Properties

Label 75.4.b.c.49.4
Level $75$
Weight $4$
Character 75.49
Analytic conductor $4.425$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,4,Mod(49,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 9x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(-2.17945 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.4.b.c.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.35890i q^{2} +3.00000i q^{3} -20.7178 q^{4} -16.0767 q^{6} +4.43560i q^{7} -68.1534i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+5.35890i q^{2} +3.00000i q^{3} -20.7178 q^{4} -16.0767 q^{6} +4.43560i q^{7} -68.1534i q^{8} -9.00000 q^{9} -3.43560 q^{11} -62.1534i q^{12} +78.7424i q^{13} -23.7699 q^{14} +199.485 q^{16} -53.1780i q^{17} -48.2301i q^{18} -20.4356 q^{19} -13.3068 q^{21} -18.4110i q^{22} +118.307i q^{23} +204.460 q^{24} -421.972 q^{26} -27.0000i q^{27} -91.8958i q^{28} -168.049 q^{29} -61.0492 q^{31} +523.792i q^{32} -10.3068i q^{33} +284.975 q^{34} +186.460 q^{36} +246.614i q^{37} -109.512i q^{38} -236.227 q^{39} +422.663 q^{41} -71.3097i q^{42} +362.436i q^{43} +71.1780 q^{44} -633.994 q^{46} +170.515i q^{47} +598.454i q^{48} +323.325 q^{49} +159.534 q^{51} -1631.37i q^{52} -546.049i q^{53} +144.690 q^{54} +302.301 q^{56} -61.3068i q^{57} -900.559i q^{58} +216.970 q^{59} +130.902 q^{61} -327.156i q^{62} -39.9204i q^{63} -1211.07 q^{64} +55.2330 q^{66} +614.890i q^{67} +1101.73i q^{68} -354.920 q^{69} +324.822 q^{71} +613.381i q^{72} -88.8712i q^{73} -1321.58 q^{74} +423.381 q^{76} -15.2389i q^{77} -1265.92i q^{78} +1137.42 q^{79} +81.0000 q^{81} +2265.01i q^{82} -758.909i q^{83} +275.687 q^{84} -1942.26 q^{86} -504.148i q^{87} +234.148i q^{88} -195.681 q^{89} -349.269 q^{91} -2451.06i q^{92} -183.148i q^{93} -913.774 q^{94} -1571.37 q^{96} -521.000i q^{97} +1732.67i q^{98} +30.9204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 48 q^{4} - 12 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 48 q^{4} - 12 q^{6} - 36 q^{9} + 56 q^{11} - 252 q^{14} + 240 q^{16} - 12 q^{19} + 156 q^{21} + 504 q^{24} - 1252 q^{26} - 184 q^{29} + 244 q^{31} + 1384 q^{34} + 432 q^{36} - 108 q^{39} + 784 q^{41} - 64 q^{44} - 1176 q^{46} - 520 q^{49} - 408 q^{51} + 108 q^{54} - 360 q^{56} - 248 q^{59} + 1500 q^{61} - 2752 q^{64} + 744 q^{66} - 792 q^{69} + 1648 q^{71} - 2392 q^{74} + 752 q^{76} + 1760 q^{79} + 324 q^{81} - 48 q^{84} - 1684 q^{86} + 1728 q^{89} - 4396 q^{91} + 1192 q^{94} - 3984 q^{96} - 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.35890i 1.89466i 0.320264 + 0.947328i \(0.396228\pi\)
−0.320264 + 0.947328i \(0.603772\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −20.7178 −2.58972
\(5\) 0 0
\(6\) −16.0767 −1.09388
\(7\) 4.43560i 0.239500i 0.992804 + 0.119750i \(0.0382092\pi\)
−0.992804 + 0.119750i \(0.961791\pi\)
\(8\) − 68.1534i − 3.01198i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −3.43560 −0.0941701 −0.0470851 0.998891i \(-0.514993\pi\)
−0.0470851 + 0.998891i \(0.514993\pi\)
\(12\) − 62.1534i − 1.49518i
\(13\) 78.7424i 1.67994i 0.542634 + 0.839970i \(0.317427\pi\)
−0.542634 + 0.839970i \(0.682573\pi\)
\(14\) −23.7699 −0.453770
\(15\) 0 0
\(16\) 199.485 3.11695
\(17\) − 53.1780i − 0.758680i −0.925257 0.379340i \(-0.876151\pi\)
0.925257 0.379340i \(-0.123849\pi\)
\(18\) − 48.2301i − 0.631552i
\(19\) −20.4356 −0.246750 −0.123375 0.992360i \(-0.539372\pi\)
−0.123375 + 0.992360i \(0.539372\pi\)
\(20\) 0 0
\(21\) −13.3068 −0.138275
\(22\) − 18.4110i − 0.178420i
\(23\) 118.307i 1.07255i 0.844043 + 0.536275i \(0.180169\pi\)
−0.844043 + 0.536275i \(0.819831\pi\)
\(24\) 204.460 1.73897
\(25\) 0 0
\(26\) −421.972 −3.18291
\(27\) − 27.0000i − 0.192450i
\(28\) − 91.8958i − 0.620238i
\(29\) −168.049 −1.07607 −0.538034 0.842923i \(-0.680833\pi\)
−0.538034 + 0.842923i \(0.680833\pi\)
\(30\) 0 0
\(31\) −61.0492 −0.353702 −0.176851 0.984238i \(-0.556591\pi\)
−0.176851 + 0.984238i \(0.556591\pi\)
\(32\) 523.792i 2.89357i
\(33\) − 10.3068i − 0.0543691i
\(34\) 284.975 1.43744
\(35\) 0 0
\(36\) 186.460 0.863242
\(37\) 246.614i 1.09576i 0.836558 + 0.547879i \(0.184564\pi\)
−0.836558 + 0.547879i \(0.815436\pi\)
\(38\) − 109.512i − 0.467506i
\(39\) −236.227 −0.969913
\(40\) 0 0
\(41\) 422.663 1.60997 0.804986 0.593294i \(-0.202173\pi\)
0.804986 + 0.593294i \(0.202173\pi\)
\(42\) − 71.3097i − 0.261984i
\(43\) 362.436i 1.28537i 0.766131 + 0.642685i \(0.222180\pi\)
−0.766131 + 0.642685i \(0.777820\pi\)
\(44\) 71.1780 0.243875
\(45\) 0 0
\(46\) −633.994 −2.03212
\(47\) 170.515i 0.529196i 0.964359 + 0.264598i \(0.0852392\pi\)
−0.964359 + 0.264598i \(0.914761\pi\)
\(48\) 598.454i 1.79957i
\(49\) 323.325 0.942640
\(50\) 0 0
\(51\) 159.534 0.438024
\(52\) − 1631.37i − 4.35058i
\(53\) − 546.049i − 1.41520i −0.706613 0.707600i \(-0.749778\pi\)
0.706613 0.707600i \(-0.250222\pi\)
\(54\) 144.690 0.364627
\(55\) 0 0
\(56\) 302.301 0.721369
\(57\) − 61.3068i − 0.142461i
\(58\) − 900.559i − 2.03878i
\(59\) 216.970 0.478763 0.239382 0.970926i \(-0.423055\pi\)
