Properties

Label 2025.4.a.be.1.6
Level $2025$
Weight $4$
Character 2025.1
Self dual yes
Analytic conductor $119.479$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2025,4,Mod(1,2025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2025.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2025.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(119.478867762\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 64 x^{10} + 241 x^{9} + 1452 x^{8} - 5130 x^{7} - 13834 x^{6} + 45247 x^{5} + 52669 x^{4} - 144610 x^{3} - 96816 x^{2} + 143136 x + 73008 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 225)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.38827\) of defining polynomial
Character \(\chi\) \(=\) 2025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.38827 q^{2} -6.07270 q^{4} -26.4332 q^{7} +19.5367 q^{8} +O(q^{10})\) \(q-1.38827 q^{2} -6.07270 q^{4} -26.4332 q^{7} +19.5367 q^{8} -62.0909 q^{11} +50.6621 q^{13} +36.6965 q^{14} +21.4593 q^{16} +40.9457 q^{17} -139.322 q^{19} +86.1990 q^{22} +60.7057 q^{23} -70.3328 q^{26} +160.521 q^{28} +175.339 q^{29} +13.9611 q^{31} -186.085 q^{32} -56.8438 q^{34} -144.558 q^{37} +193.416 q^{38} +414.449 q^{41} +191.900 q^{43} +377.059 q^{44} -84.2759 q^{46} -201.541 q^{47} +355.716 q^{49} -307.656 q^{52} +114.667 q^{53} -516.419 q^{56} -243.418 q^{58} +313.338 q^{59} -382.137 q^{61} -19.3817 q^{62} +86.6623 q^{64} +321.266 q^{67} -248.651 q^{68} +373.813 q^{71} +716.432 q^{73} +200.686 q^{74} +846.058 q^{76} +1641.26 q^{77} +609.015 q^{79} -575.368 q^{82} -222.116 q^{83} -266.410 q^{86} -1213.05 q^{88} +175.854 q^{89} -1339.16 q^{91} -368.647 q^{92} +279.794 q^{94} +546.993 q^{97} -493.831 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8} - 29 q^{11} - 24 q^{13} + 69 q^{14} + 192 q^{16} - 79 q^{17} - 75 q^{19} + 18 q^{22} - 318 q^{23} + 154 q^{26} - 96 q^{28} - 106 q^{29} + 60 q^{31} - 914 q^{32} - 108 q^{34} + 84 q^{37} - 640 q^{38} + 353 q^{41} + 426 q^{43} - 571 q^{44} + 270 q^{46} - 1210 q^{47} + 666 q^{49} + 75 q^{52} - 448 q^{53} + 570 q^{56} - 594 q^{58} - 482 q^{59} + 402 q^{61} - 2544 q^{62} + 975 q^{64} + 201 q^{67} - 3437 q^{68} + 944 q^{71} + 453 q^{73} - 10 q^{74} - 462 q^{76} - 2652 q^{77} + 258 q^{79} - 873 q^{82} - 3012 q^{83} - 1952 q^{86} + 216 q^{88} + 738 q^{89} - 618 q^{91} - 5232 q^{92} + 63 q^{94} + 318 q^{97} - 7511 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38827 −0.490828 −0.245414 0.969418i \(-0.578924\pi\)
−0.245414 + 0.969418i \(0.578924\pi\)
\(3\) 0 0
\(4\) −6.07270 −0.759088
\(5\) 0 0
\(6\) 0 0
\(7\) −26.4332 −1.42726 −0.713631 0.700522i \(-0.752950\pi\)
−0.713631 + 0.700522i \(0.752950\pi\)
\(8\) 19.5367 0.863410
\(9\) 0 0
\(10\) 0 0
\(11\) −62.0909 −1.70192 −0.850959 0.525232i \(-0.823979\pi\)
−0.850959 + 0.525232i \(0.823979\pi\)
\(12\) 0 0
\(13\) 50.6621 1.08086 0.540428 0.841390i \(-0.318262\pi\)
0.540428 + 0.841390i \(0.318262\pi\)
\(14\) 36.6965 0.700540
\(15\) 0 0
\(16\) 21.4593 0.335302
\(17\) 40.9457 0.584165 0.292082 0.956393i \(-0.405652\pi\)
0.292082 + 0.956393i \(0.405652\pi\)
\(18\) 0 0
\(19\) −139.322 −1.68224 −0.841120 0.540849i \(-0.818103\pi\)
−0.841120 + 0.540849i \(0.818103\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 86.1990 0.835349
\(23\) 60.7057 0.550348 0.275174 0.961394i \(-0.411265\pi\)
0.275174 + 0.961394i \(0.411265\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −70.3328 −0.530515
\(27\) 0 0
\(28\) 160.521 1.08342
\(29\) 175.339 1.12274 0.561372 0.827563i \(-0.310274\pi\)
0.561372 + 0.827563i \(0.310274\pi\)
\(30\) 0 0
\(31\) 13.9611 0.0808865 0.0404432 0.999182i \(-0.487123\pi\)
0.0404432 + 0.999182i \(0.487123\pi\)
\(32\) −186.085 −1.02799
\(33\) 0 0
\(34\) −56.8438 −0.286725
\(35\) 0 0
\(36\) 0 0
\(37\) −144.558 −0.642303 −0.321152 0.947028i \(-0.604070\pi\)
−0.321152 + 0.947028i \(0.604070\pi\)
\(38\) 193.416 0.825691
\(39\) 0 0
\(40\) 0 0
\(41\) 414.449 1.57868 0.789342 0.613953i \(-0.210422\pi\)
0.789342 + 0.613953i \(0.210422\pi\)
\(42\) 0 0
\(43\) 191.900 0.680570 0.340285 0.940322i \(-0.389476\pi\)
0.340285 + 0.940322i \(0.389476\pi\)
\(44\) 377.059 1.29191
\(45\) 0 0
\(46\) −84.2759 −0.270126
\(47\) −201.541 −0.625485 −0.312742 0.949838i \(-0.601248\pi\)
−0.312742 + 0.949838i \(0.601248\pi\)
\(48\) 0 0
\(49\) 355.716 1.03707
\(50\) 0 0
\(51\) 0 0
\(52\) −307.656 −0.820465
\(53\) 114.667 0.297185 0.148592 0.988899i \(-0.452526\pi\)
0.148592 + 0.988899i \(0.452526\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −516.419 −1.23231
