Properties

Label 1840.4.a.n.1.3
Level $1840$
Weight $4$
Character 1840.1
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $5$
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] = [1840,4,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1840.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.60878\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.89520 q^{3} -5.00000 q^{5} -11.4426 q^{7} -23.4082 q^{9} +O(q^{10})\) \(q+1.89520 q^{3} -5.00000 q^{5} -11.4426 q^{7} -23.4082 q^{9} -37.7245 q^{11} -8.69346 q^{13} -9.47598 q^{15} -105.687 q^{17} +128.279 q^{19} -21.6859 q^{21} +23.0000 q^{23} +25.0000 q^{25} -95.5335 q^{27} -133.383 q^{29} -106.008 q^{31} -71.4953 q^{33} +57.2128 q^{35} -248.835 q^{37} -16.4758 q^{39} +134.233 q^{41} -108.684 q^{43} +117.041 q^{45} +76.2000 q^{47} -212.068 q^{49} -200.297 q^{51} +476.207 q^{53} +188.622 q^{55} +243.114 q^{57} -608.000 q^{59} -366.273 q^{61} +267.850 q^{63} +43.4673 q^{65} -136.041 q^{67} +43.5895 q^{69} +152.874 q^{71} +1228.16 q^{73} +47.3799 q^{75} +431.664 q^{77} +364.637 q^{79} +450.968 q^{81} +762.744 q^{83} +528.433 q^{85} -252.788 q^{87} +271.222 q^{89} +99.4754 q^{91} -200.906 q^{93} -641.396 q^{95} +574.510 q^{97} +883.063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9} - 23 q^{11} + 132 q^{13} + 20 q^{15} + 23 q^{17} + 161 q^{19} - 60 q^{21} + 115 q^{23} + 125 q^{25} - 577 q^{27} + 401 q^{29} - 32 q^{31} + 189 q^{33} - 15 q^{35} - 38 q^{37} - 335 q^{39} - 12 q^{41} + 566 q^{43} - 385 q^{45} - 919 q^{47} - 738 q^{49} + 993 q^{51} + 1156 q^{53} + 115 q^{55} + 114 q^{57} - 1324 q^{59} - 1673 q^{61} - 270 q^{63} - 660 q^{65} - 558 q^{67} - 92 q^{69} + 108 q^{71} + 1173 q^{73} - 100 q^{75} + 2608 q^{77} - 656 q^{79} - 319 q^{81} + 82 q^{83} - 115 q^{85} - 2389 q^{87} + 570 q^{89} + 1589 q^{91} + 911 q^{93} - 805 q^{95} + 633 q^{97} - 2021 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.89520 0.364731 0.182365 0.983231i \(-0.441625\pi\)
0.182365 + 0.983231i \(0.441625\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −11.4426 −0.617840 −0.308920 0.951088i \(-0.599968\pi\)
−0.308920 + 0.951088i \(0.599968\pi\)
\(8\) 0 0
\(9\) −23.4082 −0.866972
\(10\) 0 0
\(11\) −37.7245 −1.03403 −0.517016 0.855976i \(-0.672957\pi\)
−0.517016 + 0.855976i \(0.672957\pi\)
\(12\) 0 0
\(13\) −8.69346 −0.185472 −0.0927358 0.995691i \(-0.529561\pi\)
−0.0927358 + 0.995691i \(0.529561\pi\)
\(14\) 0 0
\(15\) −9.47598 −0.163112
\(16\) 0 0
\(17\) −105.687 −1.50781 −0.753905 0.656983i \(-0.771832\pi\)
−0.753905 + 0.656983i \(0.771832\pi\)
\(18\) 0 0
\(19\) 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
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −95.5335 −0.680942
\(28\) 0 0
\(29\) −133.383 −0.854092 −0.427046 0.904230i \(-0.640446\pi\)
−0.427046 + 0.904230i \(0.640446\pi\)
\(30\) 0 0
\(31\) −106.008 −0.614179 −0.307090 0.951681i \(-0.599355\pi\)
−0.307090 + 0.951681i \(0.599355\pi\)
\(32\) 0 0
\(33\) −71.4953 −0.377143
\(34\) 0 0
\(35\) 57.2128 0.276306
\(36\) 0 0
\(37\) −248.835 −1.10563 −0.552814 0.833305i \(-0.686446\pi\)
−0.552814 + 0.833305i \(0.686446\pi\)
\(38\) 0 0
\(39\) −16.4758 −0.0676472
\(40\) 0 0
\(41\) 134.233 0.511308 0.255654 0.966768i \(-0.417709\pi\)
0.255654 + 0.966768i \(0.417709\pi\)
\(42\) 0 0
\(43\) −108.684 −0.385444 −0.192722 0.981253i \(-0.561732\pi\)
−0.192722 + 0.981253i \(0.561732\pi\)
\(44\) 0 0
\(45\) 117.041 0.387721
\(46\) 0 0
\(47\) 76.2000 0.236488 0.118244 0.992985i \(-0.462274\pi\)
0.118244 + 0.992985i \(0.462274\pi\)
\(48\) 0 0
\(49\) −212.068 −0.618274
\(50\) 0 0
\(51\) −200.297 −0.549945
\(52\) 0 0
\(53\) 476.207 1.23419 0.617095 0.786889i \(-0.288309\pi\)
0.617095 + 0.786889i \(0.288309\pi\)
\(54\) 0 0
\(55\) 188.622 0.462433
\(56\) 0 0
\(57\) 243.114 0.564935
\(58\) 0 0
\(59\) −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 \(0.625591\pi\)
\(62\) 0 0
\(63\) 267.850 0.535650
\(64\) 0 0
\(65\) 43.4673 0.0829454
\(66\) 0 0
