Properties

Label 2025.4.a.r.1.3
Level $2025$
Weight $4$
Character 2025.1
Self dual yes
Analytic conductor $119.479$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(119.478867762\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.7032.1
Defining polynomial: \( x^{3} - x^{2} - 14x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 405)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.52348\) of defining polynomial
Character \(\chi\) \(=\) 2025.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.52348 q^{2} +4.41489 q^{4} -25.4325 q^{7} -12.6321 q^{8} +O(q^{10})\) \(q+3.52348 q^{2} +4.41489 q^{4} -25.4325 q^{7} -12.6321 q^{8} -71.3561 q^{11} +51.3935 q^{13} -89.6107 q^{14} -79.8279 q^{16} +33.3191 q^{17} +113.372 q^{19} -251.422 q^{22} -81.9767 q^{23} +181.084 q^{26} -112.281 q^{28} -246.827 q^{29} +222.679 q^{31} -180.215 q^{32} +117.399 q^{34} -22.3910 q^{37} +399.463 q^{38} -434.225 q^{41} +236.850 q^{43} -315.029 q^{44} -288.843 q^{46} +107.984 q^{47} +303.810 q^{49} +226.896 q^{52} +123.961 q^{53} +321.265 q^{56} -869.690 q^{58} +171.091 q^{59} -79.4391 q^{61} +784.604 q^{62} +3.63945 q^{64} +611.506 q^{67} +147.100 q^{68} +511.102 q^{71} +410.012 q^{73} -78.8941 q^{74} +500.524 q^{76} +1814.76 q^{77} -793.597 q^{79} -1529.98 q^{82} -270.081 q^{83} +834.534 q^{86} +901.376 q^{88} +177.400 q^{89} -1307.06 q^{91} -361.918 q^{92} +380.478 q^{94} +881.860 q^{97} +1070.47 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8} - 58 q^{11} + 47 q^{13} - 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 260 q^{22} - 51 q^{23} - 253 q^{26} - 83 q^{28} - 350 q^{29} + 638 q^{31} + 245 q^{32} - 154 q^{34} + 414 q^{37} + 397 q^{38} - 179 q^{41} + 836 q^{43} - 332 q^{44} + 261 q^{46} + 235 q^{47} + 892 q^{49} + 1335 q^{52} + 505 q^{53} - 15 q^{56} - 1876 q^{58} - 535 q^{59} - 104 q^{61} + 348 q^{62} - 303 q^{64} + 40 q^{67} + 830 q^{68} - 452 q^{71} + 710 q^{73} - 1394 q^{74} + 849 q^{76} + 2148 q^{77} - 634 q^{79} - 613 q^{82} + 1734 q^{83} - 460 q^{86} + 768 q^{88} + 852 q^{89} - 1229 q^{91} - 1839 q^{92} + 1751 q^{94} + 38 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.52348 1.24574 0.622868 0.782327i \(-0.285967\pi\)
0.622868 + 0.782327i \(0.285967\pi\)
\(3\) 0 0
\(4\) 4.41489 0.551861
\(5\) 0 0
\(6\) 0 0
\(7\) −25.4325 −1.37322 −0.686612 0.727024i \(-0.740903\pi\)
−0.686612 + 0.727024i \(0.740903\pi\)
\(8\) −12.6321 −0.558264
\(9\) 0 0
\(10\) 0 0
\(11\) −71.3561 −1.95588 −0.977940 0.208885i \(-0.933017\pi\)
−0.977940 + 0.208885i \(0.933017\pi\)
\(12\) 0 0
\(13\) 51.3935 1.09646 0.548231 0.836327i \(-0.315302\pi\)
0.548231 + 0.836327i \(0.315302\pi\)
\(14\) −89.6107 −1.71068
\(15\) 0 0
\(16\) −79.8279 −1.24731
\(17\) 33.3191 0.475357 0.237678 0.971344i \(-0.423614\pi\)
0.237678 + 0.971344i \(0.423614\pi\)
\(18\) 0 0
\(19\) 113.372 1.36891 0.684455 0.729055i \(-0.260040\pi\)
0.684455 + 0.729055i \(0.260040\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −251.422 −2.43651
\(23\) −81.9767 −0.743188 −0.371594 0.928395i \(-0.621189\pi\)
−0.371594 + 0.928395i \(0.621189\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 181.084 1.36590
\(27\) 0 0
\(28\) −112.281 −0.757828
\(29\) −246.827 −1.58051 −0.790253 0.612781i \(-0.790051\pi\)
−0.790253 + 0.612781i \(0.790051\pi\)
\(30\) 0 0
\(31\) 222.679 1.29014 0.645070 0.764124i \(-0.276828\pi\)
0.645070 + 0.764124i \(0.276828\pi\)
\(32\) −180.215 −0.995557
\(33\) 0 0
\(34\) 117.399 0.592169
\(35\) 0 0
\(36\) 0 0
\(37\) −22.3910 −0.0994880 −0.0497440 0.998762i \(-0.515841\pi\)
−0.0497440 + 0.998762i \(0.515841\pi\)
\(38\) 399.463 1.70530
\(39\) 0 0
\(40\) 0 0
\(41\) −434.225 −1.65401 −0.827007 0.562191i \(-0.809959\pi\)
−0.827007 + 0.562191i \(0.809959\pi\)
\(42\) 0 0
\(43\) 236.850 0.839982 0.419991 0.907528i \(-0.362033\pi\)
0.419991 + 0.907528i \(0.362033\pi\)
\(44\) −315.029 −1.07937
\(45\) 0 0
\(46\) −288.843 −0.925817
\(47\) 107.984 0.335129 0.167564 0.985861i \(-0.446410\pi\)
0.167564 + 0.985861i \(0.446410\pi\)
\(48\) 0 0
\(49\) 303.810 0.885744
\(50\) 0 0
\(51\) 0 0
\(52\) 226.896 0.605094
\(53\) 123.961 0.321272 0.160636 0.987014i \(-0.448646\pi\)
0.160636 + 0.987014i \(0.448646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 321.265 0.766621
\(57\) 0 0
\(58\) −869.690 −1.96889
\(59\) 171.091 0.377528 0.188764 0.982023i \(-0.439552\pi\)
0.188764 + 0.982023i \(0.439552\pi\)
\(60\) 0 0