\(57\) 0 0
\(58\) −243.418 −0.551075
\(59\) 313.338 0.691410 0.345705 0.938343i \(-0.387640\pi\)
0.345705 + 0.938343i \(0.387640\pi\)
\(60\) 0 0
\(61\) −382.137 −0.802093 −0.401046 0.916058i \(-0.631353\pi\)
−0.401046 + 0.916058i \(0.631353\pi\)
\(62\) −19.3817 −0.0397013
\(63\) 0 0
\(64\) 86.6623 0.169262
\(65\) 0 0
\(66\) 0 0
\(67\) 321.266 0.585805 0.292903 0.956142i \(-0.405379\pi\)
0.292903 + 0.956142i \(0.405379\pi\)
\(68\) −248.651 −0.443432
\(69\) 0 0
\(70\) 0 0
\(71\) 373.813 0.624837 0.312418 0.949945i \(-0.398861\pi\)
0.312418 + 0.949945i \(0.398861\pi\)
\(72\) 0 0
\(73\) 716.432 1.14866 0.574329 0.818624i \(-0.305263\pi\)
0.574329 + 0.818624i \(0.305263\pi\)
\(74\) 200.686 0.315261
\(75\) 0 0
\(76\) 846.058 1.27697
\(77\) 1641.26 2.42908
\(78\) 0 0
\(79\) 609.015 0.867336 0.433668 0.901073i \(-0.357219\pi\)
0.433668 + 0.901073i \(0.357219\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −575.368 −0.774863
\(83\) −222.116 −0.293740 −0.146870 0.989156i \(-0.546920\pi\)
−0.146870 + 0.989156i \(0.546920\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −266.410 −0.334043
\(87\) 0 0
\(88\) −1213.05 −1.46945
\(89\) 175.854 0.209443 0.104722 0.994502i \(-0.466605\pi\)
0.104722 + 0.994502i \(0.466605\pi\)
\(90\) 0 0
\(91\) −1339.16 −1.54266
\(92\) −368.647 −0.417762
\(93\) 0 0
\(94\) 279.794 0.307006
\(95\) 0 0
\(96\) 0 0
\(97\) 546.993 0.572564 0.286282 0.958145i \(-0.407581\pi\)
0.286282 + 0.958145i \(0.407581\pi\)
\(98\) −493.831 −0.509025
\(99\) 0 0
\(100\) 0 0
\(101\) −1864.39 −1.83677 −0.918385 0.395689i \(-0.870506\pi\)
−0.918385 + 0.395689i \(0.870506\pi\)
\(102\) 0 0
\(103\) −1521.04 −1.45507 −0.727535 0.686071i \(-0.759334\pi\)
−0.727535 + 0.686071i \(0.759334\pi\)
\(104\) 989.772 0.933222
\(105\) 0 0
\(106\) −159.190 −0.145867
\(107\) 1042.82 0.942179 0.471090 0.882085i \(-0.343861\pi\)
0.471090 + 0.882085i \(0.343861\pi\)
\(108\) 0 0
\(109\) 995.777 0.875029 0.437515 0.899211i \(-0.355859\pi\)
0.437515 + 0.899211i \(0.355859\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −567.239 −0.478563
\(113\) −1760.65 −1.46573 −0.732866 0.680373i \(-0.761818\pi\)
−0.732866 + 0.680373i \(0.761818\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1064.78 −0.852262
\(117\) 0 0
\(118\) −434.999 −0.339363
\(119\) −1082.33 −0.833756
\(120\) 0 0
\(121\) 2524.28 1.89653
\(122\) 530.510 0.393690
\(123\) 0 0
\(124\) −84.7813 −0.0613999
\(125\) 0 0
\(126\) 0 0
\(127\) −156.362 −0.109251 −0.0546257 0.998507i \(-0.517397\pi\)
−0.0546257 + 0.998507i \(0.517397\pi\)
\(128\) 1368.37 0.944907
\(129\) 0 0
\(130\) 0 0
\(131\) 221.921 0.148010 0.0740050 0.997258i \(-0.476422\pi\)
0.0740050 + 0.997258i \(0.476422\pi\)
\(132\) 0 0
\(133\) 3682.72 2.40100
\(134\) −446.005 −0.287530
\(135\) 0 0
\(136\) 799.946 0.504374
\(137\) −1326.33 −0.827126 −0.413563 0.910476i \(-0.635716\pi\)
−0.413563 + 0.910476i \(0.635716\pi\)
\(138\) 0 0
\(139\) 853.570 0.520855 0.260428 0.965493i \(-0.416136\pi\)
0.260428 + 0.965493i \(0.416136\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −518.954 −0.306687
\(143\) −3145.65 −1.83953
\(144\) 0 0
\(145\) 0 0
\(146\) −994.603 −0.563794
\(147\) 0 0
\(148\) 877.859 0.487565
\(149\) 299.702 0.164782 0.0823911 0.996600i \(-0.473744\pi\)
0.0823911 + 0.996600i \(0.473744\pi\)
\(150\) 0 0
\(151\) 2892.42 1.55882 0.779411 0.626513i \(-0.215518\pi\)
0.779411 + 0.626513i \(0.215518\pi\)
\(152\) −2721.89 −1.45246
\(153\) 0 0
\(154\) −2278.52 −1.19226
\(155\) 0 0
\(156\) 0 0
\(157\) 1223.70 0.622049 0.311025 0.950402i \(-0.399328\pi\)
0.311025 + 0.950402i \(0.399328\pi\)
\(158\) −845.478 −0.425713
\(159\) 0 0
\(160\) 0 0
\(161\) −1604.65 −0.785490
\(162\) 0 0
\(163\) 1015.65 0.488047 0.244023 0.969769i \(-0.421533\pi\)
0.244023 + 0.969769i \(0.421533\pi\)
\(164\) −2516.83 −1.19836
\(165\) 0 0
\(166\) 308.357 0.144176
\(167\) −3580.19 −1.65894 −0.829472 0.558549i \(-0.811358\pi\)
−0.829472 + 0.558549i \(0.811358\pi\)
\(168\) 0 0
\(169\) 369.648 0.168251
\(170\) 0 0
\(171\) 0 0
\(172\) −1165.35 −0.516613
\(173\) −3273.54 −1.43863 −0.719314 0.694685i \(-0.755544\pi\)
−0.719314 + 0.694685i \(0.755544\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1332.43 −0.570656
\(177\) 0 0
\(178\) −244.133 −0.102801
\(179\) −401.667 −0.167720 −0.0838602 0.996478i \(-0.526725\pi\)
−0.0838602 + 0.996478i \(0.526725\pi\)
\(180\) 0 0
\(181\) −1575.07 −0.646816 −0.323408 0.946260i \(-0.604829\pi\)
