Properties

Label 575.4.b.i.24.10
Level $575$
Weight $4$
Character 575.24
Analytic conductor $33.926$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,4,Mod(24,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.24");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(33.9260982533\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 55x^{8} + 1079x^{6} + 8937x^{4} + 26936x^{2} + 8464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.10
Root \(4.60878i\) of defining polynomial
Character \(\chi\) \(=\) 575.24
Dual form 575.4.b.i.24.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.60878i q^{2} +1.89520i q^{3} -23.4584 q^{4} -10.6297 q^{6} +11.4426i q^{7} -86.7031i q^{8} +23.4082 q^{9} +O(q^{10})\) \(q+5.60878i q^{2} +1.89520i q^{3} -23.4584 q^{4} -10.6297 q^{6} +11.4426i q^{7} -86.7031i q^{8} +23.4082 q^{9} +37.7245 q^{11} -44.4584i q^{12} +8.69346i q^{13} -64.1788 q^{14} +298.631 q^{16} -105.687i q^{17} +131.292i q^{18} +128.279 q^{19} -21.6859 q^{21} +211.588i q^{22} +23.0000i q^{23} +164.319 q^{24} -48.7597 q^{26} +95.5335i q^{27} -268.425i q^{28} +133.383 q^{29} +106.008 q^{31} +981.333i q^{32} +71.4953i q^{33} +592.773 q^{34} -549.121 q^{36} -248.835i q^{37} +719.491i q^{38} -16.4758 q^{39} +134.233 q^{41} -121.631i q^{42} -108.684i q^{43} -884.957 q^{44} -129.002 q^{46} -76.2000i q^{47} +565.965i q^{48} +212.068 q^{49} +200.297 q^{51} -203.935i q^{52} -476.207i q^{53} -535.827 q^{54} +992.105 q^{56} +243.114i q^{57} +748.118i q^{58} -608.000 q^{59} -366.273 q^{61} +594.575i q^{62} +267.850i q^{63} -3115.03 q^{64} -401.001 q^{66} +136.041i q^{67} +2479.24i q^{68} -43.5895 q^{69} -152.874 q^{71} -2029.57i q^{72} -1228.16i q^{73} +1395.66 q^{74} -3009.23 q^{76} +431.664i q^{77} -92.4092i q^{78} +364.637 q^{79} +450.968 q^{81} +752.882i q^{82} +762.744i q^{83} +508.717 q^{84} +609.583 q^{86} +252.788i q^{87} -3270.83i q^{88} -271.222 q^{89} -99.4754 q^{91} -539.544i q^{92} +200.906i q^{93} +427.389 q^{94} -1859.82 q^{96} +574.510i q^{97} +1189.44i q^{98} +883.063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 44 q^{4} + 38 q^{6} - 154 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 44 q^{4} + 38 q^{6} - 154 q^{9} + 46 q^{11} - 186 q^{14} + 564 q^{16} + 322 q^{19} - 120 q^{21} - 210 q^{24} - 514 q^{26} - 802 q^{29} + 64 q^{31} + 1326 q^{34} - 1318 q^{36} - 670 q^{39} - 24 q^{41} - 94 q^{44} - 276 q^{46} + 1476 q^{49} - 1986 q^{51} + 16 q^{54} + 686 q^{56} - 2648 q^{59} - 3346 q^{61} - 4932 q^{64} - 5562 q^{66} + 184 q^{69} - 216 q^{71} - 2916 q^{74} - 6954 q^{76} - 1312 q^{79} - 638 q^{81} + 1436 q^{84} + 224 q^{86} - 1140 q^{89} - 3178 q^{91} + 1896 q^{94} - 11982 q^{96} - 4042 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.60878i 1.98300i 0.130090 + 0.991502i \(0.458473\pi\)
−0.130090 + 0.991502i \(0.541527\pi\)
\(3\) 1.89520i 0.364731i 0.983231 + 0.182365i \(0.0583753\pi\)
−0.983231 + 0.182365i \(0.941625\pi\)
\(4\) −23.4584 −2.93231
\(5\) 0 0
\(6\) −10.6297 −0.723262
\(7\) 11.4426i 0.617840i 0.951088 + 0.308920i \(0.0999675\pi\)
−0.951088 + 0.308920i \(0.900032\pi\)
\(8\) − 86.7031i − 3.83177i
\(9\) 23.4082 0.866972
\(10\) 0 0
\(11\) 37.7245 1.03403 0.517016 0.855976i \(-0.327043\pi\)
0.517016 + 0.855976i \(0.327043\pi\)
\(12\) − 44.4584i − 1.06950i
\(13\) 8.69346i 0.185472i 0.995691 + 0.0927358i \(0.0295612\pi\)
−0.995691 + 0.0927358i \(0.970439\pi\)
\(14\) −64.1788 −1.22518
\(15\) 0 0
\(16\) 298.631 4.66611
\(17\) − 105.687i − 1.50781i −0.656983 0.753905i \(-0.728168\pi\)
0.656983 0.753905i \(-0.271832\pi\)
\(18\) 131.292i 1.71921i
\(19\) 128.279 1.54891 0.774455 0.632629i \(-0.218024\pi\)
0.774455 + 0.632629i \(0.218024\pi\)
\(20\) 0 0
\(21\) −21.6859 −0.225345
\(22\) 211.588i 2.05049i
\(23\) 23.0000i 0.208514i
\(24\) 164.319 1.39756
\(25\) 0 0
\(26\) −48.7597 −0.367791
\(27\) 95.5335i 0.680942i
\(28\) − 268.425i − 1.81170i
\(29\) 133.383 0.854092 0.427046 0.904230i \(-0.359554\pi\)
0.427046 + 0.904230i \(0.359554\pi\)
\(30\) 0 0
\(31\) 106.008 0.614179 0.307090 0.951681i \(-0.400645\pi\)
0.307090 + 0.951681i \(0.400645\pi\)
\(32\) 981.333i 5.42115i
\(33\) 71.4953i 0.377143i
\(34\) 592.773 2.98999
\(35\) 0 0
\(36\) −549.121 −2.54223
\(37\) − 248.835i − 1.10563i −0.833305 0.552814i \(-0.813554\pi\)
0.833305 0.552814i \(-0.186446\pi\)
\(38\) 719.491i 3.07150i
\(39\) −16.4758 −0.0676472
\(40\) 0 0
\(41\) 134.233 0.511308 0.255654 0.966768i \(-0.417709\pi\)
0.255654 + 0.966768i \(0.417709\pi\)
\(42\) − 121.631i − 0.446860i
\(43\) − 108.684i − 0.385444i −0.981253 0.192722i \(-0.938268\pi\)
0.981253 0.192722i \(-0.0617316\pi\)
\(44\) −884.957 −3.03210
\(45\) 0 0
\(46\) −129.002 −0.413485
\(47\) − 76.2000i − 0.236488i −0.992985 0.118244i \(-0.962274\pi\)
0.992985 0.118244i \(-0.0377264\pi\)
\(48\) 565.965i 1.70187i
\(49\) 212.068 0.618274
\(50\) 0 0
\(51\) 200.297 0.549945
\(52\) − 203.935i − 0.543859i
\(53\) − 476.207i − 1.23419i −0.786889 0.617095i \(-0.788309\pi\)
0.786889 0.617095i \(-0.211691\pi\)