\(61\) −79.4391 −0.166740 −0.0833700 0.996519i \(-0.526568\pi\)
−0.0833700 + 0.996519i \(0.526568\pi\)
\(62\) 784.604 1.60717
\(63\) 0 0
\(64\) 3.63945 0.00710830
\(65\) 0 0
\(66\) 0 0
\(67\) 611.506 1.11503 0.557517 0.830165i \(-0.311754\pi\)
0.557517 + 0.830165i \(0.311754\pi\)
\(68\) 147.100 0.262331
\(69\) 0 0
\(70\) 0 0
\(71\) 511.102 0.854318 0.427159 0.904176i \(-0.359514\pi\)
0.427159 + 0.904176i \(0.359514\pi\)
\(72\) 0 0
\(73\) 410.012 0.657373 0.328686 0.944439i \(-0.393394\pi\)
0.328686 + 0.944439i \(0.393394\pi\)
\(74\) −78.8941 −0.123936
\(75\) 0 0
\(76\) 500.524 0.755447
\(77\) 1814.76 2.68586
\(78\) 0 0
\(79\) −793.597 −1.13021 −0.565105 0.825019i \(-0.691164\pi\)
−0.565105 + 0.825019i \(0.691164\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1529.98 −2.06047
\(83\) −270.081 −0.357171 −0.178586 0.983924i \(-0.557152\pi\)
−0.178586 + 0.983924i \(0.557152\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 834.534 1.04640
\(87\) 0 0
\(88\) 901.376 1.09190
\(89\) 177.400 0.211284 0.105642 0.994404i \(-0.466310\pi\)
0.105642 + 0.994404i \(0.466310\pi\)
\(90\) 0 0
\(91\) −1307.06 −1.50569
\(92\) −361.918 −0.410136
\(93\) 0 0
\(94\) 380.478 0.417482
\(95\) 0 0
\(96\) 0 0
\(97\) 881.860 0.923086 0.461543 0.887118i \(-0.347296\pi\)
0.461543 + 0.887118i \(0.347296\pi\)
\(98\) 1070.47 1.10340
\(99\) 0 0
\(100\) 0 0
\(101\) 1026.03 1.01083 0.505417 0.862875i \(-0.331339\pi\)
0.505417 + 0.862875i \(0.331339\pi\)
\(102\) 0 0
\(103\) 1790.28 1.71263 0.856317 0.516451i \(-0.172747\pi\)
0.856317 + 0.516451i \(0.172747\pi\)
\(104\) −649.206 −0.612115
\(105\) 0 0
\(106\) 436.775 0.400220
\(107\) 2007.30 1.81358 0.906788 0.421587i \(-0.138527\pi\)
0.906788 + 0.421587i \(0.138527\pi\)
\(108\) 0 0
\(109\) −211.654 −0.185989 −0.0929945 0.995667i \(-0.529644\pi\)
−0.0929945 + 0.995667i \(0.529644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2030.22 1.71284
\(113\) −1.94491 −0.00161913 −0.000809567 1.00000i \(-0.500258\pi\)
−0.000809567 1.00000i \(0.500258\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1089.71 −0.872219
\(117\) 0 0
\(118\) 602.835 0.470300
\(119\) −847.386 −0.652771
\(120\) 0 0
\(121\) 3760.70 2.82547
\(122\) −279.902 −0.207714
\(123\) 0 0
\(124\) 983.102 0.711977
\(125\) 0 0
\(126\) 0 0
\(127\) 1178.02 0.823089 0.411544 0.911390i \(-0.364989\pi\)
0.411544 + 0.911390i \(0.364989\pi\)
\(128\) 1454.54 1.00441
\(129\) 0 0
\(130\) 0 0
\(131\) 521.218 0.347626 0.173813 0.984779i \(-0.444391\pi\)
0.173813 + 0.984779i \(0.444391\pi\)
\(132\) 0 0
\(133\) −2883.32 −1.87982
\(134\) 2154.63 1.38904
\(135\) 0 0
\(136\) −420.889 −0.265374
\(137\) 449.532 0.280336 0.140168 0.990128i \(-0.455236\pi\)
0.140168 + 0.990128i \(0.455236\pi\)
\(138\) 0 0
\(139\) −2345.45 −1.43121 −0.715605 0.698505i \(-0.753849\pi\)
−0.715605 + 0.698505i \(0.753849\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1800.85 1.06426
\(143\) −3667.24 −2.14455
\(144\) 0 0
\(145\) 0 0
\(146\) 1444.67 0.818914
\(147\) 0 0
\(148\) −98.8537 −0.0549035
\(149\) −1269.38 −0.697932 −0.348966 0.937135i \(-0.613467\pi\)
−0.348966 + 0.937135i \(0.613467\pi\)
\(150\) 0 0
\(151\) 2158.32 1.16319 0.581594 0.813480i \(-0.302429\pi\)
0.581594 + 0.813480i \(0.302429\pi\)
\(152\) −1432.12 −0.764213
\(153\) 0 0
\(154\) 6394.27 3.34588
\(155\) 0 0
\(156\) 0 0
\(157\) 2343.31 1.19119 0.595593 0.803286i \(-0.296917\pi\)
0.595593 + 0.803286i \(0.296917\pi\)
\(158\) −2796.22 −1.40794
\(159\) 0 0
\(160\) 0 0
\(161\) 2084.87 1.02056
\(162\) 0 0
\(163\) −3399.94 −1.63376 −0.816882 0.576804i \(-0.804300\pi\)
−0.816882 + 0.576804i \(0.804300\pi\)
\(164\) −1917.06 −0.912786
\(165\) 0 0
\(166\) −951.623 −0.444942
\(167\) 1809.22 0.838332 0.419166 0.907910i \(-0.362323\pi\)
0.419166 + 0.907910i \(0.362323\pi\)
\(168\) 0 0
\(169\) 444.292 0.202227
\(170\) 0 0
\(171\) 0 0
\(172\) 1045.66 0.463553
\(173\) 1327.41 0.583358 0.291679 0.956516i \(-0.405786\pi\)
0.291679 + 0.956516i \(0.405786\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5696.21 2.43959
\(177\) 0 0
\(178\) 625.063 0.263205
\(179\) −448.734 −0.187374 −0.0936871 0.995602i \(-0.529865\pi\)
−0.0936871 + 0.995602i \(0.529865\pi\)
\(180\) 0 0
\(181\) −3450.55 −1.41700 −0.708502 0.705709i \(-0.750629\pi\)
−0.708502 + 0.705709i \(0.750629\pi\)
\(182\) −4605.41 −1.87569
\(183\) 0 0
\(184\) 1035.54 0.414895
\(185\) 0 0
\(186\) 0 0
