Properties

Label 1875.4.a.f.1.5
Level $1875$
Weight $4$
Character 1875.1
Self dual yes
Analytic conductor $110.629$
Analytic rank $0$
Dimension $14$
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] = [1875,4,Mod(1,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, 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("1875.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1875.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: \(110.628581261\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.73740\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73740 q^{2} -3.00000 q^{3} -4.98145 q^{4} +5.21219 q^{6} -7.66213 q^{7} +22.5539 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.73740 q^{2} -3.00000 q^{3} -4.98145 q^{4} +5.21219 q^{6} -7.66213 q^{7} +22.5539 q^{8} +9.00000 q^{9} +20.4704 q^{11} +14.9443 q^{12} +14.8079 q^{13} +13.3122 q^{14} +0.666425 q^{16} +109.000 q^{17} -15.6366 q^{18} +97.7849 q^{19} +22.9864 q^{21} -35.5652 q^{22} +80.5714 q^{23} -67.6618 q^{24} -25.7272 q^{26} -27.0000 q^{27} +38.1685 q^{28} +59.0555 q^{29} +176.318 q^{31} -181.589 q^{32} -61.4111 q^{33} -189.376 q^{34} -44.8330 q^{36} +230.536 q^{37} -169.891 q^{38} -44.4236 q^{39} +179.464 q^{41} -39.9365 q^{42} +407.459 q^{43} -101.972 q^{44} -139.985 q^{46} -477.166 q^{47} -1.99927 q^{48} -284.292 q^{49} -326.999 q^{51} -73.7647 q^{52} +515.525 q^{53} +46.9097 q^{54} -172.811 q^{56} -293.355 q^{57} -102.603 q^{58} -571.444 q^{59} -52.3159 q^{61} -306.334 q^{62} -68.9591 q^{63} +310.162 q^{64} +106.696 q^{66} +551.814 q^{67} -542.976 q^{68} -241.714 q^{69} +368.658 q^{71} +202.985 q^{72} +406.465 q^{73} -400.533 q^{74} -487.111 q^{76} -156.847 q^{77} +77.1815 q^{78} -654.750 q^{79} +81.0000 q^{81} -311.801 q^{82} -421.856 q^{83} -114.505 q^{84} -707.919 q^{86} -177.166 q^{87} +461.688 q^{88} +425.138 q^{89} -113.460 q^{91} -401.362 q^{92} -528.953 q^{93} +829.027 q^{94} +544.768 q^{96} -590.666 q^{97} +493.928 q^{98} +184.233 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 42 q^{3} + 50 q^{4} + 27 q^{7} - 21 q^{8} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 42 q^{3} + 50 q^{4} + 27 q^{7} - 21 q^{8} + 126 q^{9} - 33 q^{11} - 150 q^{12} + 188 q^{13} - 287 q^{14} + 82 q^{16} + 146 q^{17} - 184 q^{19} - 81 q^{21} + 277 q^{22} + 164 q^{23} + 63 q^{24} + 103 q^{26} - 378 q^{27} - 224 q^{28} + 252 q^{29} - 889 q^{31} + 422 q^{32} + 99 q^{33} - 230 q^{34} + 450 q^{36} + 642 q^{37} + 1833 q^{38} - 564 q^{39} - 164 q^{41} + 861 q^{42} + 696 q^{43} - 6 q^{44} - 419 q^{46} - 92 q^{47} - 246 q^{48} + 81 q^{49} - 438 q^{51} + 1956 q^{52} + 949 q^{53} - 2380 q^{56} + 552 q^{57} + 1041 q^{58} - 81 q^{59} - 496 q^{61} + 1454 q^{62} + 243 q^{63} - 3903 q^{64} - 831 q^{66} + 1926 q^{67} + 685 q^{68} - 492 q^{69} + 2498 q^{71} - 189 q^{72} - 1026 q^{73} - 707 q^{74} - 5704 q^{76} + 5434 q^{77} - 309 q^{78} - 695 q^{79} + 1134 q^{81} - 886 q^{82} + 5315 q^{83} + 672 q^{84} + 3997 q^{86} - 756 q^{87} + 2969 q^{88} - 1424 q^{89} - 1194 q^{91} + 1607 q^{92} + 2667 q^{93} - 629 q^{94} - 1266 q^{96} + 291 q^{97} - 1018 q^{98} - 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73740 −0.614263 −0.307131 0.951667i \(-0.599369\pi\)
−0.307131 + 0.951667i \(0.599369\pi\)
\(3\) −3.00000 −0.577350
\(4\) −4.98145 −0.622681
\(5\) 0 0
\(6\) 5.21219 0.354645
\(7\) −7.66213 −0.413716 −0.206858 0.978371i \(-0.566324\pi\)
−0.206858 + 0.978371i \(0.566324\pi\)
\(8\) 22.5539 0.996753
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 20.4704 0.561095 0.280548 0.959840i \(-0.409484\pi\)
0.280548 + 0.959840i \(0.409484\pi\)
\(12\) 14.9443 0.359505
\(13\) 14.8079 0.315920 0.157960 0.987445i \(-0.449508\pi\)
0.157960 + 0.987445i \(0.449508\pi\)
\(14\) 13.3122 0.254130
\(15\) 0 0
\(16\) 0.666425 0.0104129
\(17\) 109.000 1.55508 0.777539 0.628835i \(-0.216468\pi\)
0.777539 + 0.628835i \(0.216468\pi\)
\(18\) −15.6366 −0.204754
\(19\) 97.7849 1.18071 0.590353 0.807145i \(-0.298989\pi\)
0.590353 + 0.807145i \(0.298989\pi\)
\(20\) 0 0
\(21\) 22.9864 0.238859
\(22\) −35.5652 −0.344660
\(23\) 80.5714 0.730447 0.365224 0.930920i \(-0.380992\pi\)
0.365224 + 0.930920i \(0.380992\pi\)
\(24\) −67.6618 −0.575475
\(25\) 0 0
\(26\) −25.7272 −0.194058
\(27\) −27.0000 −0.192450
\(28\) 38.1685 0.257613
\(29\) 59.0555 0.378149 0.189075 0.981963i \(-0.439451\pi\)
0.189075 + 0.981963i \(0.439451\pi\)
\(30\) 0 0
\(31\) 176.318 1.02153 0.510767 0.859719i \(-0.329361\pi\)
0.510767 + 0.859719i \(0.329361\pi\)
\(32\) −181.589 −1.00315
\(33\) −61.4111 −0.323949
\(34\) −189.376 −0.955226
\(35\) 0 0
\(36\) −44.8330 −0.207560
\(37\) 230.536 1.02432 0.512161 0.858889i \(-0.328845\pi\)
0.512161 + 0.858889i \(0.328845\pi\)
\(38\) −169.891 −0.725264
\(39\) −44.4236 −0.182397
\(40\) 0 0
\(41\) 179.464 0.683600 0.341800 0.939773i \(-0.388963\pi\)
0.341800 + 0.939773i \(0.388963\pi\)
\(42\) −39.9365 −0.146722