\(54\) −535.827 −1.35031
\(55\) 0 0
\(56\) 992.105 2.36742
\(57\) 243.114i 0.564935i
\(58\) 748.118i 1.69367i
\(59\) −608.000 −1.34161 −0.670804 0.741635i \(-0.734051\pi\)
−0.670804 + 0.741635i \(0.734051\pi\)
\(60\) 0 0
\(61\) −366.273 −0.768794 −0.384397 0.923168i \(-0.625591\pi\)
−0.384397 + 0.923168i \(0.625591\pi\)
\(62\) 594.575i 1.21792i
\(63\) 267.850i 0.535650i
\(64\) −3115.03 −6.08405
\(65\) 0 0
\(66\) −401.001 −0.747877
\(67\) 136.041i 0.248060i 0.992278 + 0.124030i \(0.0395819\pi\)
−0.992278 + 0.124030i \(0.960418\pi\)
\(68\) 2479.24i 4.42136i
\(69\) −43.5895 −0.0760516
\(70\) 0 0
\(71\) −152.874 −0.255533 −0.127766 0.991804i \(-0.540781\pi\)
−0.127766 + 0.991804i \(0.540781\pi\)
\(72\) − 2029.57i − 3.32204i
\(73\) − 1228.16i − 1.96911i −0.175070 0.984556i \(-0.556015\pi\)
0.175070 0.984556i \(-0.443985\pi\)
\(74\) 1395.66 2.19246
\(75\) 0 0
\(76\) −3009.23 −4.54188
\(77\) 431.664i 0.638867i
\(78\) − 92.4092i − 0.134145i
\(79\) 364.637 0.519302 0.259651 0.965702i \(-0.416392\pi\)
0.259651 + 0.965702i \(0.416392\pi\)
\(80\) 0 0
\(81\) 450.968 0.618611
\(82\) 752.882i 1.01393i
\(83\) 762.744i 1.00870i 0.863499 + 0.504350i \(0.168268\pi\)
−0.863499 + 0.504350i \(0.831732\pi\)
\(84\) 508.717 0.660781
\(85\) 0 0
\(86\) 609.583 0.764338
\(87\) 252.788i 0.311514i
\(88\) − 3270.83i − 3.96217i
\(89\) −271.222 −0.323028 −0.161514 0.986870i \(-0.551638\pi\)
−0.161514 + 0.986870i \(0.551638\pi\)
\(90\) 0 0
\(91\) −99.4754 −0.114592
\(92\) − 539.544i − 0.611428i
\(93\) 200.906i 0.224010i
\(94\) 427.389 0.468956
\(95\) 0 0
\(96\) −1859.82 −1.97726
\(97\) 574.510i 0.601367i 0.953724 + 0.300684i \(0.0972148\pi\)
−0.953724 + 0.300684i \(0.902785\pi\)
\(98\) 1189.44i 1.22604i
\(99\) 883.063 0.896477
\(100\) 0 0
\(101\) 1372.25 1.35192 0.675958 0.736940i \(-0.263730\pi\)
0.675958 + 0.736940i \(0.263730\pi\)
\(102\) 1123.42i 1.09054i
\(103\) 242.428i 0.231914i 0.993254 + 0.115957i \(0.0369935\pi\)
−0.993254 + 0.115957i \(0.963007\pi\)
\(104\) 753.749 0.710685
\(105\) 0 0
\(106\) 2670.94 2.44740
\(107\) 650.896i 0.588079i 0.955793 + 0.294039i \(0.0949997\pi\)
−0.955793 + 0.294039i \(0.905000\pi\)
\(108\) − 2241.07i − 1.99673i
\(109\) 1230.43 1.08123 0.540613 0.841271i \(-0.318192\pi\)
0.540613 + 0.841271i \(0.318192\pi\)
\(110\) 0 0
\(111\) 471.591 0.403256
\(112\) 3417.10i 2.88291i
\(113\) − 238.959i − 0.198932i −0.995041 0.0994662i \(-0.968286\pi\)
0.995041 0.0994662i \(-0.0317135\pi\)
\(114\) −1363.58 −1.12027
\(115\) 0 0
\(116\) −3128.97 −2.50446
\(117\) 203.498i 0.160799i
\(118\) − 3410.14i − 2.66041i
\(119\) 1209.33 0.931586
\(120\) 0 0
\(121\) 92.1354 0.0692227
\(122\) − 2054.34i − 1.52452i
\(123\) 254.397i 0.186490i
\(124\) −2486.78 −1.80096
\(125\) 0 0
\(126\) −1502.31 −1.06220
\(127\) 2608.48i 1.82256i 0.411788 + 0.911280i \(0.364904\pi\)
−0.411788 + 0.911280i \(0.635096\pi\)
\(128\) − 9620.89i − 6.64355i
\(129\) 205.977 0.140583
\(130\) 0 0
\(131\) −936.409 −0.624538 −0.312269 0.949994i \(-0.601089\pi\)
−0.312269 + 0.949994i \(0.601089\pi\)
\(132\) − 1677.17i − 1.10590i
\(133\) 1467.84i 0.956979i
\(134\) −763.023 −0.491904
\(135\) 0 0
\(136\) −9163.36 −5.77758
\(137\) 415.511i 0.259121i 0.991572 + 0.129560i \(0.0413566\pi\)
−0.991572 + 0.129560i \(0.958643\pi\)
\(138\) − 244.484i − 0.150811i
\(139\) 949.629 0.579471 0.289736 0.957107i \(-0.406433\pi\)
0.289736 + 0.957107i \(0.406433\pi\)
\(140\) 0 0
\(141\) 144.414 0.0862542
\(142\) − 857.439i − 0.506723i
\(143\) 327.956i 0.191784i
\(144\) 6990.43 4.04539
\(145\) 0 0
\(146\) 6888.48 3.90476
\(147\) 401.910i 0.225503i
\(148\) 5837.28i 3.24204i
\(149\) −2209.84 −1.21502 −0.607508 0.794314i \(-0.707831\pi\)
−0.607508 + 0.794314i \(0.707831\pi\)
\(150\) 0 0
\(151\) −1384.25 −0.746018 −0.373009 0.927828i \(-0.621674\pi\)
−0.373009 + 0.927828i \(0.621674\pi\)
\(152\) − 11122.2i − 5.93507i
\(153\) − 2473.94i − 1.30723i
\(154\) −2421.11 −1.26688
\(155\) 0 0
\(156\) 386.497 0.198362
\(157\) 561.399i 0.285379i 0.989767 + 0.142690i \(0.0455751\pi\)
−0.989767 + 0.142690i \(0.954425\pi\)
\(158\) 2045.17i 1.02978i
\(159\) 902.506 0.450147
\(160\) 0 0
\(161\) −263.179 −0.128829
\(162\) 2529.38i 1.22671i
\(163\) − 2134.47i − 1.02567i −0.858487 0.512836i \(-0.828595\pi\)
0.858487 0.512836i \(-0.171405\pi\)
\(164\) −3148.89 −1.49931
\(165\) 0 0
\(166\) −4278.07 −2.00026
\(167\) − 1315.64i − 0.609623i −0.952413 0.304812i \(-0.901406\pi\)
0.952413 0.304812i \(-0.0985935\pi\)
\(168\) 1880.23i 0.863471i
\(169\) 2121.42 0.965600
\(170\) 0 0
\(171\) 3002.79 1.34286
\(172\) 2549.55i 1.13024i
\(173\) − 676.565i − 0.297331i −0.988888 0.148666i \(-0.952502\pi\)
0.988888 0.148666i \(-0.0474978\pi\)
\(174\) −1417.83 −0.617733
\(175\) 0 0
\(176\) 11265.7 4.82491
\(177\) − 1152.28i − 0.489325i
\(178\) − 1521.23i − 0.640566i
\(179\) 3737.96 1.56083 0.780414 0.625263i \(-0.215008\pi\)
0.780414 + 0.625263i \(0.215008\pi\)
\(180\) 0 0
\(181\) −1873.40 −0.769330 −0.384665 0.923056i \(-0.625683\pi\)
−0.384665 + 0.923056i \(0.625683\pi\)