\(187\) −2377.52 −0.929741
\(188\) 476.736 0.184944
\(189\) 0 0
\(190\) 0 0
\(191\) −1592.09 −0.603141 −0.301570 0.953444i \(-0.597511\pi\)
−0.301570 + 0.953444i \(0.597511\pi\)
\(192\) 0 0
\(193\) −443.299 −0.165334 −0.0826668 0.996577i \(-0.526344\pi\)
−0.0826668 + 0.996577i \(0.526344\pi\)
\(194\) 3107.21 1.14992
\(195\) 0 0
\(196\) 1341.29 0.488807
\(197\) 208.992 0.0755839 0.0377919 0.999286i \(-0.487968\pi\)
0.0377919 + 0.999286i \(0.487968\pi\)
\(198\) 0 0
\(199\) −479.916 −0.170957 −0.0854783 0.996340i \(-0.527242\pi\)
−0.0854783 + 0.996340i \(0.527242\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 3615.21 1.25923
\(203\) 6277.43 2.17039
\(204\) 0 0
\(205\) 0 0
\(206\) 6308.00 2.13349
\(207\) 0 0
\(208\) −4102.63 −1.36763
\(209\) −8089.78 −2.67742
\(210\) 0 0
\(211\) −2632.14 −0.858786 −0.429393 0.903118i \(-0.641273\pi\)
−0.429393 + 0.903118i \(0.641273\pi\)
\(212\) 547.275 0.177297
\(213\) 0 0
\(214\) 7072.66 2.25924
\(215\) 0 0
\(216\) 0 0
\(217\) −5663.28 −1.77165
\(218\) −745.759 −0.231693
\(219\) 0 0
\(220\) 0 0
\(221\) 1712.38 0.521210
\(222\) 0 0
\(223\) −3353.22 −1.00694 −0.503471 0.864012i \(-0.667944\pi\)
−0.503471 + 0.864012i \(0.667944\pi\)
\(224\) 4583.31 1.36712
\(225\) 0 0
\(226\) −6.85286 −0.00201701
\(227\) −690.764 −0.201972 −0.100986 0.994888i \(-0.532200\pi\)
−0.100986 + 0.994888i \(0.532200\pi\)
\(228\) 0 0
\(229\) −668.131 −0.192801 −0.0964003 0.995343i \(-0.530733\pi\)
−0.0964003 + 0.995343i \(0.530733\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3117.94 0.882339
\(233\) 940.537 0.264449 0.132225 0.991220i \(-0.457788\pi\)
0.132225 + 0.991220i \(0.457788\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 755.347 0.208343
\(237\) 0 0
\(238\) −2985.75 −0.813181
\(239\) 1107.40 0.299715 0.149858 0.988708i \(-0.452118\pi\)
0.149858 + 0.988708i \(0.452118\pi\)
\(240\) 0 0
\(241\) 1228.86 0.328456 0.164228 0.986422i \(-0.447487\pi\)
0.164228 + 0.986422i \(0.447487\pi\)
\(242\) 13250.7 3.51979
\(243\) 0 0
\(244\) −350.715 −0.0920172
\(245\) 0 0
\(246\) 0 0
\(247\) 5826.58 1.50096
\(248\) −2812.90 −0.720238
\(249\) 0 0
\(250\) 0 0
\(251\) 738.480 0.185707 0.0928534 0.995680i \(-0.470401\pi\)
0.0928534 + 0.995680i \(0.470401\pi\)
\(252\) 0 0
\(253\) 5849.54 1.45359
\(254\) 4150.72 1.02535
\(255\) 0 0
\(256\) 5095.94 1.24412
\(257\) −6312.57 −1.53217 −0.766084 0.642740i \(-0.777797\pi\)
−0.766084 + 0.642740i \(0.777797\pi\)
\(258\) 0 0
\(259\) 569.458 0.136619
\(260\) 0 0
\(261\) 0 0
\(262\) 1836.50 0.433051
\(263\) −5993.62 −1.40526 −0.702628 0.711558i \(-0.747990\pi\)
−0.702628 + 0.711558i \(0.747990\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −10159.3 −2.34176
\(267\) 0 0
\(268\) 2699.73 0.615344
\(269\) −1749.70 −0.396584 −0.198292 0.980143i \(-0.563539\pi\)
−0.198292 + 0.980143i \(0.563539\pi\)
\(270\) 0 0
\(271\) −1931.68 −0.432994 −0.216497 0.976283i \(-0.569463\pi\)
−0.216497 + 0.976283i \(0.569463\pi\)
\(272\) −2659.79 −0.592917
\(273\) 0 0
\(274\) 1583.91 0.349225
\(275\) 0 0
\(276\) 0 0
\(277\) 3791.55 0.822427 0.411213 0.911539i \(-0.365105\pi\)
0.411213 + 0.911539i \(0.365105\pi\)
\(278\) −8264.13 −1.78291
\(279\) 0 0
\(280\) 0 0
\(281\) 4489.36 0.953071 0.476536 0.879155i \(-0.341892\pi\)
0.476536 + 0.879155i \(0.341892\pi\)
\(282\) 0 0
\(283\) −3133.86 −0.658264 −0.329132 0.944284i \(-0.606756\pi\)
−0.329132 + 0.944284i \(0.606756\pi\)
\(284\) 2256.45 0.471465
\(285\) 0 0
\(286\) −12921.4 −2.67154
\(287\) 11043.4 2.27133
\(288\) 0 0
\(289\) −3802.84 −0.774036
\(290\) 0 0
\(291\) 0 0
\(292\) 1810.15 0.362778
\(293\) −2797.50 −0.557787 −0.278894 0.960322i \(-0.589968\pi\)
−0.278894 + 0.960322i \(0.589968\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 282.845 0.0555406
\(297\) 0 0
\(298\) −4472.64 −0.869439
\(299\) −4213.07 −0.814877
\(300\) 0 0
\(301\) −6023.67 −1.15348
\(302\) 7604.77 1.44903
\(303\) 0 0
\(304\) −9050.23 −1.70746
\(305\) 0 0
\(306\) 0 0
\(307\) 6839.91 1.27158 0.635789 0.771863i \(-0.280675\pi\)
0.635789 + 0.771863i \(0.280675\pi\)
\(308\) 8011.97 1.48222
\(309\) 0 0
\(310\) 0 0
\(311\) 2419.41 0.441132 0.220566 0.975372i \(-0.429210\pi\)
0.220566 + 0.975372i \(0.429210\pi\)
\(312\) 0 0
\(313\) −3903.18 −0.704858 −0.352429 0.935838i \(-0.614644\pi\)
−0.352429 + 0.935838i \(0.614644\pi\)
\(314\) 8256.59 1.48391
\(315\) 0 0
\(316\) −3503.64 −0.623718
\(317\) −9894.73 −1.75313 −0.876567 0.481280i \(-0.840172\pi\)