\(43\) 407.459 1.44505 0.722523 0.691347i \(-0.242982\pi\)
0.722523 + 0.691347i \(0.242982\pi\)
\(44\) −101.972 −0.349384
\(45\) 0 0
\(46\) −139.985 −0.448687
\(47\) −477.166 −1.48089 −0.740445 0.672117i \(-0.765385\pi\)
−0.740445 + 0.672117i \(0.765385\pi\)
\(48\) −1.99927 −0.00601188
\(49\) −284.292 −0.828839
\(50\) 0 0
\(51\) −326.999 −0.897824
\(52\) −73.7647 −0.196718
\(53\) 515.525 1.33609 0.668046 0.744120i \(-0.267131\pi\)
0.668046 + 0.744120i \(0.267131\pi\)
\(54\) 46.9097 0.118215
\(55\) 0 0
\(56\) −172.811 −0.412372
\(57\) −293.355 −0.681681
\(58\) −102.603 −0.232283
\(59\) −571.444 −1.26094 −0.630472 0.776212i \(-0.717139\pi\)
−0.630472 + 0.776212i \(0.717139\pi\)
\(60\) 0 0
\(61\) −52.3159 −0.109809 −0.0549046 0.998492i \(-0.517485\pi\)
−0.0549046 + 0.998492i \(0.517485\pi\)
\(62\) −306.334 −0.627491
\(63\) −68.9591 −0.137905
\(64\) 310.162 0.605784
\(65\) 0 0
\(66\) 106.696 0.198990
\(67\) 551.814 1.00619 0.503096 0.864231i \(-0.332194\pi\)
0.503096 + 0.864231i \(0.332194\pi\)
\(68\) −542.976 −0.968317
\(69\) −241.714 −0.421724
\(70\) 0 0
\(71\) 368.658 0.616220 0.308110 0.951351i \(-0.400304\pi\)
0.308110 + 0.951351i \(0.400304\pi\)
\(72\) 202.985 0.332251
\(73\) 406.465 0.651687 0.325843 0.945424i \(-0.394352\pi\)
0.325843 + 0.945424i \(0.394352\pi\)
\(74\) −400.533 −0.629203
\(75\) 0 0
\(76\) −487.111 −0.735203
\(77\) −156.847 −0.232134
\(78\) 77.1815 0.112040
\(79\) −654.750 −0.932470 −0.466235 0.884661i \(-0.654390\pi\)
−0.466235 + 0.884661i \(0.654390\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −311.801 −0.419910
\(83\) −421.856 −0.557887 −0.278944 0.960307i \(-0.589984\pi\)
−0.278944 + 0.960307i \(0.589984\pi\)
\(84\) −114.505 −0.148733
\(85\) 0 0
\(86\) −707.919 −0.887638
\(87\) −177.166 −0.218325
\(88\) 461.688 0.559273
\(89\) 425.138 0.506343 0.253171 0.967421i \(-0.418526\pi\)
0.253171 + 0.967421i \(0.418526\pi\)
\(90\) 0 0
\(91\) −113.460 −0.130701
\(92\) −401.362 −0.454836
\(93\) −528.953 −0.589783
\(94\) 829.027 0.909655
\(95\) 0 0
\(96\) 544.768 0.579168
\(97\) −590.666 −0.618279 −0.309139 0.951017i \(-0.600041\pi\)
−0.309139 + 0.951017i \(0.600041\pi\)
\(98\) 493.928 0.509125
\(99\) 184.233 0.187032
\(100\) 0 0
\(101\) −1533.55 −1.51083 −0.755417 0.655244i \(-0.772566\pi\)
−0.755417 + 0.655244i \(0.772566\pi\)
\(102\) 568.128 0.551500
\(103\) 718.897 0.687719 0.343859 0.939021i \(-0.388266\pi\)
0.343859 + 0.939021i \(0.388266\pi\)
\(104\) 333.976 0.314895
\(105\) 0 0
\(106\) −895.672 −0.820711
\(107\) 944.844 0.853659 0.426830 0.904332i \(-0.359630\pi\)
0.426830 + 0.904332i \(0.359630\pi\)
\(108\) 134.499 0.119835
\(109\) 1472.33 1.29380 0.646899 0.762576i \(-0.276066\pi\)
0.646899 + 0.762576i \(0.276066\pi\)
\(110\) 0 0
\(111\) −691.609 −0.591393
\(112\) −5.10623 −0.00430798
\(113\) 820.255 0.682859 0.341429 0.939907i \(-0.389089\pi\)
0.341429 + 0.939907i \(0.389089\pi\)
\(114\) 509.674 0.418731
\(115\) 0 0
\(116\) −294.182 −0.235466
\(117\) 133.271 0.105307
\(118\) 992.826 0.774551
\(119\) −835.170 −0.643360
\(120\) 0 0
\(121\) −911.964 −0.685172
\(122\) 90.8935 0.0674517
\(123\) −538.393 −0.394677
\(124\) −878.317 −0.636090
\(125\) 0 0
\(126\) 119.809 0.0847101
\(127\) −1663.86 −1.16255 −0.581274 0.813708i \(-0.697446\pi\)
−0.581274 + 0.813708i \(0.697446\pi\)
\(128\) 913.841 0.631038
\(129\) −1222.38 −0.834297
\(130\) 0 0
\(131\) −2281.74 −1.52180 −0.760901 0.648868i \(-0.775243\pi\)
−0.760901 + 0.648868i \(0.775243\pi\)
\(132\) 305.916 0.201717
\(133\) −749.241 −0.488477
\(134\) −958.721 −0.618066
\(135\) 0 0
\(136\) 2458.37 1.55003
\(137\) −934.992 −0.583078 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(138\) 419.954 0.259049
\(139\) −2874.22 −1.75387 −0.876937 0.480605i \(-0.840417\pi\)
−0.876937 + 0.480605i \(0.840417\pi\)
\(140\) 0 0
\(141\) 1431.50 0.854992
\(142\) −640.505 −0.378521
\(143\) 303.123 0.177262
\(144\) 5.99782 0.00347096
\(145\) 0 0
\(146\) −706.192 −0.400307
\(147\) 852.875 0.478530
\(148\) −1148.40 −0.637826
\(149\) 2275.13 1.25091 0.625457 0.780259i \(-0.284913\pi\)
0.625457 + 0.780259i \(0.284913\pi\)
\(150\) 0 0
\(151\) 2396.11 1.29134 0.645670 0.763617i \(-0.276578\pi\)
0.645670 + 0.763617i \(0.276578\pi\)
\(152\) 2205.44 1.17687
\(153\) 980.997 0.518359
\(154\) 272.505 0.142591
\(155\) 0 0
\(156\) 221.294 0.113575
\(157\) −902.424 −0.458734 −0.229367 0.973340i \(-0.573666\pi\)
−0.229367 + 0.973340i \(0.573666\pi\)
\(158\) 1137.56 0.572782
\(159\) −1546.58 −0.771393
\(160\) 0 0
\(161\) −617.348 −0.302198
\(162\) −140.729 −0.0682514
\(163\) 4028.26 1.93569 0.967846 0.251543i \(-0.0809380\pi\)
0.967846 + 0.251543i \(0.0809380\pi\)
\(164\) −893.992 −0.425665
\(165\) 0 0
\(166\) 732.931 0.342690
\(167\) 982.313 0.455172 0.227586 0.973758i \(-0.426917\pi\)
0.227586 + 0.973758i \(0.426917\pi\)
\(168\) 518.433 0.238083
\(169\) −1977.73 −0.900194
\(170\) 0 0
\(171\) 880.065 0.393569
\(172\) −2029.74 −0.899802