−0.323408 + 0.946260i \(0.604829\pi\)
\(182\) 1859.12 0.757183
\(183\) 0 0
\(184\) 1185.99 0.475176
\(185\) 0 0
\(186\) 0 0
\(187\) −2542.36 −0.994201
\(188\) 1223.90 0.474798
\(189\) 0 0
\(190\) 0 0
\(191\) 1152.15 0.436476 0.218238 0.975896i \(-0.429969\pi\)
0.218238 + 0.975896i \(0.429969\pi\)
\(192\) 0 0
\(193\) 3515.12 1.31101 0.655503 0.755193i \(-0.272457\pi\)
0.655503 + 0.755193i \(0.272457\pi\)
\(194\) −759.375 −0.281031
\(195\) 0 0
\(196\) −2160.16 −0.787230
\(197\) 2854.38 1.03232 0.516158 0.856493i \(-0.327362\pi\)
0.516158 + 0.856493i \(0.327362\pi\)
\(198\) 0 0
\(199\) 2961.76 1.05504 0.527521 0.849542i \(-0.323122\pi\)
0.527521 + 0.849542i \(0.323122\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2588.28 0.901538
\(203\) −4634.77 −1.60245
\(204\) 0 0
\(205\) 0 0
\(206\) 2111.61 0.714189
\(207\) 0 0
\(208\) 1087.17 0.362413
\(209\) 8650.60 2.86303
\(210\) 0 0
\(211\) −4802.41 −1.56688 −0.783440 0.621467i \(-0.786537\pi\)
−0.783440 + 0.621467i \(0.786537\pi\)
\(212\) −696.341 −0.225589
\(213\) 0 0
\(214\) −1447.72 −0.462448
\(215\) 0 0
\(216\) 0 0
\(217\) −369.036 −0.115446
\(218\) −1382.41 −0.429489
\(219\) 0 0
\(220\) 0 0
\(221\) 2074.40 0.631398
\(222\) 0 0
\(223\) −2114.30 −0.634906 −0.317453 0.948274i \(-0.602828\pi\)
−0.317453 + 0.948274i \(0.602828\pi\)
\(224\) 4918.84 1.46720
\(225\) 0 0
\(226\) 2444.26 0.719422
\(227\) −3769.35 −1.10212 −0.551058 0.834467i \(-0.685776\pi\)
−0.551058 + 0.834467i \(0.685776\pi\)
\(228\) 0 0
\(229\) −1373.72 −0.396412 −0.198206 0.980160i \(-0.563511\pi\)
−0.198206 + 0.980160i \(0.563511\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3425.55 0.969389
\(233\) −3045.69 −0.856352 −0.428176 0.903695i \(-0.640844\pi\)
−0.428176 + 0.903695i \(0.640844\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1902.81 −0.524840
\(237\) 0 0
\(238\) 1502.57 0.409231
\(239\) 3032.36 0.820699 0.410349 0.911928i \(-0.365407\pi\)
0.410349 + 0.911928i \(0.365407\pi\)
\(240\) 0 0
\(241\) −2529.56 −0.676113 −0.338057 0.941126i \(-0.609770\pi\)
−0.338057 + 0.941126i \(0.609770\pi\)
\(242\) −3504.38 −0.930868
\(243\) 0 0
\(244\) 2320.61 0.608859
\(245\) 0 0
\(246\) 0 0
\(247\) −7058.32 −1.81826
\(248\) 272.753 0.0698382
\(249\) 0 0
\(250\) 0 0
\(251\) −2473.85 −0.622104 −0.311052 0.950393i \(-0.600681\pi\)
−0.311052 + 0.950393i \(0.600681\pi\)
\(252\) 0 0
\(253\) −3769.27 −0.936647
\(254\) 217.073 0.0536236
\(255\) 0 0
\(256\) −2592.97 −0.633049
\(257\) −4504.78 −1.09339 −0.546694 0.837333i \(-0.684114\pi\)
−0.546694 + 0.837333i \(0.684114\pi\)
\(258\) 0 0
\(259\) 3821.14 0.916735
\(260\) 0 0
\(261\) 0 0
\(262\) −308.086 −0.0726475
\(263\) −4193.64 −0.983235 −0.491617 0.870811i \(-0.663594\pi\)
−0.491617 + 0.870811i \(0.663594\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −5112.62 −1.17848
\(267\) 0 0
\(268\) −1950.96 −0.444677
\(269\) 4727.04 1.07142 0.535711 0.844401i \(-0.320044\pi\)
0.535711 + 0.844401i \(0.320044\pi\)
\(270\) 0 0
\(271\) 4692.61 1.05187 0.525933 0.850526i \(-0.323716\pi\)
0.525933 + 0.850526i \(0.323716\pi\)
\(272\) 878.668 0.195872
\(273\) 0 0
\(274\) 1841.31 0.405977
\(275\) 0 0
\(276\) 0 0
\(277\) −2865.79 −0.621619 −0.310809 0.950472i \(-0.600600\pi\)
−0.310809 + 0.950472i \(0.600600\pi\)
\(278\) −1184.99 −0.255650
\(279\) 0 0
\(280\) 0 0
\(281\) 8407.98 1.78498 0.892488 0.451071i \(-0.148958\pi\)
0.892488 + 0.451071i \(0.148958\pi\)
\(282\) 0 0
\(283\) −8549.38 −1.79579 −0.897893 0.440213i \(-0.854903\pi\)
−0.897893 + 0.440213i \(0.854903\pi\)
\(284\) −2270.05 −0.474306
\(285\) 0 0
\(286\) 4367.02 0.902893
\(287\) −10955.2 −2.25320
\(288\) 0 0
\(289\) −3236.45 −0.658752
\(290\) 0 0
\(291\) 0 0
\(292\) −4350.68 −0.871933
\(293\) 5122.05 1.02128 0.510638 0.859796i \(-0.329409\pi\)
0.510638 + 0.859796i \(0.329409\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2824.20 −0.554571
\(297\) 0 0
\(298\) −416.068 −0.0808797
\(299\) 3075.48 0.594847
\(300\) 0 0
\(301\) −5072.55 −0.971352
\(302\) −4015.47 −0.765114
\(303\) 0 0
\(304\) −2989.75 −0.564058
\(305\) 0 0
\(306\) 0 0
\(307\) 5427.96 1.00909 0.504544 0.863386i \(-0.331661\pi\)
0.504544 + 0.863386i \(0.331661\pi\)
\(308\) −9966.90 −1.84389
\(309\) 0 0
\(310\) 0 0
\(311\) −10418.0 −1.89952 −0.949758 0.312985i \(-0.898671\pi\)
−0.949758 + 0.312985i \(0.898671\pi\)
\(312\) 0 0
\(313\) −612.798 −0.110663 −0.0553313 0.998468i \(-0.517621\pi\)
−0.0553313 + 0.998468i \(0.517621\pi\)
\(314\) −1698.83 −0.305319
\(315\) 0 0
\(316\) −3698.37 −0.658384