−0.876567 + 0.481280i \(0.840172\pi\)
\(318\) 0 0
\(319\) 17612.6 3.09128
\(320\) 0 0
\(321\) 0 0
\(322\) 7345.99 1.27135
\(323\) 3777.44 0.650720
\(324\) 0 0
\(325\) 0 0
\(326\) −11979.6 −2.03524
\(327\) 0 0
\(328\) 5485.16 0.923377
\(329\) −2746.29 −0.460207
\(330\) 0 0
\(331\) 4163.04 0.691303 0.345651 0.938363i \(-0.387658\pi\)
0.345651 + 0.938363i \(0.387658\pi\)
\(332\) −1192.38 −0.197109
\(333\) 0 0
\(334\) 6374.73 1.04434
\(335\) 0 0
\(336\) 0 0
\(337\) −9704.93 −1.56873 −0.784364 0.620301i \(-0.787011\pi\)
−0.784364 + 0.620301i \(0.787011\pi\)
\(338\) 1565.45 0.251921
\(339\) 0 0
\(340\) 0 0
\(341\) −15889.5 −2.52336
\(342\) 0 0
\(343\) 996.691 0.156899
\(344\) −2991.90 −0.468931
\(345\) 0 0
\(346\) 4677.09 0.726711
\(347\) 5432.38 0.840420 0.420210 0.907427i \(-0.361956\pi\)
0.420210 + 0.907427i \(0.361956\pi\)
\(348\) 0 0
\(349\) 10571.9 1.62150 0.810749 0.585394i \(-0.199060\pi\)
0.810749 + 0.585394i \(0.199060\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 12859.5 1.94719
\(353\) 9642.98 1.45395 0.726975 0.686664i \(-0.240926\pi\)
0.726975 + 0.686664i \(0.240926\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 783.198 0.116600
\(357\) 0 0
\(358\) −1581.10 −0.233419
\(359\) −3002.18 −0.441362 −0.220681 0.975346i \(-0.570828\pi\)
−0.220681 + 0.975346i \(0.570828\pi\)
\(360\) 0 0
\(361\) 5994.17 0.873913
\(362\) −12157.9 −1.76521
\(363\) 0 0
\(364\) −5770.54 −0.830929
\(365\) 0 0
\(366\) 0 0
\(367\) 1149.28 0.163465 0.0817326 0.996654i \(-0.473955\pi\)
0.0817326 + 0.996654i \(0.473955\pi\)
\(368\) 6544.03 0.926986
\(369\) 0 0
\(370\) 0 0
\(371\) −3152.64 −0.441178
\(372\) 0 0
\(373\) 513.214 0.0712418 0.0356209 0.999365i \(-0.488659\pi\)
0.0356209 + 0.999365i \(0.488659\pi\)
\(374\) −8377.14 −1.15821
\(375\) 0 0
\(376\) −1364.06 −0.187090
\(377\) −12685.3 −1.73296
\(378\) 0 0
\(379\) 12560.0 1.70228 0.851138 0.524942i \(-0.175913\pi\)
0.851138 + 0.524942i \(0.175913\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −5609.70 −0.751355
\(383\) 273.990 0.0365542 0.0182771 0.999833i \(-0.494182\pi\)
0.0182771 + 0.999833i \(0.494182\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1561.95 −0.205962
\(387\) 0 0
\(388\) 3893.31 0.509415
\(389\) −6846.43 −0.892359 −0.446179 0.894944i \(-0.647216\pi\)
−0.446179 + 0.894944i \(0.647216\pi\)
\(390\) 0 0
\(391\) −2731.39 −0.353279
\(392\) −3837.75 −0.494479
\(393\) 0 0
\(394\) 736.377 0.0941577
\(395\) 0 0
\(396\) 0 0
\(397\) 12117.5 1.53189 0.765943 0.642909i \(-0.222273\pi\)
0.765943 + 0.642909i \(0.222273\pi\)
\(398\) −1690.97 −0.212967
\(399\) 0 0
\(400\) 0 0
\(401\) −3016.80 −0.375691 −0.187845 0.982199i \(-0.560150\pi\)
−0.187845 + 0.982199i \(0.560150\pi\)
\(402\) 0 0
\(403\) 11444.3 1.41459
\(404\) 4529.82 0.557839
\(405\) 0 0
\(406\) 22118.4 2.70373
\(407\) 1597.74 0.194587
\(408\) 0 0
\(409\) 6535.13 0.790076 0.395038 0.918665i \(-0.370731\pi\)
0.395038 + 0.918665i \(0.370731\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 7903.87 0.945135
\(413\) −4351.26 −0.518430
\(414\) 0 0
\(415\) 0 0
\(416\) −9261.88 −1.09159
\(417\) 0 0
\(418\) −28504.1 −3.33537
\(419\) 7148.45 0.833471 0.416736 0.909028i \(-0.363174\pi\)
0.416736 + 0.909028i \(0.363174\pi\)
\(420\) 0 0
\(421\) 14801.5 1.71350 0.856749 0.515733i \(-0.172480\pi\)
0.856749 + 0.515733i \(0.172480\pi\)
\(422\) −9274.28 −1.06982
\(423\) 0 0
\(424\) −1565.89 −0.179354
\(425\) 0 0
\(426\) 0 0
\(427\) 2020.33 0.228971
\(428\) 8861.98 1.00084
\(429\) 0 0
\(430\) 0 0
\(431\) 2284.19 0.255280 0.127640 0.991821i \(-0.459260\pi\)
0.127640 + 0.991821i \(0.459260\pi\)
\(432\) 0 0
\(433\) −5529.26 −0.613670 −0.306835 0.951763i \(-0.599270\pi\)
−0.306835 + 0.951763i \(0.599270\pi\)
\(434\) −19954.4 −2.20701
\(435\) 0 0
\(436\) −934.429 −0.102640
\(437\) −9293.85 −1.01736
\(438\) 0 0
\(439\) −11861.6 −1.28958 −0.644788 0.764361i \(-0.723054\pi\)
−0.644788 + 0.764361i \(0.723054\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6033.55 0.649291
\(443\) 15293.6 1.64023 0.820115 0.572199i \(-0.193910\pi\)
0.820115 + 0.572199i \(0.193910\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −11815.0 −1.25439
\(447\) 0 0
\(448\) −92.5602 −0.00976129
\(449\) 12998.8 1.36626 0.683129 0.730297i \(-0.260619\pi\)
0.683129 + 0.730297i \(0.260619\pi\)
\(450\) 0 0
\(451\) 30984.6 3.23506
\(452\) −8.58657 −0.000893536 0
\(453\) 0 0