\(173\) −1265.10 −0.555975 −0.277988 0.960585i \(-0.589667\pi\)
−0.277988 + 0.960585i \(0.589667\pi\)
\(174\) 307.809 0.134109
\(175\) 0 0
\(176\) 13.6420 0.00584262
\(177\) 1714.33 0.728007
\(178\) −738.634 −0.311028
\(179\) −1471.61 −0.614488 −0.307244 0.951631i \(-0.599407\pi\)
−0.307244 + 0.951631i \(0.599407\pi\)
\(180\) 0 0
\(181\) −3742.99 −1.53709 −0.768547 0.639794i \(-0.779020\pi\)
−0.768547 + 0.639794i \(0.779020\pi\)
\(182\) 197.125 0.0802850
\(183\) 156.948 0.0633984
\(184\) 1817.20 0.728075
\(185\) 0 0
\(186\) 919.001 0.362282
\(187\) 2231.26 0.872547
\(188\) 2376.98 0.922122
\(189\) 206.877 0.0796197
\(190\) 0 0
\(191\) 3953.76 1.49782 0.748911 0.662670i \(-0.230577\pi\)
0.748911 + 0.662670i \(0.230577\pi\)
\(192\) −930.485 −0.349750
\(193\) 4739.26 1.76756 0.883780 0.467903i \(-0.154990\pi\)
0.883780 + 0.467903i \(0.154990\pi\)
\(194\) 1026.22 0.379786
\(195\) 0 0
\(196\) 1416.19 0.516102
\(197\) 1912.59 0.691708 0.345854 0.938288i \(-0.387589\pi\)
0.345854 + 0.938288i \(0.387589\pi\)
\(198\) −320.087 −0.114887
\(199\) −3291.81 −1.17262 −0.586308 0.810089i \(-0.699419\pi\)
−0.586308 + 0.810089i \(0.699419\pi\)
\(200\) 0 0
\(201\) −1655.44 −0.580925
\(202\) 2664.39 0.928050
\(203\) −452.491 −0.156446
\(204\) 1628.93 0.559058
\(205\) 0 0
\(206\) −1249.01 −0.422440
\(207\) 725.142 0.243482
\(208\) 9.86833 0.00328964
\(209\) 2001.69 0.662489
\(210\) 0 0
\(211\) 713.739 0.232871 0.116436 0.993198i \(-0.462853\pi\)
0.116436 + 0.993198i \(0.462853\pi\)
\(212\) −2568.06 −0.831959
\(213\) −1105.97 −0.355775
\(214\) −1641.57 −0.524371
\(215\) 0 0
\(216\) −608.956 −0.191825
\(217\) −1350.97 −0.422625
\(218\) −2558.03 −0.794732
\(219\) −1219.40 −0.376252
\(220\) 0 0
\(221\) 1614.05 0.491281
\(222\) 1201.60 0.363271
\(223\) 4536.63 1.36231 0.681155 0.732140i \(-0.261478\pi\)
0.681155 + 0.732140i \(0.261478\pi\)
\(224\) 1391.36 0.415019
\(225\) 0 0
\(226\) −1425.11 −0.419455
\(227\) −49.5501 −0.0144879 −0.00724396 0.999974i \(-0.502306\pi\)
−0.00724396 + 0.999974i \(0.502306\pi\)
\(228\) 1461.33 0.424470
\(229\) −2354.73 −0.679498 −0.339749 0.940516i \(-0.610342\pi\)
−0.339749 + 0.940516i \(0.610342\pi\)
\(230\) 0 0
\(231\) 470.540 0.134023
\(232\) 1331.93 0.376921
\(233\) 2899.04 0.815116 0.407558 0.913179i \(-0.366380\pi\)
0.407558 + 0.913179i \(0.366380\pi\)
\(234\) −231.545 −0.0646861
\(235\) 0 0
\(236\) 2846.62 0.785166
\(237\) 1964.25 0.538362
\(238\) 1451.02 0.395192
\(239\) −1054.21 −0.285319 −0.142660 0.989772i \(-0.545565\pi\)
−0.142660 + 0.989772i \(0.545565\pi\)
\(240\) 0 0
\(241\) 6842.40 1.82887 0.914435 0.404732i \(-0.132635\pi\)
0.914435 + 0.404732i \(0.132635\pi\)
\(242\) 1584.44 0.420876
\(243\) −243.000 −0.0641500
\(244\) 260.609 0.0683761
\(245\) 0 0
\(246\) 935.403 0.242435
\(247\) 1447.99 0.373009
\(248\) 3976.66 1.01822
\(249\) 1265.57 0.322096
\(250\) 0 0
\(251\) −6021.53 −1.51424 −0.757122 0.653273i \(-0.773395\pi\)
−0.757122 + 0.653273i \(0.773395\pi\)
\(252\) 343.516 0.0858710
\(253\) 1649.33 0.409851
\(254\) 2890.78 0.714110
\(255\) 0 0
\(256\) −4069.00 −0.993408
\(257\) 539.470 0.130939 0.0654693 0.997855i \(-0.479146\pi\)
0.0654693 + 0.997855i \(0.479146\pi\)
\(258\) 2123.76 0.512478
\(259\) −1766.40 −0.423779
\(260\) 0 0
\(261\) 531.499 0.126050
\(262\) 3964.28 0.934787
\(263\) 707.750 0.165938 0.0829690 0.996552i \(-0.473560\pi\)
0.0829690 + 0.996552i \(0.473560\pi\)
\(264\) −1385.06 −0.322897
\(265\) 0 0
\(266\) 1301.73 0.300053
\(267\) −1275.41 −0.292337
\(268\) −2748.83 −0.626536
\(269\) −2877.92 −0.652304 −0.326152 0.945317i \(-0.605752\pi\)
−0.326152 + 0.945317i \(0.605752\pi\)
\(270\) 0 0
\(271\) 3569.27 0.800065 0.400033 0.916501i \(-0.368999\pi\)
0.400033 + 0.916501i \(0.368999\pi\)
\(272\) 72.6401 0.0161928
\(273\) 340.379 0.0754604
\(274\) 1624.45 0.358163
\(275\) 0 0
\(276\) 1204.09 0.262600
\(277\) −6956.58 −1.50895 −0.754477 0.656327i \(-0.772109\pi\)
−0.754477 + 0.656327i \(0.772109\pi\)
\(278\) 4993.67 1.07734
\(279\) 1586.86 0.340512
\(280\) 0 0
\(281\) 3329.41 0.706819 0.353409 0.935469i \(-0.385022\pi\)
0.353409 + 0.935469i \(0.385022\pi\)
\(282\) −2487.08 −0.525190
\(283\) 613.980 0.128966 0.0644829 0.997919i \(-0.479460\pi\)
0.0644829 + 0.997919i \(0.479460\pi\)
\(284\) −1836.45 −0.383709
\(285\) 0 0
\(286\) −526.645 −0.108885
\(287\) −1375.08 −0.282816
\(288\) −1634.30 −0.334383
\(289\) 6967.94 1.41827
\(290\) 0 0
\(291\) 1772.00 0.356963
\(292\) −2024.79 −0.405793
\(293\) 6494.46 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(294\) −1481.78 −0.293944
\(295\) 0 0
\(296\) 5199.50 1.02100
\(297\) −552.700 −0.107983
\(298\) −3952.81 −0.768390
\(299\) 1193.09 0.230763
\(300\) 0 0
\(301\) −3122.00 −0.597838
\(302\) −4162.99 −0.793222
\(303\) 4600.66 0.872281
\(304\) 65.1663 0.0122946
\(305\) 0 0
\(306\) −1704.38 −0.318409
\(307\) 967.468 0.179858 0.0899289 0.995948i \(-0.471336\pi\)
0.0899289 + 0.995948i \(0.471336\pi\)
\(308\) 781.323 0.144546
