \(-0.417560\pi\)
0.256107 + 0.966649i \(0.417560\pi\)
\(828\) 0 0
\(829\) −17370.0 −0.727724 −0.363862 0.931453i \(-0.618542\pi\)
−0.363862 + 0.931453i \(0.618542\pi\)
\(830\) 12344.7 0.516253
\(831\) 0 0
\(832\) −34731.3 −1.44723
\(833\) 8875.58 0.369172
\(834\) 0 0
\(835\) −10174.9 −0.421695
\(836\) −105823. −4.37795
\(837\) 0 0
\(838\) −10729.5 −0.442296
\(839\) 31363.0 1.29055 0.645275 0.763950i \(-0.276743\pi\)
0.645275 + 0.763950i \(0.276743\pi\)
\(840\) 0 0
\(841\) −21011.4 −0.861509
\(842\) 58245.8 2.38395
\(843\) 0 0
\(844\) 100216. 4.08717
\(845\) 4456.43 0.181427
\(846\) 0 0
\(847\) −11701.0 −0.474676
\(848\) −88283.5 −3.57508
\(849\) 0 0
\(850\) 7256.62 0.292823
\(851\) 1908.12 0.0768621
\(852\) 0 0
\(853\) −39677.5 −1.59265 −0.796326 0.604867i \(-0.793226\pi\)
−0.796326 + 0.604867i \(0.793226\pi\)
\(854\) 3429.37 0.137413
\(855\) 0 0
\(856\) 65149.0 2.60134
\(857\) 33209.3 1.32370 0.661849 0.749637i \(-0.269772\pi\)
0.661849 + 0.749637i \(0.269772\pi\)
\(858\) 0 0
\(859\) 34404.9 1.36656 0.683282 0.730154i \(-0.260552\pi\)
0.683282 + 0.730154i \(0.260552\pi\)
\(860\) 16626.2 0.659242
\(861\) 0 0
\(862\) 11294.1 0.446263
\(863\) 12999.7 0.512765 0.256382 0.966575i \(-0.417469\pi\)
0.256382 + 0.966575i \(0.417469\pi\)
\(864\) 0 0
\(865\) −7734.58 −0.304027
\(866\) 85947.7 3.37254
\(867\) 0 0
\(868\) 80449.5 3.14589
\(869\) 9560.51 0.373208
\(870\) 0 0
\(871\) 1044.95 0.0406509
\(872\) 59636.4 2.31599
\(873\) 0 0
\(874\) 21290.2 0.823972
\(875\) 1679.22 0.0648775
\(876\) 0 0
\(877\) −22831.5 −0.879093 −0.439546 0.898220i \(-0.644861\pi\)
−0.439546 + 0.898220i \(0.644861\pi\)
\(878\) −3679.38 −0.141427
\(879\) 0 0
\(880\) 43221.5 1.65568
\(881\) −1889.31 −0.0722502 −0.0361251 0.999347i \(-0.511501\pi\)
−0.0361251 + 0.999347i \(0.511501\pi\)
\(882\) 0 0
\(883\) −1778.37 −0.0677768 −0.0338884 0.999426i \(-0.510789\pi\)
−0.0338884 + 0.999426i \(0.510789\pi\)
\(884\) 39966.7 1.52062
\(885\) 0 0
\(886\) 35457.2 1.34448
\(887\) 22984.1 0.870046 0.435023 0.900419i \(-0.356740\pi\)
0.435023 + 0.900419i \(0.356740\pi\)
\(888\) 0 0
\(889\) 13348.2 0.503582
\(890\) 26412.3 0.994766
\(891\) 0 0
\(892\) 22363.4 0.839441
\(893\) 9779.08 0.366455
\(894\) 0 0
\(895\) 4119.87 0.153868
\(896\) 19405.3 0.723535
\(897\) 0 0
\(898\) 51789.9 1.92456
\(899\) −17183.4 −0.637485
\(900\) 0 0
\(901\) 26170.3 0.967658
\(902\) 32017.1 1.18188
\(903\) 0 0
\(904\) 86303.6 3.17524
\(905\) 22017.7 0.808723
\(906\) 0 0
\(907\) 45410.1 1.66242 0.831211 0.555957i \(-0.187648\pi\)
0.831211 + 0.555957i \(0.187648\pi\)
\(908\) −77377.5 −2.82804
\(909\) 0 0
\(910\) 12901.4 0.469973
\(911\) −30781.8 −1.11948 −0.559740 0.828669i \(-0.689099\pi\)
−0.559740 + 0.828669i \(0.689099\pi\)
\(912\) 0 0
\(913\) −21795.9 −0.790076
\(914\) −23615.3 −0.854623
\(915\) 0 0
\(916\) 2932.25 0.105769
\(917\) −18592.1 −0.669538
\(918\) 0 0
\(919\) −22457.3 −0.806093 −0.403046 0.915180i \(-0.632049\pi\)
−0.403046 + 0.915180i \(0.632049\pi\)
\(920\) −11716.7 −0.419877
\(921\) 0 0
\(922\) −100673. −3.59596
\(923\) −20820.2 −0.742475
\(924\) 0 0
\(925\) −1326.04 −0.0471352
\(926\) 89259.1 3.16764
\(927\) 0 0
\(928\) −26621.8 −0.941706
\(929\) −43654.2 −1.54171 −0.770855 0.637011i \(-0.780171\pi\)