0.239382 + 0.970926i \(0.423055\pi\)
\(60\) 0 0
\(61\) 130.902 0.274758 0.137379 0.990519i \(-0.456132\pi\)
0.137379 + 0.990519i \(0.456132\pi\)
\(62\) − 327.156i − 0.670143i
\(63\) − 39.9204i − 0.0798332i
\(64\) −1211.07 −2.36537
\(65\) 0 0
\(66\) 55.2330 0.103011
\(67\) 614.890i 1.12121i 0.828085 + 0.560603i \(0.189430\pi\)
−0.828085 + 0.560603i \(0.810570\pi\)
\(68\) 1101.73i 1.96477i
\(69\) −354.920 −0.619238
\(70\) 0 0
\(71\) 324.822 0.542948 0.271474 0.962446i \(-0.412489\pi\)
0.271474 + 0.962446i \(0.412489\pi\)
\(72\) 613.381i 1.00399i
\(73\) − 88.8712i − 0.142487i −0.997459 0.0712437i \(-0.977303\pi\)
0.997459 0.0712437i \(-0.0226968\pi\)
\(74\) −1321.58 −2.07608
\(75\) 0 0
\(76\) 423.381 0.639014
\(77\) − 15.2389i − 0.0225537i
\(78\) − 1265.92i − 1.83765i
\(79\) 1137.42 1.61988 0.809938 0.586516i \(-0.199501\pi\)
0.809938 + 0.586516i \(0.199501\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 2265.01i 3.05034i
\(83\) − 758.909i − 1.00363i −0.864976 0.501813i \(-0.832666\pi\)
0.864976 0.501813i \(-0.167334\pi\)
\(84\) 275.687 0.358095
\(85\) 0 0
\(86\) −1942.26 −2.43534
\(87\) − 504.148i − 0.621268i
\(88\) 234.148i 0.283639i
\(89\) −195.681 −0.233058 −0.116529 0.993187i \(-0.537177\pi\)
−0.116529 + 0.993187i \(0.537177\pi\)
\(90\) 0 0
\(91\) −349.269 −0.402345
\(92\) − 2451.06i − 2.77761i
\(93\) − 183.148i − 0.204210i
\(94\) −913.774 −1.00264
\(95\) 0 0
\(96\) −1571.37 −1.67060
\(97\) − 521.000i − 0.545356i −0.962105 0.272678i \(-0.912091\pi\)
0.962105 0.272678i \(-0.0879094\pi\)
\(98\) 1732.67i 1.78598i
\(99\) 30.9204 0.0313900
\(100\) 0 0
\(101\) 660.920 0.651129 0.325565 0.945520i \(-0.394446\pi\)
0.325565 + 0.945520i \(0.394446\pi\)
\(102\) 854.926i 0.829905i
\(103\) 1530.75i 1.46436i 0.681110 + 0.732181i \(0.261497\pi\)
−0.681110 + 0.732181i \(0.738503\pi\)
\(104\) 5366.56 5.05995
\(105\) 0 0
\(106\) 2926.22 2.68132
\(107\) 264.625i 0.239087i 0.992829 + 0.119543i \(0.0381431\pi\)
−0.992829 + 0.119543i \(0.961857\pi\)
\(108\) 559.381i 0.498393i
\(109\) −1117.61 −0.982091 −0.491046 0.871134i \(-0.663385\pi\)
−0.491046 + 0.871134i \(0.663385\pi\)
\(110\) 0 0
\(111\) −739.841 −0.632636
\(112\) 884.834i 0.746508i
\(113\) − 934.061i − 0.777602i −0.921322 0.388801i \(-0.872889\pi\)
0.921322 0.388801i \(-0.127111\pi\)
\(114\) 328.537 0.269915
\(115\) 0 0
\(116\) 3481.61 2.78672
\(117\) − 708.681i − 0.559980i
\(118\) 1162.72i 0.907092i
\(119\) 235.876 0.181704
\(120\) 0 0
\(121\) −1319.20 −0.991132
\(122\) 701.489i 0.520572i
\(123\) 1267.99i 0.929517i
\(124\) 1264.80 0.915990
\(125\) 0 0
\(126\) 213.929 0.151257
\(127\) − 630.356i − 0.440433i −0.975451 0.220217i \(-0.929324\pi\)
0.975451 0.220217i \(-0.0706765\pi\)
\(128\) − 2299.66i − 1.58799i
\(129\) −1087.31 −0.742109
\(130\) 0 0
\(131\) −2163.06 −1.44265 −0.721325 0.692597i \(-0.756466\pi\)
−0.721325 + 0.692597i \(0.756466\pi\)
\(132\) 213.534i 0.140801i
\(133\) − 90.6440i − 0.0590965i
\(134\) −3295.13 −2.12430
\(135\) 0 0
\(136\) −3624.26 −2.28513
\(137\) − 1118.61i − 0.697588i −0.937199 0.348794i \(-0.886591\pi\)
0.937199 0.348794i \(-0.113409\pi\)
\(138\) − 1901.98i − 1.17324i
\(139\) 166.478 0.101586 0.0507930 0.998709i \(-0.483825\pi\)
0.0507930 + 0.998709i \(0.483825\pi\)
\(140\) 0 0
\(141\) −511.546 −0.305531
\(142\) 1740.69i 1.02870i
\(143\) − 270.527i − 0.158200i
\(144\) −1795.36 −1.03898
\(145\) 0 0
\(146\) 476.252 0.269965
\(147\) 969.976i 0.544233i
\(148\) − 5109.29i − 2.83771i
\(149\) 653.143 0.359111 0.179555 0.983748i \(-0.442534\pi\)
0.179555 + 0.983748i \(0.442534\pi\)
\(150\) 0 0
\(151\) −1929.38 −1.03981 −0.519903 0.854225i \(-0.674032\pi\)
−0.519903 + 0.854225i \(0.674032\pi\)
\(152\) 1392.76i 0.743206i
\(153\) 478.602i 0.252893i
\(154\) 81.6638 0.0427315
\(155\) 0 0
\(156\) 4894.11 2.51181
\(157\) − 2169.75i − 1.10296i −0.834188 0.551480i \(-0.814063\pi\)
0.834188 0.551480i \(-0.185937\pi\)
\(158\) 6095.34i 3.06911i
\(159\) 1638.15 0.817066
\(160\) 0 0
\(161\) −524.761 −0.256876
\(162\) 434.071i 0.210517i
\(163\) − 763.738i − 0.366997i −0.983020 0.183499i \(-0.941258\pi\)
0.983020 0.183499i \(-0.0587423\pi\)
\(164\) −8756.64 −4.16938
\(165\) 0 0
\(166\) 4066.91 1.90153
\(167\) 2564.28i 1.18820i 0.804389 + 0.594102i \(0.202493\pi\)
−0.804389 + 0.594102i \(0.797507\pi\)
\(168\) 906.903i 0.416483i
\(169\) −4003.36 −1.82220
\(170\) 0 0
\(171\) 183.920 0.0822500
\(172\) − 7508.87i − 3.32875i
\(173\) 51.8290i 0.0227774i 0.999935 + 0.0113887i \(0.00362521\pi\)
−0.999935 + 0.0113887i \(0.996375\pi\)
\(174\) 2701.68 1.17709
\(175\) 0 0
\(176\) −685.349 −0.293523
\(177\) 650.909i 0.276414i
\(178\) − 1048.64i − 0.441566i
\(179\) 3956.63 1.65214 0.826068 0.563571i \(-0.190573\pi\)
0.826068 + 0.563571i \(0.190573\pi\)
\(180\) 0 0
\(181\) 1804.04 0.740848 0.370424 0.928863i \(-0.379212\pi\)
0.370424 + 0.928863i \(0.379212\pi\)
\(182\) − 1871.70i − 0.762305i
\(183\) 392.705i 0.158632i
\(184\) 8063.01 3.23050
\(185\) 0 0
\(186\) 981.469 0.386908
\(187\) 182.698i 0.0714449i
\(188\) − 3532.70i − 1.37047i