\(309\) −2156.69 −0.397055
\(310\) 0 0
\(311\) −1530.33 −0.279026 −0.139513 0.990220i \(-0.544554\pi\)
−0.139513 + 0.990220i \(0.544554\pi\)
\(312\) −1001.93 −0.181804
\(313\) −7983.61 −1.44173 −0.720863 0.693078i \(-0.756254\pi\)
−0.720863 + 0.693078i \(0.756254\pi\)
\(314\) 1567.87 0.281783
\(315\) 0 0
\(316\) 3261.61 0.580632
\(317\) 5281.05 0.935689 0.467844 0.883811i \(-0.345031\pi\)
0.467844 + 0.883811i \(0.345031\pi\)
\(318\) 2687.02 0.473838
\(319\) 1208.89 0.212178
\(320\) 0 0
\(321\) −2834.53 −0.492860
\(322\) 1072.58 0.185629
\(323\) 10658.5 1.83609
\(324\) −403.497 −0.0691868
\(325\) 0 0
\(326\) −6998.69 −1.18902
\(327\) −4417.00 −0.746975
\(328\) 4047.63 0.681381
\(329\) 3656.11 0.612667
\(330\) 0 0
\(331\) −2082.60 −0.345830 −0.172915 0.984937i \(-0.555319\pi\)
−0.172915 + 0.984937i \(0.555319\pi\)
\(332\) 2101.45 0.347386
\(333\) 2074.83 0.341441
\(334\) −1706.67 −0.279595
\(335\) 0 0
\(336\) 15.3187 0.00248721
\(337\) −6924.52 −1.11930 −0.559648 0.828730i \(-0.689064\pi\)
−0.559648 + 0.828730i \(0.689064\pi\)
\(338\) 3436.10 0.552956
\(339\) −2460.76 −0.394249
\(340\) 0 0
\(341\) 3609.29 0.573179
\(342\) −1529.02 −0.241755
\(343\) 4806.39 0.756620
\(344\) 9189.81 1.44035
\(345\) 0 0
\(346\) 2197.98 0.341515
\(347\) 2159.18 0.334037 0.167019 0.985954i \(-0.446586\pi\)
0.167019 + 0.985954i \(0.446586\pi\)
\(348\) 882.546 0.135947
\(349\) −11809.8 −1.81136 −0.905679 0.423963i \(-0.860639\pi\)
−0.905679 + 0.423963i \(0.860639\pi\)
\(350\) 0 0
\(351\) −399.813 −0.0607989
\(352\) −3717.20 −0.562862
\(353\) −8267.86 −1.24661 −0.623306 0.781978i \(-0.714211\pi\)
−0.623306 + 0.781978i \(0.714211\pi\)
\(354\) −2978.48 −0.447187
\(355\) 0 0
\(356\) −2117.80 −0.315290
\(357\) 2505.51 0.371444
\(358\) 2556.77 0.377457
\(359\) 4656.29 0.684539 0.342269 0.939602i \(-0.388804\pi\)
0.342269 + 0.939602i \(0.388804\pi\)
\(360\) 0 0
\(361\) 2702.90 0.394066
\(362\) 6503.05 0.944179
\(363\) 2735.89 0.395584
\(364\) 565.194 0.0813852
\(365\) 0 0
\(366\) −272.680 −0.0389433
\(367\) 519.600 0.0739044 0.0369522 0.999317i \(-0.488235\pi\)
0.0369522 + 0.999317i \(0.488235\pi\)
\(368\) 53.6947 0.00760606
\(369\) 1615.18 0.227867
\(370\) 0 0
\(371\) −3950.02 −0.552762
\(372\) 2634.95 0.367247
\(373\) 4202.26 0.583337 0.291669 0.956519i \(-0.405790\pi\)
0.291669 + 0.956519i \(0.405790\pi\)
\(374\) −3876.59 −0.535973
\(375\) 0 0
\(376\) −10762.0 −1.47608
\(377\) 874.486 0.119465
\(378\) −359.428 −0.0489074
\(379\) 2855.16 0.386964 0.193482 0.981104i \(-0.438022\pi\)
0.193482 + 0.981104i \(0.438022\pi\)
\(380\) 0 0
\(381\) 4991.58 0.671197
\(382\) −6869.26 −0.920057
\(383\) 5756.86 0.768046 0.384023 0.923324i \(-0.374538\pi\)
0.384023 + 0.923324i \(0.374538\pi\)
\(384\) −2741.52 −0.364330
\(385\) 0 0
\(386\) −8233.97 −1.08575
\(387\) 3667.13 0.481682
\(388\) 2942.37 0.384990
\(389\) 10994.1 1.43296 0.716481 0.697607i \(-0.245752\pi\)
0.716481 + 0.697607i \(0.245752\pi\)
\(390\) 0 0
\(391\) 8782.25 1.13590
\(392\) −6411.90 −0.826148
\(393\) 6845.21 0.878613
\(394\) −3322.93 −0.424890
\(395\) 0 0
\(396\) −917.749 −0.116461
\(397\) 163.743 0.0207004 0.0103502 0.999946i \(-0.496705\pi\)
0.0103502 + 0.999946i \(0.496705\pi\)
\(398\) 5719.19 0.720294
\(399\) 2247.72 0.282022
\(400\) 0 0
\(401\) 8096.74 1.00831 0.504154 0.863614i \(-0.331804\pi\)
0.504154 + 0.863614i \(0.331804\pi\)
\(402\) 2876.16 0.356841
\(403\) 2610.89 0.322724
\(404\) 7639.32 0.940768
\(405\) 0 0
\(406\) 786.156 0.0960992
\(407\) 4719.16 0.574743
\(408\) −7375.12 −0.894909
\(409\) 5493.78 0.664181 0.332091 0.943247i \(-0.392246\pi\)
0.332091 + 0.943247i \(0.392246\pi\)
\(410\) 0 0
\(411\) 2804.98 0.336640
\(412\) −3581.15 −0.428229
\(413\) 4378.48 0.521673
\(414\) −1259.86 −0.149562
\(415\) 0 0
\(416\) −2688.95 −0.316915
\(417\) 8622.67 1.01260
\(418\) −3477.74 −0.406942
\(419\) 11739.7 1.36879 0.684393 0.729114i \(-0.260067\pi\)
0.684393 + 0.729114i \(0.260067\pi\)
\(420\) 0 0
\(421\) 8506.36 0.984738 0.492369 0.870387i \(-0.336131\pi\)
0.492369 + 0.870387i \(0.336131\pi\)
\(422\) −1240.05 −0.143044
\(423\) −4294.49 −0.493630
\(424\) 11627.1 1.33175
\(425\) 0 0
\(426\) 1921.52 0.218539
\(427\) 400.851 0.0454298
\(428\) −4706.69 −0.531557
\(429\) −909.368 −0.102342
\(430\) 0 0
\(431\) −15772.5 −1.76272 −0.881362 0.472441i \(-0.843373\pi\)
−0.881362 + 0.472441i \(0.843373\pi\)
\(432\) −17.9935 −0.00200396
\(433\) 5269.35 0.584824 0.292412 0.956292i \(-0.405542\pi\)
0.292412 + 0.956292i \(0.405542\pi\)
\(434\) 2347.17 0.259603
\(435\) 0 0
\(436\) −7334.35 −0.805624
\(437\) 7878.67 0.862443
\(438\) 2118.58 0.231117
\(439\) 6863.61 0.746201 0.373100 0.927791i \(-0.378295\pi\)
0.373100 + 0.927791i \(0.378295\pi\)
\(440\) 0 0
\(441\) −2558.63 −0.276280
\(442\) −2804.25 −0.301775
\(443\) −16108.2 −1.72759 −0.863795 0.503843i \(-0.831919\pi\)
−0.863795 + 0.503843i \(0.831919\pi\)
\(444\) 3445.21 0.368249
\(445\) 0 0
\(446\) −7881.92 −0.836816