−0.770855 + 0.637011i \(0.780171\pi\)
\(930\) 0 0
\(931\) −18096.4 −0.637043
\(932\) 26053.8 0.915686
\(933\) 0 0
\(934\) −7271.26 −0.254736
\(935\) −12812.4 −0.448138
\(936\) 0 0
\(937\) −41123.3 −1.43377 −0.716884 0.697193i \(-0.754432\pi\)
−0.716884 + 0.697193i \(0.754432\pi\)
\(938\) −2064.97 −0.0718803
\(939\) 0 0
\(940\) −8895.01 −0.308642
\(941\) 22077.7 0.764839 0.382419 0.923989i \(-0.375091\pi\)
0.382419 + 0.923989i \(0.375091\pi\)
\(942\) 0 0
\(943\) −4617.59 −0.159459
\(944\) 117088. 4.03697
\(945\) 0 0
\(946\) −40949.9 −1.40740
\(947\) 8616.41 0.295666 0.147833 0.989012i \(-0.452770\pi\)
0.147833 + 0.989012i \(0.452770\pi\)
\(948\) 0 0
\(949\) −30174.5 −1.03215
\(950\) −14795.5 −0.505295
\(951\) 0 0
\(952\) −47784.9 −1.62680
\(953\) −29558.0 −1.00470 −0.502350 0.864664i \(-0.667531\pi\)
−0.502350 + 0.864664i \(0.667531\pi\)
\(954\) 0 0
\(955\) 15234.6 0.516211
\(956\) 85463.4 2.89130
\(957\) 0 0
\(958\) −1077.67 −0.0363445
\(959\) −20783.8 −0.699837
\(960\) 0 0
\(961\) 57628.0 1.93441
\(962\) −10187.9 −0.341448
\(963\) 0 0
\(964\) 46244.0 1.54504
\(965\) −9454.52 −0.315391
\(966\) 0 0
\(967\) 44575.9 1.48238 0.741192 0.671293i \(-0.234261\pi\)
0.741192 + 0.671293i \(0.234261\pi\)
\(968\) −56737.6 −1.88390
\(969\) 0 0
\(970\) 23425.8 0.775421
\(971\) 43702.8 1.44438 0.722188 0.691696i \(-0.243136\pi\)
0.722188 + 0.691696i \(0.243136\pi\)
\(972\) 0 0
\(973\) 7247.56 0.238793
\(974\) −100789. −3.31569
\(975\) 0 0
\(976\) 8846.95 0.290147
\(977\) 45341.8 1.48476 0.742381 0.669978i \(-0.233696\pi\)
0.742381 + 0.669978i \(0.233696\pi\)
\(978\) 0 0
\(979\) −46633.8 −1.52239
\(980\) 16460.4 0.536540
\(981\) 0 0
\(982\) −62256.8 −2.02311
\(983\) 7360.52 0.238824 0.119412 0.992845i \(-0.461899\pi\)
0.119412 + 0.992845i \(0.461899\pi\)
\(984\) 0 0
\(985\) −18260.0 −0.590673
\(986\) 16869.5 0.544861
\(987\) 0 0
\(988\) −81488.1 −2.62397
\(989\) 5905.91 0.189886
\(990\) 0 0
\(991\) 33328.4 1.06833 0.534164 0.845381i \(-0.320627\pi\)
0.534164 + 0.845381i \(0.320627\pi\)
\(992\) 135436. 4.33477
\(993\) 0 0
\(994\) 41143.5 1.31287
\(995\) −16086.3 −0.512532
\(996\) 0 0
\(997\) −35331.7 −1.12233 −0.561166 0.827703i \(-0.689647\pi\)
−0.561166 + 0.827703i \(0.689647\pi\)
\(998\) −45845.6 −1.45412
\(999\) 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.1.1 7
3.2 odd 2 405.4.a.n.1.7 7
5.4 even 2 2025.4.a.bb.1.7 7
9.2 odd 6 135.4.e.c.91.1 14
9.4 even 3 45.4.e.c.16.7 14
9.5 odd 6 135.4.e.c.46.1 14
9.7 even 3 45.4.e.c.31.7 yes 14
15.14 odd 2 2025.4.a.ba.1.1 7
45.4 even 6 225.4.e.d.151.1 14
45.7 odd 12 225.4.k.d.49.2 28
45.13 odd 12 225.4.k.d.124.2 28
45.22 odd 12 225.4.k.d.124.13 28
45.34 even 6 225.4.e.d.76.1 14
45.43 odd 12 225.4.k.d.49.13 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.e.c.16.7 14 9.4 even 3
45.4.e.c.31.7 yes 14 9.7 even 3
135.4.e.c.46.1 14 9.5 odd 6
135.4.e.c.91.1 14 9.2 odd 6
225.4.e.d.76.1 14 45.34 even 6
225.4.e.d.151.1 14 45.4 even 6
225.4.k.d.49.2 28 45.7 odd 12
225.4.k.d.49.13 28 45.43 odd 12
225.4.k.d.124.2 28 45.13 odd 12
225.4.k.d.124.13 28 45.22 odd 12
405.4.a.m.1.1 7 1.1 even 1 trivial
405.4.a.n.1.7 7 3.2 odd 2
2025.4.a.ba.1.1 7 15.14 odd 2
2025.4.a.bb.1.7 7 5.4 even 2