\(182\) − 557.936i − 0.227236i
\(183\) − 694.158i − 0.280403i
\(184\) 1994.17 0.798979
\(185\) 0 0
\(186\) −1126.84 −0.444213
\(187\) − 3986.97i − 1.55912i
\(188\) 1787.53i 0.693454i
\(189\) −1093.15 −0.420713
\(190\) 0 0
\(191\) −5158.92 −1.95438 −0.977190 0.212366i \(-0.931883\pi\)
−0.977190 + 0.212366i \(0.931883\pi\)
\(192\) − 5903.60i − 2.21904i
\(193\) 4806.41i 1.79261i 0.443442 + 0.896303i \(0.353757\pi\)
−0.443442 + 0.896303i \(0.646243\pi\)
\(194\) −3222.30 −1.19251
\(195\) 0 0
\(196\) −4974.78 −1.81297
\(197\) 3202.59i 1.15825i 0.815239 + 0.579125i \(0.196606\pi\)
−0.815239 + 0.579125i \(0.803394\pi\)
\(198\) 4952.91i 1.77772i
\(199\) 2210.46 0.787415 0.393707 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(200\) 0 0
\(201\) −257.824 −0.0904751
\(202\) 7696.62i 2.68085i
\(203\) 1526.25i 0.527692i
\(204\) −4698.65 −1.61261
\(205\) 0 0
\(206\) −1359.73 −0.459886
\(207\) 538.389i 0.180776i
\(208\) 2596.14i 0.865431i
\(209\) 4839.27 1.60162
\(210\) 0 0
\(211\) 153.164 0.0499726 0.0249863 0.999688i \(-0.492046\pi\)
0.0249863 + 0.999688i \(0.492046\pi\)
\(212\) 11171.1i 3.61902i
\(213\) − 289.727i − 0.0932007i
\(214\) −3650.73 −1.16616
\(215\) 0 0
\(216\) 8283.05 2.60921
\(217\) 1213.00i 0.379465i
\(218\) 6901.21i 2.14408i
\(219\) 2327.60 0.718195
\(220\) 0 0
\(221\) 918.782 0.279656
\(222\) 2645.05i 0.799659i
\(223\) 3068.41i 0.921416i 0.887552 + 0.460708i \(0.152405\pi\)
−0.887552 + 0.460708i \(0.847595\pi\)
\(224\) −11229.0 −3.34940
\(225\) 0 0
\(226\) 1340.27 0.394484
\(227\) 4540.20i 1.32750i 0.747953 + 0.663752i \(0.231037\pi\)
−0.747953 + 0.663752i \(0.768963\pi\)
\(228\) − 5703.09i − 1.65656i
\(229\) −1476.25 −0.425996 −0.212998 0.977053i \(-0.568323\pi\)
−0.212998 + 0.977053i \(0.568323\pi\)
\(230\) 0 0
\(231\) −818.089 −0.233014
\(232\) − 11564.7i − 3.27269i
\(233\) 90.7306i 0.0255106i 0.999919 + 0.0127553i \(0.00406024\pi\)
−0.999919 + 0.0127553i \(0.995940\pi\)
\(234\) −1141.38 −0.318864
\(235\) 0 0
\(236\) 14262.7 3.93400
\(237\) 691.059i 0.189405i
\(238\) 6782.85i 1.84734i
\(239\) 1619.16 0.438221 0.219111 0.975700i \(-0.429684\pi\)
0.219111 + 0.975700i \(0.429684\pi\)
\(240\) 0 0
\(241\) −6447.48 −1.72331 −0.861657 0.507491i \(-0.830573\pi\)
−0.861657 + 0.507491i \(0.830573\pi\)
\(242\) 516.768i 0.137269i
\(243\) 3434.08i 0.906568i
\(244\) 8592.19 2.25434
\(245\) 0 0
\(246\) −1426.86 −0.369810
\(247\) 1115.19i 0.287279i
\(248\) − 9191.20i − 2.35339i
\(249\) −1445.55 −0.367904
\(250\) 0 0
\(251\) 3428.17 0.862089 0.431044 0.902331i \(-0.358145\pi\)
0.431044 + 0.902331i \(0.358145\pi\)
\(252\) − 6283.35i − 1.57069i
\(253\) 867.663i 0.215611i
\(254\) −14630.4 −3.61414
\(255\) 0 0
\(256\) 29041.2 7.09014
\(257\) 2261.08i 0.548803i 0.961615 + 0.274402i \(0.0884798\pi\)
−0.961615 + 0.274402i \(0.911520\pi\)
\(258\) 1155.28i 0.278777i
\(259\) 2847.31 0.683101
\(260\) 0 0
\(261\) 3122.27 0.740474
\(262\) − 5252.11i − 1.23846i
\(263\) − 5319.64i − 1.24724i −0.781729 0.623618i \(-0.785662\pi\)
0.781729 0.623618i \(-0.214338\pi\)
\(264\) 6198.86 1.44513
\(265\) 0 0
\(266\) −8232.81 −1.89769
\(267\) − 514.019i − 0.117818i
\(268\) − 3191.31i − 0.727388i
\(269\) 1992.51 0.451620 0.225810 0.974171i \(-0.427497\pi\)
0.225810 + 0.974171i \(0.427497\pi\)
\(270\) 0 0
\(271\) 3950.62 0.885546 0.442773 0.896634i \(-0.353995\pi\)
0.442773 + 0.896634i \(0.353995\pi\)
\(272\) − 31561.3i − 7.03561i
\(273\) − 188.525i − 0.0417951i
\(274\) −2330.51 −0.513837
\(275\) 0 0
\(276\) 1022.54 0.223007
\(277\) − 178.126i − 0.0386375i −0.999813 0.0193187i \(-0.993850\pi\)
0.999813 0.0193187i \(-0.00614973\pi\)
\(278\) 5326.27i 1.14909i
\(279\) 2481.46 0.532476
\(280\) 0 0
\(281\) 3523.49 0.748020 0.374010 0.927425i \(-0.377983\pi\)
0.374010 + 0.927425i \(0.377983\pi\)
\(282\) 809.986i 0.171043i
\(283\) 699.284i 0.146884i 0.997300 + 0.0734419i \(0.0233983\pi\)
−0.997300 + 0.0734419i \(0.976602\pi\)
\(284\) 3586.19 0.749301
\(285\) 0 0
\(286\) −1839.43 −0.380308
\(287\) 1535.96i 0.315906i
\(288\) 22971.3i 4.69998i
\(289\) −6256.67 −1.27349
\(290\) 0 0
\(291\) −1088.81 −0.219337
\(292\) 28810.7i 5.77404i
\(293\) − 8552.97i − 1.70536i −0.522436 0.852678i \(-0.674977\pi\)
0.522436 0.852678i \(-0.325023\pi\)
\(294\) −2254.23 −0.447174
\(295\) 0 0
\(296\) −21574.8 −4.23651
\(297\) 3603.95i 0.704116i
\(298\) − 12394.5i − 2.40938i
\(299\) −199.949 −0.0386735
\(300\) 0 0
\(301\) 1243.62 0.238143
\(302\) − 7763.96i − 1.47936i
\(303\) 2600.67i 0.493085i
\(304\) 38308.2 7.22739
\(305\) 0 0
\(306\) 13875.8 2.59224
\(307\) − 5621.73i − 1.04511i −0.852606 0.522555i \(-0.824979\pi\)
0.852606 0.522555i \(-0.175021\pi\)
\(308\) − 10126.2i − 1.87335i
\(309\) −459.448 −0.0845861
\(310\) 0 0
\(311\) −6533.62 −1.19128 −0.595639 0.803252i \(-0.703101\pi\)
−0.595639 + 0.803252i \(0.703101\pi\)
\(312\) 1428.50i 0.259208i
\(313\) 2713.24i 0.489973i 0.969526 + 0.244987i \(0.0787836\pi\)
−0.969526 + 0.244987i \(0.921216\pi\)
\(314\) −3148.77 −0.565908
\(315\) 0 0