\(447\) −6825.40 −0.722215
\(448\) −2376.50 −0.250623
\(449\) 12908.2 1.35674 0.678370 0.734720i \(-0.262687\pi\)
0.678370 + 0.734720i \(0.262687\pi\)
\(450\) 0 0
\(451\) 3673.70 0.383565
\(452\) −4086.06 −0.425203
\(453\) −7188.32 −0.745556
\(454\) 86.0883 0.00889939
\(455\) 0 0
\(456\) −6616.31 −0.679467
\(457\) −10742.8 −1.09962 −0.549812 0.835288i \(-0.685301\pi\)
−0.549812 + 0.835288i \(0.685301\pi\)
\(458\) 4091.11 0.417390
\(459\) −2942.99 −0.299275
\(460\) 0 0
\(461\) 14421.4 1.45698 0.728492 0.685054i \(-0.240222\pi\)
0.728492 + 0.685054i \(0.240222\pi\)
\(462\) −817.515 −0.0823252
\(463\) −11600.6 −1.16442 −0.582209 0.813039i \(-0.697812\pi\)
−0.582209 + 0.813039i \(0.697812\pi\)
\(464\) 39.3560 0.00393762
\(465\) 0 0
\(466\) −5036.78 −0.500696
\(467\) 3093.46 0.306527 0.153264 0.988185i \(-0.451022\pi\)
0.153264 + 0.988185i \(0.451022\pi\)
\(468\) −663.882 −0.0655726
\(469\) −4228.07 −0.416277
\(470\) 0 0
\(471\) 2707.27 0.264850
\(472\) −12888.3 −1.25685
\(473\) 8340.84 0.810808
\(474\) −3412.69 −0.330696
\(475\) 0 0
\(476\) 4160.35 0.400608
\(477\) 4639.73 0.445364
\(478\) 1831.59 0.175261
\(479\) −17950.9 −1.71231 −0.856156 0.516717i \(-0.827154\pi\)
−0.856156 + 0.516717i \(0.827154\pi\)
\(480\) 0 0
\(481\) 3413.75 0.323604
\(482\) −11888.0 −1.12341
\(483\) 1852.04 0.174474
\(484\) 4542.90 0.426644
\(485\) 0 0
\(486\) 422.188 0.0394050
\(487\) −17260.9 −1.60609 −0.803043 0.595921i \(-0.796787\pi\)
−0.803043 + 0.595921i \(0.796787\pi\)
\(488\) −1179.93 −0.109453
\(489\) −12084.8 −1.11757
\(490\) 0 0
\(491\) 8189.74 0.752745 0.376372 0.926468i \(-0.377171\pi\)
0.376372 + 0.926468i \(0.377171\pi\)
\(492\) 2681.98 0.245758
\(493\) 6437.03 0.588051
\(494\) −2515.73 −0.229126
\(495\) 0 0
\(496\) 117.502 0.0106371
\(497\) −2824.70 −0.254940
\(498\) −2198.79 −0.197852
\(499\) −10653.8 −0.955767 −0.477884 0.878423i \(-0.658596\pi\)
−0.477884 + 0.878423i \(0.658596\pi\)
\(500\) 0 0
\(501\) −2946.94 −0.262793
\(502\) 10461.8 0.930144
\(503\) −6067.02 −0.537803 −0.268902 0.963168i \(-0.586661\pi\)
−0.268902 + 0.963168i \(0.586661\pi\)
\(504\) −1555.30 −0.137457
\(505\) 0 0
\(506\) −2865.54 −0.251756
\(507\) 5933.18 0.519727
\(508\) 8288.43 0.723897
\(509\) −3141.27 −0.273545 −0.136773 0.990602i \(-0.543673\pi\)
−0.136773 + 0.990602i \(0.543673\pi\)
\(510\) 0 0
\(511\) −3114.39 −0.269613
\(512\) −241.260 −0.0208247
\(513\) −2640.19 −0.227227
\(514\) −937.275 −0.0804308
\(515\) 0 0
\(516\) 6089.21 0.519501
\(517\) −9767.76 −0.830920
\(518\) 3068.94 0.260311
\(519\) 3795.30 0.320992
\(520\) 0 0
\(521\) 15548.5 1.30747 0.653737 0.756722i \(-0.273200\pi\)
0.653737 + 0.756722i \(0.273200\pi\)
\(522\) −923.426 −0.0774277
\(523\) −19848.9 −1.65952 −0.829761 0.558119i \(-0.811523\pi\)
−0.829761 + 0.558119i \(0.811523\pi\)
\(524\) 11366.3 0.947598
\(525\) 0 0
\(526\) −1229.64 −0.101930
\(527\) 19218.6 1.58857
\(528\) −40.9259 −0.00337324
\(529\) −5675.26 −0.466447
\(530\) 0 0
\(531\) −5143.00 −0.420315
\(532\) 3732.30 0.304165
\(533\) 2657.49 0.215963
\(534\) 2215.90 0.179572
\(535\) 0 0
\(536\) 12445.6 1.00292
\(537\) 4414.83 0.354775
\(538\) 5000.09 0.400686
\(539\) −5819.56 −0.465058
\(540\) 0 0
\(541\) −13213.7 −1.05010 −0.525048 0.851072i \(-0.675953\pi\)
−0.525048 + 0.851072i \(0.675953\pi\)
\(542\) −6201.24 −0.491451
\(543\) 11229.0 0.887441
\(544\) −19793.2 −1.55997
\(545\) 0 0
\(546\) −591.374 −0.0463525
\(547\) 6622.55 0.517660 0.258830 0.965923i \(-0.416663\pi\)
0.258830 + 0.965923i \(0.416663\pi\)
\(548\) 4657.61 0.363072
\(549\) −470.843 −0.0366031
\(550\) 0 0
\(551\) 5774.74 0.446483
\(552\) −5451.61 −0.420355
\(553\) 5016.78 0.385778
\(554\) 12086.3 0.926894
\(555\) 0 0
\(556\) 14317.8 1.09210
\(557\) 15042.5 1.14430 0.572148 0.820150i \(-0.306110\pi\)
0.572148 + 0.820150i \(0.306110\pi\)
\(558\) −2757.00 −0.209164
\(559\) 6033.61 0.456519
\(560\) 0 0
\(561\) −6693.79 −0.503765
\(562\) −5784.51 −0.434173
\(563\) 3495.16 0.261641 0.130820 0.991406i \(-0.458239\pi\)
0.130820 + 0.991406i \(0.458239\pi\)
\(564\) −7130.93 −0.532387
\(565\) 0 0
\(566\) −1066.73 −0.0792189
\(567\) −620.632 −0.0459684
\(568\) 8314.68 0.614219
\(569\) 7467.34 0.550171 0.275085 0.961420i \(-0.411294\pi\)
0.275085 + 0.961420i \(0.411294\pi\)
\(570\) 0 0
\(571\) −15504.9 −1.13635 −0.568177 0.822906i \(-0.692351\pi\)
−0.568177 + 0.822906i \(0.692351\pi\)
\(572\) −1509.99 −0.110377
\(573\) −11861.3 −0.864768
\(574\) 2389.06 0.173724
\(575\) 0 0
\(576\) 2791.45 0.201928
\(577\) 25877.8 1.86708 0.933542 0.358468i \(-0.116701\pi\)
0.933542 + 0.358468i \(0.116701\pi\)
\(578\) −12106.1 −0.871188
\(579\) −14217.8 −1.02050
\(580\) 0 0
\(581\) 3232.31 0.230807
\(582\) −3078.66 −0.219269
\(583\) 10553.0 0.749675
\(584\) 9167.39 0.649571
\(585\) 0 0
\(586\) −11283.5 −0.795418
\(587\) 498.887 0.0350788 0.0175394 0.999846i \(-0.494417\pi\)
0.0175394 + 0.999846i \(0.494417\pi\)
\(588\) −4248.56 −0.297972