\(454\) −2433.89 −0.251604
\(455\) 0 0
\(456\) 0 0
\(457\) −14311.9 −1.46495 −0.732477 0.680792i \(-0.761636\pi\)
−0.732477 + 0.680792i \(0.761636\pi\)
\(458\) −2354.14 −0.240179
\(459\) 0 0
\(460\) 0 0
\(461\) 4290.34 0.433451 0.216725 0.976233i \(-0.430462\pi\)
0.216725 + 0.976233i \(0.430462\pi\)
\(462\) 0 0
\(463\) 14510.3 1.45648 0.728240 0.685322i \(-0.240339\pi\)
0.728240 + 0.685322i \(0.240339\pi\)
\(464\) 19703.7 1.97138
\(465\) 0 0
\(466\) 3313.96 0.329434
\(467\) 7393.05 0.732569 0.366285 0.930503i \(-0.380630\pi\)
0.366285 + 0.930503i \(0.380630\pi\)
\(468\) 0 0
\(469\) −15552.1 −1.53119
\(470\) 0 0
\(471\) 0 0
\(472\) −2161.23 −0.210760
\(473\) −16900.7 −1.64290
\(474\) 0 0
\(475\) 0 0
\(476\) −3741.11 −0.360239
\(477\) 0 0
\(478\) 3901.91 0.373366
\(479\) 13509.9 1.28869 0.644344 0.764736i \(-0.277131\pi\)
0.644344 + 0.764736i \(0.277131\pi\)
\(480\) 0 0
\(481\) −1150.75 −0.109085
\(482\) 4329.87 0.409170
\(483\) 0 0
\(484\) 16603.1 1.55926
\(485\) 0 0
\(486\) 0 0
\(487\) 11140.0 1.03655 0.518277 0.855213i \(-0.326573\pi\)
0.518277 + 0.855213i \(0.326573\pi\)
\(488\) 1003.48 0.0930849
\(489\) 0 0
\(490\) 0 0
\(491\) 2012.74 0.184998 0.0924988 0.995713i \(-0.470515\pi\)
0.0924988 + 0.995713i \(0.470515\pi\)
\(492\) 0 0
\(493\) −8224.06 −0.751304
\(494\) 20529.8 1.86980
\(495\) 0 0
\(496\) −17776.0 −1.60920
\(497\) −12998.6 −1.17317
\(498\) 0 0
\(499\) 7352.98 0.659648 0.329824 0.944042i \(-0.393011\pi\)
0.329824 + 0.944042i \(0.393011\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2602.02 0.231342
\(503\) −16898.3 −1.49793 −0.748965 0.662609i \(-0.769449\pi\)
−0.748965 + 0.662609i \(0.769449\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 20610.7 1.81079
\(507\) 0 0
\(508\) 5200.82 0.454230
\(509\) −19358.6 −1.68577 −0.842884 0.538095i \(-0.819144\pi\)
−0.842884 + 0.538095i \(0.819144\pi\)
\(510\) 0 0
\(511\) −10427.6 −0.902720
\(512\) 6319.06 0.545440
\(513\) 0 0
\(514\) −22242.2 −1.90868
\(515\) 0 0
\(516\) 0 0
\(517\) −7705.30 −0.655471
\(518\) 2006.47 0.170192
\(519\) 0 0
\(520\) 0 0
\(521\) −146.772 −0.0123420 −0.00617100 0.999981i \(-0.501964\pi\)
−0.00617100 + 0.999981i \(0.501964\pi\)
\(522\) 0 0
\(523\) 2872.03 0.240125 0.120062 0.992766i \(-0.461691\pi\)
0.120062 + 0.992766i \(0.461691\pi\)
\(524\) 2301.12 0.191841
\(525\) 0 0
\(526\) −21118.4 −1.75058
\(527\) 7419.46 0.613277
\(528\) 0 0
\(529\) −5446.82 −0.447671
\(530\) 0 0
\(531\) 0 0
\(532\) −12729.5 −1.03740
\(533\) −22316.4 −1.81356
\(534\) 0 0
\(535\) 0 0
\(536\) −7724.58 −0.622484
\(537\) 0 0
\(538\) −6165.03 −0.494039
\(539\) −21678.7 −1.73241
\(540\) 0 0
\(541\) −3810.28 −0.302804 −0.151402 0.988472i \(-0.548379\pi\)
−0.151402 + 0.988472i \(0.548379\pi\)
\(542\) −6806.24 −0.539397
\(543\) 0 0
\(544\) −6004.60 −0.473245
\(545\) 0 0
\(546\) 0 0
\(547\) −5945.59 −0.464744 −0.232372 0.972627i \(-0.574649\pi\)
−0.232372 + 0.972627i \(0.574649\pi\)
\(548\) 1984.63 0.154707
\(549\) 0 0
\(550\) 0 0
\(551\) −27983.3 −2.16357
\(552\) 0 0
\(553\) 20183.1 1.55203
\(554\) 13359.4 1.02453
\(555\) 0 0
\(556\) −10354.9 −0.789828
\(557\) 10872.2 0.827059 0.413530 0.910491i \(-0.364296\pi\)
0.413530 + 0.910491i \(0.364296\pi\)
\(558\) 0 0
\(559\) 12172.5 0.921007
\(560\) 0 0
\(561\) 0 0
\(562\) 15818.2 1.18728
\(563\) −13575.1 −1.01620 −0.508100 0.861298i \(-0.669652\pi\)
−0.508100 + 0.861298i \(0.669652\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −11042.1 −0.820024
\(567\) 0 0
\(568\) −6456.27 −0.476935
\(569\) 20642.6 1.52088 0.760442 0.649406i \(-0.224982\pi\)
0.760442 + 0.649406i \(0.224982\pi\)
\(570\) 0 0
\(571\) −2730.77 −0.200139 −0.100069 0.994980i \(-0.531906\pi\)
−0.100069 + 0.994980i \(0.531906\pi\)
\(572\) −16190.5 −1.18349
\(573\) 0 0
\(574\) 38911.2 2.82948
\(575\) 0 0
\(576\) 0 0
\(577\) −21953.4 −1.58394 −0.791970 0.610560i \(-0.790945\pi\)
−0.791970 + 0.610560i \(0.790945\pi\)
\(578\) −13399.2 −0.964245
\(579\) 0 0
\(580\) 0 0
\(581\) 6868.82 0.490476
\(582\) 0 0
\(583\) −8845.40 −0.628369
\(584\) −5179.29 −0.366988
\(585\) 0 0
\(586\) −9856.92 −0.694856
\(587\) −11055.3 −0.777348 −0.388674 0.921375i \(-0.627067\pi\)
−0.388674 + 0.921375i \(0.627067\pi\)
\(588\) 0 0
\(589\) 25245.5 1.76608
\(590\) 0 0
\(591\) 0 0
\(592\) 1787.43 0.124092
\(593\) −16538.0 −1.14525 −0.572625 0.819817i \(-0.694075\pi\)
−0.572625 + 0.819817i \(0.694075\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5604.18 −0.385161