\(316\) −8553.82 −1.52275
\(317\) − 7544.32i − 1.33669i −0.743851 0.668346i \(-0.767003\pi\)
0.743851 0.668346i \(-0.232997\pi\)
\(318\) 5061.96i 0.892643i
\(319\) 5031.82 0.883159
\(320\) 0 0
\(321\) −1233.57 −0.214490
\(322\) − 1476.11i − 0.255468i
\(323\) − 13557.4i − 2.33546i
\(324\) −10579.0 −1.81396
\(325\) 0 0
\(326\) 11971.8 2.03391
\(327\) 2331.90i 0.394356i
\(328\) − 11638.4i − 1.95921i
\(329\) 871.923 0.146111
\(330\) 0 0
\(331\) 5991.64 0.994956 0.497478 0.867477i \(-0.334260\pi\)
0.497478 + 0.867477i \(0.334260\pi\)
\(332\) − 17892.8i − 2.95782i
\(333\) − 5824.79i − 0.958548i
\(334\) 7379.13 1.20889
\(335\) 0 0
\(336\) −6476.08 −1.05149
\(337\) 5665.46i 0.915778i 0.889009 + 0.457889i \(0.151394\pi\)
−0.889009 + 0.457889i \(0.848606\pi\)
\(338\) 11898.6i 1.91479i
\(339\) 452.874 0.0725567
\(340\) 0 0
\(341\) 3999.09 0.635081
\(342\) 16842.0i 2.66290i
\(343\) 6351.40i 0.999834i
\(344\) −9423.21 −1.47693
\(345\) 0 0
\(346\) 3794.71 0.589609
\(347\) 5593.30i 0.865315i 0.901558 + 0.432657i \(0.142424\pi\)
−0.901558 + 0.432657i \(0.857576\pi\)
\(348\) − 5930.00i − 0.913453i
\(349\) −4304.61 −0.660230 −0.330115 0.943941i \(-0.607088\pi\)
−0.330115 + 0.943941i \(0.607088\pi\)
\(350\) 0 0
\(351\) −830.516 −0.126295
\(352\) 37020.3i 5.60564i
\(353\) − 1056.64i − 0.159318i −0.996822 0.0796592i \(-0.974617\pi\)
0.996822 0.0796592i \(-0.0253832\pi\)
\(354\) 6462.88 0.970334
\(355\) 0 0
\(356\) 6362.45 0.947217
\(357\) 2291.91i 0.339778i
\(358\) 20965.4i 3.09513i
\(359\) −5186.43 −0.762477 −0.381238 0.924477i \(-0.624502\pi\)
−0.381238 + 0.924477i \(0.624502\pi\)
\(360\) 0 0
\(361\) 9596.58 1.39912
\(362\) − 10507.5i − 1.52558i
\(363\) 174.615i 0.0252476i
\(364\) 2333.54 0.336018
\(365\) 0 0
\(366\) 3893.38 0.556039
\(367\) 178.772i 0.0254274i 0.999919 + 0.0127137i \(0.00404700\pi\)
−0.999919 + 0.0127137i \(0.995953\pi\)
\(368\) 6868.52i 0.972952i
\(369\) 3142.15 0.443289
\(370\) 0 0
\(371\) 5449.03 0.762532
\(372\) − 4712.93i − 0.656866i
\(373\) − 7463.86i − 1.03610i −0.855351 0.518049i \(-0.826659\pi\)
0.855351 0.518049i \(-0.173341\pi\)
\(374\) 22362.1 3.09175
\(375\) 0 0
\(376\) −6606.78 −0.906166
\(377\) 1159.56i 0.158410i
\(378\) − 6131.23i − 0.834276i
\(379\) 6075.99 0.823490 0.411745 0.911299i \(-0.364919\pi\)
0.411745 + 0.911299i \(0.364919\pi\)
\(380\) 0 0
\(381\) −4943.58 −0.664743
\(382\) − 28935.3i − 3.87554i
\(383\) 580.709i 0.0774747i 0.999249 + 0.0387374i \(0.0123336\pi\)
−0.999249 + 0.0387374i \(0.987666\pi\)
\(384\) 18233.5 2.42311
\(385\) 0 0
\(386\) −26958.1 −3.55475
\(387\) − 2544.09i − 0.334169i
\(388\) − 13477.1i − 1.76339i
\(389\) −11373.5 −1.48241 −0.741205 0.671278i \(-0.765745\pi\)
−0.741205 + 0.671278i \(0.765745\pi\)
\(390\) 0 0
\(391\) 2430.79 0.314400
\(392\) − 18386.9i − 2.36908i
\(393\) − 1774.68i − 0.227788i
\(394\) −17962.6 −2.29681
\(395\) 0 0
\(396\) −20715.3 −2.62874
\(397\) 3701.46i 0.467937i 0.972244 + 0.233969i \(0.0751713\pi\)
−0.972244 + 0.233969i \(0.924829\pi\)
\(398\) 12398.0i 1.56145i
\(399\) −2781.85 −0.349039
\(400\) 0 0
\(401\) −7615.01 −0.948317 −0.474159 0.880439i \(-0.657248\pi\)
−0.474159 + 0.880439i \(0.657248\pi\)
\(402\) − 1446.08i − 0.179413i
\(403\) 921.574i 0.113913i
\(404\) −32190.7 −3.96423
\(405\) 0 0
\(406\) −8560.39 −1.04642
\(407\) − 9387.17i − 1.14325i
\(408\) − 17366.4i − 2.10726i
\(409\) 15423.6 1.86466 0.932330 0.361608i \(-0.117772\pi\)
0.932330 + 0.361608i \(0.117772\pi\)
\(410\) 0 0
\(411\) −787.475 −0.0945092
\(412\) − 5686.98i − 0.680042i
\(413\) − 6957.07i − 0.828899i
\(414\) −3019.71 −0.358480
\(415\) 0 0
\(416\) −8531.17 −1.00547
\(417\) 1799.73i 0.211351i
\(418\) 27142.4i 3.17603i
\(419\) −4273.20 −0.498233 −0.249116 0.968474i \(-0.580140\pi\)
−0.249116 + 0.968474i \(0.580140\pi\)
\(420\) 0 0
\(421\) −6076.38 −0.703432 −0.351716 0.936107i \(-0.614402\pi\)
−0.351716 + 0.936107i \(0.614402\pi\)
\(422\) 859.061i 0.0990958i
\(423\) − 1783.71i − 0.205028i
\(424\) −41288.6 −4.72913
\(425\) 0 0
\(426\) 1625.01 0.184817
\(427\) − 4191.10i − 0.474991i
\(428\) − 15269.0i − 1.72443i
\(429\) −621.541 −0.0699494
\(430\) 0 0
\(431\) −3386.04 −0.378422 −0.189211 0.981936i \(-0.560593\pi\)
−0.189211 + 0.981936i \(0.560593\pi\)
\(432\) 28529.3i 3.17735i
\(433\) − 7924.63i − 0.879523i −0.898114 0.439762i \(-0.855063\pi\)
0.898114 0.439762i \(-0.144937\pi\)
\(434\) −6803.46 −0.752480
\(435\) 0 0
\(436\) −28863.9 −3.17049
\(437\) 2950.42i 0.322970i
\(438\) 13055.0i 1.42418i
\(439\) −13530.9 −1.47106 −0.735530 0.677492i \(-0.763067\pi\)
−0.735530 + 0.677492i \(0.763067\pi\)
\(440\) 0 0
\(441\) 4964.13 0.536026
\(442\) 5153.25i 0.554559i
\(443\) − 996.052i − 0.106826i −0.998573 0.0534129i \(-0.982990\pi\)
0.998573 0.0534129i \(-0.0170100\pi\)
\(444\) −11062.8 −1.18247
\(445\) 0 0
\(446\) −17210.0 −1.82717
\(447\) − 4188.08i − 0.443153i
\(448\) − 35644.0i − 3.75897i
\(449\) 5079.36 0.533875 0.266938 0.963714i \(-0.413988\pi\)
0.266938 + 0.963714i \(0.413988\pi\)
\(450\) 0 0
\(451\) 5063.85 0.528709