\(589\) 17241.2 1.20613
\(590\) 0 0
\(591\) −5737.77 −0.399358
\(592\) 153.635 0.0106662
\(593\) −16095.6 −1.11462 −0.557309 0.830305i \(-0.688166\pi\)
−0.557309 + 0.830305i \(0.688166\pi\)
\(594\) 960.260 0.0663299
\(595\) 0 0
\(596\) −11333.5 −0.778920
\(597\) 9875.44 0.677010
\(598\) −2072.87 −0.141749
\(599\) −998.977 −0.0681421 −0.0340710 0.999419i \(-0.510847\pi\)
−0.0340710 + 0.999419i \(0.510847\pi\)
\(600\) 0 0
\(601\) −12876.3 −0.873937 −0.436968 0.899477i \(-0.643948\pi\)
−0.436968 + 0.899477i \(0.643948\pi\)
\(602\) 5424.16 0.367230
\(603\) 4966.33 0.335397
\(604\) −11936.1 −0.804093
\(605\) 0 0
\(606\) −7993.18 −0.535810
\(607\) 12236.9 0.818254 0.409127 0.912477i \(-0.365833\pi\)
0.409127 + 0.912477i \(0.365833\pi\)
\(608\) −17756.7 −1.18442
\(609\) 1357.47 0.0903244
\(610\) 0 0
\(611\) −7065.81 −0.467843
\(612\) −4886.79 −0.322772
\(613\) 1839.73 0.121217 0.0606085 0.998162i \(-0.480696\pi\)
0.0606085 + 0.998162i \(0.480696\pi\)
\(614\) −1680.88 −0.110480
\(615\) 0 0
\(616\) −3537.51 −0.231380
\(617\) 3461.32 0.225847 0.112924 0.993604i \(-0.463978\pi\)
0.112924 + 0.993604i \(0.463978\pi\)
\(618\) 3747.03 0.243896
\(619\) −4899.75 −0.318155 −0.159077 0.987266i \(-0.550852\pi\)
−0.159077 + 0.987266i \(0.550852\pi\)
\(620\) 0 0
\(621\) −2175.43 −0.140575
\(622\) 2658.80 0.171396
\(623\) −3257.46 −0.209482
\(624\) −29.6050 −0.00189928
\(625\) 0 0
\(626\) 13870.7 0.885599
\(627\) −6005.08 −0.382488
\(628\) 4495.38 0.285645
\(629\) 25128.4 1.59290
\(630\) 0 0
\(631\) −2129.24 −0.134332 −0.0671661 0.997742i \(-0.521396\pi\)
−0.0671661 + 0.997742i \(0.521396\pi\)
\(632\) −14767.2 −0.929443
\(633\) −2141.22 −0.134448
\(634\) −9175.29 −0.574759
\(635\) 0 0
\(636\) 7704.19 0.480332
\(637\) −4209.76 −0.261847
\(638\) −2100.32 −0.130333
\(639\) 3317.92 0.205407
\(640\) 0 0
\(641\) −20797.9 −1.28154 −0.640771 0.767732i \(-0.721385\pi\)
−0.640771 + 0.767732i \(0.721385\pi\)
\(642\) 4924.71 0.302746
\(643\) −6395.45 −0.392243 −0.196121 0.980580i \(-0.562835\pi\)
−0.196121 + 0.980580i \(0.562835\pi\)
\(644\) 3075.29 0.188173
\(645\) 0 0
\(646\) −18518.1 −1.12784
\(647\) −9964.69 −0.605491 −0.302745 0.953071i \(-0.597903\pi\)
−0.302745 + 0.953071i \(0.597903\pi\)
\(648\) 1826.87 0.110750
\(649\) −11697.7 −0.707510
\(650\) 0 0
\(651\) 4052.90 0.244003
\(652\) −20066.6 −1.20532
\(653\) −29629.0 −1.77561 −0.887803 0.460223i \(-0.847769\pi\)
−0.887803 + 0.460223i \(0.847769\pi\)
\(654\) 7674.09 0.458839
\(655\) 0 0
\(656\) 119.599 0.00711825
\(657\) 3658.19 0.217229
\(658\) −6352.11 −0.376339
\(659\) 18698.2 1.10528 0.552640 0.833420i \(-0.313621\pi\)
0.552640 + 0.833420i \(0.313621\pi\)
\(660\) 0 0
\(661\) 16576.7 0.975430 0.487715 0.873003i \(-0.337831\pi\)
0.487715 + 0.873003i \(0.337831\pi\)
\(662\) 3618.30 0.212431
\(663\) −4842.16 −0.283641
\(664\) −9514.50 −0.556076
\(665\) 0 0
\(666\) −3604.80 −0.209734
\(667\) 4758.18 0.276218
\(668\) −4893.34 −0.283427
\(669\) −13609.9 −0.786530
\(670\) 0 0
\(671\) −1070.93 −0.0616135
\(672\) −4174.08 −0.239611
\(673\) 8889.49 0.509160 0.254580 0.967052i \(-0.418063\pi\)
0.254580 + 0.967052i \(0.418063\pi\)
\(674\) 12030.7 0.687542
\(675\) 0 0
\(676\) 9851.95 0.560534
\(677\) −22803.3 −1.29454 −0.647270 0.762261i \(-0.724089\pi\)
−0.647270 + 0.762261i \(0.724089\pi\)
\(678\) 4275.33 0.242172
\(679\) 4525.76 0.255792
\(680\) 0 0
\(681\) 148.650 0.00836460
\(682\) −6270.77 −0.352082
\(683\) 7599.47 0.425748 0.212874 0.977080i \(-0.431718\pi\)
0.212874 + 0.977080i \(0.431718\pi\)
\(684\) −4384.00 −0.245068
\(685\) 0 0
\(686\) −8350.61 −0.464764
\(687\) 7064.20 0.392308
\(688\) 271.541 0.0150471
\(689\) 7633.83 0.422099
\(690\) 0 0
\(691\) −12622.0 −0.694883 −0.347441 0.937702i \(-0.612949\pi\)
−0.347441 + 0.937702i \(0.612949\pi\)
\(692\) 6302.03 0.346195
\(693\) −1411.62 −0.0773780
\(694\) −3751.36 −0.205187
\(695\) 0 0
\(696\) −3995.80 −0.217616
\(697\) 19561.6 1.06305
\(698\) 20518.3 1.11265
\(699\) −8697.11 −0.470608
\(700\) 0 0
\(701\) −4783.64 −0.257740 −0.128870 0.991662i \(-0.541135\pi\)
−0.128870 + 0.991662i \(0.541135\pi\)
\(702\) 694.634 0.0373465
\(703\) 22543.0 1.20942
\(704\) 6349.12 0.339903
\(705\) 0 0
\(706\) 14364.6 0.765747
\(707\) 11750.3 0.625056
\(708\) −8539.86 −0.453316
\(709\) −33251.4 −1.76133 −0.880664 0.473741i \(-0.842903\pi\)
−0.880664 + 0.473741i \(0.842903\pi\)
\(710\) 0 0
\(711\) −5892.75 −0.310823
\(712\) 9588.53 0.504699
\(713\) 14206.1 0.746177
\(714\) −4353.06 −0.228164
\(715\) 0 0
\(716\) 7330.76 0.382630
\(717\) 3162.64 0.164729
\(718\) −8089.82 −0.420487
\(719\) 12428.5 0.644652 0.322326 0.946629i \(-0.395535\pi\)
0.322326 + 0.946629i \(0.395535\pi\)
\(720\) 0 0
\(721\) −5508.28 −0.284520
\(722\) −4696.01 −0.242060
\(723\) −20527.2 −1.05590
\(724\) 18645.5 0.957119
\(725\) 0 0
\(726\) −4753.33 −0.242993
\(727\) −8782.11 −0.448020 −0.224010 0.974587i \(-0.571915\pi\)
−0.224010 + 0.974587i \(0.571915\pi\)