\(67\) −136.041 −0.248060 −0.124030 0.992278i \(-0.539582\pi\)
−0.124030 + 0.992278i \(0.539582\pi\)
\(68\) 0 0
\(69\) 43.5895 0.0760516
\(70\) 0 0
\(71\) 152.874 0.255533 0.127766 0.991804i \(-0.459219\pi\)
0.127766 + 0.991804i \(0.459219\pi\)
\(72\) 0 0
\(73\) 1228.16 1.96911 0.984556 0.175070i \(-0.0560151\pi\)
0.984556 + 0.175070i \(0.0560151\pi\)
\(74\) 0 0
\(75\) 47.3799 0.0729461
\(76\) 0 0
\(77\) 431.664 0.638867
\(78\) 0 0
\(79\) 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
\(82\) 0 0
\(83\) 762.744 1.00870 0.504350 0.863499i \(-0.331732\pi\)
0.504350 + 0.863499i \(0.331732\pi\)
\(84\) 0 0
\(85\) 528.433 0.674313
\(86\) 0 0
\(87\) −252.788 −0.311514
\(88\) 0 0
\(89\) 271.222 0.323028 0.161514 0.986870i \(-0.448362\pi\)
0.161514 + 0.986870i \(0.448362\pi\)
\(90\) 0 0
\(91\) 99.4754 0.114592
\(92\) 0 0
\(93\) −200.906 −0.224010
\(94\) 0 0
\(95\) −641.396 −0.692694
\(96\) 0 0
\(97\) 574.510 0.601367 0.300684 0.953724i \(-0.402785\pi\)
0.300684 + 0.953724i \(0.402785\pi\)
\(98\) 0 0
\(99\) 883.063 0.896477
\(100\) 0 0
\(101\) 1372.25 1.35192 0.675958 0.736940i \(-0.263730\pi\)
0.675958 + 0.736940i \(0.263730\pi\)
\(102\) 0 0
\(103\) 242.428 0.231914 0.115957 0.993254i \(-0.463007\pi\)
0.115957 + 0.993254i \(0.463007\pi\)
\(104\) 0 0
\(105\) 108.429 0.100777
\(106\) 0 0
\(107\) −650.896 −0.588079 −0.294039 0.955793i \(-0.595000\pi\)
−0.294039 + 0.955793i \(0.595000\pi\)
\(108\) 0 0
\(109\) −1230.43 −1.08123 −0.540613 0.841271i \(-0.681808\pi\)
−0.540613 + 0.841271i \(0.681808\pi\)
\(110\) 0 0
\(111\) −471.591 −0.403256
\(112\) 0 0
\(113\) 238.959 0.198932 0.0994662 0.995041i \(-0.468286\pi\)
0.0994662 + 0.995041i \(0.468286\pi\)
\(114\) 0 0
\(115\) −115.000 −0.0932505
\(116\) 0 0
\(117\) 203.498 0.160799
\(118\) 0 0
\(119\) 1209.33 0.931586
\(120\) 0 0
\(121\) 92.1354 0.0692227
\(122\) 0 0
\(123\) 254.397 0.186490
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −2608.48 −1.82256 −0.911280 0.411788i \(-0.864904\pi\)
−0.911280 + 0.411788i \(0.864904\pi\)
\(128\) 0 0
\(129\) −205.977 −0.140583
\(130\) 0 0
\(131\) 936.409 0.624538 0.312269 0.949994i \(-0.398911\pi\)
0.312269 + 0.949994i \(0.398911\pi\)
\(132\) 0 0
\(133\) −1467.84 −0.956979
\(134\) 0 0
\(135\) 477.667 0.304526
\(136\) 0 0
\(137\) 415.511 0.259121 0.129560 0.991572i \(-0.458643\pi\)
0.129560 + 0.991572i \(0.458643\pi\)
\(138\) 0 0
\(139\) 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
\(142\) 0 0
\(143\) 327.956 0.191784
\(144\) 0 0
\(145\) 666.917 0.381962
\(146\) 0 0
\(147\) −401.910 −0.225503
\(148\) 0 0
\(149\) 2209.84 1.21502 0.607508 0.794314i \(-0.292169\pi\)
0.607508 + 0.794314i \(0.292169\pi\)
\(150\) 0 0
\(151\) 1384.25 0.746018 0.373009 0.927828i \(-0.378326\pi\)
0.373009 + 0.927828i \(0.378326\pi\)
\(152\) 0 0
\(153\) 2473.94 1.30723
\(154\) 0 0
\(155\) 530.039 0.274669
\(156\) 0 0
\(157\) 561.399 0.285379 0.142690 0.989767i \(-0.454425\pi\)
0.142690 + 0.989767i \(0.454425\pi\)
\(158\) 0 0
\(159\) 902.506 0.450147
\(160\) 0 0
\(161\) −263.179 −0.128829
\(162\) 0 0
\(163\) −2134.47 −1.02567 −0.512836 0.858487i \(-0.671405\pi\)
−0.512836 + 0.858487i \(0.671405\pi\)
\(164\) 0 0
\(165\) 357.476 0.168664
\(166\) 0 0
\(167\) 1315.64 0.609623 0.304812 0.952413i \(-0.401406\pi\)
0.304812 + 0.952413i \(0.401406\pi\)
\(168\) 0 0
\(169\) −2121.42 −0.965600
\(170\) 0 0
\(171\) −3002.79 −1.34286
\(172\) 0 0
\(173\) 676.565 0.297331 0.148666 0.988888i \(-0.452502\pi\)
0.148666 + 0.988888i \(0.452502\pi\)
\(174\) 0 0
\(175\) −286.064 −0.123568
\(176\) 0 0
\(177\) −1152.28 −0.489325
\(178\) 0 0
\(179\) 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 \(0.625683\pi\)
\(182\) 0 0
\(183\) −694.158 −0.280403
\(184\) 0 0
\(185\) 1244.18 0.494452
\(186\) 0 0
\(187\) 3986.97 1.55912
\(188\) 0 0
\(189\) 1093.15 0.420713
\(190\) 0 0
\(191\) 5158.92 1.95438 0.977190 0.212366i \(-0.0681169\pi\)
0.977190 + 0.212366i \(0.0681169\pi\)
\(192\) 0 0
\(193\) −4806.41 −1.79261 −0.896303 0.443442i \(-0.853757\pi\)