\(317\) −3945.19 −0.699004 −0.349502 0.936936i \(-0.613649\pi\)
−0.349502 + 0.936936i \(0.613649\pi\)
\(318\) 0 0
\(319\) −10886.9 −1.91082
\(320\) 0 0
\(321\) 0 0
\(322\) 2227.69 0.385541
\(323\) −5704.62 −0.982705
\(324\) 0 0
\(325\) 0 0
\(326\) −1409.99 −0.239547
\(327\) 0 0
\(328\) 8096.98 1.36305
\(329\) 5327.38 0.892730
\(330\) 0 0
\(331\) −1569.58 −0.260641 −0.130320 0.991472i \(-0.541601\pi\)
−0.130320 + 0.991472i \(0.541601\pi\)
\(332\) 1348.84 0.222974
\(333\) 0 0
\(334\) 4970.28 0.814256
\(335\) 0 0
\(336\) 0 0
\(337\) −162.355 −0.0262435 −0.0131217 0.999914i \(-0.504177\pi\)
−0.0131217 + 0.999914i \(0.504177\pi\)
\(338\) −513.172 −0.0825825
\(339\) 0 0
\(340\) 0 0
\(341\) −866.854 −0.137662
\(342\) 0 0
\(343\) −336.135 −0.0529142
\(344\) 3749.11 0.587611
\(345\) 0 0
\(346\) 4544.57 0.706120
\(347\) 2240.16 0.346566 0.173283 0.984872i \(-0.444563\pi\)
0.173283 + 0.984872i \(0.444563\pi\)
\(348\) 0 0
\(349\) 6442.75 0.988174 0.494087 0.869413i \(-0.335503\pi\)
0.494087 + 0.869413i \(0.335503\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 11554.2 1.74955
\(353\) −6723.97 −1.01383 −0.506914 0.861997i \(-0.669214\pi\)
−0.506914 + 0.861997i \(0.669214\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1067.91 −0.158986
\(357\) 0 0
\(358\) 557.622 0.0823219
\(359\) −13327.2 −1.95929 −0.979643 0.200746i \(-0.935663\pi\)
−0.979643 + 0.200746i \(0.935663\pi\)
\(360\) 0 0
\(361\) 12551.5 1.82993
\(362\) 2186.62 0.317476
\(363\) 0 0
\(364\) 8132.34 1.17102
\(365\) 0 0
\(366\) 0 0
\(367\) 180.672 0.0256975 0.0128487 0.999917i \(-0.495910\pi\)
0.0128487 + 0.999917i \(0.495910\pi\)
\(368\) 1302.70 0.184533
\(369\) 0 0
\(370\) 0 0
\(371\) −3031.03 −0.424160
\(372\) 0 0
\(373\) 9876.50 1.37101 0.685504 0.728069i \(-0.259582\pi\)
0.685504 + 0.728069i \(0.259582\pi\)
\(374\) 3529.48 0.487982
\(375\) 0 0
\(376\) −3937.45 −0.540050
\(377\) 8883.03 1.21353
\(378\) 0 0
\(379\) −7885.55 −1.06874 −0.534371 0.845250i \(-0.679452\pi\)
−0.534371 + 0.845250i \(0.679452\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1599.50 −0.214235
\(383\) −5056.75 −0.674642 −0.337321 0.941390i \(-0.609521\pi\)
−0.337321 + 0.941390i \(0.609521\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4879.95 −0.643479
\(387\) 0 0
\(388\) −3321.73 −0.434627
\(389\) −143.433 −0.0186950 −0.00934748 0.999956i \(-0.502975\pi\)
−0.00934748 + 0.999956i \(0.502975\pi\)
\(390\) 0 0
\(391\) 2485.64 0.321494
\(392\) 6949.54 0.895420
\(393\) 0 0
\(394\) −3962.66 −0.506690
\(395\) 0 0
\(396\) 0 0
\(397\) −4718.38 −0.596496 −0.298248 0.954488i \(-0.596402\pi\)
−0.298248 + 0.954488i \(0.596402\pi\)
\(398\) −4111.72 −0.517844
\(399\) 0 0
\(400\) 0 0
\(401\) −8004.58 −0.996832 −0.498416 0.866938i \(-0.666085\pi\)
−0.498416 + 0.866938i \(0.666085\pi\)
\(402\) 0 0
\(403\) 707.297 0.0874267
\(404\) 11321.9 1.39427
\(405\) 0 0
\(406\) 6434.32 0.786528
\(407\) 8975.75 1.09315
\(408\) 0 0
\(409\) −14396.5 −1.74049 −0.870245 0.492618i \(-0.836040\pi\)
−0.870245 + 0.492618i \(0.836040\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 9236.80 1.10453
\(413\) −8282.55 −0.986822
\(414\) 0 0
\(415\) 0 0
\(416\) −9427.47 −1.11110
\(417\) 0 0
\(418\) −12009.4 −1.40526
\(419\) −7082.65 −0.825799 −0.412900 0.910777i \(-0.635484\pi\)
−0.412900 + 0.910777i \(0.635484\pi\)
\(420\) 0 0
\(421\) 11764.3 1.36189 0.680946 0.732333i \(-0.261569\pi\)
0.680946 + 0.732333i \(0.261569\pi\)
\(422\) 6667.06 0.769069
\(423\) 0 0
\(424\) 2240.23 0.256592
\(425\) 0 0
\(426\) 0 0
\(427\) 10101.1 1.14480
\(428\) −6332.73 −0.715197
\(429\) 0 0
\(430\) 0 0
\(431\) −11183.1 −1.24981 −0.624906 0.780700i \(-0.714863\pi\)
−0.624906 + 0.780700i \(0.714863\pi\)
\(432\) 0 0
\(433\) 2138.50 0.237344 0.118672 0.992934i \(-0.462136\pi\)
0.118672 + 0.992934i \(0.462136\pi\)
\(434\) 512.322 0.0566642
\(435\) 0 0
\(436\) −6047.06 −0.664224
\(437\) −8457.61 −0.925817
\(438\) 0 0
\(439\) −3478.39 −0.378166 −0.189083 0.981961i \(-0.560551\pi\)
−0.189083 + 0.981961i \(0.560551\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2879.83 −0.309908
\(443\) 11835.2 1.26932 0.634658 0.772793i \(-0.281141\pi\)
0.634658 + 0.772793i \(0.281141\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2935.23 0.311630
\(447\) 0 0
\(448\) −2290.77 −0.241582
\(449\) 11395.8 1.19778 0.598890 0.800832i \(-0.295609\pi\)
0.598890 + 0.800832i \(0.295609\pi\)
\(450\) 0 0
\(451\) −25733.5 −2.68679
\(452\) 10691.9 1.11262
\(453\) 0 0
\(454\) 5232.88 0.540950