\(189\) 119.761 0.0460917
\(190\) 0 0
\(191\) 3666.75 1.38909 0.694547 0.719448i \(-0.255605\pi\)
0.694547 + 0.719448i \(0.255605\pi\)
\(192\) − 3633.20i − 1.36565i
\(193\) − 2716.98i − 1.01333i −0.862144 0.506664i \(-0.830879\pi\)
0.862144 0.506664i \(-0.169121\pi\)
\(194\) 2791.99 1.03326
\(195\) 0 0
\(196\) −6698.59 −2.44118
\(197\) − 2034.30i − 0.735723i −0.929881 0.367862i \(-0.880090\pi\)
0.929881 0.367862i \(-0.119910\pi\)
\(198\) 165.699i 0.0594733i
\(199\) 1551.27 0.552596 0.276298 0.961072i \(-0.410892\pi\)
0.276298 + 0.961072i \(0.410892\pi\)
\(200\) 0 0
\(201\) −1844.67 −0.647328
\(202\) 3541.81i 1.23367i
\(203\) − 745.398i − 0.257718i
\(204\) −3305.19 −1.13436
\(205\) 0 0
\(206\) −8203.13 −2.77446
\(207\) − 1064.76i − 0.357517i
\(208\) 15707.9i 5.23629i
\(209\) 70.2084 0.0232365
\(210\) 0 0
\(211\) 3192.51 1.04162 0.520809 0.853673i \(-0.325630\pi\)
0.520809 + 0.853673i \(0.325630\pi\)
\(212\) 11312.9i 3.66498i
\(213\) 974.466i 0.313471i
\(214\) −1418.10 −0.452988
\(215\) 0 0
\(216\) −1840.14 −0.579656
\(217\) − 270.789i − 0.0847115i
\(218\) − 5989.18i − 1.86073i
\(219\) 266.614 0.0822652
\(220\) 0 0
\(221\) 4187.36 1.27454
\(222\) − 3964.73i − 1.19863i
\(223\) − 1555.55i − 0.467120i −0.972342 0.233560i \(-0.924963\pi\)
0.972342 0.233560i \(-0.0750374\pi\)
\(224\) −2323.33 −0.693008
\(225\) 0 0
\(226\) 5005.54 1.47329
\(227\) 6206.86i 1.81482i 0.420248 + 0.907409i \(0.361943\pi\)
−0.420248 + 0.907409i \(0.638057\pi\)
\(228\) 1270.14i 0.368935i
\(229\) −4679.51 −1.35035 −0.675176 0.737657i \(-0.735932\pi\)
−0.675176 + 0.737657i \(0.735932\pi\)
\(230\) 0 0
\(231\) 45.7167 0.0130214
\(232\) 11453.1i 3.24110i
\(233\) − 3244.53i − 0.912259i −0.889913 0.456129i \(-0.849235\pi\)
0.889913 0.456129i \(-0.150765\pi\)
\(234\) 3797.75 1.06097
\(235\) 0 0
\(236\) −4495.13 −1.23986
\(237\) 3412.27i 0.935236i
\(238\) 1264.04i 0.344266i
\(239\) 3658.62 0.990193 0.495097 0.868838i \(-0.335133\pi\)
0.495097 + 0.868838i \(0.335133\pi\)
\(240\) 0 0
\(241\) 1931.01 0.516131 0.258065 0.966127i \(-0.416915\pi\)
0.258065 + 0.966127i \(0.416915\pi\)
\(242\) − 7069.44i − 1.87786i
\(243\) 243.000i 0.0641500i
\(244\) −2711.99 −0.711548
\(245\) 0 0
\(246\) −6795.02 −1.76112
\(247\) − 1609.15i − 0.414525i
\(248\) 4160.71i 1.06534i
\(249\) 2276.73 0.579444
\(250\) 0 0
\(251\) −5843.34 −1.46944 −0.734718 0.678373i \(-0.762685\pi\)
−0.734718 + 0.678373i \(0.762685\pi\)
\(252\) 827.062i 0.206746i
\(253\) − 406.454i − 0.101002i
\(254\) 3378.01 0.834470
\(255\) 0 0
\(256\) 2635.09 0.643333
\(257\) − 4506.11i − 1.09371i −0.837227 0.546855i \(-0.815825\pi\)
0.837227 0.546855i \(-0.184175\pi\)
\(258\) − 5826.77i − 1.40604i
\(259\) −1093.88 −0.262434
\(260\) 0 0
\(261\) 1512.44 0.358689
\(262\) − 11591.6i − 2.73333i
\(263\) 5340.16i 1.25205i 0.779804 + 0.626024i \(0.215319\pi\)
−0.779804 + 0.626024i \(0.784681\pi\)
\(264\) −702.443 −0.163759
\(265\) 0 0
\(266\) 485.752 0.111968
\(267\) − 587.044i − 0.134556i
\(268\) − 12739.2i − 2.90361i
\(269\) −2809.79 −0.636863 −0.318431 0.947946i \(-0.603156\pi\)
−0.318431 + 0.947946i \(0.603156\pi\)
\(270\) 0 0
\(271\) 3102.95 0.695537 0.347769 0.937580i \(-0.386940\pi\)
0.347769 + 0.937580i \(0.386940\pi\)
\(272\) − 10608.2i − 2.36477i
\(273\) − 1047.81i − 0.232294i
\(274\) 5994.54 1.32169
\(275\) 0 0
\(276\) 7353.17 1.60365
\(277\) 4598.93i 0.997555i 0.866730 + 0.498777i \(0.166217\pi\)
−0.866730 + 0.498777i \(0.833783\pi\)
\(278\) 892.138i 0.192471i
\(279\) 549.443 0.117901
\(280\) 0 0
\(281\) 2571.83 0.545987 0.272994 0.962016i \(-0.411986\pi\)
0.272994 + 0.962016i \(0.411986\pi\)
\(282\) − 2741.32i − 0.578877i
\(283\) 5575.31i 1.17109i 0.810641 + 0.585544i \(0.199119\pi\)
−0.810641 + 0.585544i \(0.800881\pi\)
\(284\) −6729.60 −1.40608
\(285\) 0 0
\(286\) 1449.73 0.299735
\(287\) 1874.76i 0.385588i
\(288\) − 4714.12i − 0.964522i
\(289\) 2085.10 0.424405
\(290\) 0 0
\(291\) 1563.00 0.314861
\(292\) 1841.22i 0.369003i
\(293\) 5794.27i 1.15531i 0.816282 + 0.577654i \(0.196032\pi\)
−0.816282 + 0.577654i \(0.803968\pi\)
\(294\) −5198.01 −1.03114
\(295\) 0 0
\(296\) 16807.6 3.30040
\(297\) 92.7611i 0.0181230i
\(298\) 3500.13i 0.680392i
\(299\) −9315.76 −1.80182
\(300\) 0 0
\(301\) −1607.62 −0.307846
\(302\) − 10339.4i − 1.97008i
\(303\) 1982.76i 0.375930i
\(304\) −4076.59 −0.769107
\(305\) 0 0
\(306\) −2564.78 −0.479146
\(307\) 1404.47i 0.261099i 0.991442 + 0.130550i \(0.0416742\pi\)
−0.991442 + 0.130550i \(0.958326\pi\)
\(308\) 315.717i 0.0584079i
\(309\) −4592.25 −0.845449
\(310\) 0 0
\(311\) −4096.75 −0.746963 −0.373481 0.927638i \(-0.621836\pi\)
−0.373481 + 0.927638i \(0.621836\pi\)
\(312\) 16099.7i 2.92136i
\(313\) 974.611i 0.176001i 0.996120 + 0.0880004i \(0.0280477\pi\)
−0.996120 + 0.0880004i \(0.971952\pi\)
\(314\) 11627.5 2.08973
\(315\) 0 0
\(316\) −23564.9 −4.19503
\(317\) 2071.69i 0.367058i 0.983014 + 0.183529i \(0.0587522\pi\)
−0.983014 + 0.183529i \(0.941248\pi\)
\(318\) 8778.67i 1.54806i
\(319\) 577.349 0.101333
\(320\) 0 0
\(321\) −793.876 −0.138037