−0.896303 + 0.443442i \(0.853757\pi\)
\(194\) 0 0
\(195\) 82.3790 0.0302527
\(196\) 0 0
\(197\) 3202.59 1.15825 0.579125 0.815239i \(-0.303394\pi\)
0.579125 + 0.815239i \(0.303394\pi\)
\(198\) 0 0
\(199\) 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
\(202\) 0 0
\(203\) 1526.25 0.527692
\(204\) 0 0
\(205\) −671.163 −0.228664
\(206\) 0 0
\(207\) −538.389 −0.180776
\(208\) 0 0
\(209\) −4839.27 −1.60162
\(210\) 0 0
\(211\) −153.164 −0.0499726 −0.0249863 0.999688i \(-0.507954\pi\)
−0.0249863 + 0.999688i \(0.507954\pi\)
\(212\) 0 0
\(213\) 289.727 0.0932007
\(214\) 0 0
\(215\) 543.419 0.172376
\(216\) 0 0
\(217\) 1213.00 0.379465
\(218\) 0 0
\(219\) 2327.60 0.718195
\(220\) 0 0
\(221\) 918.782 0.279656
\(222\) 0 0
\(223\) 3068.41 0.921416 0.460708 0.887552i \(-0.347595\pi\)
0.460708 + 0.887552i \(0.347595\pi\)
\(224\) 0 0
\(225\) −585.206 −0.173394
\(226\) 0 0
\(227\) −4540.20 −1.32750 −0.663752 0.747953i \(-0.731037\pi\)
−0.663752 + 0.747953i \(0.731037\pi\)
\(228\) 0 0
\(229\) 1476.25 0.425996 0.212998 0.977053i \(-0.431677\pi\)
0.212998 + 0.977053i \(0.431677\pi\)
\(230\) 0 0
\(231\) 818.089 0.233014
\(232\) 0 0
\(233\) −90.7306 −0.0255106 −0.0127553 0.999919i \(-0.504060\pi\)
−0.0127553 + 0.999919i \(0.504060\pi\)
\(234\) 0 0
\(235\) −381.000 −0.105760
\(236\) 0 0
\(237\) 691.059 0.189405
\(238\) 0 0
\(239\) 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 \(0.830573\pi\)
\(242\) 0 0
\(243\) 3434.08 0.906568
\(244\) 0 0
\(245\) 1060.34 0.276500
\(246\) 0 0
\(247\) −1115.19 −0.287279
\(248\) 0 0
\(249\) 1445.55 0.367904
\(250\) 0 0
\(251\) −3428.17 −0.862089 −0.431044 0.902331i \(-0.641855\pi\)
−0.431044 + 0.902331i \(0.641855\pi\)
\(252\) 0 0
\(253\) −867.663 −0.215611
\(254\) 0 0
\(255\) 1001.48 0.245943
\(256\) 0 0
\(257\) 2261.08 0.548803 0.274402 0.961615i \(-0.411520\pi\)
0.274402 + 0.961615i \(0.411520\pi\)
\(258\) 0 0
\(259\) 2847.31 0.683101
\(260\) 0 0
\(261\) 3122.27 0.740474
\(262\) 0 0
\(263\) −5319.64 −1.24724 −0.623618 0.781729i \(-0.714338\pi\)
−0.623618 + 0.781729i \(0.714338\pi\)
\(264\) 0 0
\(265\) −2381.04 −0.551947
\(266\) 0 0
\(267\) 514.019 0.117818
\(268\) 0 0
\(269\) −1992.51 −0.451620 −0.225810 0.974171i \(-0.572503\pi\)
−0.225810 + 0.974171i \(0.572503\pi\)
\(270\) 0 0
\(271\) −3950.62 −0.885546 −0.442773 0.896634i \(-0.646005\pi\)
−0.442773 + 0.896634i \(0.646005\pi\)
\(272\) 0 0
\(273\) 188.525 0.0417951
\(274\) 0 0
\(275\) −943.112 −0.206806
\(276\) 0 0
\(277\) −178.126 −0.0386375 −0.0193187 0.999813i \(-0.506150\pi\)
−0.0193187 + 0.999813i \(0.506150\pi\)
\(278\) 0 0
\(279\) 2481.46 0.532476
\(280\) 0 0
\(281\) 3523.49 0.748020 0.374010 0.927425i \(-0.377983\pi\)
0.374010 + 0.927425i \(0.377983\pi\)
\(282\) 0 0
\(283\) 699.284 0.146884 0.0734419 0.997300i \(-0.476602\pi\)
0.0734419 + 0.997300i \(0.476602\pi\)
\(284\) 0 0
\(285\) −1215.57 −0.252647
\(286\) 0 0
\(287\) −1535.96 −0.315906
\(288\) 0 0
\(289\) 6256.67 1.27349
\(290\) 0 0
\(291\) 1088.81 0.219337
\(292\) 0 0
\(293\) 8552.97 1.70536 0.852678 0.522436i \(-0.174977\pi\)
0.852678 + 0.522436i \(0.174977\pi\)
\(294\) 0 0
\(295\) 3040.00 0.599985
\(296\) 0 0
\(297\) 3603.95 0.704116
\(298\) 0 0
\(299\) −199.949 −0.0386735
\(300\) 0 0
\(301\) 1243.62 0.238143
\(302\) 0 0
\(303\) 2600.67 0.493085
\(304\) 0 0
\(305\) 1831.36 0.343815
\(306\) 0 0
\(307\) 5621.73 1.04511 0.522555 0.852606i \(-0.324979\pi\)
0.522555 + 0.852606i \(0.324979\pi\)
\(308\) 0 0
\(309\) 459.448 0.0845861
\(310\) 0 0
\(311\) 6533.62 1.19128 0.595639 0.803252i \(-0.296899\pi\)
0.595639 + 0.803252i \(0.296899\pi\)
\(312\) 0 0
\(313\) −2713.24 −0.489973 −0.244987 0.969526i \(-0.578784\pi\)
−0.244987 + 0.969526i \(0.578784\pi\)
\(314\) 0 0
\(315\) −1339.25 −0.239550
\(316\) 0 0
\(317\) −7544.32 −1.33669 −0.668346 0.743851i \(-0.732997\pi\)
−0.668346 + 0.743851i \(0.732997\pi\)
\(318\) 0 0
\(319\) 5031.82 0.883159
\(320\) 0 0
\(321\) −1233.57 −0.214490
\(322\) 0 0