\(597\) 0 0
\(598\) −14844.7 −1.01512
\(599\) −6403.59 −0.436800 −0.218400 0.975859i \(-0.570084\pi\)
−0.218400 + 0.975859i \(0.570084\pi\)
\(600\) 0 0
\(601\) −14210.4 −0.964480 −0.482240 0.876039i \(-0.660177\pi\)
−0.482240 + 0.876039i \(0.660177\pi\)
\(602\) −21224.3 −1.43694
\(603\) 0 0
\(604\) 9528.72 0.641917
\(605\) 0 0
\(606\) 0 0
\(607\) 8915.72 0.596174 0.298087 0.954539i \(-0.403651\pi\)
0.298087 + 0.954539i \(0.403651\pi\)
\(608\) −20431.3 −1.36283
\(609\) 0 0
\(610\) 0 0
\(611\) 5549.66 0.367455
\(612\) 0 0
\(613\) 15372.5 1.01287 0.506436 0.862277i \(-0.330963\pi\)
0.506436 + 0.862277i \(0.330963\pi\)
\(614\) 24100.3 1.58405
\(615\) 0 0
\(616\) −22924.2 −1.49942
\(617\) −17327.5 −1.13060 −0.565298 0.824887i \(-0.691239\pi\)
−0.565298 + 0.824887i \(0.691239\pi\)
\(618\) 0 0
\(619\) −28787.3 −1.86924 −0.934621 0.355645i \(-0.884261\pi\)
−0.934621 + 0.355645i \(0.884261\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 8524.73 0.549535
\(623\) −4511.71 −0.290141
\(624\) 0 0
\(625\) 0 0
\(626\) −13752.8 −0.878068
\(627\) 0 0
\(628\) 10345.4 0.657369
\(629\) −746.047 −0.0472923
\(630\) 0 0
\(631\) 18091.0 1.14135 0.570675 0.821176i \(-0.306682\pi\)
0.570675 + 0.821176i \(0.306682\pi\)
\(632\) 10024.8 0.630955
\(633\) 0 0
\(634\) −34863.8 −2.18394
\(635\) 0 0
\(636\) 0 0
\(637\) 15613.9 0.971184
\(638\) 62057.7 3.85092
\(639\) 0 0
\(640\) 0 0
\(641\) 19577.5 1.20634 0.603171 0.797612i \(-0.293904\pi\)
0.603171 + 0.797612i \(0.293904\pi\)
\(642\) 0 0
\(643\) 29971.7 1.83821 0.919106 0.394011i \(-0.128913\pi\)
0.919106 + 0.394011i \(0.128913\pi\)
\(644\) 9204.46 0.563209
\(645\) 0 0
\(646\) 13309.7 0.810626
\(647\) 3635.64 0.220915 0.110457 0.993881i \(-0.464768\pi\)
0.110457 + 0.993881i \(0.464768\pi\)
\(648\) 0 0
\(649\) −12208.4 −0.738399
\(650\) 0 0
\(651\) 0 0
\(652\) −15010.3 −0.901610
\(653\) 19407.3 1.16304 0.581520 0.813532i \(-0.302458\pi\)
0.581520 + 0.813532i \(0.302458\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 34663.3 2.06307
\(657\) 0 0
\(658\) −9676.49 −0.573296
\(659\) −5627.86 −0.332671 −0.166335 0.986069i \(-0.553193\pi\)
−0.166335 + 0.986069i \(0.553193\pi\)
\(660\) 0 0
\(661\) −16663.0 −0.980507 −0.490253 0.871580i \(-0.663096\pi\)
−0.490253 + 0.871580i \(0.663096\pi\)
\(662\) 14668.4 0.861182
\(663\) 0 0
\(664\) 3411.68 0.199396
\(665\) 0 0
\(666\) 0 0
\(667\) 20234.1 1.17461
\(668\) 7987.48 0.462642
\(669\) 0 0
\(670\) 0 0
\(671\) 5668.47 0.326124
\(672\) 0 0
\(673\) 12638.8 0.723906 0.361953 0.932196i \(-0.382110\pi\)
0.361953 + 0.932196i \(0.382110\pi\)
\(674\) −34195.1 −1.95422
\(675\) 0 0
\(676\) 1961.50 0.111601
\(677\) 25316.0 1.43718 0.718591 0.695433i \(-0.244787\pi\)
0.718591 + 0.695433i \(0.244787\pi\)
\(678\) 0 0
\(679\) −22427.9 −1.26760
\(680\) 0 0
\(681\) 0 0
\(682\) −55986.3 −3.14344
\(683\) −24818.4 −1.39041 −0.695204 0.718812i \(-0.744686\pi\)
−0.695204 + 0.718812i \(0.744686\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 3511.82 0.195455
\(687\) 0 0
\(688\) −18907.2 −1.04772
\(689\) 6370.81 0.352262
\(690\) 0 0
\(691\) 19958.0 1.09875 0.549375 0.835576i \(-0.314866\pi\)
0.549375 + 0.835576i \(0.314866\pi\)
\(692\) 5860.35 0.321932
\(693\) 0 0
\(694\) 19140.9 1.04694
\(695\) 0 0
\(696\) 0 0
\(697\) −14468.0 −0.786247
\(698\) 37250.0 2.01996
\(699\) 0 0
\(700\) 0 0
\(701\) −5624.21 −0.303029 −0.151515 0.988455i \(-0.548415\pi\)
−0.151515 + 0.988455i \(0.548415\pi\)
\(702\) 0 0
\(703\) −2538.51 −0.136190
\(704\) −259.697 −0.0139030
\(705\) 0 0
\(706\) 33976.8 1.81124
\(707\) −26094.6 −1.38810
\(708\) 0 0
\(709\) −7715.43 −0.408687 −0.204344 0.978899i \(-0.565506\pi\)
−0.204344 + 0.978899i \(0.565506\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −2240.92 −0.117952
\(713\) −18254.5 −0.958817
\(714\) 0 0
\(715\) 0 0
\(716\) −1981.11 −0.103404
\(717\) 0 0
\(718\) −10578.1 −0.549821
\(719\) 21647.8 1.12285 0.561424 0.827528i \(-0.310254\pi\)
0.561424 + 0.827528i \(0.310254\pi\)
\(720\) 0 0
\(721\) −45531.2 −2.35183
\(722\) 21120.3 1.08867
\(723\) 0 0
\(724\) −15233.8 −0.781989
\(725\) 0 0
\(726\) 0 0
\(727\) 3071.30 0.156682 0.0783412 0.996927i \(-0.475038\pi\)
0.0783412 + 0.996927i \(0.475038\pi\)
\(728\) 16510.9 0.840570
\(729\) 0 0
\(730\) 0 0
\(731\) 7891.61 0.399291
\(732\) 0 0
\(733\) −14652.0 −0.738314 −0.369157 0.929367i \(-0.620354\pi\)