\(455\) 0 0
\(456\) 0 0
\(457\) −3548.95 −0.363267 −0.181633 0.983366i \(-0.558138\pi\)
−0.181633 + 0.983366i \(0.558138\pi\)
\(458\) 1907.10 0.194570
\(459\) 0 0
\(460\) 0 0
\(461\) 782.186 0.0790239 0.0395120 0.999219i \(-0.487420\pi\)
0.0395120 + 0.999219i \(0.487420\pi\)
\(462\) 0 0
\(463\) −483.665 −0.0485483 −0.0242741 0.999705i \(-0.507727\pi\)
−0.0242741 + 0.999705i \(0.507727\pi\)
\(464\) 3762.65 0.376458
\(465\) 0 0
\(466\) 4228.25 0.420322
\(467\) −1463.20 −0.144987 −0.0724934 0.997369i \(-0.523096\pi\)
−0.0724934 + 0.997369i \(0.523096\pi\)
\(468\) 0 0
\(469\) −8492.11 −0.836097
\(470\) 0 0
\(471\) 0 0
\(472\) 6121.61 0.596970
\(473\) −11915.3 −1.15828
\(474\) 0 0
\(475\) 0 0
\(476\) 6572.66 0.632894
\(477\) 0 0
\(478\) −4209.74 −0.402822
\(479\) −2813.55 −0.268380 −0.134190 0.990956i \(-0.542843\pi\)
−0.134190 + 0.990956i \(0.542843\pi\)
\(480\) 0 0
\(481\) −7323.62 −0.694238
\(482\) 3511.72 0.331855
\(483\) 0 0
\(484\) −15329.2 −1.43963
\(485\) 0 0
\(486\) 0 0
\(487\) −763.032 −0.0709985 −0.0354993 0.999370i \(-0.511302\pi\)
−0.0354993 + 0.999370i \(0.511302\pi\)
\(488\) −7465.72 −0.692535
\(489\) 0 0
\(490\) 0 0
\(491\) 6818.64 0.626723 0.313362 0.949634i \(-0.398545\pi\)
0.313362 + 0.949634i \(0.398545\pi\)
\(492\) 0 0
\(493\) 7179.38 0.655868
\(494\) 9798.87 0.892453
\(495\) 0 0
\(496\) 299.595 0.0271214
\(497\) −9881.08 −0.891805
\(498\) 0 0
\(499\) 15367.3 1.37863 0.689315 0.724462i \(-0.257911\pi\)
0.689315 + 0.724462i \(0.257911\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3434.38 0.305346
\(503\) −10126.5 −0.897652 −0.448826 0.893619i \(-0.648158\pi\)
−0.448826 + 0.893619i \(0.648158\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 5232.77 0.459733
\(507\) 0 0
\(508\) 949.542 0.0829313
\(509\) 379.594 0.0330554 0.0165277 0.999863i \(-0.494739\pi\)
0.0165277 + 0.999863i \(0.494739\pi\)
\(510\) 0 0
\(511\) −18937.6 −1.63944
\(512\) −7347.22 −0.634188
\(513\) 0 0
\(514\) 6253.86 0.536665
\(515\) 0 0
\(516\) 0 0
\(517\) 12513.9 1.06452
\(518\) −5304.78 −0.449959
\(519\) 0 0
\(520\) 0 0
\(521\) −175.365 −0.0147464 −0.00737320 0.999973i \(-0.502347\pi\)
−0.00737320 + 0.999973i \(0.502347\pi\)
\(522\) 0 0
\(523\) −6412.16 −0.536107 −0.268054 0.963404i \(-0.586380\pi\)
−0.268054 + 0.963404i \(0.586380\pi\)
\(524\) −1347.66 −0.112353
\(525\) 0 0
\(526\) 5821.91 0.482599
\(527\) 571.646 0.0472510
\(528\) 0 0
\(529\) −8481.82 −0.697117
\(530\) 0 0
\(531\) 0 0
\(532\) −22364.1 −1.82257
\(533\) 20996.9 1.70633
\(534\) 0 0
\(535\) 0 0
\(536\) 6276.50 0.505790
\(537\) 0 0
\(538\) −6562.41 −0.525884
\(539\) −22086.7 −1.76502
\(540\) 0 0
\(541\) −3097.02 −0.246120 −0.123060 0.992399i \(-0.539271\pi\)
−0.123060 + 0.992399i \(0.539271\pi\)
\(542\) −6514.62 −0.516286
\(543\) 0 0
\(544\) −7619.40 −0.600513
\(545\) 0 0
\(546\) 0 0
\(547\) −19366.9 −1.51384 −0.756918 0.653510i \(-0.773296\pi\)
−0.756918 + 0.653510i \(0.773296\pi\)
\(548\) 8054.42 0.627861
\(549\) 0 0
\(550\) 0 0
\(551\) −24428.5 −1.88873
\(552\) 0 0
\(553\) −16098.2 −1.23791
\(554\) 3978.49 0.305108
\(555\) 0 0
\(556\) −5183.47 −0.395375
\(557\) 2285.09 0.173828 0.0869142 0.996216i \(-0.472299\pi\)
0.0869142 + 0.996216i \(0.472299\pi\)
\(558\) 0 0
\(559\) 9722.07 0.735599
\(560\) 0 0
\(561\) 0 0
\(562\) −11672.6 −0.876117
\(563\) −9294.62 −0.695775 −0.347888 0.937536i \(-0.613101\pi\)
−0.347888 + 0.937536i \(0.613101\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 11868.9 0.881423
\(567\) 0 0
\(568\) 7303.08 0.539490
\(569\) 4799.80 0.353635 0.176817 0.984244i \(-0.443420\pi\)
0.176817 + 0.984244i \(0.443420\pi\)
\(570\) 0 0
\(571\) 7340.74 0.538005 0.269002 0.963140i \(-0.413306\pi\)
0.269002 + 0.963140i \(0.413306\pi\)
\(572\) 19102.6 1.39636
\(573\) 0 0
\(574\) 15208.8 1.10593
\(575\) 0 0
\(576\) 0 0
\(577\) −4603.40 −0.332136 −0.166068 0.986114i \(-0.553107\pi\)
−0.166068 + 0.986114i \(0.553107\pi\)
\(578\) 4493.07 0.323334
\(579\) 0 0
\(580\) 0 0
\(581\) 5871.24 0.419243
\(582\) 0 0
\(583\) −7119.80 −0.505784
\(584\) 13996.8 0.991763
\(585\) 0 0
\(586\) −7110.80 −0.501271
\(587\) −26036.4 −1.83073 −0.915364 0.402628i \(-0.868097\pi\)
−0.915364 + 0.402628i \(0.868097\pi\)
\(588\) 0 0
\(589\) −1945.08 −0.136070
\(590\) 0 0
\(591\) 0 0
\(592\) −3102.12 −0.215366
\(593\) 9879.23 0.684134 0.342067 0.939676i \(-0.388873\pi\)
0.342067 + 0.939676i \(0.388873\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1820.00 −0.125084
\(597\) 0 0
\(598\) −4269.60 −0.291968