\(323\) −13557.4 −2.33546
\(324\) 0 0
\(325\) −217.336 −0.0370943
\(326\) 0 0
\(327\) −2331.90 −0.394356
\(328\) 0 0
\(329\) −871.923 −0.146111
\(330\) 0 0
\(331\) −5991.64 −0.994956 −0.497478 0.867477i \(-0.665740\pi\)
−0.497478 + 0.867477i \(0.665740\pi\)
\(332\) 0 0
\(333\) 5824.79 0.958548
\(334\) 0 0
\(335\) 680.204 0.110936
\(336\) 0 0
\(337\) 5665.46 0.915778 0.457889 0.889009i \(-0.348606\pi\)
0.457889 + 0.889009i \(0.348606\pi\)
\(338\) 0 0
\(339\) 452.874 0.0725567
\(340\) 0 0
\(341\) 3999.09 0.635081
\(342\) 0 0
\(343\) 6351.40 0.999834
\(344\) 0 0
\(345\) −217.948 −0.0340113
\(346\) 0 0
\(347\) −5593.30 −0.865315 −0.432657 0.901558i \(-0.642424\pi\)
−0.432657 + 0.901558i \(0.642424\pi\)
\(348\) 0 0
\(349\) 4304.61 0.660230 0.330115 0.943941i \(-0.392912\pi\)
0.330115 + 0.943941i \(0.392912\pi\)
\(350\) 0 0
\(351\) 830.516 0.126295
\(352\) 0 0
\(353\) 1056.64 0.159318 0.0796592 0.996822i \(-0.474617\pi\)
0.0796592 + 0.996822i \(0.474617\pi\)
\(354\) 0 0
\(355\) −764.371 −0.114278
\(356\) 0 0
\(357\) 2291.91 0.339778
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) 174.615 0.0252476
\(364\) 0 0
\(365\) −6140.80 −0.880614
\(366\) 0 0
\(367\) −178.772 −0.0254274 −0.0127137 0.999919i \(-0.504047\pi\)
−0.0127137 + 0.999919i \(0.504047\pi\)
\(368\) 0 0
\(369\) −3142.15 −0.443289
\(370\) 0 0
\(371\) −5449.03 −0.762532
\(372\) 0 0
\(373\) 7463.86 1.03610 0.518049 0.855351i \(-0.326659\pi\)
0.518049 + 0.855351i \(0.326659\pi\)
\(374\) 0 0
\(375\) −236.899 −0.0326225
\(376\) 0 0
\(377\) 1159.56 0.158410
\(378\) 0 0
\(379\) 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
\(382\) 0 0
\(383\) 580.709 0.0774747 0.0387374 0.999249i \(-0.487666\pi\)
0.0387374 + 0.999249i \(0.487666\pi\)
\(384\) 0 0
\(385\) −2158.32 −0.285710
\(386\) 0 0
\(387\) 2544.09 0.334169
\(388\) 0 0
\(389\) 11373.5 1.48241 0.741205 0.671278i \(-0.234255\pi\)
0.741205 + 0.671278i \(0.234255\pi\)
\(390\) 0 0
\(391\) −2430.79 −0.314400
\(392\) 0 0
\(393\) 1774.68 0.227788
\(394\) 0 0
\(395\) −1823.19 −0.232239
\(396\) 0 0
\(397\) 3701.46 0.467937 0.233969 0.972244i \(-0.424829\pi\)
0.233969 + 0.972244i \(0.424829\pi\)
\(398\) 0 0
\(399\) −2781.85 −0.349039
\(400\) 0 0
\(401\) −7615.01 −0.948317 −0.474159 0.880439i \(-0.657248\pi\)
−0.474159 + 0.880439i \(0.657248\pi\)
\(402\) 0 0
\(403\) 921.574 0.113913
\(404\) 0 0
\(405\) −2254.84 −0.276651
\(406\) 0 0
\(407\) 9387.17 1.14325
\(408\) 0 0
\(409\) −15423.6 −1.86466 −0.932330 0.361608i \(-0.882228\pi\)
−0.932330 + 0.361608i \(0.882228\pi\)
\(410\) 0 0
\(411\) 787.475 0.0945092
\(412\) 0 0
\(413\) 6957.07 0.828899
\(414\) 0 0
\(415\) −3813.72 −0.451104
\(416\) 0 0
\(417\) 1799.73 0.211351
\(418\) 0 0
\(419\) −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 \(0.614402\pi\)
\(422\) 0 0
\(423\) −1783.71 −0.205028
\(424\) 0 0
\(425\) −2642.17 −0.301562
\(426\) 0 0
\(427\) 4191.10 0.474991
\(428\) 0 0
\(429\) 621.541 0.0699494
\(430\) 0 0
\(431\) 3386.04 0.378422 0.189211 0.981936i \(-0.439407\pi\)
0.189211 + 0.981936i \(0.439407\pi\)
\(432\) 0 0
\(433\) 7924.63 0.879523 0.439762 0.898114i \(-0.355063\pi\)
0.439762 + 0.898114i \(0.355063\pi\)
\(434\) 0 0
\(435\) 1263.94 0.139313
\(436\) 0 0
\(437\) 2950.42 0.322970
\(438\) 0 0
\(439\) −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
\(442\) 0 0
\(443\) −996.052 −0.106826 −0.0534129 0.998573i \(-0.517010\pi\)
−0.0534129 + 0.998573i \(0.517010\pi\)
\(444\) 0 0
\(445\) −1356.11 −0.144463
\(446\) 0 0
\(447\) 4188.08 0.443153
\(448\) 0 0
\(449\) −5079.36 −0.533875 −0.266938 0.963714i \(-0.586012\pi\)
−0.266938 + 0.963714i \(0.586012\pi\)
\(450\) 0 0
\(451\) −5063.85 −0.528709
\(452\) 0 0
\(453\) 2623.43 0.272096
\(454\) 0 0
\(455\) −497.377 −0.0512470
\(456\) 0 0
\(457\) −11190.9 −1.14549 −0.572747 0.819732i \(-0.694122\pi\)
−0.572747 + 0.819732i \(0.694122\pi\)
\(458\) 0 0
\(459\) 10096.6 1.02673
\(460\) 0 0
\(461\) 2426.07 0.245105 0.122553 0.992462i \(-0.460892\pi\)