\(728\) −2558.97 −0.130277
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 44412.9 2.24716
\(732\) −781.827 −0.0394770
\(733\) −26084.3 −1.31438 −0.657192 0.753723i \(-0.728256\pi\)
−0.657192 + 0.753723i \(0.728256\pi\)
\(734\) −902.753 −0.0453967
\(735\) 0 0
\(736\) −14630.9 −0.732748
\(737\) 11295.8 0.564569
\(738\) −2806.21 −0.139970
\(739\) −23385.1 −1.16405 −0.582027 0.813170i \(-0.697740\pi\)
−0.582027 + 0.813170i \(0.697740\pi\)
\(740\) 0 0
\(741\) −4343.96 −0.215357
\(742\) 6862.75 0.339541
\(743\) −26389.6 −1.30302 −0.651508 0.758642i \(-0.725863\pi\)
−0.651508 + 0.758642i \(0.725863\pi\)
\(744\) −11930.0 −0.587868
\(745\) 0 0
\(746\) −7301.00 −0.358322
\(747\) −3796.70 −0.185962
\(748\) −11114.9 −0.543318
\(749\) −7239.52 −0.353172
\(750\) 0 0
\(751\) 17725.3 0.861257 0.430629 0.902529i \(-0.358292\pi\)
0.430629 + 0.902529i \(0.358292\pi\)
\(752\) −317.995 −0.0154203
\(753\) 18064.6 0.874250
\(754\) −1519.33 −0.0733830
\(755\) 0 0
\(756\) −1030.55 −0.0495777
\(757\) 4478.94 0.215046 0.107523 0.994203i \(-0.465708\pi\)
0.107523 + 0.994203i \(0.465708\pi\)
\(758\) −4960.54 −0.237698
\(759\) −4947.98 −0.236627
\(760\) 0 0
\(761\) 1368.09 0.0651683 0.0325841 0.999469i \(-0.489626\pi\)
0.0325841 + 0.999469i \(0.489626\pi\)
\(762\) −8672.35 −0.412292
\(763\) −11281.2 −0.535265
\(764\) −19695.5 −0.932666
\(765\) 0 0
\(766\) −10001.9 −0.471782
\(767\) −8461.88 −0.398358
\(768\) 12207.0 0.573544
\(769\) 16530.6 0.775176 0.387588 0.921833i \(-0.373308\pi\)
0.387588 + 0.921833i \(0.373308\pi\)
\(770\) 0 0
\(771\) −1618.41 −0.0755975
\(772\) −23608.4 −1.10063
\(773\) 8002.04 0.372333 0.186166 0.982518i \(-0.440394\pi\)
0.186166 + 0.982518i \(0.440394\pi\)
\(774\) −6371.27 −0.295879
\(775\) 0 0
\(776\) −13321.8 −0.616271
\(777\) 5299.19 0.244669
\(778\) −19101.1 −0.880215
\(779\) 17548.9 0.807131
\(780\) 0 0
\(781\) 7546.56 0.345758
\(782\) −15258.3 −0.697742
\(783\) −1594.50 −0.0727749
\(784\) −189.459 −0.00863061
\(785\) 0 0
\(786\) −11892.8 −0.539699
\(787\) 19432.4 0.880166 0.440083 0.897957i \(-0.354949\pi\)
0.440083 + 0.897957i \(0.354949\pi\)
\(788\) −9527.47 −0.430713
\(789\) −2123.25 −0.0958044
\(790\) 0 0
\(791\) −6284.89 −0.282510
\(792\) 4155.19 0.186424
\(793\) −774.687 −0.0346910
\(794\) −284.487 −0.0127155
\(795\) 0 0
\(796\) 16398.0 0.730165
\(797\) 39528.1 1.75678 0.878391 0.477942i \(-0.158617\pi\)
0.878391 + 0.477942i \(0.158617\pi\)
\(798\) −3905.19 −0.173236
\(799\) −52010.9 −2.30290
\(800\) 0 0
\(801\) 3826.24 0.168781
\(802\) −14067.3 −0.619366
\(803\) 8320.49 0.365659
\(804\) 8246.50 0.361731
\(805\) 0 0
\(806\) −4536.15 −0.198237
\(807\) 8633.76 0.376608
\(808\) −34587.7 −1.50593
\(809\) −2308.05 −0.100305 −0.0501526 0.998742i \(-0.515971\pi\)
−0.0501526 + 0.998742i \(0.515971\pi\)
\(810\) 0 0
\(811\) −4243.29 −0.183726 −0.0918632 0.995772i \(-0.529282\pi\)
−0.0918632 + 0.995772i \(0.529282\pi\)
\(812\) 2254.06 0.0974162
\(813\) −10707.8 −0.461918
\(814\) −8199.07 −0.353043
\(815\) 0 0
\(816\) −217.920 −0.00934894
\(817\) 39843.4 1.70617
\(818\) −9544.89 −0.407982
\(819\) −1021.14 −0.0435671
\(820\) 0 0
\(821\) 28705.4 1.22025 0.610124 0.792306i \(-0.291120\pi\)
0.610124 + 0.792306i \(0.291120\pi\)
\(822\) −4873.36 −0.206786
\(823\) −28010.8 −1.18639 −0.593193 0.805061i \(-0.702133\pi\)
−0.593193 + 0.805061i \(0.702133\pi\)
\(824\) 16214.0 0.685486
\(825\) 0 0
\(826\) −7607.16 −0.320444
\(827\) −8868.42 −0.372896 −0.186448 0.982465i \(-0.559698\pi\)
−0.186448 + 0.982465i \(0.559698\pi\)
\(828\) −3612.26 −0.151612
\(829\) −24216.6 −1.01457 −0.507283 0.861779i \(-0.669350\pi\)
−0.507283 + 0.861779i \(0.669350\pi\)
\(830\) 0 0
\(831\) 20869.7 0.871195
\(832\) 4592.83 0.191380
\(833\) −30987.7 −1.28891
\(834\) −14981.0 −0.622003
\(835\) 0 0
\(836\) −9971.34 −0.412519
\(837\) −4760.58 −0.196594
\(838\) −20396.5 −0.840794
\(839\) 33351.0 1.37235 0.686177 0.727435i \(-0.259288\pi\)
0.686177 + 0.727435i \(0.259288\pi\)
\(840\) 0 0
\(841\) −20901.4 −0.857003
\(842\) −14778.9 −0.604888
\(843\) −9988.24 −0.408082
\(844\) −3555.45 −0.145004
\(845\) 0 0
\(846\) 7461.24 0.303218
\(847\) 6987.58 0.283467
\(848\) 343.559 0.0139126
\(849\) −1841.94 −0.0744585
\(850\) 0 0
\(851\) 18574.6 0.748214
\(852\) 5509.35 0.221534
\(853\) 25395.6 1.01938 0.509688 0.860359i \(-0.329761\pi\)
0.509688 + 0.860359i \(0.329761\pi\)
\(854\) −696.437 −0.0279059
\(855\) 0 0
\(856\) 21310.0 0.850887
\(857\) −28013.4 −1.11659 −0.558296 0.829642i \(-0.688545\pi\)
−0.558296 + 0.829642i \(0.688545\pi\)
\(858\) 1579.93 0.0628649
\(859\) 7094.28 0.281786 0.140893 0.990025i \(-0.455003\pi\)
0.140893 + 0.990025i \(0.455003\pi\)
\(860\) 0 0
\(861\) 4125.23 0.163284
\(862\) 27403.1 1.08278
\(863\) 12168.3 0.479969 0.239984 0.970777i \(-0.422858\pi\)
0.239984 + 0.970777i \(0.422858\pi\)
\(864\) 4902.91 0.193056
\(865\) 0 0
\(866\) −9154.96 −0.359236
\(867\) −20903.8 −0.818836
\(868\) 6729.78 0.263161