−0.369157 + 0.929367i \(0.620354\pi\)
\(734\) 4049.45 0.203635
\(735\) 0 0
\(736\) 14773.4 0.739886
\(737\) −43634.7 −2.18087
\(738\) 0 0
\(739\) 17610.9 0.876629 0.438314 0.898822i \(-0.355576\pi\)
0.438314 + 0.898822i \(0.355576\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −11108.3 −0.549592
\(743\) 16788.2 0.828935 0.414468 0.910064i \(-0.363968\pi\)
0.414468 + 0.910064i \(0.363968\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1808.30 0.0887485
\(747\) 0 0
\(748\) −10496.5 −0.513087
\(749\) −51050.5 −2.49045
\(750\) 0 0
\(751\) 23514.2 1.14254 0.571269 0.820763i \(-0.306451\pi\)
0.571269 + 0.820763i \(0.306451\pi\)
\(752\) −8620.11 −0.418009
\(753\) 0 0
\(754\) −44696.4 −2.15882
\(755\) 0 0
\(756\) 0 0
\(757\) −24817.4 −1.19155 −0.595775 0.803151i \(-0.703155\pi\)
−0.595775 + 0.803151i \(0.703155\pi\)
\(758\) 44254.8 2.12059
\(759\) 0 0
\(760\) 0 0
\(761\) 32517.7 1.54897 0.774485 0.632592i \(-0.218009\pi\)
0.774485 + 0.632592i \(0.218009\pi\)
\(762\) 0 0
\(763\) 5382.89 0.255404
\(764\) −7028.91 −0.332850
\(765\) 0 0
\(766\) 965.398 0.0455369
\(767\) 8792.96 0.413944
\(768\) 0 0
\(769\) 18057.8 0.846791 0.423395 0.905945i \(-0.360838\pi\)
0.423395 + 0.905945i \(0.360838\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1957.11 −0.0912411
\(773\) −38147.2 −1.77498 −0.887491 0.460825i \(-0.847553\pi\)
−0.887491 + 0.460825i \(0.847553\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −11139.7 −0.515326
\(777\) 0 0
\(778\) −24123.2 −1.11164
\(779\) −49228.9 −2.26420
\(780\) 0 0
\(781\) −36470.2 −1.67094
\(782\) −9623.98 −0.440093
\(783\) 0 0
\(784\) −24252.5 −1.10480
\(785\) 0 0
\(786\) 0 0
\(787\) 4801.76 0.217489 0.108745 0.994070i \(-0.465317\pi\)
0.108745 + 0.994070i \(0.465317\pi\)
\(788\) 922.673 0.0417118
\(789\) 0 0
\(790\) 0 0
\(791\) 49.4640 0.00222343
\(792\) 0 0
\(793\) −4082.66 −0.182824
\(794\) 42695.6 1.90833
\(795\) 0 0
\(796\) −2118.77 −0.0943442
\(797\) 15471.0 0.687592 0.343796 0.939044i \(-0.388287\pi\)
0.343796 + 0.939044i \(0.388287\pi\)
\(798\) 0 0
\(799\) 3597.92 0.159306
\(800\) 0 0
\(801\) 0 0
\(802\) −10629.6 −0.468012
\(803\) −29256.8 −1.28574
\(804\) 0 0
\(805\) 0 0
\(806\) 40323.6 1.76220
\(807\) 0 0
\(808\) −12960.9 −0.564312
\(809\) −17007.6 −0.739129 −0.369564 0.929205i \(-0.620493\pi\)
−0.369564 + 0.929205i \(0.620493\pi\)
\(810\) 0 0
\(811\) −7552.52 −0.327010 −0.163505 0.986543i \(-0.552280\pi\)
−0.163505 + 0.986543i \(0.552280\pi\)
\(812\) 27714.1 1.19775
\(813\) 0 0
\(814\) 5629.58 0.242404
\(815\) 0 0
\(816\) 0 0
\(817\) 26852.1 1.14986
\(818\) 23026.4 0.984227
\(819\) 0 0
\(820\) 0 0
\(821\) 3745.20 0.159206 0.0796031 0.996827i \(-0.474635\pi\)
0.0796031 + 0.996827i \(0.474635\pi\)
\(822\) 0 0
\(823\) 30021.0 1.27153 0.635764 0.771884i \(-0.280685\pi\)
0.635764 + 0.771884i \(0.280685\pi\)
\(824\) −22614.9 −0.956101
\(825\) 0 0
\(826\) −15331.6 −0.645828
\(827\) 7918.68 0.332962 0.166481 0.986045i \(-0.446760\pi\)
0.166481 + 0.986045i \(0.446760\pi\)
\(828\) 0 0
\(829\) −6976.17 −0.292271 −0.146135 0.989265i \(-0.546683\pi\)
−0.146135 + 0.989265i \(0.546683\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 187.044 0.00779398
\(833\) 10122.7 0.421044
\(834\) 0 0
\(835\) 0 0
\(836\) −35715.4 −1.47756
\(837\) 0 0
\(838\) 25187.4 1.03829
\(839\) 8657.91 0.356263 0.178131 0.984007i \(-0.442995\pi\)
0.178131 + 0.984007i \(0.442995\pi\)
\(840\) 0 0
\(841\) 36534.7 1.49800
\(842\) 52152.9 2.13457
\(843\) 0 0
\(844\) −11620.6 −0.473930
\(845\) 0 0
\(846\) 0 0
\(847\) −95643.8 −3.88000
\(848\) −9895.57 −0.400726
\(849\) 0 0
\(850\) 0 0
\(851\) 1835.54 0.0739383
\(852\) 0 0
\(853\) −18984.8 −0.762047 −0.381024 0.924565i \(-0.624428\pi\)
−0.381024 + 0.924565i \(0.624428\pi\)
\(854\) 7118.60 0.285238
\(855\) 0 0
\(856\) −25356.3 −1.01245
\(857\) 36434.4 1.45225 0.726124 0.687564i \(-0.241320\pi\)
0.726124 + 0.687564i \(0.241320\pi\)
\(858\) 0 0
\(859\) −17485.0 −0.694507 −0.347253 0.937771i \(-0.612886\pi\)
−0.347253 + 0.937771i \(0.612886\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8048.29 0.318011
\(863\) 10423.6 0.411152 0.205576 0.978641i \(-0.434093\pi\)
0.205576 + 0.978641i \(0.434093\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −19482.2 −0.764472
\(867\) 0 0
\(868\) −25002.7 −0.977704
\(869\) 56628.0 2.21056
\(870\) 0 0
\(871\) 31427.4 1.22259
\(872\) 2673.63 0.103831
\(873\) 0 0