\(322\) − 2812.14i − 0.486691i
\(323\) 1086.72i 0.187204i
\(324\) −1678.14 −0.287747
\(325\) 0 0
\(326\) 4092.79 0.695334
\(327\) − 3352.84i − 0.567011i
\(328\) − 28805.9i − 4.84921i
\(329\) −756.337 −0.126742
\(330\) 0 0
\(331\) −6159.17 −1.02278 −0.511388 0.859350i \(-0.670868\pi\)
−0.511388 + 0.859350i \(0.670868\pi\)
\(332\) 15722.9i 2.59912i
\(333\) − 2219.52i − 0.365252i
\(334\) −13741.7 −2.25124
\(335\) 0 0
\(336\) −2654.50 −0.430997
\(337\) − 2791.26i − 0.451186i −0.974222 0.225593i \(-0.927568\pi\)
0.974222 0.225593i \(-0.0724320\pi\)
\(338\) − 21453.6i − 3.45243i
\(339\) 2802.18 0.448949
\(340\) 0 0
\(341\) 209.740 0.0333081
\(342\) 985.611i 0.155835i
\(343\) 2955.55i 0.465262i
\(344\) 24701.2 3.87151
\(345\) 0 0
\(346\) −277.746 −0.0431553
\(347\) − 940.848i − 0.145554i −0.997348 0.0727772i \(-0.976814\pi\)
0.997348 0.0727772i \(-0.0231862\pi\)
\(348\) 10444.8i 1.60891i
\(349\) 3519.62 0.539831 0.269915 0.962884i \(-0.413004\pi\)
0.269915 + 0.962884i \(0.413004\pi\)
\(350\) 0 0
\(351\) 2126.04 0.323304
\(352\) − 1799.54i − 0.272487i
\(353\) 5021.60i 0.757147i 0.925571 + 0.378573i \(0.123585\pi\)
−0.925571 + 0.378573i \(0.876415\pi\)
\(354\) −3488.15 −0.523710
\(355\) 0 0
\(356\) 4054.09 0.603557
\(357\) 707.628i 0.104907i
\(358\) 21203.2i 3.13023i
\(359\) −6811.99 −1.00146 −0.500728 0.865604i \(-0.666934\pi\)
−0.500728 + 0.865604i \(0.666934\pi\)
\(360\) 0 0
\(361\) −6441.39 −0.939115
\(362\) 9667.69i 1.40365i
\(363\) − 3957.59i − 0.572230i
\(364\) 7236.09 1.04196
\(365\) 0 0
\(366\) −2104.47 −0.300553
\(367\) − 3748.07i − 0.533099i −0.963821 0.266550i \(-0.914116\pi\)
0.963821 0.266550i \(-0.0858836\pi\)
\(368\) 23600.4i 3.34309i
\(369\) −3803.96 −0.536657
\(370\) 0 0
\(371\) 2422.05 0.338940
\(372\) 3794.41i 0.528847i
\(373\) − 898.302i − 0.124698i −0.998054 0.0623489i \(-0.980141\pi\)
0.998054 0.0623489i \(-0.0198592\pi\)
\(374\) −979.060 −0.135364
\(375\) 0 0
\(376\) 11621.2 1.59393
\(377\) − 13232.6i − 1.80773i
\(378\) 641.788i 0.0873280i
\(379\) 9378.99 1.27115 0.635576 0.772038i \(-0.280763\pi\)
0.635576 + 0.772038i \(0.280763\pi\)
\(380\) 0 0
\(381\) 1891.07 0.254284
\(382\) 19649.8i 2.63186i
\(383\) − 9446.29i − 1.26027i −0.776486 0.630134i \(-0.783000\pi\)
0.776486 0.630134i \(-0.217000\pi\)
\(384\) 6898.97 0.916828
\(385\) 0 0
\(386\) 14560.0 1.91991
\(387\) − 3261.92i − 0.428457i
\(388\) 10794.0i 1.41232i
\(389\) 7643.23 0.996214 0.498107 0.867116i \(-0.334029\pi\)
0.498107 + 0.867116i \(0.334029\pi\)
\(390\) 0 0
\(391\) 6291.32 0.813723
\(392\) − 22035.7i − 2.83922i
\(393\) − 6489.17i − 0.832914i
\(394\) 10901.6 1.39394
\(395\) 0 0
\(396\) −640.602 −0.0812915
\(397\) 12013.6i 1.51876i 0.650650 + 0.759378i \(0.274497\pi\)
−0.650650 + 0.759378i \(0.725503\pi\)
\(398\) 8313.10i 1.04698i
\(399\) 271.932 0.0341194
\(400\) 0 0
\(401\) −8538.51 −1.06332 −0.531662 0.846957i \(-0.678432\pi\)
−0.531662 + 0.846957i \(0.678432\pi\)
\(402\) − 9885.40i − 1.22646i
\(403\) − 4807.16i − 0.594197i
\(404\) −13692.8 −1.68625
\(405\) 0 0
\(406\) 3994.51 0.488287
\(407\) − 847.265i − 0.103188i
\(408\) − 10872.8i − 1.31932i
\(409\) 12267.6 1.48312 0.741558 0.670889i \(-0.234087\pi\)
0.741558 + 0.670889i \(0.234087\pi\)
\(410\) 0 0
\(411\) 3355.84 0.402753
\(412\) − 31713.8i − 3.79229i
\(413\) 962.389i 0.114664i
\(414\) 5705.95 0.677372
\(415\) 0 0
\(416\) −41244.6 −4.86102
\(417\) 499.433i 0.0586508i
\(418\) 376.240i 0.0440251i
\(419\) −15493.0 −1.80641 −0.903204 0.429212i \(-0.858791\pi\)
−0.903204 + 0.429212i \(0.858791\pi\)
\(420\) 0 0
\(421\) 7510.67 0.869472 0.434736 0.900558i \(-0.356842\pi\)
0.434736 + 0.900558i \(0.356842\pi\)
\(422\) 17108.3i 1.97351i
\(423\) − 1534.64i − 0.176399i
\(424\) −37215.1 −4.26256
\(425\) 0 0
\(426\) −5222.07 −0.593920
\(427\) 580.627i 0.0658045i
\(428\) − 5482.45i − 0.619169i
\(429\) 811.581 0.0913368
\(430\) 0 0
\(431\) −15675.3 −1.75186 −0.875932 0.482434i \(-0.839753\pi\)
−0.875932 + 0.482434i \(0.839753\pi\)
\(432\) − 5386.09i − 0.599857i
\(433\) − 9604.78i − 1.06600i −0.846117 0.532998i \(-0.821065\pi\)
0.846117 0.532998i \(-0.178935\pi\)
\(434\) 1451.13 0.160499
\(435\) 0 0
\(436\) 23154.5 2.54335
\(437\) − 2417.67i − 0.264652i
\(438\) 1428.76i 0.155864i
\(439\) 6362.06 0.691673 0.345837 0.938295i \(-0.387595\pi\)
0.345837 + 0.938295i \(0.387595\pi\)
\(440\) 0 0
\(441\) −2909.93 −0.314213
\(442\) 22439.6i 2.41481i
\(443\) 931.658i 0.0999196i 0.998751 + 0.0499598i \(0.0159093\pi\)
−0.998751 + 0.0499598i \(0.984091\pi\)
\(444\) 15327.9 1.63835
\(445\) 0 0
\(446\) 8336.06 0.885031
\(447\) 1959.43i 0.207333i
\(448\) − 5371.81i − 0.566505i
\(449\) 18684.1 1.96383 0.981914 0.189329i \(-0.0606314\pi\)
0.981914 + 0.189329i \(0.0606314\pi\)
\(450\) 0 0
\(451\) −1452.10 −0.151611
\(452\) 19351.7i 2.01378i
\(453\) − 5788.14i − 0.600333i
\(454\) −33261.9 −3.43846
\(455\) 0 0
\(456\) −4178.27 −0.429090
\(457\) 11565.9i 1.18387i 0.805985 + 0.591936i \(0.201636\pi\)
−0.805985 + 0.591936i \(0.798364\pi\)
\(458\) − 25077.0i − 2.55845i