\(599\) 10145.0 0.692011 0.346005 0.938233i \(-0.387538\pi\)
0.346005 + 0.938233i \(0.387538\pi\)
\(600\) 0 0
\(601\) −11375.1 −0.772047 −0.386023 0.922489i \(-0.626152\pi\)
−0.386023 + 0.922489i \(0.626152\pi\)
\(602\) 7042.08 0.476767
\(603\) 0 0
\(604\) −17564.8 −1.18328
\(605\) 0 0
\(606\) 0 0
\(607\) 136.064 0.00909827 0.00454914 0.999990i \(-0.498552\pi\)
0.00454914 + 0.999990i \(0.498552\pi\)
\(608\) 25925.7 1.72932
\(609\) 0 0
\(610\) 0 0
\(611\) −10210.5 −0.676059
\(612\) 0 0
\(613\) 16261.8 1.07146 0.535731 0.844389i \(-0.320036\pi\)
0.535731 + 0.844389i \(0.320036\pi\)
\(614\) −7535.48 −0.495289
\(615\) 0 0
\(616\) 32064.9 2.09729
\(617\) −25423.1 −1.65883 −0.829413 0.558636i \(-0.811325\pi\)
−0.829413 + 0.558636i \(0.811325\pi\)
\(618\) 0 0
\(619\) 14794.0 0.960618 0.480309 0.877099i \(-0.340525\pi\)
0.480309 + 0.877099i \(0.340525\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 14463.0 0.932336
\(623\) −4648.39 −0.298930
\(624\) 0 0
\(625\) 0 0
\(626\) 850.730 0.0543163
\(627\) 0 0
\(628\) −7431.15 −0.472190
\(629\) −5919.04 −0.375211
\(630\) 0 0
\(631\) −25793.0 −1.62726 −0.813632 0.581381i \(-0.802513\pi\)
−0.813632 + 0.581381i \(0.802513\pi\)
\(632\) 11898.2 0.748866
\(633\) 0 0
\(634\) 5477.00 0.343091
\(635\) 0 0
\(636\) 0 0
\(637\) 18021.3 1.12093
\(638\) 15114.0 0.937884
\(639\) 0 0
\(640\) 0 0
\(641\) −1300.58 −0.0801399 −0.0400700 0.999197i \(-0.512758\pi\)
−0.0400700 + 0.999197i \(0.512758\pi\)
\(642\) 0 0
\(643\) −3758.25 −0.230499 −0.115250 0.993337i \(-0.536767\pi\)
−0.115250 + 0.993337i \(0.536767\pi\)
\(644\) 9744.54 0.596256
\(645\) 0 0
\(646\) 7919.57 0.482339
\(647\) −22398.4 −1.36101 −0.680505 0.732743i \(-0.738240\pi\)
−0.680505 + 0.732743i \(0.738240\pi\)
\(648\) 0 0
\(649\) −19455.4 −1.17672
\(650\) 0 0
\(651\) 0 0
\(652\) −6167.72 −0.370470
\(653\) 20852.4 1.24964 0.624822 0.780767i \(-0.285171\pi\)
0.624822 + 0.780767i \(0.285171\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 8893.80 0.529336
\(657\) 0 0
\(658\) −7395.86 −0.438177
\(659\) −2275.07 −0.134483 −0.0672414 0.997737i \(-0.521420\pi\)
−0.0672414 + 0.997737i \(0.521420\pi\)
\(660\) 0 0
\(661\) 1840.62 0.108308 0.0541542 0.998533i \(-0.482754\pi\)
0.0541542 + 0.998533i \(0.482754\pi\)
\(662\) 2179.01 0.127930
\(663\) 0 0
\(664\) −4339.42 −0.253618
\(665\) 0 0
\(666\) 0 0
\(667\) 10644.1 0.617900
\(668\) 21741.4 1.25928
\(669\) 0 0
\(670\) 0 0
\(671\) 23727.2 1.36510
\(672\) 0 0
\(673\) 31183.1 1.78606 0.893030 0.449996i \(-0.148575\pi\)
0.893030 + 0.449996i \(0.148575\pi\)
\(674\) 225.393 0.0128810
\(675\) 0 0
\(676\) −2244.76 −0.127718
\(677\) −16071.3 −0.912362 −0.456181 0.889887i \(-0.650783\pi\)
−0.456181 + 0.889887i \(0.650783\pi\)
\(678\) 0 0
\(679\) −14458.8 −0.817199
\(680\) 0 0
\(681\) 0 0
\(682\) 1203.43 0.0675684
\(683\) −2603.13 −0.145836 −0.0729180 0.997338i \(-0.523231\pi\)
−0.0729180 + 0.997338i \(0.523231\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 466.647 0.0259718
\(687\) 0 0
\(688\) 4118.05 0.228197
\(689\) 5809.30 0.321214
\(690\) 0 0
\(691\) 31931.3 1.75792 0.878961 0.476894i \(-0.158237\pi\)
0.878961 + 0.476894i \(0.158237\pi\)
\(692\) 19879.2 1.09205
\(693\) 0 0
\(694\) −3109.96 −0.170104
\(695\) 0 0
\(696\) 0 0
\(697\) 16969.9 0.922212
\(698\) −8944.29 −0.485023
\(699\) 0 0
\(700\) 0 0
\(701\) 28906.8 1.55748 0.778740 0.627347i \(-0.215859\pi\)
0.778740 + 0.627347i \(0.215859\pi\)
\(702\) 0 0
\(703\) 20140.1 1.08051
\(704\) −5380.94 −0.288071
\(705\) 0 0
\(706\) 9334.70 0.497615
\(707\) 49281.9 2.62155
\(708\) 0 0
\(709\) 4665.85 0.247151 0.123575 0.992335i \(-0.460564\pi\)
0.123575 + 0.992335i \(0.460564\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3435.61 0.180836
\(713\) 847.515 0.0445157
\(714\) 0 0
\(715\) 0 0
\(716\) 2439.20 0.127315
\(717\) 0 0
\(718\) 18501.8 0.961673
\(719\) −11419.5 −0.592318 −0.296159 0.955139i \(-0.595706\pi\)
−0.296159 + 0.955139i \(0.595706\pi\)
\(720\) 0 0
\(721\) 40205.9 2.07676
\(722\) −17424.9 −0.898182
\(723\) 0 0
\(724\) 9564.90 0.490990
\(725\) 0 0
\(726\) 0 0
\(727\) 21207.8 1.08192 0.540960 0.841049i \(-0.318061\pi\)
0.540960 + 0.841049i \(0.318061\pi\)
\(728\) −26162.9 −1.33195
\(729\) 0 0
\(730\) 0 0
\(731\) 7857.50 0.397565
\(732\) 0 0
\(733\) −5089.94 −0.256482 −0.128241 0.991743i \(-0.540933\pi\)
−0.128241 + 0.991743i \(0.540933\pi\)
\(734\) −250.821 −0.0126130
\(735\) 0 0
\(736\) −11296.4 −0.565750
\(737\) −19947.7 −0.996992
\(738\) 0 0