0.122553 + 0.992462i \(0.460892\pi\)
\(462\) 0 0
\(463\) 16349.2 1.64106 0.820531 0.571602i \(-0.193678\pi\)
0.820531 + 0.571602i \(0.193678\pi\)
\(464\) 0 0
\(465\) 1004.53 0.100180
\(466\) 0 0
\(467\) 2880.80 0.285455 0.142728 0.989762i \(-0.454413\pi\)
0.142728 + 0.989762i \(0.454413\pi\)
\(468\) 0 0
\(469\) 1556.65 0.153261
\(470\) 0 0
\(471\) 1063.96 0.104087
\(472\) 0 0
\(473\) 4100.03 0.398562
\(474\) 0 0
\(475\) 3206.98 0.309782
\(476\) 0 0
\(477\) −11147.2 −1.07001
\(478\) 0 0
\(479\) −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
\(482\) 0 0
\(483\) −498.775 −0.0469877
\(484\) 0 0
\(485\) −2872.55 −0.268940
\(486\) 0 0
\(487\) 11262.4 1.04794 0.523971 0.851736i \(-0.324450\pi\)
0.523971 + 0.851736i \(0.324450\pi\)
\(488\) 0 0
\(489\) −4045.24 −0.374094
\(490\) 0 0
\(491\) −15810.7 −1.45321 −0.726607 0.687054i \(-0.758904\pi\)
−0.726607 + 0.687054i \(0.758904\pi\)
\(492\) 0 0
\(493\) 14096.8 1.28781
\(494\) 0 0
\(495\) −4415.32 −0.400917
\(496\) 0 0
\(497\) −1749.27 −0.157878
\(498\) 0 0
\(499\) 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
\(502\) 0 0
\(503\) 5425.35 0.480923 0.240462 0.970659i \(-0.422701\pi\)
0.240462 + 0.970659i \(0.422701\pi\)
\(504\) 0 0
\(505\) −6861.23 −0.604595
\(506\) 0 0
\(507\) −4020.51 −0.352184
\(508\) 0 0
\(509\) 8098.38 0.705215 0.352608 0.935771i \(-0.385295\pi\)
0.352608 + 0.935771i \(0.385295\pi\)
\(510\) 0 0
\(511\) −14053.3 −1.21660
\(512\) 0 0
\(513\) −12255.0 −1.05472
\(514\) 0 0
\(515\) −1212.14 −0.103715
\(516\) 0 0
\(517\) −2874.60 −0.244536
\(518\) 0 0
\(519\) 1282.22 0.108446
\(520\) 0 0
\(521\) 14691.8 1.23543 0.617717 0.786400i \(-0.288058\pi\)
0.617717 + 0.786400i \(0.288058\pi\)
\(522\) 0 0
\(523\) 19263.6 1.61059 0.805297 0.592872i \(-0.202006\pi\)
0.805297 + 0.592872i \(0.202006\pi\)
\(524\) 0 0
\(525\) −542.147 −0.0450690
\(526\) 0 0
\(527\) 11203.6 0.926066
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) 14232.2 1.16314
\(532\) 0 0
\(533\) −1166.95 −0.0948331
\(534\) 0 0
\(535\) 3254.48 0.262997
\(536\) 0 0
\(537\) 7084.17 0.569282
\(538\) 0 0
\(539\) 8000.15 0.639315
\(540\) 0 0
\(541\) −18737.4 −1.48906 −0.744531 0.667588i \(-0.767327\pi\)
−0.744531 + 0.667588i \(0.767327\pi\)
\(542\) 0 0
\(543\) −3550.46 −0.280598
\(544\) 0 0
\(545\) 6152.14 0.483539
\(546\) 0 0
\(547\) 14180.2 1.10841 0.554207 0.832379i \(-0.313022\pi\)
0.554207 + 0.832379i \(0.313022\pi\)
\(548\) 0 0
\(549\) 8573.80 0.666522
\(550\) 0 0
\(551\) −17110.3 −1.32291
\(552\) 0 0
\(553\) −4172.38 −0.320846
\(554\) 0 0
\(555\) 2357.96 0.180342
\(556\) 0 0
\(557\) 15154.2 1.15279 0.576397 0.817170i \(-0.304458\pi\)
0.576397 + 0.817170i \(0.304458\pi\)
\(558\) 0 0
\(559\) 944.837 0.0714890
\(560\) 0 0
\(561\) 7556.09 0.568660
\(562\) 0 0
\(563\) −14444.7 −1.08130 −0.540651 0.841247i \(-0.681822\pi\)
−0.540651 + 0.841247i \(0.681822\pi\)
\(564\) 0 0
\(565\) −1194.79 −0.0889653
\(566\) 0 0
\(567\) −5160.22 −0.382203
\(568\) 0 0
\(569\) −8851.15 −0.652125 −0.326063 0.945348i \(-0.605722\pi\)
−0.326063 + 0.945348i \(0.605722\pi\)
\(570\) 0 0
\(571\) 22318.9 1.63576 0.817879 0.575391i \(-0.195150\pi\)
0.817879 + 0.575391i \(0.195150\pi\)
\(572\) 0 0
\(573\) 9777.17 0.712822
\(574\) 0 0
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) −4530.99 −0.326911 −0.163455 0.986551i \(-0.552264\pi\)
−0.163455 + 0.986551i \(0.552264\pi\)
\(578\) 0 0
\(579\) −9109.09 −0.653818
\(580\) 0 0
\(581\) −8727.75 −0.623215
\(582\) 0 0
\(583\) −17964.7 −1.27619
\(584\) 0 0
\(585\) −1017.49 −0.0719113
\(586\) 0 0
\(587\) 5092.89 0.358103 0.179051 0.983840i \(-0.442697\pi\)
0.179051 + 0.983840i \(0.442697\pi\)
\(588\) 0 0
\(589\) −13598.6 −0.951309
\(590\) 0 0
\(591\) 6069.54 0.422449
\(592\) 0 0
\(593\) 4193.92 0.290427 0.145214 0.989400i \(-0.453613\pi\)
0.145214 + 0.989400i \(0.453613\pi\)
\(594\) 0 0
\(595\) −6046.63 −0.416618
\(596\) 0 0
\(597\) 4189.26 0.287194
\(598\) 0 0
\(599\) −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 \(0.680361\pi\)