\(459\) −1435.81 −0.146008
\(460\) 0 0
\(461\) 19401.0 1.96008 0.980039 0.198806i \(-0.0637062\pi\)
0.980039 + 0.198806i \(0.0637062\pi\)
\(462\) 244.991i 0.0246711i
\(463\) 1576.28i 0.158220i 0.996866 + 0.0791099i \(0.0252078\pi\)
−0.996866 + 0.0791099i \(0.974792\pi\)
\(464\) −33523.2 −3.35405
\(465\) 0 0
\(466\) 17387.1 1.72842
\(467\) − 3256.55i − 0.322687i −0.986898 0.161344i \(-0.948417\pi\)
0.986898 0.161344i \(-0.0515827\pi\)
\(468\) 14682.3i 1.45019i
\(469\) −2727.40 −0.268528
\(470\) 0 0
\(471\) 6509.25 0.636795
\(472\) − 14787.2i − 1.44203i
\(473\) − 1245.18i − 0.121043i
\(474\) −18286.0 −1.77195
\(475\) 0 0
\(476\) −4886.83 −0.470562
\(477\) 4914.44i 0.471733i
\(478\) 19606.2i 1.87608i
\(479\) −8291.59 −0.790924 −0.395462 0.918482i \(-0.629415\pi\)
−0.395462 + 0.918482i \(0.629415\pi\)
\(480\) 0 0
\(481\) −19418.9 −1.84081
\(482\) 10348.1i 0.977891i
\(483\) − 1574.28i − 0.148307i
\(484\) 27330.9 2.56676
\(485\) 0 0
\(486\) −1302.21 −0.121542
\(487\) − 4758.55i − 0.442773i −0.975186 0.221387i \(-0.928942\pi\)
0.975186 0.221387i \(-0.0710583\pi\)
\(488\) − 8921.39i − 0.827567i
\(489\) 2291.21 0.211886
\(490\) 0 0
\(491\) 3906.46 0.359055 0.179528 0.983753i \(-0.442543\pi\)
0.179528 + 0.983753i \(0.442543\pi\)
\(492\) − 26269.9i − 2.40719i
\(493\) 8936.52i 0.816390i
\(494\) 8623.26 0.785382
\(495\) 0 0
\(496\) −12178.4 −1.10247
\(497\) 1440.78i 0.130036i
\(498\) 12200.7i 1.09785i
\(499\) 3093.31 0.277506 0.138753 0.990327i \(-0.455691\pi\)
0.138753 + 0.990327i \(0.455691\pi\)
\(500\) 0 0
\(501\) −7692.85 −0.686010
\(502\) − 31313.9i − 2.78408i
\(503\) − 18153.9i − 1.60923i −0.593796 0.804616i \(-0.702371\pi\)
0.593796 0.804616i \(-0.297629\pi\)
\(504\) −2720.71 −0.240456
\(505\) 0 0
\(506\) 2178.15 0.191365
\(507\) − 12010.1i − 1.05204i
\(508\) 13059.6i 1.14060i
\(509\) −2281.32 −0.198660 −0.0993298 0.995055i \(-0.531670\pi\)
−0.0993298 + 0.995055i \(0.531670\pi\)
\(510\) 0 0
\(511\) 394.197 0.0341257
\(512\) − 4276.07i − 0.369097i
\(513\) 551.761i 0.0474870i
\(514\) 24147.8 2.07221
\(515\) 0 0
\(516\) 22526.6 1.92186
\(517\) − 585.821i − 0.0498344i
\(518\) − 5861.98i − 0.497221i
\(519\) −155.487 −0.0131505
\(520\) 0 0
\(521\) −16691.9 −1.40362 −0.701809 0.712366i \(-0.747624\pi\)
−0.701809 + 0.712366i \(0.747624\pi\)
\(522\) 8105.03i 0.679593i
\(523\) − 17090.4i − 1.42889i −0.699690 0.714446i \(-0.746679\pi\)
0.699690 0.714446i \(-0.253321\pi\)
\(524\) 44813.8 3.73607
\(525\) 0 0
\(526\) −28617.4 −2.37220
\(527\) 3246.47i 0.268346i
\(528\) − 2056.05i − 0.169466i
\(529\) −1829.50 −0.150365
\(530\) 0 0
\(531\) −1952.73 −0.159588
\(532\) 1877.94i 0.153044i
\(533\) 33281.5i 2.70465i
\(534\) 3145.91 0.254938
\(535\) 0 0
\(536\) 41906.8 3.37705
\(537\) 11869.9i 0.953861i
\(538\) − 15057.4i − 1.20664i
\(539\) −1110.82 −0.0887685
\(540\) 0 0
\(541\) 5271.40 0.418919 0.209459 0.977817i \(-0.432830\pi\)
0.209459 + 0.977817i \(0.432830\pi\)
\(542\) 16628.4i 1.31780i
\(543\) 5412.13i 0.427729i
\(544\) 27854.2 2.19529
\(545\) 0 0
\(546\) 5615.10 0.440117
\(547\) − 15182.2i − 1.18673i −0.804933 0.593366i \(-0.797799\pi\)
0.804933 0.593366i \(-0.202201\pi\)
\(548\) 23175.2i 1.80656i
\(549\) −1178.11 −0.0915860
\(550\) 0 0
\(551\) 3434.18 0.265519
\(552\) 24189.0i 1.86513i
\(553\) 5045.15i 0.387960i
\(554\) −24645.2 −1.89002
\(555\) 0 0
\(556\) −3449.05 −0.263080
\(557\) − 12241.2i − 0.931198i −0.884996 0.465599i \(-0.845839\pi\)
0.884996 0.465599i \(-0.154161\pi\)
\(558\) 2944.41i 0.223381i
\(559\) −28539.0 −2.15934
\(560\) 0 0
\(561\) −548.094 −0.0412488
\(562\) 13782.2i 1.03446i
\(563\) 14196.4i 1.06271i 0.847149 + 0.531355i \(0.178317\pi\)
−0.847149 + 0.531355i \(0.821683\pi\)
\(564\) 10598.1 0.791242
\(565\) 0 0
\(566\) −29877.5 −2.21881
\(567\) 359.283i 0.0266111i
\(568\) − 22137.7i − 1.63535i
\(569\) −9150.05 −0.674148 −0.337074 0.941478i \(-0.609437\pi\)
−0.337074 + 0.941478i \(0.609437\pi\)
\(570\) 0 0
\(571\) 23582.1 1.72833 0.864167 0.503206i \(-0.167846\pi\)
0.864167 + 0.503206i \(0.167846\pi\)
\(572\) 5604.72i 0.409695i
\(573\) 11000.3i 0.801993i
\(574\) −10046.7 −0.730556
\(575\) 0 0
\(576\) 10899.6 0.788456
\(577\) − 3906.22i − 0.281834i −0.990021 0.140917i \(-0.954995\pi\)
0.990021 0.140917i \(-0.0450050\pi\)
\(578\) 11173.9i 0.804102i
\(579\) 8150.93 0.585045
\(580\) 0 0
\(581\) 3366.21 0.240368
\(582\) 8375.96i 0.596554i
\(583\) 1876.00i 0.133270i
\(584\) −6056.87 −0.429170
\(585\) 0 0
\(586\) −31050.9 −2.18891
\(587\) 25938.0i 1.82381i 0.410401 + 0.911905i \(0.365389\pi\)
−0.410401 + 0.911905i \(0.634611\pi\)
\(588\) − 20095.8i − 1.40941i
\(589\) 1247.58 0.0872759
\(590\) 0 0
\(591\) 6102.89 0.424770
\(592\) 49195.7i 3.41542i
\(593\) 1908.23i 0.132145i 0.997815 + 0.0660723i \(0.0210468\pi\)
−0.997815 + 0.0660723i \(0.978953\pi\)
\(594\) −497.097 −0.0343370
\(595\) 0 0
\(596\) −13531.7 −0.929999
\(597\) 4653.81i 0.319041i
\(598\) − 49922.2i − 3.41383i
\(599\) 3495.41 0.238429 0.119214 0.992869i \(-0.461962\pi\)
0.119214 + 0.992869i \(0.461962\pi\)