\(739\) 22403.1 1.11517 0.557586 0.830119i \(-0.311728\pi\)
0.557586 + 0.830119i \(0.311728\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4207.90 0.208190
\(743\) 9518.12 0.469968 0.234984 0.971999i \(-0.424496\pi\)
0.234984 + 0.971999i \(0.424496\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −13711.3 −0.672929
\(747\) 0 0
\(748\) 15439.0 0.754685
\(749\) −27565.1 −1.34474
\(750\) 0 0
\(751\) −7550.19 −0.366858 −0.183429 0.983033i \(-0.558720\pi\)
−0.183429 + 0.983033i \(0.558720\pi\)
\(752\) −4324.93 −0.209726
\(753\) 0 0
\(754\) −12332.1 −0.595633
\(755\) 0 0
\(756\) 0 0
\(757\) 2455.82 0.117911 0.0589554 0.998261i \(-0.481223\pi\)
0.0589554 + 0.998261i \(0.481223\pi\)
\(758\) 10947.3 0.524569
\(759\) 0 0
\(760\) 0 0
\(761\) 22130.3 1.05417 0.527086 0.849812i \(-0.323285\pi\)
0.527086 + 0.849812i \(0.323285\pi\)
\(762\) 0 0
\(763\) −26321.6 −1.24889
\(764\) −6996.69 −0.331324
\(765\) 0 0
\(766\) 7020.15 0.331134
\(767\) 15874.4 0.747315
\(768\) 0 0
\(769\) 24572.2 1.15227 0.576136 0.817354i \(-0.304560\pi\)
0.576136 + 0.817354i \(0.304560\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −21346.3 −0.995168
\(773\) 4726.47 0.219922 0.109961 0.993936i \(-0.464927\pi\)
0.109961 + 0.993936i \(0.464927\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10686.5 0.494358
\(777\) 0 0
\(778\) 199.124 0.00917601
\(779\) −57741.7 −2.65573
\(780\) 0 0
\(781\) −23210.4 −1.06342
\(782\) −3450.74 −0.157798
\(783\) 0 0
\(784\) 7633.43 0.347733
\(785\) 0 0
\(786\) 0 0
\(787\) −8034.51 −0.363913 −0.181956 0.983307i \(-0.558243\pi\)
−0.181956 + 0.983307i \(0.558243\pi\)
\(788\) −17333.8 −0.783618
\(789\) 0 0
\(790\) 0 0
\(791\) 46539.6 2.09198
\(792\) 0 0
\(793\) −19359.9 −0.866948
\(794\) 6550.40 0.292777
\(795\) 0 0
\(796\) −17985.9 −0.800869
\(797\) −28316.0 −1.25847 −0.629237 0.777214i \(-0.716632\pi\)
−0.629237 + 0.777214i \(0.716632\pi\)
\(798\) 0 0
\(799\) −8252.25 −0.365386
\(800\) 0 0
\(801\) 0 0
\(802\) 11112.5 0.489273
\(803\) −44483.9 −1.95492
\(804\) 0 0
\(805\) 0 0
\(806\) −981.920 −0.0429115
\(807\) 0 0
\(808\) −36424.1 −1.58588
\(809\) −25667.5 −1.11548 −0.557739 0.830016i \(-0.688331\pi\)
−0.557739 + 0.830016i \(0.688331\pi\)
\(810\) 0 0
\(811\) −7069.35 −0.306089 −0.153045 0.988219i \(-0.548908\pi\)
−0.153045 + 0.988219i \(0.548908\pi\)
\(812\) 28145.6 1.21640
\(813\) 0 0
\(814\) −12460.8 −0.536548
\(815\) 0 0
\(816\) 0 0
\(817\) −26735.9 −1.14488
\(818\) 19986.2 0.854282
\(819\) 0 0
\(820\) 0 0
\(821\) 40718.4 1.73092 0.865458 0.500982i \(-0.167028\pi\)
0.865458 + 0.500982i \(0.167028\pi\)
\(822\) 0 0
\(823\) 27978.1 1.18500 0.592501 0.805570i \(-0.298141\pi\)
0.592501 + 0.805570i \(0.298141\pi\)
\(824\) −29716.1 −1.25632
\(825\) 0 0
\(826\) 11498.4 0.484360
\(827\) 33072.0 1.39060 0.695299 0.718720i \(-0.255272\pi\)
0.695299 + 0.718720i \(0.255272\pi\)
\(828\) 0 0
\(829\) 8834.66 0.370133 0.185067 0.982726i \(-0.440750\pi\)
0.185067 + 0.982726i \(0.440750\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4390.49 0.182948
\(833\) 14565.1 0.605822
\(834\) 0 0
\(835\) 0 0
\(836\) −52532.5 −2.17329
\(837\) 0 0
\(838\) 9832.64 0.405326
\(839\) 7157.16 0.294508 0.147254 0.989099i \(-0.452956\pi\)
0.147254 + 0.989099i \(0.452956\pi\)
\(840\) 0 0
\(841\) 6354.70 0.260556
\(842\) −16332.0 −0.668455
\(843\) 0 0
\(844\) 29163.6 1.18940
\(845\) 0 0
\(846\) 0 0
\(847\) −66724.8 −2.70684
\(848\) 2460.69 0.0996466
\(849\) 0 0
\(850\) 0 0
\(851\) −8775.50 −0.353490
\(852\) 0 0
\(853\) 17887.7 0.718012 0.359006 0.933335i \(-0.383116\pi\)
0.359006 + 0.933335i \(0.383116\pi\)
\(854\) −14023.1 −0.561898
\(855\) 0 0
\(856\) 20373.3 0.813487
\(857\) 7275.68 0.290003 0.145001 0.989431i \(-0.453681\pi\)
0.145001 + 0.989431i \(0.453681\pi\)
\(858\) 0 0
\(859\) 14124.1 0.561011 0.280505 0.959852i \(-0.409498\pi\)
0.280505 + 0.959852i \(0.409498\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 15525.1 0.613443
\(863\) 46912.8 1.85044 0.925221 0.379430i \(-0.123880\pi\)
0.925221 + 0.379430i \(0.123880\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2968.82 −0.116495
\(867\) 0 0
\(868\) 2241.05 0.0876337
\(869\) −37814.3 −1.47613
\(870\) 0 0
\(871\) 16276.0 0.633171
\(872\) 19454.2 0.755509
\(873\) 0 0
\(874\) 11741.5 0.454417
\(875\) 0 0
\(876\) 0 0
\(877\) −23062.3 −0.887982 −0.443991 0.896031i \(-0.646438\pi\)
−0.443991 + 0.896031i \(0.646438\pi\)
\(878\) 4828.96 0.185614
\(879\) 0 0
\(880\) 0 0
\(881\) 22672.9 0.867047 0.433523 0.901142i \(-0.357270\pi\)