\(602\) 0 0
\(603\) 3184.47 0.215061
\(604\) 0 0
\(605\) −460.677 −0.0309573
\(606\) 0 0
\(607\) 740.284 0.0495011 0.0247506 0.999694i \(-0.492121\pi\)
0.0247506 + 0.999694i \(0.492121\pi\)
\(608\) 0 0
\(609\) 2892.54 0.192466
\(610\) 0 0
\(611\) −662.441 −0.0438617
\(612\) 0 0
\(613\) −12412.1 −0.817815 −0.408908 0.912576i \(-0.634090\pi\)
−0.408908 + 0.912576i \(0.634090\pi\)
\(614\) 0 0
\(615\) −1271.99 −0.0834007
\(616\) 0 0
\(617\) −27047.9 −1.76484 −0.882422 0.470459i \(-0.844089\pi\)
−0.882422 + 0.470459i \(0.844089\pi\)
\(618\) 0 0
\(619\) −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
\(622\) 0 0
\(623\) −3103.48 −0.199580
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −9171.36 −0.584161
\(628\) 0 0
\(629\) 26298.5 1.66708
\(630\) 0 0
\(631\) −11459.7 −0.722982 −0.361491 0.932376i \(-0.617732\pi\)
−0.361491 + 0.932376i \(0.617732\pi\)
\(632\) 0 0
\(633\) −290.275 −0.0182265
\(634\) 0 0
\(635\) 13042.4 0.815073
\(636\) 0 0
\(637\) 1843.60 0.114672
\(638\) 0 0
\(639\) −3578.52 −0.221540
\(640\) 0 0
\(641\) 2261.29 0.139338 0.0696689 0.997570i \(-0.477806\pi\)
0.0696689 + 0.997570i \(0.477806\pi\)
\(642\) 0 0
\(643\) −10224.8 −0.627101 −0.313551 0.949572i \(-0.601519\pi\)
−0.313551 + 0.949572i \(0.601519\pi\)
\(644\) 0 0
\(645\) 1029.88 0.0628708
\(646\) 0 0
\(647\) 16733.4 1.01678 0.508390 0.861127i \(-0.330241\pi\)
0.508390 + 0.861127i \(0.330241\pi\)
\(648\) 0 0
\(649\) 22936.5 1.38727
\(650\) 0 0
\(651\) 2298.87 0.138402
\(652\) 0 0
\(653\) −9106.08 −0.545710 −0.272855 0.962055i \(-0.587968\pi\)
−0.272855 + 0.962055i \(0.587968\pi\)
\(654\) 0 0
\(655\) −4682.04 −0.279302
\(656\) 0 0
\(657\) −28749.0 −1.70716
\(658\) 0 0
\(659\) 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 \(0.338319\pi\)
\(662\) 0 0
\(663\) 1741.27 0.101999
\(664\) 0 0
\(665\) 7339.22 0.427974
\(666\) 0 0
\(667\) −3067.82 −0.178091
\(668\) 0 0
\(669\) 5815.24 0.336069
\(670\) 0 0
\(671\) 13817.4 0.794957
\(672\) 0 0
\(673\) −10268.9 −0.588169 −0.294084 0.955779i \(-0.595015\pi\)
−0.294084 + 0.955779i \(0.595015\pi\)
\(674\) 0 0
\(675\) −2388.34 −0.136188
\(676\) 0 0
\(677\) 4729.79 0.268509 0.134254 0.990947i \(-0.457136\pi\)
0.134254 + 0.990947i \(0.457136\pi\)
\(678\) 0 0
\(679\) −6573.86 −0.371549
\(680\) 0 0
\(681\) −8604.56 −0.484182
\(682\) 0 0
\(683\) −11551.0 −0.647123 −0.323561 0.946207i \(-0.604880\pi\)
−0.323561 + 0.946207i \(0.604880\pi\)
\(684\) 0 0
\(685\) −2077.56 −0.115882
\(686\) 0 0
\(687\) 2797.77 0.155374
\(688\) 0 0
\(689\) −4139.89 −0.228907
\(690\) 0 0
\(691\) −26575.2 −1.46305 −0.731526 0.681813i \(-0.761192\pi\)
−0.731526 + 0.681813i \(0.761192\pi\)
\(692\) 0 0
\(693\) −10104.5 −0.553879
\(694\) 0 0
\(695\) −4748.15 −0.259147
\(696\) 0 0
\(697\) −14186.6 −0.770955
\(698\) 0 0
\(699\) −171.952 −0.00930448
\(700\) 0 0
\(701\) 5886.27 0.317149 0.158574 0.987347i \(-0.449310\pi\)
0.158574 + 0.987347i \(0.449310\pi\)
\(702\) 0 0
\(703\) −31920.4 −1.71252
\(704\) 0 0
\(705\) −722.070 −0.0385741
\(706\) 0 0
\(707\) −15702.0 −0.835268
\(708\) 0 0
\(709\) −2588.26 −0.137100 −0.0685501 0.997648i \(-0.521837\pi\)
−0.0685501 + 0.997648i \(0.521837\pi\)
\(710\) 0 0
\(711\) −8535.51 −0.450220
\(712\) 0 0
\(713\) −2438.18 −0.128065
\(714\) 0 0
\(715\) −1639.78 −0.0857682
\(716\) 0 0
\(717\) 3068.63 0.159833
\(718\) 0 0
\(719\) −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
\(722\) 0 0
\(723\) −12219.2 −0.628545
\(724\) 0 0
\(725\) −3334.58 −0.170818
\(726\) 0 0
\(727\) 3135.70 0.159968 0.0799838 0.996796i \(-0.474513\pi\)
0.0799838 + 0.996796i \(0.474513\pi\)
\(728\) 0 0
\(729\) −5667.88 −0.287958
\(730\) 0 0
\(731\) 11486.4 0.581177
\(732\) 0 0
\(733\) −25792.4 −1.29968 −0.649839 0.760072i \(-0.725164\pi\)
−0.649839 + 0.760072i \(0.725164\pi\)
\(734\) 0 0
\(735\) 2009.55 0.100848
\(736\) 0 0
\(737\) 5132.07 0.256502
\(738\) 0 0
\(739\) 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
\(742\) 0 0