\(600\) 0 0
\(601\) −18267.2 −1.23983 −0.619913 0.784671i \(-0.712832\pi\)
−0.619913 + 0.784671i \(0.712832\pi\)
\(602\) − 8615.06i − 0.583262i
\(603\) − 5534.01i − 0.373735i
\(604\) 39972.5 2.69281
\(605\) 0 0
\(606\) −10625.4 −0.712257
\(607\) − 11538.2i − 0.771534i −0.922596 0.385767i \(-0.873937\pi\)
0.922596 0.385767i \(-0.126063\pi\)
\(608\) − 10704.0i − 0.713987i
\(609\) 2236.19 0.148793
\(610\) 0 0
\(611\) −13426.8 −0.889017
\(612\) − 9915.58i − 0.654924i
\(613\) 21136.9i 1.39268i 0.717713 + 0.696340i \(0.245189\pi\)
−0.717713 + 0.696340i \(0.754811\pi\)
\(614\) −7526.43 −0.494694
\(615\) 0 0
\(616\) −1038.58 −0.0679314
\(617\) − 15673.2i − 1.02266i −0.859385 0.511329i \(-0.829153\pi\)
0.859385 0.511329i \(-0.170847\pi\)
\(618\) − 24609.4i − 1.60184i
\(619\) −22923.7 −1.48850 −0.744249 0.667902i \(-0.767193\pi\)
−0.744249 + 0.667902i \(0.767193\pi\)
\(620\) 0 0
\(621\) 3194.28 0.206413
\(622\) − 21954.1i − 1.41524i
\(623\) − 867.964i − 0.0558174i
\(624\) −47123.7 −3.02317
\(625\) 0 0
\(626\) −5222.84 −0.333461
\(627\) 210.625i 0.0134156i
\(628\) 44952.4i 2.85636i
\(629\) 13114.4 0.831329
\(630\) 0 0
\(631\) 9108.23 0.574632 0.287316 0.957836i \(-0.407237\pi\)
0.287316 + 0.957836i \(0.407237\pi\)
\(632\) − 77519.3i − 4.87904i
\(633\) 9577.53i 0.601379i
\(634\) −11102.0 −0.695449
\(635\) 0 0
\(636\) −33938.8 −2.11598
\(637\) 25459.4i 1.58358i
\(638\) 3093.96i 0.191992i
\(639\) −2923.40 −0.180983
\(640\) 0 0
\(641\) 20103.5 1.23875 0.619375 0.785095i \(-0.287386\pi\)
0.619375 + 0.785095i \(0.287386\pi\)
\(642\) − 4254.30i − 0.261533i
\(643\) 5934.92i 0.363997i 0.983299 + 0.181999i \(0.0582567\pi\)
−0.983299 + 0.181999i \(0.941743\pi\)
\(644\) 10871.9 0.665237
\(645\) 0 0
\(646\) −5823.64 −0.354688
\(647\) 14193.7i 0.862460i 0.902242 + 0.431230i \(0.141920\pi\)
−0.902242 + 0.431230i \(0.858080\pi\)
\(648\) − 5520.42i − 0.334665i
\(649\) −745.420 −0.0450852
\(650\) 0 0
\(651\) 812.368 0.0489082
\(652\) 15823.0i 0.950422i
\(653\) − 4795.80i − 0.287403i −0.989621 0.143701i \(-0.954100\pi\)
0.989621 0.143701i \(-0.0459005\pi\)
\(654\) 17967.5 1.07429
\(655\) 0 0
\(656\) 84314.8 5.01820
\(657\) 799.841i 0.0474958i
\(658\) − 4053.13i − 0.240133i
\(659\) −4399.57 −0.260065 −0.130032 0.991510i \(-0.541508\pi\)
−0.130032 + 0.991510i \(0.541508\pi\)
\(660\) 0 0
\(661\) 24096.0 1.41789 0.708945 0.705263i \(-0.249171\pi\)
0.708945 + 0.705263i \(0.249171\pi\)
\(662\) − 33006.4i − 1.93781i
\(663\) 12562.1i 0.735853i
\(664\) −51722.2 −3.02291
\(665\) 0 0
\(666\) 11894.2 0.692028
\(667\) − 19881.4i − 1.15414i
\(668\) − 53126.3i − 3.07712i
\(669\) 4666.66 0.269692
\(670\) 0 0
\(671\) −449.725 −0.0258740
\(672\) − 6969.98i − 0.400109i
\(673\) − 27648.3i − 1.58360i −0.610781 0.791800i \(-0.709144\pi\)
0.610781 0.791800i \(-0.290856\pi\)
\(674\) 14958.1 0.854843
\(675\) 0 0
\(676\) 82940.9 4.71898
\(677\) 27605.5i 1.56716i 0.621292 + 0.783580i \(0.286608\pi\)
−0.621292 + 0.783580i \(0.713392\pi\)
\(678\) 15016.6i 0.850604i
\(679\) 2310.95 0.130613
\(680\) 0 0
\(681\) −18620.6 −1.04779
\(682\) 1123.98i 0.0631075i
\(683\) − 14949.4i − 0.837513i −0.908099 0.418756i \(-0.862466\pi\)
0.908099 0.418756i \(-0.137534\pi\)
\(684\) −3810.42 −0.213005
\(685\) 0 0
\(686\) −15838.5 −0.881511
\(687\) − 14038.5i − 0.779626i
\(688\) 72300.4i 4.00643i
\(689\) 42997.2 2.37745
\(690\) 0 0
\(691\) 8884.30 0.489110 0.244555 0.969635i \(-0.421358\pi\)
0.244555 + 0.969635i \(0.421358\pi\)
\(692\) − 1073.78i − 0.0589871i
\(693\) 137.150i 0.00751790i
\(694\) 5041.91 0.275775
\(695\) 0 0
\(696\) −34359.4 −1.87125
\(697\) − 22476.4i − 1.22145i
\(698\) 18861.3i 1.02279i
\(699\) 9733.59 0.526693
\(700\) 0 0
\(701\) 10556.9 0.568798 0.284399 0.958706i \(-0.408206\pi\)
0.284399 + 0.958706i \(0.408206\pi\)
\(702\) 11393.3i 0.612551i
\(703\) − 5039.70i − 0.270378i
\(704\) 4160.74 0.222747
\(705\) 0 0
\(706\) −26910.2 −1.43453
\(707\) 2931.58i 0.155945i
\(708\) − 13485.4i − 0.715836i
\(709\) −25351.9 −1.34289 −0.671445 0.741055i \(-0.734326\pi\)
−0.671445 + 0.741055i \(0.734326\pi\)
\(710\) 0 0
\(711\) −10236.8 −0.539959
\(712\) 13336.4i 0.701968i
\(713\) − 7222.53i − 0.379363i
\(714\) −3792.11 −0.198762
\(715\) 0 0
\(716\) −81972.6 −4.27858
\(717\) 10975.8i 0.571688i
\(718\) − 36504.8i − 1.89742i
\(719\) −9719.94 −0.504162 −0.252081 0.967706i \(-0.581115\pi\)
−0.252081 + 0.967706i \(0.581115\pi\)
\(720\) 0 0
\(721\) −6789.79 −0.350714
\(722\) − 34518.7i − 1.77930i
\(723\) 5793.04i 0.297988i
\(724\) −37375.8 −1.91859
\(725\) 0 0
\(726\) 21208.3 1.08418
\(727\) 27509.3i 1.40339i 0.712479 + 0.701694i \(0.247572\pi\)
−0.712479 + 0.701694i \(0.752428\pi\)
\(728\) 23803.9i 1.21186i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 19273.6 0.975184
\(732\) − 8135.98i − 0.410812i
\(733\) − 7240.49i − 0.364848i −0.983220 0.182424i \(-0.941606\pi\)
0.983220 0.182424i \(-0.0583944\pi\)
\(734\) 20085.5 1.01004
\(735\) 0 0
\(736\) −61968.1 −3.10350
\(737\) − 2112.51i − 0.105584i
\(738\) − 20385.1i − 1.01678i