0.433523 + 0.901142i \(0.357270\pi\)
\(882\) 0 0
\(883\) −24676.9 −0.940481 −0.470241 0.882538i \(-0.655833\pi\)
−0.470241 + 0.882538i \(0.655833\pi\)
\(884\) −12597.2 −0.479287
\(885\) 0 0
\(886\) −16430.5 −0.623016
\(887\) −8579.09 −0.324755 −0.162378 0.986729i \(-0.551916\pi\)
−0.162378 + 0.986729i \(0.551916\pi\)
\(888\) 0 0
\(889\) 4133.16 0.155930
\(890\) 0 0
\(891\) 0 0
\(892\) 12839.5 0.481950
\(893\) 28079.0 1.05222
\(894\) 0 0
\(895\) 0 0
\(896\) −36170.5 −1.34863
\(897\) 0 0
\(898\) −15820.5 −0.587904
\(899\) 2447.92 0.0908148
\(900\) 0 0
\(901\) 4695.15 0.173605
\(902\) 35725.1 1.31875
\(903\) 0 0
\(904\) −34397.3 −1.26553
\(905\) 0 0
\(906\) 0 0
\(907\) −10650.0 −0.389888 −0.194944 0.980814i \(-0.562453\pi\)
−0.194944 + 0.980814i \(0.562453\pi\)
\(908\) 22890.1 0.836603
\(909\) 0 0
\(910\) 0 0
\(911\) 4222.07 0.153549 0.0767747 0.997048i \(-0.475538\pi\)
0.0767747 + 0.997048i \(0.475538\pi\)
\(912\) 0 0
\(913\) 13791.4 0.499921
\(914\) 4926.91 0.178302
\(915\) 0 0
\(916\) 8342.22 0.300911
\(917\) −5866.09 −0.211249
\(918\) 0 0
\(919\) −5262.94 −0.188910 −0.0944550 0.995529i \(-0.530111\pi\)
−0.0944550 + 0.995529i \(0.530111\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1085.89 −0.0387872
\(923\) 18938.1 0.675359
\(924\) 0 0
\(925\) 0 0
\(926\) 671.459 0.0238289
\(927\) 0 0
\(928\) −32628.0 −1.15417
\(929\) −9699.02 −0.342535 −0.171267 0.985225i \(-0.554786\pi\)
−0.171267 + 0.985225i \(0.554786\pi\)
\(930\) 0 0
\(931\) −49559.0 −1.74461
\(932\) 18495.6 0.650046
\(933\) 0 0
\(934\) 2031.32 0.0711636
\(935\) 0 0
\(936\) 0 0
\(937\) 26451.8 0.922243 0.461121 0.887337i \(-0.347447\pi\)
0.461121 + 0.887337i \(0.347447\pi\)
\(938\) 11789.4 0.410380
\(939\) 0 0
\(940\) 0 0
\(941\) −177.394 −0.00614547 −0.00307273 0.999995i \(-0.500978\pi\)
−0.00307273 + 0.999995i \(0.500978\pi\)
\(942\) 0 0
\(943\) 25159.4 0.868826
\(944\) 6724.02 0.231831
\(945\) 0 0
\(946\) 16541.6 0.568514
\(947\) 41170.5 1.41274 0.706368 0.707845i \(-0.250333\pi\)
0.706368 + 0.707845i \(0.250333\pi\)
\(948\) 0 0
\(949\) 36296.0 1.24154
\(950\) 0 0
\(951\) 0 0
\(952\) −21145.2 −0.719873
\(953\) −95.6454 −0.00325106 −0.00162553 0.999999i \(-0.500517\pi\)
−0.00162553 + 0.999999i \(0.500517\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −18414.6 −0.622982
\(957\) 0 0
\(958\) 3905.97 0.131729
\(959\) 35059.3 1.18052
\(960\) 0 0
\(961\) −29596.1 −0.993457
\(962\) 10167.2 0.340752
\(963\) 0 0
\(964\) 15361.3 0.513229
\(965\) 0 0
\(966\) 0 0
\(967\) −38487.1 −1.27990 −0.639950 0.768417i \(-0.721045\pi\)
−0.639950 + 0.768417i \(0.721045\pi\)
\(968\) 49316.1 1.63748
\(969\) 0 0
\(970\) 0 0
\(971\) −28964.1 −0.957263 −0.478632 0.878016i \(-0.658867\pi\)
−0.478632 + 0.878016i \(0.658867\pi\)
\(972\) 0 0
\(973\) −22562.6 −0.743396
\(974\) 1059.30 0.0348481
\(975\) 0 0
\(976\) −8200.41 −0.268943
\(977\) −2029.70 −0.0664647 −0.0332323 0.999448i \(-0.510580\pi\)
−0.0332323 + 0.999448i \(0.510580\pi\)
\(978\) 0 0
\(979\) −10918.9 −0.356456
\(980\) 0 0
\(981\) 0 0
\(982\) −9466.13 −0.307613
\(983\) 44554.5 1.44564 0.722821 0.691035i \(-0.242845\pi\)
0.722821 + 0.691035i \(0.242845\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −9966.93 −0.321918
\(987\) 0 0
\(988\) 42863.1 1.38022
\(989\) 11649.4 0.374551
\(990\) 0 0
\(991\) −3215.90 −0.103084 −0.0515420 0.998671i \(-0.516414\pi\)
−0.0515420 + 0.998671i \(0.516414\pi\)
\(992\) −2597.95 −0.0831501
\(993\) 0 0
\(994\) 13717.6 0.437723
\(995\) 0 0
\(996\) 0 0
\(997\) 19664.3 0.624649 0.312325 0.949975i \(-0.398892\pi\)
0.312325 + 0.949975i \(0.398892\pi\)
\(998\) −21334.0 −0.676671
\(999\) 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.be.1.6 12
3.2 odd 2 2025.4.a.bi.1.7 12
5.4 even 2 2025.4.a.bj.1.7 12
9.2 odd 6 225.4.e.e.76.6 24
9.5 odd 6 225.4.e.e.151.6 yes 24
15.14 odd 2 2025.4.a.bf.1.6 12
45.2 even 12 225.4.k.e.49.15 48
45.14 odd 6 225.4.e.f.151.7 yes 24
45.23 even 12 225.4.k.e.124.15 48
45.29 odd 6 225.4.e.f.76.7 yes 24
45.32 even 12 225.4.k.e.124.10 48
45.38 even 12 225.4.k.e.49.10 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.4.e.e.76.6 24 9.2 odd 6
225.4.e.e.151.6 yes 24 9.5 odd 6
225.4.e.f.76.7 yes 24 45.29 odd 6
225.4.e.f.151.7 yes 24 45.14 odd 6
225.4.k.e.49.10 48 45.38 even 12
225.4.k.e.49.15 48 45.2 even 12
225.4.k.e.124.10 48 45.32 even 12
225.4.k.e.124.15 48 45.23 even 12
2025.4.a.be.1.6 12 1.1 even 1 trivial
2025.4.a.bf.1.6 12 15.14 odd 2
2025.4.a.bi.1.7 12 3.2 odd 2
2025.4.a.bj.1.7 12 5.4 even 2