\(743\) −1825.32 −0.0901270 −0.0450635 0.998984i \(-0.514349\pi\)
−0.0450635 + 0.998984i \(0.514349\pi\)
\(744\) 0 0
\(745\) −11049.2 −0.543371
\(746\) 0 0
\(747\) −17854.5 −0.874514
\(748\) 0 0
\(749\) 7447.91 0.363339
\(750\) 0 0
\(751\) −4310.27 −0.209433 −0.104716 0.994502i \(-0.533394\pi\)
−0.104716 + 0.994502i \(0.533394\pi\)
\(752\) 0 0
\(753\) −6497.06 −0.314430
\(754\) 0 0
\(755\) −6921.25 −0.333629
\(756\) 0 0
\(757\) 19302.2 0.926750 0.463375 0.886162i \(-0.346638\pi\)
0.463375 + 0.886162i \(0.346638\pi\)
\(758\) 0 0
\(759\) −1644.39 −0.0786398
\(760\) 0 0
\(761\) −30144.8 −1.43594 −0.717968 0.696076i \(-0.754927\pi\)
−0.717968 + 0.696076i \(0.754927\pi\)
\(762\) 0 0
\(763\) 14079.3 0.668025
\(764\) 0 0
\(765\) −12369.7 −0.584610
\(766\) 0 0
\(767\) 5285.62 0.248830
\(768\) 0 0
\(769\) 37297.7 1.74901 0.874506 0.485015i \(-0.161186\pi\)
0.874506 + 0.485015i \(0.161186\pi\)
\(770\) 0 0
\(771\) 4285.19 0.200165
\(772\) 0 0
\(773\) −13602.8 −0.632933 −0.316467 0.948604i \(-0.602497\pi\)
−0.316467 + 0.948604i \(0.602497\pi\)
\(774\) 0 0
\(775\) −2650.19 −0.122836
\(776\) 0 0
\(777\) 5396.21 0.249148
\(778\) 0 0
\(779\) 17219.3 0.791970
\(780\) 0 0
\(781\) −5767.10 −0.264229
\(782\) 0 0
\(783\) 12742.6 0.581587
\(784\) 0 0
\(785\) −2807.00 −0.127625
\(786\) 0 0
\(787\) −14129.9 −0.639997 −0.319998 0.947418i \(-0.603682\pi\)
−0.319998 + 0.947418i \(0.603682\pi\)
\(788\) 0 0
\(789\) −10081.8 −0.454905
\(790\) 0 0
\(791\) −2734.30 −0.122908
\(792\) 0 0
\(793\) 3184.18 0.142589
\(794\) 0 0
\(795\) −4512.53 −0.201312
\(796\) 0 0
\(797\) 16171.3 0.718715 0.359357 0.933200i \(-0.382996\pi\)
0.359357 + 0.933200i \(0.382996\pi\)
\(798\) 0 0
\(799\) −8053.32 −0.356578
\(800\) 0 0
\(801\) −6348.83 −0.280056
\(802\) 0 0
\(803\) −46331.7 −2.03613
\(804\) 0 0
\(805\) 1315.89 0.0576139
\(806\) 0 0
\(807\) −3776.20 −0.164720
\(808\) 0 0
\(809\) −41744.9 −1.81418 −0.907090 0.420937i \(-0.861701\pi\)
−0.907090 + 0.420937i \(0.861701\pi\)
\(810\) 0 0
\(811\) 12210.2 0.528679 0.264340 0.964430i \(-0.414846\pi\)
0.264340 + 0.964430i \(0.414846\pi\)
\(812\) 0 0
\(813\) −7487.19 −0.322986
\(814\) 0 0
\(815\) 10672.3 0.458694
\(816\) 0 0
\(817\) −13941.9 −0.597019
\(818\) 0 0
\(819\) −2328.54 −0.0993478
\(820\) 0 0
\(821\) −22326.2 −0.949074 −0.474537 0.880236i \(-0.657385\pi\)
−0.474537 + 0.880236i \(0.657385\pi\)
\(822\) 0 0
\(823\) 29128.6 1.23373 0.616864 0.787069i \(-0.288403\pi\)
0.616864 + 0.787069i \(0.288403\pi\)
\(824\) 0 0
\(825\) −1787.38 −0.0754286
\(826\) 0 0
\(827\) 13540.8 0.569357 0.284679 0.958623i \(-0.408113\pi\)
0.284679 + 0.958623i \(0.408113\pi\)
\(828\) 0 0
\(829\) 93.8298 0.00393105 0.00196553 0.999998i \(-0.499374\pi\)
0.00196553 + 0.999998i \(0.499374\pi\)
\(830\) 0 0
\(831\) −337.585 −0.0140923
\(832\) 0 0
\(833\) 22412.7 0.932239
\(834\) 0 0
\(835\) −6578.19 −0.272632
\(836\) 0 0
\(837\) 10127.3 0.418220
\(838\) 0 0
\(839\) 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
\(842\) 0 0
\(843\) 6677.70 0.272826
\(844\) 0 0
\(845\) 10607.1 0.431830
\(846\) 0 0
\(847\) −1054.26 −0.0427686
\(848\) 0 0
\(849\) 1325.28 0.0535730
\(850\) 0 0
\(851\) −5723.21 −0.230539
\(852\) 0 0
\(853\) −2680.73 −0.107604 −0.0538022 0.998552i \(-0.517134\pi\)
−0.0538022 + 0.998552i \(0.517134\pi\)
\(854\) 0 0
\(855\) 15014.0 0.600546
\(856\) 0 0
\(857\) 29267.7 1.16659 0.583295 0.812261i \(-0.301763\pi\)
0.583295 + 0.812261i \(0.301763\pi\)
\(858\) 0 0
\(859\) −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
\(862\) 0 0
\(863\) 4041.96 0.159432 0.0797161 0.996818i \(-0.474599\pi\)
0.0797161 + 0.996818i \(0.474599\pi\)
\(864\) 0 0
\(865\) −3382.83 −0.132971
\(866\) 0 0
\(867\) 11857.6 0.464482
\(868\) 0 0
\(869\) −13755.7 −0.536975
\(870\) 0 0
\(871\) 1182.66 0.0460081
\(872\) 0 0
\(873\) −13448.3 −0.521368
\(874\) 0 0
\(875\) 1430.32 0.0552613
\(876\) 0 0
\(877\) 34918.5 1.34448 0.672242 0.740331i \(-0.265331\pi\)
0.672242 + 0.740331i \(0.265331\pi\)
\(878\) 0 0
\(879\) 16209.5 0.621996