\(869\) −13403.0 −0.523205
\(870\) 0 0
\(871\) 8171.20 0.317876
\(872\) 33206.9 1.28960
\(873\) −5315.99 −0.206093
\(874\) −13688.4 −0.529767
\(875\) 0 0
\(876\) 6074.36 0.234285
\(877\) 5576.23 0.214704 0.107352 0.994221i \(-0.465763\pi\)
0.107352 + 0.994221i \(0.465763\pi\)
\(878\) −11924.8 −0.458364
\(879\) −19483.4 −0.747620
\(880\) 0 0
\(881\) 44427.2 1.69897 0.849483 0.527616i \(-0.176914\pi\)
0.849483 + 0.527616i \(0.176914\pi\)
\(882\) 4445.35 0.169708
\(883\) −28781.0 −1.09689 −0.548447 0.836186i \(-0.684781\pi\)
−0.548447 + 0.836186i \(0.684781\pi\)
\(884\) −8040.33 −0.305911
\(885\) 0 0
\(886\) 27986.3 1.06119
\(887\) −13229.3 −0.500785 −0.250392 0.968144i \(-0.580560\pi\)
−0.250392 + 0.968144i \(0.580560\pi\)
\(888\) −15598.5 −0.589472
\(889\) 12748.7 0.480965
\(890\) 0 0
\(891\) 1658.10 0.0623439
\(892\) −22599.0 −0.848284
\(893\) −46659.6 −1.74849
\(894\) 11858.4 0.443630
\(895\) 0 0
\(896\) −7001.96 −0.261071
\(897\) −3579.27 −0.133231
\(898\) −22426.7 −0.833395
\(899\) 10412.5 0.386293
\(900\) 0 0
\(901\) 56192.1 2.07772
\(902\) −6382.68 −0.235610
\(903\) 9366.01 0.345162
\(904\) 18500.0 0.680642
\(905\) 0 0
\(906\) 12489.0 0.457967
\(907\) 18849.7 0.690071 0.345035 0.938590i \(-0.387867\pi\)
0.345035 + 0.938590i \(0.387867\pi\)
\(908\) 246.831 0.00902135
\(909\) −13802.0 −0.503611
\(910\) 0 0
\(911\) −20189.5 −0.734258 −0.367129 0.930170i \(-0.619659\pi\)
−0.367129 + 0.930170i \(0.619659\pi\)
\(912\) −195.499 −0.00709826
\(913\) −8635.54 −0.313028
\(914\) 18664.6 0.675459
\(915\) 0 0
\(916\) 11730.0 0.423111
\(917\) 17482.9 0.629594
\(918\) 5113.15 0.183833
\(919\) −9727.34 −0.349157 −0.174579 0.984643i \(-0.555856\pi\)
−0.174579 + 0.984643i \(0.555856\pi\)
\(920\) 0 0
\(921\) −2902.41 −0.103841
\(922\) −25055.6 −0.894971
\(923\) 5459.04 0.194676
\(924\) −2343.97 −0.0834534
\(925\) 0 0
\(926\) 20154.9 0.715259
\(927\) 6470.07 0.229240
\(928\) −10723.8 −0.379340
\(929\) −39516.5 −1.39558 −0.697790 0.716303i \(-0.745833\pi\)
−0.697790 + 0.716303i \(0.745833\pi\)
\(930\) 0 0
\(931\) −27799.5 −0.978615
\(932\) −14441.4 −0.507558
\(933\) 4591.00 0.161096
\(934\) −5374.57 −0.188288
\(935\) 0 0
\(936\) 3005.78 0.104965
\(937\) −32692.4 −1.13982 −0.569912 0.821706i \(-0.693023\pi\)
−0.569912 + 0.821706i \(0.693023\pi\)
\(938\) 7345.84 0.255704
\(939\) 23950.8 0.832381
\(940\) 0 0
\(941\) −11482.8 −0.397797 −0.198899 0.980020i \(-0.563736\pi\)
−0.198899 + 0.980020i \(0.563736\pi\)
\(942\) −4703.61 −0.162688
\(943\) 14459.7 0.499334
\(944\) −380.825 −0.0131301
\(945\) 0 0
\(946\) −14491.4 −0.498049
\(947\) −43381.4 −1.48860 −0.744300 0.667845i \(-0.767217\pi\)
−0.744300 + 0.667845i \(0.767217\pi\)
\(948\) −9784.82 −0.335228
\(949\) 6018.89 0.205881
\(950\) 0 0
\(951\) −15843.2 −0.540220
\(952\) −18836.4 −0.641271
\(953\) −23222.7 −0.789358 −0.394679 0.918819i \(-0.629144\pi\)
−0.394679 + 0.918819i \(0.629144\pi\)
\(954\) −8061.05 −0.273570
\(955\) 0 0
\(956\) 5251.51 0.177663
\(957\) −3626.66 −0.122501
\(958\) 31187.8 1.05181
\(959\) 7164.03 0.241229
\(960\) 0 0
\(961\) 1296.90 0.0435333
\(962\) −5931.05 −0.198778
\(963\) 8503.60 0.284553
\(964\) −34085.1 −1.13880
\(965\) 0 0
\(966\) −3217.74 −0.107173
\(967\) −16334.6 −0.543211 −0.271605 0.962409i \(-0.587555\pi\)
−0.271605 + 0.962409i \(0.587555\pi\)
\(968\) −20568.4 −0.682947
\(969\) −31975.6 −1.06007
\(970\) 0 0
\(971\) −39066.5 −1.29115 −0.645574 0.763698i \(-0.723382\pi\)
−0.645574 + 0.763698i \(0.723382\pi\)
\(972\) 1210.49 0.0399450
\(973\) 22022.7 0.725606
\(974\) 29989.0 0.986559
\(975\) 0 0
\(976\) −34.8646 −0.00114343
\(977\) 39942.7 1.30796 0.653982 0.756510i \(-0.273097\pi\)
0.653982 + 0.756510i \(0.273097\pi\)
\(978\) 20996.1 0.686483
\(979\) 8702.73 0.284107
\(980\) 0 0
\(981\) 13251.0 0.431266
\(982\) −14228.8 −0.462383
\(983\) 23347.1 0.757534 0.378767 0.925492i \(-0.376348\pi\)
0.378767 + 0.925492i \(0.376348\pi\)
\(984\) −12142.9 −0.393395
\(985\) 0 0
\(986\) −11183.7 −0.361218
\(987\) −10968.3 −0.353724
\(988\) −7213.08 −0.232266
\(989\) 32829.5 1.05553
\(990\) 0 0
\(991\) 1661.91 0.0532717 0.0266359 0.999645i \(-0.491521\pi\)
0.0266359 + 0.999645i \(0.491521\pi\)
\(992\) −32017.4 −1.02475
\(993\) 6247.79 0.199665
\(994\) 4907.63 0.156600
\(995\) 0 0
\(996\) −6304.36 −0.200563
\(997\) −4547.81 −0.144464 −0.0722320 0.997388i \(-0.523012\pi\)
−0.0722320 + 0.997388i \(0.523012\pi\)
\(998\) 18509.8 0.587092
\(999\) −6224.48 −0.197131
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 1875.4.a.f.1.5 14
5.4 even 2 1875.4.a.g.1.10 14
25.9 even 10 75.4.g.b.31.5 28
25.14 even 10 75.4.g.b.46.5 yes 28
75.14 odd 10 225.4.h.a.46.3 28
75.59 odd 10 225.4.h.a.181.3 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.g.b.31.5 28 25.9 even 10
75.4.g.b.46.5 yes 28 25.14 even 10
225.4.h.a.46.3 28 75.14 odd 10
225.4.h.a.181.3 28 75.59 odd 10
1875.4.a.f.1.5 14 1.1 even 1 trivial
1875.4.a.g.1.10 14 5.4 even 2