\(452\) 5605.60i 0.583331i
\(453\) − 2623.43i − 0.272096i
\(454\) −25465.0 −2.63245
\(455\) 0 0
\(456\) 21078.8 2.16470
\(457\) − 11190.9i − 1.14549i −0.819732 0.572747i \(-0.805878\pi\)
0.819732 0.572747i \(-0.194122\pi\)
\(458\) − 8279.94i − 0.844752i
\(459\) 10096.6 1.02673
\(460\) 0 0
\(461\) 2426.07 0.245105 0.122553 0.992462i \(-0.460892\pi\)
0.122553 + 0.992462i \(0.460892\pi\)
\(462\) − 4588.48i − 0.462068i
\(463\) 16349.2i 1.64106i 0.571602 + 0.820531i \(0.306322\pi\)
−0.571602 + 0.820531i \(0.693678\pi\)
\(464\) 39832.4 3.98529
\(465\) 0 0
\(466\) −508.889 −0.0505876
\(467\) − 2880.80i − 0.285455i −0.989762 0.142728i \(-0.954413\pi\)
0.989762 0.142728i \(-0.0455873\pi\)
\(468\) − 4773.76i − 0.471511i
\(469\) −1556.65 −0.153261
\(470\) 0 0
\(471\) −1063.96 −0.104087
\(472\) 52715.4i 5.14073i
\(473\) − 4100.03i − 0.398562i
\(474\) −3876.00 −0.375592
\(475\) 0 0
\(476\) −28368.9 −2.73169
\(477\) − 11147.2i − 1.07001i
\(478\) 9081.53i 0.868994i
\(479\) −7850.31 −0.748831 −0.374415 0.927261i \(-0.622157\pi\)
−0.374415 + 0.927261i \(0.622157\pi\)
\(480\) 0 0
\(481\) 2163.24 0.205063
\(482\) − 36162.5i − 3.41734i
\(483\) − 498.775i − 0.0469877i
\(484\) −2161.35 −0.202982
\(485\) 0 0
\(486\) −19261.0 −1.79773
\(487\) − 11262.4i − 1.04794i −0.851736 0.523971i \(-0.824450\pi\)
0.851736 0.523971i \(-0.175550\pi\)
\(488\) 31757.0i 2.94584i
\(489\) 4045.24 0.374094
\(490\) 0 0
\(491\) 15810.7 1.45321 0.726607 0.687054i \(-0.241096\pi\)
0.726607 + 0.687054i \(0.241096\pi\)
\(492\) − 5967.76i − 0.546844i
\(493\) − 14096.8i − 1.28781i
\(494\) −6254.86 −0.569675
\(495\) 0 0
\(496\) 31657.2 2.86583
\(497\) − 1749.27i − 0.157878i
\(498\) − 8107.78i − 0.729555i
\(499\) 10470.4 0.939322 0.469661 0.882847i \(-0.344376\pi\)
0.469661 + 0.882847i \(0.344376\pi\)
\(500\) 0 0
\(501\) 2493.39 0.222348
\(502\) 19227.9i 1.70953i
\(503\) 5425.35i 0.480923i 0.970659 + 0.240462i \(0.0772988\pi\)
−0.970659 + 0.240462i \(0.922701\pi\)
\(504\) 23223.4 2.05249
\(505\) 0 0
\(506\) −4866.53 −0.427557
\(507\) 4020.51i 0.352184i
\(508\) − 61190.9i − 5.34430i
\(509\) −8098.38 −0.705215 −0.352608 0.935771i \(-0.614705\pi\)
−0.352608 + 0.935771i \(0.614705\pi\)
\(510\) 0 0
\(511\) 14053.3 1.21660
\(512\) 85918.7i 7.41622i
\(513\) 12255.0i 1.05472i
\(514\) −12681.9 −1.08828
\(515\) 0 0
\(516\) −4831.90 −0.412233
\(517\) − 2874.60i − 0.244536i
\(518\) 15969.9i 1.35459i
\(519\) 1282.22 0.108446
\(520\) 0 0
\(521\) 14691.8 1.23543 0.617717 0.786400i \(-0.288058\pi\)
0.617717 + 0.786400i \(0.288058\pi\)
\(522\) 17512.1i 1.46836i
\(523\) 19263.6i 1.61059i 0.592872 + 0.805297i \(0.297994\pi\)
−0.592872 + 0.805297i \(0.702006\pi\)
\(524\) 21966.7 1.83134
\(525\) 0 0
\(526\) 29836.7 2.47327
\(527\) − 11203.6i − 0.926066i
\(528\) 21350.7i 1.75979i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) −14232.2 −1.16314
\(532\) − 34433.3i − 2.80615i
\(533\) 1166.95i 0.0948331i
\(534\) 2883.02 0.233634
\(535\) 0 0
\(536\) 11795.2 0.950510
\(537\) 7084.17i 0.569282i
\(538\) 11175.6i 0.895564i
\(539\) 8000.15 0.639315
\(540\) 0 0
\(541\) −18737.4 −1.48906 −0.744531 0.667588i \(-0.767327\pi\)
−0.744531 + 0.667588i \(0.767327\pi\)
\(542\) 22158.2i 1.75604i
\(543\) − 3550.46i − 0.280598i
\(544\) 103714. 8.17407
\(545\) 0 0
\(546\) 1057.40 0.0828799
\(547\) − 14180.2i − 1.10841i −0.832379 0.554207i \(-0.813022\pi\)
0.832379 0.554207i \(-0.186978\pi\)
\(548\) − 9747.25i − 0.759821i
\(549\) −8573.80 −0.666522
\(550\) 0 0
\(551\) 17110.3 1.32291
\(552\) 3779.34i 0.291412i
\(553\) 4172.38i 0.320846i
\(554\) 999.073 0.0766183
\(555\) 0 0
\(556\) −22276.8 −1.69919
\(557\) 15154.2i 1.15279i 0.817170 + 0.576397i \(0.195542\pi\)
−0.817170 + 0.576397i \(0.804458\pi\)
\(558\) 13917.9i 1.05590i
\(559\) 944.837 0.0714890
\(560\) 0 0
\(561\) 7556.09 0.568660
\(562\) 19762.5i 1.48333i
\(563\) − 14444.7i − 1.08130i −0.841247 0.540651i \(-0.818178\pi\)
0.841247 0.540651i \(-0.181822\pi\)
\(564\) −3387.73 −0.252924
\(565\) 0 0
\(566\) −3922.13 −0.291271
\(567\) 5160.22i 0.382203i
\(568\) 13254.7i 0.979144i
\(569\) 8851.15 0.652125 0.326063 0.945348i \(-0.394278\pi\)
0.326063 + 0.945348i \(0.394278\pi\)
\(570\) 0 0
\(571\) −22318.9 −1.63576 −0.817879 0.575391i \(-0.804850\pi\)
−0.817879 + 0.575391i \(0.804850\pi\)
\(572\) − 7693.34i − 0.562368i
\(573\) − 9777.17i − 0.712822i
\(574\) −8614.89 −0.626444
\(575\) 0 0
\(576\) −72917.4 −5.27470
\(577\) − 4530.99i − 0.326911i −0.986551 0.163455i \(-0.947736\pi\)
0.986551 0.163455i \(-0.0522639\pi\)
\(578\) − 35092.3i − 2.52534i
\(579\) −9109.09 −0.653818
\(580\) 0 0
\(581\) −8727.75 −0.623215
\(582\) − 6106.89i − 0.434946i
\(583\) − 17964.7i − 1.27619i
\(584\) −106485. −7.54519
\(585\) 0 0
\(586\) 47971.7 3.38173
\(587\) − 5092.89i − 0.358103i −0.983840 0.179051i \(-0.942697\pi\)
0.983840 0.179051i \(-0.0573028\pi\)
\(588\) − 9428.19i − 0.661245i
\(589\) 13598.6 0.951309
\(590\) 0 0
\(591\) −6069.54 −0.422449
\(592\) − 74309.9i − 5.15898i