\(739\) 15875.3 0.790234 0.395117 0.918631i \(-0.370704\pi\)
0.395117 + 0.918631i \(0.370704\pi\)
\(740\) 0 0
\(741\) 4827.44 0.239326
\(742\) 12979.5i 0.642175i
\(743\) − 25714.3i − 1.26967i −0.772647 0.634836i \(-0.781068\pi\)
0.772647 0.634836i \(-0.218932\pi\)
\(744\) −12482.1 −0.615076
\(745\) 0 0
\(746\) 4813.91 0.236260
\(747\) 6830.18i 0.334542i
\(748\) − 3785.10i − 0.185023i
\(749\) −1173.77 −0.0572612
\(750\) 0 0
\(751\) −9709.09 −0.471757 −0.235879 0.971783i \(-0.575797\pi\)
−0.235879 + 0.971783i \(0.575797\pi\)
\(752\) 34015.2i 1.64948i
\(753\) − 17530.0i − 0.848379i
\(754\) 70912.1 3.42502
\(755\) 0 0
\(756\) −2481.19 −0.119365
\(757\) 9567.13i 0.459344i 0.973268 + 0.229672i \(0.0737653\pi\)
−0.973268 + 0.229672i \(0.926235\pi\)
\(758\) 50261.1i 2.40840i
\(759\) 1219.36 0.0583137
\(760\) 0 0
\(761\) −12322.5 −0.586980 −0.293490 0.955962i \(-0.594817\pi\)
−0.293490 + 0.955962i \(0.594817\pi\)
\(762\) 10134.0i 0.481782i
\(763\) − 4957.28i − 0.235211i
\(764\) −75967.0 −3.59737
\(765\) 0 0
\(766\) 50621.7 2.38778
\(767\) 17084.7i 0.804293i
\(768\) 7905.27i 0.371428i
\(769\) −2575.56 −0.120776 −0.0603881 0.998175i \(-0.519234\pi\)
−0.0603881 + 0.998175i \(0.519234\pi\)
\(770\) 0 0
\(771\) 13518.3 0.631454
\(772\) 56289.8i 2.62424i
\(773\) − 6606.23i − 0.307386i −0.988119 0.153693i \(-0.950883\pi\)
0.988119 0.153693i \(-0.0491167\pi\)
\(774\) 17480.3 0.811778
\(775\) 0 0
\(776\) −35507.9 −1.64260
\(777\) − 3281.63i − 0.151516i
\(778\) 40959.3i 1.88748i
\(779\) −8637.37 −0.397260
\(780\) 0 0
\(781\) −1115.96 −0.0511294
\(782\) 33714.5i 1.54173i
\(783\) 4537.33i 0.207089i
\(784\) 64498.5 2.93816
\(785\) 0 0
\(786\) 34774.8 1.57809
\(787\) − 16417.0i − 0.743587i −0.928315 0.371793i \(-0.878743\pi\)
0.928315 0.371793i \(-0.121257\pi\)
\(788\) 42146.1i 1.90532i
\(789\) −16020.5 −0.722870
\(790\) 0 0
\(791\) 4143.12 0.186235
\(792\) − 2107.33i − 0.0945462i
\(793\) 10307.5i 0.461577i
\(794\) −64379.8 −2.87752
\(795\) 0 0
\(796\) −32138.9 −1.43107
\(797\) 3944.19i 0.175295i 0.996152 + 0.0876477i \(0.0279350\pi\)
−0.996152 + 0.0876477i \(0.972065\pi\)
\(798\) 1457.26i 0.0646445i
\(799\) 9067.66 0.401490
\(800\) 0 0
\(801\) 1761.13 0.0776861
\(802\) − 45757.0i − 2.01463i
\(803\) 305.325i 0.0134181i
\(804\) 38217.5 1.67640
\(805\) 0 0
\(806\) 25761.1 1.12580
\(807\) − 8429.37i − 0.367693i
\(808\) − 45044.0i − 1.96119i
\(809\) 17960.7 0.780549 0.390275 0.920699i \(-0.372380\pi\)
0.390275 + 0.920699i \(0.372380\pi\)
\(810\) 0 0
\(811\) −13162.5 −0.569912 −0.284956 0.958541i \(-0.591979\pi\)
−0.284956 + 0.958541i \(0.591979\pi\)
\(812\) 15443.0i 0.667418i
\(813\) 9308.84i 0.401569i
\(814\) 4540.41 0.195505
\(815\) 0 0
\(816\) 31824.6 1.36530
\(817\) − 7406.59i − 0.317165i
\(818\) 65740.9i 2.81000i
\(819\) 3143.42 0.134115
\(820\) 0 0
\(821\) 26502.4 1.12660 0.563302 0.826251i \(-0.309531\pi\)
0.563302 + 0.826251i \(0.309531\pi\)
\(822\) 17983.6i 0.763078i
\(823\) 6937.86i 0.293850i 0.989148 + 0.146925i \(0.0469376\pi\)
−0.989148 + 0.146925i \(0.953062\pi\)
\(824\) 104326. 4.41063
\(825\) 0 0
\(826\) −5157.35 −0.217248
\(827\) − 41197.9i − 1.73228i −0.499805 0.866138i \(-0.666595\pi\)
0.499805 0.866138i \(-0.333405\pi\)
\(828\) 22059.5i 0.925871i
\(829\) 693.324 0.0290472 0.0145236 0.999895i \(-0.495377\pi\)
0.0145236 + 0.999895i \(0.495377\pi\)
\(830\) 0 0
\(831\) −13796.8 −0.575938
\(832\) − 95362.4i − 3.97367i
\(833\) − 17193.8i − 0.715162i
\(834\) −2676.41 −0.111123
\(835\) 0 0
\(836\) −1454.56 −0.0601760
\(837\) 1648.33i 0.0680699i
\(838\) − 83025.7i − 3.42252i
\(839\) 6491.28 0.267108 0.133554 0.991042i \(-0.457361\pi\)
0.133554 + 0.991042i \(0.457361\pi\)
\(840\) 0 0
\(841\) 3851.52 0.157921
\(842\) 40248.9i 1.64735i
\(843\) 7715.49i 0.315226i
\(844\) −66141.8 −2.69750
\(845\) 0 0
\(846\) 8223.97 0.334215
\(847\) − 5851.42i − 0.237376i
\(848\) − 108928.i − 4.41111i
\(849\) −16725.9 −0.676128
\(850\) 0 0
\(851\) −29176.1 −1.17526
\(852\) − 20188.8i − 0.811803i
\(853\) − 1116.68i − 0.0448233i −0.999749 0.0224117i \(-0.992866\pi\)
0.999749 0.0224117i \(-0.00713445\pi\)
\(854\) −3111.52 −0.124677
\(855\) 0 0
\(856\) 18035.1 0.720126
\(857\) − 44383.9i − 1.76911i −0.466438 0.884554i \(-0.654463\pi\)
0.466438 0.884554i \(-0.345537\pi\)
\(858\) 4349.18i 0.173052i
\(859\) 25579.3 1.01601 0.508006 0.861354i \(-0.330383\pi\)
0.508006 + 0.861354i \(0.330383\pi\)
\(860\) 0 0
\(861\) −5624.28 −0.222619
\(862\) − 84002.5i − 3.31918i
\(863\) − 11194.8i − 0.441570i −0.975323 0.220785i \(-0.929138\pi\)
0.975323 0.220785i \(-0.0708619\pi\)
\(864\) 14142.4 0.556867
\(865\) 0 0
\(866\) 51471.0 2.01970
\(867\) 6255.31i 0.245030i
\(868\) 5610.16i 0.219379i
\(869\) −3907.73 −0.152544
\(870\) 0 0
\(871\) −48417.9 −1.88356
\(872\) 76169.2i 2.95804i
\(873\) 4689.00i 0.181785i
\(874\) 12956.0 0.501424
\(875\) 0 0
\(876\) −5523.65 −0.213044
\(877\) 5721.75i 0.220308i 0.993915 + 0.110154i \(0.0351344\pi\)
−0.993915 + 0.110154i \(0.964866\pi\)
\(878\) 34093.6i 1.31048i
\(879\) −17382.8 −0.667017
\(880\) 0 0