\(874\) −32746.7 −1.26736
\(875\) 0 0
\(876\) 0 0
\(877\) −7873.24 −0.303148 −0.151574 0.988446i \(-0.548434\pi\)
−0.151574 + 0.988446i \(0.548434\pi\)
\(878\) −41794.1 −1.60647
\(879\) 0 0
\(880\) 0 0
\(881\) −35135.0 −1.34362 −0.671809 0.740725i \(-0.734482\pi\)
−0.671809 + 0.740725i \(0.734482\pi\)
\(882\) 0 0
\(883\) −33069.4 −1.26033 −0.630167 0.776460i \(-0.717013\pi\)
−0.630167 + 0.776460i \(0.717013\pi\)
\(884\) 7559.98 0.287635
\(885\) 0 0
\(886\) 53886.7 2.04329
\(887\) 51479.1 1.94870 0.974351 0.225031i \(-0.0722485\pi\)
0.974351 + 0.225031i \(0.0722485\pi\)
\(888\) 0 0
\(889\) −29959.9 −1.13029
\(890\) 0 0
\(891\) 0 0
\(892\) −14804.1 −0.555692
\(893\) 12242.3 0.458761
\(894\) 0 0
\(895\) 0 0
\(896\) −36992.6 −1.37928
\(897\) 0 0
\(898\) 45800.9 1.70200
\(899\) −54963.3 −2.03907
\(900\) 0 0
\(901\) 4130.28 0.152719
\(902\) 109174. 4.03003
\(903\) 0 0
\(904\) 24.5683 0.000903904 0
\(905\) 0 0
\(906\) 0 0
\(907\) 10809.8 0.395737 0.197868 0.980229i \(-0.436598\pi\)
0.197868 + 0.980229i \(0.436598\pi\)
\(908\) −3049.64 −0.111460
\(909\) 0 0
\(910\) 0 0
\(911\) 31539.8 1.14705 0.573523 0.819190i \(-0.305576\pi\)
0.573523 + 0.819190i \(0.305576\pi\)
\(912\) 0 0
\(913\) 19271.9 0.698585
\(914\) −50427.8 −1.82495
\(915\) 0 0
\(916\) −2949.72 −0.106399
\(917\) −13255.9 −0.477368
\(918\) 0 0
\(919\) 3592.53 0.128952 0.0644759 0.997919i \(-0.479462\pi\)
0.0644759 + 0.997919i \(0.479462\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 15116.9 0.539966
\(923\) 26267.3 0.936727
\(924\) 0 0
\(925\) 0 0
\(926\) 51126.7 1.81439
\(927\) 0 0
\(928\) 44482.0 1.57348
\(929\) 31614.7 1.11652 0.558259 0.829667i \(-0.311470\pi\)
0.558259 + 0.829667i \(0.311470\pi\)
\(930\) 0 0
\(931\) 34443.5 1.21250
\(932\) 4152.36 0.145939
\(933\) 0 0
\(934\) 26049.2 0.912588
\(935\) 0 0
\(936\) 0 0
\(937\) 20891.5 0.728382 0.364191 0.931324i \(-0.381346\pi\)
0.364191 + 0.931324i \(0.381346\pi\)
\(938\) −54797.5 −1.90746
\(939\) 0 0
\(940\) 0 0
\(941\) 37769.2 1.30844 0.654219 0.756305i \(-0.272998\pi\)
0.654219 + 0.756305i \(0.272998\pi\)
\(942\) 0 0
\(943\) 35596.4 1.22924
\(944\) −13657.8 −0.470894
\(945\) 0 0
\(946\) −59549.1 −2.04663
\(947\) 10430.8 0.357927 0.178963 0.983856i \(-0.442726\pi\)
0.178963 + 0.983856i \(0.442726\pi\)
\(948\) 0 0
\(949\) 21071.9 0.720784
\(950\) 0 0
\(951\) 0 0
\(952\) 10704.2 0.364419
\(953\) 33865.3 1.15111 0.575553 0.817764i \(-0.304787\pi\)
0.575553 + 0.817764i \(0.304787\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 4889.06 0.165401
\(957\) 0 0
\(958\) 47601.7 1.60537
\(959\) −11432.7 −0.384965
\(960\) 0 0
\(961\) 19794.9 0.664461
\(962\) −4054.65 −0.135891
\(963\) 0 0
\(964\) 5425.28 0.181262
\(965\) 0 0
\(966\) 0 0
\(967\) 46581.8 1.54909 0.774544 0.632520i \(-0.217979\pi\)
0.774544 + 0.632520i \(0.217979\pi\)
\(968\) −47505.4 −1.57736
\(969\) 0 0
\(970\) 0 0
\(971\) 13829.5 0.457065 0.228532 0.973536i \(-0.426607\pi\)
0.228532 + 0.973536i \(0.426607\pi\)
\(972\) 0 0
\(973\) 59650.5 1.96537
\(974\) 39251.6 1.29127
\(975\) 0 0
\(976\) 6341.46 0.207977
\(977\) −15943.6 −0.522089 −0.261045 0.965327i \(-0.584067\pi\)
−0.261045 + 0.965327i \(0.584067\pi\)
\(978\) 0 0
\(979\) −12658.5 −0.413247
\(980\) 0 0
\(981\) 0 0
\(982\) 7091.85 0.230458
\(983\) 29111.6 0.944572 0.472286 0.881445i \(-0.343429\pi\)
0.472286 + 0.881445i \(0.343429\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −28977.3 −0.935927
\(987\) 0 0
\(988\) 25723.7 0.828318
\(989\) −19416.1 −0.624265
\(990\) 0 0
\(991\) 8745.85 0.280344 0.140172 0.990127i \(-0.455234\pi\)
0.140172 + 0.990127i \(0.455234\pi\)
\(992\) −40130.1 −1.28441
\(993\) 0 0
\(994\) −45800.2 −1.46146
\(995\) 0 0
\(996\) 0 0
\(997\) −45026.9 −1.43031 −0.715153 0.698968i \(-0.753643\pi\)
−0.715153 + 0.698968i \(0.753643\pi\)
\(998\) 25908.0 0.821748
\(999\) 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 2025.4.a.r.1.3 3
3.2 odd 2 2025.4.a.p.1.1 3
5.4 even 2 405.4.a.g.1.1 3
15.14 odd 2 405.4.a.i.1.3 yes 3
45.4 even 6 405.4.e.u.136.3 6
45.14 odd 6 405.4.e.s.136.1 6
45.29 odd 6 405.4.e.s.271.1 6
45.34 even 6 405.4.e.u.271.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.g.1.1 3 5.4 even 2
405.4.a.i.1.3 yes 3 15.14 odd 2
405.4.e.s.136.1 6 45.14 odd 6
405.4.e.s.271.1 6 45.29 odd 6
405.4.e.u.136.3 6 45.4 even 6
405.4.e.u.271.3 6 45.34 even 6
2025.4.a.p.1.1 3 3.2 odd 2
2025.4.a.r.1.3 3 1.1 even 1 trivial