\(593\) − 4193.92i − 0.290427i −0.989400 0.145214i \(-0.953613\pi\)
0.989400 0.145214i \(-0.0463870\pi\)
\(594\) −20213.8 −1.39626
\(595\) 0 0
\(596\) 51839.5 3.56280
\(597\) 4189.26i 0.287194i
\(598\) − 1121.47i − 0.0766897i
\(599\) −17663.0 −1.20483 −0.602414 0.798184i \(-0.705794\pi\)
−0.602414 + 0.798184i \(0.705794\pi\)
\(600\) 0 0
\(601\) −15817.7 −1.07357 −0.536785 0.843719i \(-0.680361\pi\)
−0.536785 + 0.843719i \(0.680361\pi\)
\(602\) 6975.19i 0.472239i
\(603\) 3184.47i 0.215061i
\(604\) 32472.4 2.18755
\(605\) 0 0
\(606\) −14586.6 −0.977790
\(607\) − 740.284i − 0.0495011i −0.999694 0.0247506i \(-0.992121\pi\)
0.999694 0.0247506i \(-0.00787916\pi\)
\(608\) 125885.i 8.39687i
\(609\) −2892.54 −0.192466
\(610\) 0 0
\(611\) 662.441 0.0438617
\(612\) 58034.7i 3.83319i
\(613\) 12412.1i 0.817815i 0.912576 + 0.408908i \(0.134090\pi\)
−0.912576 + 0.408908i \(0.865910\pi\)
\(614\) 31531.0 2.07246
\(615\) 0 0
\(616\) 37426.6 2.44799
\(617\) − 27047.9i − 1.76484i −0.470459 0.882422i \(-0.655911\pi\)
0.470459 0.882422i \(-0.344089\pi\)
\(618\) − 2576.95i − 0.167735i
\(619\) −193.644 −0.0125738 −0.00628691 0.999980i \(-0.502001\pi\)
−0.00628691 + 0.999980i \(0.502001\pi\)
\(620\) 0 0
\(621\) −2197.27 −0.141986
\(622\) − 36645.6i − 2.36231i
\(623\) − 3103.48i − 0.199580i
\(624\) −4920.19 −0.315649
\(625\) 0 0
\(626\) −15218.0 −0.971619
\(627\) 9171.36i 0.584161i
\(628\) − 13169.6i − 0.836819i
\(629\) −26298.5 −1.66708
\(630\) 0 0
\(631\) 11459.7 0.722982 0.361491 0.932376i \(-0.382268\pi\)
0.361491 + 0.932376i \(0.382268\pi\)
\(632\) − 31615.2i − 1.98985i
\(633\) 290.275i 0.0182265i
\(634\) 42314.4 2.65066
\(635\) 0 0
\(636\) −21171.4 −1.31997
\(637\) 1843.60i 0.114672i
\(638\) 28222.4i 1.75131i
\(639\) −3578.52 −0.221540
\(640\) 0 0
\(641\) 2261.29 0.139338 0.0696689 0.997570i \(-0.477806\pi\)
0.0696689 + 0.997570i \(0.477806\pi\)
\(642\) − 6918.85i − 0.425335i
\(643\) − 10224.8i − 0.627101i −0.949572 0.313551i \(-0.898481\pi\)
0.949572 0.313551i \(-0.101519\pi\)
\(644\) 6173.77 0.377765
\(645\) 0 0
\(646\) 76040.6 4.63123
\(647\) − 16733.4i − 1.01678i −0.861127 0.508390i \(-0.830241\pi\)
0.861127 0.508390i \(-0.169759\pi\)
\(648\) − 39100.3i − 2.37038i
\(649\) −22936.5 −1.38727
\(650\) 0 0
\(651\) −2298.87 −0.138402
\(652\) 50071.3i 3.00758i
\(653\) 9106.08i 0.545710i 0.962055 + 0.272855i \(0.0879679\pi\)
−0.962055 + 0.272855i \(0.912032\pi\)
\(654\) −13079.1 −0.782010
\(655\) 0 0
\(656\) 40086.0 2.38582
\(657\) − 28749.0i − 1.70716i
\(658\) 4890.43i 0.289740i
\(659\) 19951.5 1.17936 0.589680 0.807637i \(-0.299254\pi\)
0.589680 + 0.807637i \(0.299254\pi\)
\(660\) 0 0
\(661\) 16531.2 0.972750 0.486375 0.873750i \(-0.338319\pi\)
0.486375 + 0.873750i \(0.338319\pi\)
\(662\) 33605.8i 1.97300i
\(663\) 1741.27i 0.101999i
\(664\) 66132.3 3.86511
\(665\) 0 0
\(666\) 32670.0 1.90080
\(667\) 3067.82i 0.178091i
\(668\) 30862.8i 1.78760i
\(669\) −5815.24 −0.336069
\(670\) 0 0
\(671\) −13817.4 −0.794957
\(672\) − 21281.1i − 1.22163i
\(673\) 10268.9i 0.588169i 0.955779 + 0.294084i \(0.0950147\pi\)
−0.955779 + 0.294084i \(0.904985\pi\)
\(674\) −31776.3 −1.81599
\(675\) 0 0
\(676\) −49765.3 −2.83144
\(677\) 4729.79i 0.268509i 0.990947 + 0.134254i \(0.0428639\pi\)
−0.990947 + 0.134254i \(0.957136\pi\)
\(678\) 2540.07i 0.143880i
\(679\) −6573.86 −0.371549
\(680\) 0 0
\(681\) −8604.56 −0.484182
\(682\) 22430.0i 1.25937i
\(683\) − 11551.0i − 0.647123i −0.946207 0.323561i \(-0.895120\pi\)
0.946207 0.323561i \(-0.104880\pi\)
\(684\) −70440.8 −3.93768
\(685\) 0 0
\(686\) −35623.6 −1.98268
\(687\) − 2797.77i − 0.155374i
\(688\) − 32456.3i − 1.79853i
\(689\) 4139.89 0.228907
\(690\) 0 0
\(691\) 26575.2 1.46305 0.731526 0.681813i \(-0.238808\pi\)
0.731526 + 0.681813i \(0.238808\pi\)
\(692\) 15871.2i 0.871866i
\(693\) 10104.5i 0.553879i
\(694\) −31371.6 −1.71592
\(695\) 0 0
\(696\) 21917.5 1.19365
\(697\) − 14186.6i − 0.770955i
\(698\) − 24143.6i − 1.30924i
\(699\) −171.952 −0.00930448
\(700\) 0 0
\(701\) 5886.27 0.317149 0.158574 0.987347i \(-0.449310\pi\)
0.158574 + 0.987347i \(0.449310\pi\)
\(702\) − 4658.18i − 0.250444i
\(703\) − 31920.4i − 1.71252i
\(704\) −117513. −6.29110
\(705\) 0 0
\(706\) 5926.48 0.315929
\(707\) 15702.0i 0.835268i
\(708\) 27030.7i 1.43485i
\(709\) 2588.26 0.137100 0.0685501 0.997648i \(-0.478163\pi\)
0.0685501 + 0.997648i \(0.478163\pi\)
\(710\) 0 0
\(711\) 8535.51 0.450220
\(712\) 23515.8i 1.23777i
\(713\) 2438.18i 0.128065i
\(714\) −12854.8 −0.673781
\(715\) 0 0
\(716\) −87686.8 −4.57683
\(717\) 3068.63i 0.159833i
\(718\) − 29089.5i − 1.51199i
\(719\) −1170.46 −0.0607106 −0.0303553 0.999539i \(-0.509664\pi\)
−0.0303553 + 0.999539i \(0.509664\pi\)
\(720\) 0 0
\(721\) −2774.00 −0.143286
\(722\) 53825.1i 2.77447i
\(723\) − 12219.2i − 0.628545i
\(724\) 43947.0 2.25591
\(725\) 0 0
\(726\) −979.376 −0.0500662
\(727\) − 3135.70i − 0.159968i −0.996796 0.0799838i \(-0.974513\pi\)
0.996796 0.0799838i \(-0.0254869\pi\)
\(728\) 8624.82i 0.439089i
\(729\) 5667.88 0.287958