\(881\) −34682.8 −1.32633 −0.663163 0.748475i \(-0.730786\pi\)
−0.663163 + 0.748475i \(0.730786\pi\)
\(882\) − 15594.0i − 0.595326i
\(883\) 37990.4i 1.44788i 0.689862 + 0.723941i \(0.257671\pi\)
−0.689862 + 0.723941i \(0.742329\pi\)
\(884\) −86752.9 −3.30070
\(885\) 0 0
\(886\) −4992.66 −0.189313
\(887\) 28299.0i 1.07124i 0.844460 + 0.535618i \(0.179921\pi\)
−0.844460 + 0.535618i \(0.820079\pi\)
\(888\) 50422.7i 1.90549i
\(889\) 2796.00 0.105484
\(890\) 0 0
\(891\) −278.283 −0.0104633
\(892\) 32227.7i 1.20971i
\(893\) − 3484.58i − 0.130579i
\(894\) −10500.4 −0.392825
\(895\) 0 0
\(896\) 10200.4 0.380324
\(897\) − 27947.3i − 1.04028i
\(898\) 100126.i 3.72078i
\(899\) 10259.3 0.380607
\(900\) 0 0
\(901\) −29037.8 −1.07368
\(902\) − 7781.65i − 0.287251i
\(903\) − 4822.85i − 0.177735i
\(904\) −63659.4 −2.34212
\(905\) 0 0
\(906\) 31018.1 1.13742
\(907\) 17388.0i 0.636559i 0.947997 + 0.318280i \(0.103105\pi\)
−0.947997 + 0.318280i \(0.896895\pi\)
\(908\) − 128592.i − 4.69988i
\(909\) −5948.28 −0.217043
\(910\) 0 0
\(911\) 23555.3 0.856663 0.428332 0.903622i \(-0.359101\pi\)
0.428332 + 0.903622i \(0.359101\pi\)
\(912\) − 12229.8i − 0.444044i
\(913\) 2607.30i 0.0945117i
\(914\) −61980.4 −2.24303
\(915\) 0 0
\(916\) 96949.1 3.49704
\(917\) − 9594.44i − 0.345514i
\(918\) − 7694.34i − 0.276635i
\(919\) 5983.09 0.214760 0.107380 0.994218i \(-0.465754\pi\)
0.107380 + 0.994218i \(0.465754\pi\)
\(920\) 0 0
\(921\) −4213.42 −0.150746
\(922\) 103968.i 3.71368i
\(923\) 25577.3i 0.912119i
\(924\) −947.150 −0.0337218
\(925\) 0 0
\(926\) −8447.10 −0.299772
\(927\) − 13776.7i − 0.488120i
\(928\) − 88022.7i − 3.11367i
\(929\) −20576.7 −0.726694 −0.363347 0.931654i \(-0.618366\pi\)
−0.363347 + 0.931654i \(0.618366\pi\)
\(930\) 0 0
\(931\) −6607.35 −0.232596
\(932\) 67219.5i 2.36250i
\(933\) − 12290.2i − 0.431259i
\(934\) 17451.5 0.611382
\(935\) 0 0
\(936\) −48299.0 −1.68665
\(937\) 11228.6i 0.391485i 0.980655 + 0.195743i \(0.0627117\pi\)
−0.980655 + 0.195743i \(0.937288\pi\)
\(938\) − 14615.9i − 0.508769i
\(939\) −2923.83 −0.101614
\(940\) 0 0
\(941\) 38567.6 1.33610 0.668049 0.744118i \(-0.267130\pi\)
0.668049 + 0.744118i \(0.267130\pi\)
\(942\) 34882.4i 1.20651i
\(943\) 50003.9i 1.72678i
\(944\) 43282.1 1.49228
\(945\) 0 0
\(946\) 6672.81 0.229336
\(947\) − 4606.17i − 0.158057i −0.996872 0.0790287i \(-0.974818\pi\)
0.996872 0.0790287i \(-0.0251819\pi\)
\(948\) − 70694.8i − 2.42200i
\(949\) 6997.93 0.239370
\(950\) 0 0
\(951\) −6215.06 −0.211921
\(952\) − 16075.8i − 0.547288i
\(953\) − 25559.7i − 0.868795i −0.900721 0.434397i \(-0.856961\pi\)
0.900721 0.434397i \(-0.143039\pi\)
\(954\) −26336.0 −0.893773
\(955\) 0 0
\(956\) −75798.5 −2.56433
\(957\) 1732.05i 0.0585048i
\(958\) − 44433.8i − 1.49853i
\(959\) 4961.72 0.167072
\(960\) 0 0
\(961\) −26064.0 −0.874895
\(962\) − 104064.i − 3.48769i
\(963\) − 2381.63i − 0.0796956i
\(964\) −40006.4 −1.33664
\(965\) 0 0
\(966\) 8436.42 0.280991
\(967\) − 37895.8i − 1.26023i −0.776500 0.630117i \(-0.783007\pi\)
0.776500 0.630117i \(-0.216993\pi\)
\(968\) 89907.7i 2.98527i
\(969\) −3260.17 −0.108082
\(970\) 0 0
\(971\) −46761.0 −1.54545 −0.772726 0.634740i \(-0.781107\pi\)
−0.772726 + 0.634740i \(0.781107\pi\)
\(972\) − 5034.42i − 0.166131i
\(973\) 738.428i 0.0243298i
\(974\) 25500.6 0.838903
\(975\) 0 0
\(976\) 26112.9 0.856407
\(977\) − 3070.29i − 0.100540i −0.998736 0.0502698i \(-0.983992\pi\)
0.998736 0.0502698i \(-0.0160081\pi\)
\(978\) 12278.4i 0.401451i
\(979\) 672.282 0.0219471
\(980\) 0 0
\(981\) 10058.5 0.327364
\(982\) 20934.3i 0.680287i
\(983\) 16319.0i 0.529498i 0.964317 + 0.264749i \(0.0852891\pi\)
−0.964317 + 0.264749i \(0.914711\pi\)
\(984\) 86417.7 2.79969
\(985\) 0 0
\(986\) −47889.9 −1.54678
\(987\) − 2269.01i − 0.0731747i
\(988\) 33338.0i 1.07350i
\(989\) −42878.6 −1.37862
\(990\) 0 0
\(991\) 5105.79 0.163664 0.0818319 0.996646i \(-0.473923\pi\)
0.0818319 + 0.996646i \(0.473923\pi\)
\(992\) − 31977.0i − 1.02346i
\(993\) − 18477.5i − 0.590500i
\(994\) −7720.99 −0.246373
\(995\) 0 0
\(996\) −47168.7 −1.50060
\(997\) − 7206.97i − 0.228934i −0.993427 0.114467i \(-0.963484\pi\)
0.993427 0.114467i \(-0.0365160\pi\)
\(998\) 16576.7i 0.525779i
\(999\) 6658.57 0.210879
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 75.4.b.c.49.4 4
3.2 odd 2 225.4.b.h.199.1 4
4.3 odd 2 1200.4.f.v.49.1 4
5.2 odd 4 75.4.a.d.1.1 2
5.3 odd 4 75.4.a.e.1.2 yes 2
5.4 even 2 inner 75.4.b.c.49.1 4
15.2 even 4 225.4.a.n.1.2 2
15.8 even 4 225.4.a.j.1.1 2
15.14 odd 2 225.4.b.h.199.4 4
20.3 even 4 1200.4.a.bu.1.1 2
20.7 even 4 1200.4.a.bl.1.2 2
20.19 odd 2 1200.4.f.v.49.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.a.d.1.1 2 5.2 odd 4
75.4.a.e.1.2 yes 2 5.3 odd 4
75.4.b.c.49.1 4 5.4 even 2 inner
75.4.b.c.49.4 4 1.1 even 1 trivial
225.4.a.j.1.1 2 15.8 even 4
225.4.a.n.1.2 2 15.2 even 4
225.4.b.h.199.1 4 3.2 odd 2
225.4.b.h.199.4 4 15.14 odd 2
1200.4.a.bl.1.2 2 20.7 even 4
1200.4.a.bu.1.1 2 20.3 even 4
1200.4.f.v.49.1 4 4.3 odd 2
1200.4.f.v.49.4 4 20.19 odd 2