\(880\) 0 0
\(881\) 2473.77 0.0946008 0.0473004 0.998881i \(-0.484938\pi\)
0.0473004 + 0.998881i \(0.484938\pi\)
\(882\) 0 0
\(883\) −16956.8 −0.646252 −0.323126 0.946356i \(-0.604734\pi\)
−0.323126 + 0.946356i \(0.604734\pi\)
\(884\) 0 0
\(885\) 5761.39 0.218833
\(886\) 0 0
\(887\) −582.058 −0.0220334 −0.0110167 0.999939i \(-0.503507\pi\)
−0.0110167 + 0.999939i \(0.503507\pi\)
\(888\) 0 0
\(889\) 29847.7 1.12605
\(890\) 0 0
\(891\) −17012.5 −0.639664
\(892\) 0 0
\(893\) 9774.88 0.366298
\(894\) 0 0
\(895\) −18689.8 −0.698024
\(896\) 0 0
\(897\) −378.943 −0.0141054
\(898\) 0 0
\(899\) 14139.7 0.524566
\(900\) 0 0
\(901\) −50328.7 −1.86092
\(902\) 0 0
\(903\) 2356.90 0.0868580
\(904\) 0 0
\(905\) 9367.00 0.344055
\(906\) 0 0
\(907\) −26224.8 −0.960065 −0.480033 0.877251i \(-0.659375\pi\)
−0.480033 + 0.877251i \(0.659375\pi\)
\(908\) 0 0
\(909\) −32121.8 −1.17207
\(910\) 0 0
\(911\) 12266.6 0.446117 0.223058 0.974805i \(-0.428396\pi\)
0.223058 + 0.974805i \(0.428396\pi\)
\(912\) 0 0
\(913\) −28774.1 −1.04303
\(914\) 0 0
\(915\) 3470.79 0.125400
\(916\) 0 0
\(917\) −10714.9 −0.385864
\(918\) 0 0
\(919\) −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
\(922\) 0 0
\(923\) −1329.01 −0.0473941
\(924\) 0 0
\(925\) −6220.88 −0.221126
\(926\) 0 0
\(927\) −5674.81 −0.201063
\(928\) 0 0
\(929\) −2418.31 −0.0854059 −0.0427029 0.999088i \(-0.513597\pi\)
−0.0427029 + 0.999088i \(0.513597\pi\)
\(930\) 0 0
\(931\) −27203.9 −0.957650
\(932\) 0 0
\(933\) 12382.5 0.434496
\(934\) 0 0
\(935\) −19934.9 −0.697262
\(936\) 0 0
\(937\) 16448.7 0.573486 0.286743 0.958008i \(-0.407427\pi\)
0.286743 + 0.958008i \(0.407427\pi\)
\(938\) 0 0
\(939\) −5142.13 −0.178708
\(940\) 0 0
\(941\) −17373.2 −0.601859 −0.300929 0.953646i \(-0.597297\pi\)
−0.300929 + 0.953646i \(0.597297\pi\)
\(942\) 0 0
\(943\) 3087.35 0.106615
\(944\) 0 0
\(945\) −5465.74 −0.188149
\(946\) 0 0
\(947\) −18638.9 −0.639579 −0.319790 0.947489i \(-0.603612\pi\)
−0.319790 + 0.947489i \(0.603612\pi\)
\(948\) 0 0
\(949\) −10676.9 −0.365214
\(950\) 0 0
\(951\) −14298.0 −0.487532
\(952\) 0 0
\(953\) −31188.1 −1.06011 −0.530053 0.847965i \(-0.677828\pi\)
−0.530053 + 0.847965i \(0.677828\pi\)
\(954\) 0 0
\(955\) −25794.6 −0.874025
\(956\) 0 0
\(957\) 9536.28 0.322115
\(958\) 0 0
\(959\) −4754.51 −0.160095
\(960\) 0 0
\(961\) −18553.3 −0.622784
\(962\) 0 0
\(963\) 15236.3 0.509848
\(964\) 0 0
\(965\) 24032.1 0.801678
\(966\) 0 0
\(967\) −43197.8 −1.43655 −0.718277 0.695757i \(-0.755069\pi\)
−0.718277 + 0.695757i \(0.755069\pi\)
\(968\) 0 0
\(969\) −25693.9 −0.851815
\(970\) 0 0
\(971\) −13497.0 −0.446074 −0.223037 0.974810i \(-0.571597\pi\)
−0.223037 + 0.974810i \(0.571597\pi\)
\(972\) 0 0
\(973\) −10866.2 −0.358021
\(974\) 0 0
\(975\) −411.895 −0.0135294
\(976\) 0 0
\(977\) −34955.4 −1.14465 −0.572325 0.820027i \(-0.693958\pi\)
−0.572325 + 0.820027i \(0.693958\pi\)
\(978\) 0 0
\(979\) −10231.7 −0.334021
\(980\) 0 0
\(981\) 28802.2 0.937393
\(982\) 0 0
\(983\) 27521.6 0.892984 0.446492 0.894788i \(-0.352673\pi\)
0.446492 + 0.894788i \(0.352673\pi\)
\(984\) 0 0
\(985\) −16013.0 −0.517985
\(986\) 0 0
\(987\) −1652.46 −0.0532913
\(988\) 0 0
\(989\) −2499.73 −0.0803707
\(990\) 0 0
\(991\) −24567.9 −0.787513 −0.393756 0.919215i \(-0.628825\pi\)
−0.393756 + 0.919215i \(0.628825\pi\)
\(992\) 0 0
\(993\) −11355.3 −0.362891
\(994\) 0 0
\(995\) −11052.3 −0.352143
\(996\) 0 0
\(997\) 2293.84 0.0728652 0.0364326 0.999336i \(-0.488401\pi\)
0.0364326 + 0.999336i \(0.488401\pi\)
\(998\) 0 0
\(999\) 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 1840.4.a.n.1.3 5
4.3 odd 2 115.4.a.e.1.5 5
12.11 even 2 1035.4.a.k.1.1 5
20.3 even 4 575.4.b.i.24.1 10
20.7 even 4 575.4.b.i.24.10 10
20.19 odd 2 575.4.a.j.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.5 5 4.3 odd 2
575.4.a.j.1.1 5 20.19 odd 2
575.4.b.i.24.1 10 20.3 even 4
575.4.b.i.24.10 10 20.7 even 4
1035.4.a.k.1.1 5 12.11 even 2
1840.4.a.n.1.3 5 1.1 even 1 trivial