\(730\) 0 0
\(731\) −11486.4 −0.581177
\(732\) 16283.9i 0.822226i
\(733\) 25792.4i 1.29968i 0.760072 + 0.649839i \(0.225164\pi\)
−0.760072 + 0.649839i \(0.774836\pi\)
\(734\) −1002.70 −0.0504225
\(735\) 0 0
\(736\) −22570.7 −1.13039
\(737\) 5132.07i 0.256502i
\(738\) 17623.6i 0.879044i
\(739\) 809.931 0.0403164 0.0201582 0.999797i \(-0.493583\pi\)
0.0201582 + 0.999797i \(0.493583\pi\)
\(740\) 0 0
\(741\) −2113.50 −0.104779
\(742\) 30562.4i 1.51210i
\(743\) − 1825.32i − 0.0901270i −0.998984 0.0450635i \(-0.985651\pi\)
0.998984 0.0450635i \(-0.0143490\pi\)
\(744\) 17419.1 0.858355
\(745\) 0 0
\(746\) 41863.2 2.05459
\(747\) 17854.5i 0.874514i
\(748\) 93528.2i 4.57183i
\(749\) −7447.91 −0.363339
\(750\) 0 0
\(751\) 4310.27 0.209433 0.104716 0.994502i \(-0.466606\pi\)
0.104716 + 0.994502i \(0.466606\pi\)
\(752\) − 22755.7i − 1.10348i
\(753\) 6497.06i 0.314430i
\(754\) −6503.73 −0.314127
\(755\) 0 0
\(756\) 25643.5 1.23366
\(757\) 19302.2i 0.926750i 0.886162 + 0.463375i \(0.153362\pi\)
−0.886162 + 0.463375i \(0.846638\pi\)
\(758\) 34078.9i 1.63298i
\(759\) −1644.39 −0.0786398
\(760\) 0 0
\(761\) −30144.8 −1.43594 −0.717968 0.696076i \(-0.754927\pi\)
−0.717968 + 0.696076i \(0.754927\pi\)
\(762\) − 27727.5i − 1.31819i
\(763\) 14079.3i 0.668025i
\(764\) 121020. 5.73084
\(765\) 0 0
\(766\) −3257.07 −0.153633
\(767\) − 5285.62i − 0.248830i
\(768\) 55038.8i 2.58599i
\(769\) −37297.7 −1.74901 −0.874506 0.485015i \(-0.838814\pi\)
−0.874506 + 0.485015i \(0.838814\pi\)
\(770\) 0 0
\(771\) −4285.19 −0.200165
\(772\) − 112751.i − 5.25647i
\(773\) 13602.8i 0.632933i 0.948604 + 0.316467i \(0.102497\pi\)
−0.948604 + 0.316467i \(0.897503\pi\)
\(774\) 14269.3 0.662659
\(775\) 0 0
\(776\) 49811.8 2.30430
\(777\) 5396.21i 0.249148i
\(778\) − 63791.3i − 2.93963i
\(779\) 17219.3 0.791970
\(780\) 0 0
\(781\) −5767.10 −0.264229
\(782\) 13633.8i 0.623457i
\(783\) 12742.6i 0.581587i
\(784\) 63330.1 2.88493
\(785\) 0 0
\(786\) 9953.79 0.451705
\(787\) 14129.9i 0.639997i 0.947418 + 0.319998i \(0.103682\pi\)
−0.947418 + 0.319998i \(0.896318\pi\)
\(788\) − 75127.8i − 3.39634i
\(789\) 10081.8 0.454905
\(790\) 0 0
\(791\) 2734.30 0.122908
\(792\) − 76564.3i − 3.43509i
\(793\) − 3184.18i − 0.142589i
\(794\) −20760.7 −0.927922
\(795\) 0 0
\(796\) −51854.0 −2.30894
\(797\) 16171.3i 0.718715i 0.933200 + 0.359357i \(0.117004\pi\)
−0.933200 + 0.359357i \(0.882996\pi\)
\(798\) − 15602.8i − 0.692147i
\(799\) −8053.32 −0.356578
\(800\) 0 0
\(801\) −6348.83 −0.280056
\(802\) − 42710.9i − 1.88052i
\(803\) − 46331.7i − 2.03613i
\(804\) 6048.15 0.265301
\(805\) 0 0
\(806\) −5168.91 −0.225890
\(807\) 3776.20i 0.164720i
\(808\) − 118978.i − 5.18023i
\(809\) 41744.9 1.81418 0.907090 0.420937i \(-0.138299\pi\)
0.907090 + 0.420937i \(0.138299\pi\)
\(810\) 0 0
\(811\) −12210.2 −0.528679 −0.264340 0.964430i \(-0.585154\pi\)
−0.264340 + 0.964430i \(0.585154\pi\)
\(812\) − 35803.4i − 1.54736i
\(813\) 7487.19i 0.322986i
\(814\) 52650.6 2.26708
\(815\) 0 0
\(816\) 59814.9 2.56610
\(817\) − 13941.9i − 0.597019i
\(818\) 86507.4i 3.69763i
\(819\) −2328.54 −0.0993478
\(820\) 0 0
\(821\) −22326.2 −0.949074 −0.474537 0.880236i \(-0.657385\pi\)
−0.474537 + 0.880236i \(0.657385\pi\)
\(822\) − 4416.78i − 0.187412i
\(823\) 29128.6i 1.23373i 0.787069 + 0.616864i \(0.211597\pi\)
−0.787069 + 0.616864i \(0.788403\pi\)
\(824\) 21019.2 0.888641
\(825\) 0 0
\(826\) 39020.7 1.64371
\(827\) − 13540.8i − 0.569357i −0.958623 0.284679i \(-0.908113\pi\)
0.958623 0.284679i \(-0.0918869\pi\)
\(828\) − 12629.8i − 0.530091i
\(829\) −93.8298 −0.00393105 −0.00196553 0.999998i \(-0.500626\pi\)
−0.00196553 + 0.999998i \(0.500626\pi\)
\(830\) 0 0
\(831\) 337.585 0.0140923
\(832\) − 27080.4i − 1.12842i
\(833\) − 22412.7i − 0.932239i
\(834\) −10094.3 −0.419110
\(835\) 0 0
\(836\) −113522. −4.69645
\(837\) 10127.3i 0.418220i
\(838\) − 23967.4i − 0.987997i
\(839\) 33750.6 1.38880 0.694398 0.719591i \(-0.255671\pi\)
0.694398 + 0.719591i \(0.255671\pi\)
\(840\) 0 0
\(841\) −6597.88 −0.270527
\(842\) − 34081.1i − 1.39491i
\(843\) 6677.70i 0.272826i
\(844\) −3592.98 −0.146535
\(845\) 0 0
\(846\) 10004.4 0.406571
\(847\) 1054.26i 0.0427686i
\(848\) − 142210.i − 5.75887i
\(849\) −1325.28 −0.0535730
\(850\) 0 0
\(851\) 5723.21 0.230539
\(852\) 6796.54i 0.273293i
\(853\) 2680.73i 0.107604i 0.998552 + 0.0538022i \(0.0171340\pi\)
−0.998552 + 0.0538022i \(0.982866\pi\)
\(854\) 23506.9 0.941910
\(855\) 0 0
\(856\) 56434.7 2.25338
\(857\) 29267.7i 1.16659i 0.812261 + 0.583295i \(0.198237\pi\)
−0.812261 + 0.583295i \(0.801763\pi\)
\(858\) − 3486.09i − 0.138710i
\(859\) −34842.9 −1.38396 −0.691982 0.721914i \(-0.743262\pi\)
−0.691982 + 0.721914i \(0.743262\pi\)
\(860\) 0 0
\(861\) −2910.95 −0.115221
\(862\) − 18991.6i − 0.750412i
\(863\) 4041.96i 0.159432i 0.996818 + 0.0797161i \(0.0254014\pi\)
−0.996818 + 0.0797161i \(0.974599\pi\)
\(864\) −93750.1 −3.69149
\(865\) 0 0
\(866\) 44447.6 1.74410
\(867\) − 11857.6i − 0.464482i
\(868\) − 28455.1i − 1.11271i
\(869\) 13755.7 0.536975
\(870\) 0 0
\(871\) −1182.66 −0.0460081
\(872\) − 106682.i − 4.14301i
\(873\) 13448.3i 0.521368i
\(874\) −16548.3 −0.640451
\(875\) 0 0
\(876\) −54601.9 −2.10597
\(877\) 34918.5i 1.34448i 0.740331 + 0.672242i \(0.234669\pi\)
−0.740331 + 0.672242i \(0.765331\pi\)
\(878\) − 75891.9i − 2.91712i
\(879\) 16209.5 0.621996
\(880\) 0 0
\(881\) 2473.77 0.0946008 0.0473004 0.998881i \(-0.484938\pi\)
0.0473004 + 0.998881i \(0.484938\pi\)
\(882\) 27842.8i 1.06294i
\(883\) − 16956.8i − 0.646252i −0.946356 0.323126i \(-0.895266\pi\)
0.946356 0.323126i \(-0.104734\pi\)
\(884\) −21553.2 −0.820037
\(885\) 0 0
\(886\) 5586.64 0.211836
\(887\) 582.058i 0.0220334i 0.999939 + 0.0110167i \(0.00350679\pi\)
−0.999939 + 0.0110167i \(0.996493\pi\)
\(888\) − 40888.4i − 1.54519i
\(889\) −29847.7 −1.12605
\(890\) 0 0
\(891\) 17012.5 0.639664
\(892\) − 71980.1i − 2.70188i
\(893\) − 9774.88i − 0.366298i
\(894\) 23490.1 0.878775
\(895\) 0 0
\(896\) 110088. 4.10465
\(897\) − 378.943i − 0.0141054i
\(898\) 28489.0i 1.05868i
\(899\) 14139.7 0.524566
\(900\) 0 0
\(901\) −50328.7 −1.86092
\(902\) 28402.1i 1.04843i
\(903\) 2356.90i 0.0868580i
\(904\) −20718.5 −0.762263
\(905\) 0 0
\(906\) 14714.2 0.539567
\(907\) 26224.8i 0.960065i 0.877251 + 0.480033i \(0.159375\pi\)
−0.877251 + 0.480033i \(0.840625\pi\)
\(908\) − 106506.i − 3.89265i
\(909\) 32121.8 1.17207
\(910\) 0 0
\(911\) −12266.6 −0.446117 −0.223058 0.974805i \(-0.571604\pi\)
−0.223058 + 0.974805i \(0.571604\pi\)
\(912\) 72601.5i 2.63605i
\(913\) 28774.1i 1.04303i
\(914\) 62767.6 2.27152
\(915\) 0 0
\(916\) 34630.4 1.24915
\(917\) − 10714.9i − 0.385864i
\(918\) 56629.7i 2.03601i
\(919\) −12114.7 −0.434850 −0.217425 0.976077i \(-0.569766\pi\)
−0.217425 + 0.976077i \(0.569766\pi\)
\(920\) 0 0
\(921\) 10654.3 0.381184
\(922\) 13607.3i 0.486045i
\(923\) − 1329.01i − 0.0473941i
\(924\) 19191.1 0.683269
\(925\) 0 0
\(926\) −91699.1 −3.25423
\(927\) 5674.81i 0.201063i
\(928\) 130893.i 4.63016i
\(929\) 2418.31 0.0854059 0.0427029 0.999088i \(-0.486403\pi\)
0.0427029 + 0.999088i \(0.486403\pi\)
\(930\) 0 0
\(931\) 27203.9 0.957650
\(932\) − 2128.40i − 0.0748048i
\(933\) − 12382.5i − 0.434496i
\(934\) 16157.8 0.566059
\(935\) 0 0
\(936\) 17643.9 0.616143
\(937\) 16448.7i 0.573486i 0.958008 + 0.286743i \(0.0925726\pi\)
−0.958008 + 0.286743i \(0.907427\pi\)
\(938\) − 8730.94i − 0.303918i
\(939\) −5142.13 −0.178708
\(940\) 0 0
\(941\) −17373.2 −0.601859 −0.300929 0.953646i \(-0.597297\pi\)
−0.300929 + 0.953646i \(0.597297\pi\)
\(942\) − 5967.53i − 0.206404i
\(943\) 3087.35i 0.106615i
\(944\) −181568. −6.26009
\(945\) 0 0
\(946\) 22996.2 0.790350
\(947\) 18638.9i 0.639579i 0.947489 + 0.319790i \(0.103612\pi\)
−0.947489 + 0.319790i \(0.896388\pi\)
\(948\) − 16211.2i − 0.555395i
\(949\) 10676.9 0.365214
\(950\) 0 0
\(951\) 14298.0 0.487532
\(952\) − 104852.i − 3.56962i
\(953\) 31188.1i 1.06011i 0.847965 + 0.530053i \(0.177828\pi\)
−0.847965 + 0.530053i \(0.822172\pi\)
\(954\) 62522.0 2.12183
\(955\) 0 0
\(956\) −37983.0 −1.28500
\(957\) 9536.28i 0.322115i
\(958\) − 44030.7i − 1.48493i
\(959\) −4754.51 −0.160095
\(960\) 0 0
\(961\) −18553.3 −0.622784
\(962\) 12133.1i 0.406640i
\(963\) 15236.3i 0.509848i
\(964\) 151248. 5.05328
\(965\) 0 0
\(966\) 2797.52 0.0931768
\(967\) 43197.8i 1.43655i 0.695757 + 0.718277i \(0.255069\pi\)
−0.695757 + 0.718277i \(0.744931\pi\)
\(968\) − 7988.42i − 0.265246i
\(969\) 25693.9 0.851815
\(970\) 0 0
\(971\) 13497.0 0.446074 0.223037 0.974810i \(-0.428403\pi\)
0.223037 + 0.974810i \(0.428403\pi\)
\(972\) − 80558.1i − 2.65834i
\(973\) 10866.2i 0.358021i
\(974\) 63168.4 2.07808
\(975\) 0 0
\(976\) −109380. −3.58728
\(977\) − 34955.4i − 1.14465i −0.820027 0.572325i \(-0.806042\pi\)
0.820027 0.572325i \(-0.193958\pi\)
\(978\) 22688.8i 0.741830i
\(979\) −10231.7 −0.334021
\(980\) 0 0
\(981\) 28802.2 0.937393
\(982\) 88678.9i 2.88173i
\(983\) 27521.6i 0.892984i 0.894788 + 0.446492i \(0.147327\pi\)
−0.894788 + 0.446492i \(0.852673\pi\)
\(984\) 22057.0 0.714585
\(985\) 0 0
\(986\) 79066.1 2.55373
\(987\) 1652.46i 0.0532913i
\(988\) − 26160.6i − 0.842389i
\(989\) 2499.73 0.0803707
\(990\) 0 0
\(991\) 24567.9 0.787513 0.393756 0.919215i \(-0.371175\pi\)
0.393756 + 0.919215i \(0.371175\pi\)
\(992\) 104029.i 3.32956i
\(993\) 11355.3i 0.362891i
\(994\) 9811.29 0.313074
\(995\) 0 0
\(996\) 33910.4 1.07881
\(997\) 2293.84i 0.0728652i 0.999336 + 0.0364326i \(0.0115994\pi\)
−0.999336 + 0.0364326i \(0.988401\pi\)
\(998\) 58726.5i 1.86268i
\(999\) 23772.1 0.752868
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 575.4.b.i.24.10 10
5.2 odd 4 575.4.a.j.1.1 5
5.3 odd 4 115.4.a.e.1.5 5
5.4 even 2 inner 575.4.b.i.24.1 10
15.8 even 4 1035.4.a.k.1.1 5
20.3 even 4 1840.4.a.n.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.5 5 5.3 odd 4
575.4.a.j.1.1 5 5.2 odd 4
575.4.b.i.24.1 10 5.4 even 2 inner
575.4.b.i.24.10 10 1.1 even 1 trivial
1035.4.a.k.1.1 5 15.8 even 4
1840.4.a.n.1.3 5 20.3 even 4