Properties

Label 1088.4.a.w.1.2
Level $1088$
Weight $4$
Character 1088.1
Self dual yes
Analytic conductor $64.194$
Analytic rank $0$
Dimension $3$
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] = [1088,4,Mod(1,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1088.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1088.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: \(64.1940780862\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 68)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.13264\) of defining polynomial
Character \(\chi\) \(=\) 1088.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.36156 q^{3} +2.53055 q^{5} -32.2301 q^{7} +1.74633 q^{9} +O(q^{10})\) \(q+5.36156 q^{3} +2.53055 q^{5} -32.2301 q^{7} +1.74633 q^{9} -6.42266 q^{11} -14.3612 q^{13} +13.5677 q^{15} +17.0000 q^{17} +72.4138 q^{19} -172.804 q^{21} +167.198 q^{23} -118.596 q^{25} -135.399 q^{27} +113.377 q^{29} +159.556 q^{31} -34.4355 q^{33} -81.5599 q^{35} +39.5931 q^{37} -76.9984 q^{39} +380.135 q^{41} -192.903 q^{43} +4.41917 q^{45} +309.261 q^{47} +695.779 q^{49} +91.1465 q^{51} -26.9125 q^{53} -16.2529 q^{55} +388.251 q^{57} +214.256 q^{59} -140.744 q^{61} -56.2844 q^{63} -36.3417 q^{65} +676.207 q^{67} +896.440 q^{69} +966.999 q^{71} +639.471 q^{73} -635.861 q^{75} +207.003 q^{77} -1319.92 q^{79} -773.101 q^{81} +1418.37 q^{83} +43.0193 q^{85} +607.876 q^{87} -387.832 q^{89} +462.862 q^{91} +855.467 q^{93} +183.247 q^{95} -359.767 q^{97} -11.2161 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{3} - 26 q^{5} - 8 q^{7} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{3} - 26 q^{5} - 8 q^{7} + 63 q^{9} + 60 q^{11} - 82 q^{13} - 40 q^{15} + 51 q^{17} - 124 q^{19} + 144 q^{21} + 120 q^{23} + 85 q^{25} - 296 q^{27} + 46 q^{29} + 272 q^{31} - 48 q^{33} - 640 q^{35} - 186 q^{37} + 728 q^{39} - 82 q^{41} + 300 q^{43} - 770 q^{45} + 224 q^{47} + 1275 q^{49} + 68 q^{51} - 202 q^{53} - 984 q^{55} + 624 q^{57} - 612 q^{59} - 786 q^{61} + 232 q^{63} + 444 q^{65} + 916 q^{67} + 240 q^{69} + 1272 q^{71} - 306 q^{73} + 1308 q^{75} + 1104 q^{77} - 1568 q^{79} - 1797 q^{81} + 948 q^{83} - 442 q^{85} + 2264 q^{87} + 478 q^{89} + 1856 q^{91} + 4688 q^{93} + 2920 q^{95} + 1318 q^{97} + 1980 q^{99}+O(q^{100}) \) 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
\(3\) 5.36156 1.03183 0.515916 0.856639i \(-0.327452\pi\)
0.515916 + 0.856639i \(0.327452\pi\)
\(4\) 0 0
\(5\) 2.53055 0.226339 0.113170 0.993576i \(-0.463900\pi\)
0.113170 + 0.993576i \(0.463900\pi\)
\(6\) 0 0
\(7\) −32.2301 −1.74026 −0.870131 0.492821i \(-0.835966\pi\)
−0.870131 + 0.492821i \(0.835966\pi\)
\(8\) 0 0
\(9\) 1.74633 0.0646789
\(10\) 0 0
\(11\) −6.42266 −0.176046 −0.0880230 0.996118i \(-0.528055\pi\)
−0.0880230 + 0.996118i \(0.528055\pi\)
\(12\) 0 0
\(13\) −14.3612 −0.306391 −0.153195 0.988196i \(-0.548956\pi\)
−0.153195 + 0.988196i \(0.548956\pi\)
\(14\) 0 0
\(15\) 13.5677 0.233544
\(16\) 0 0
\(17\) 17.0000 0.242536
\(18\) 0 0
\(19\) 72.4138 0.874361 0.437180 0.899374i \(-0.355977\pi\)
0.437180 + 0.899374i \(0.355977\pi\)
\(20\) 0 0
\(21\) −172.804 −1.79566
\(22\) 0 0
\(23\) 167.198 1.51579 0.757894 0.652378i \(-0.226229\pi\)
0.757894 + 0.652378i \(0.226229\pi\)
\(24\) 0 0
\(25\) −118.596 −0.948771
\(26\) 0 0
\(27\) −135.399 −0.965095
\(28\) 0 0
\(29\) 113.377 0.725983 0.362992 0.931792i \(-0.381755\pi\)
0.362992 + 0.931792i \(0.381755\pi\)
\(30\) 0 0
\(31\) 159.556 0.924420 0.462210 0.886770i \(-0.347057\pi\)
0.462210 + 0.886770i \(0.347057\pi\)
\(32\) 0 0
\(33\) −34.4355 −0.181650
\(34\) 0 0
\(35\) −81.5599 −0.393889
\(36\) 0 0
\(37\) 39.5931 0.175921 0.0879604 0.996124i \(-0.471965\pi\)
0.0879604 + 0.996124i \(0.471965\pi\)
\(38\) 0 0
\(39\) −76.9984 −0.316144
\(40\) 0 0
\(41\) 380.135 1.44798 0.723988 0.689812i \(-0.242307\pi\)
0.723988 + 0.689812i \(0.242307\pi\)
\(42\) 0 0
\(43\) −192.903 −0.684125 −0.342062 0.939677i \(-0.611125\pi\)
−0.342062 + 0.939677i \(0.611125\pi\)
\(44\) 0 0
\(45\) 4.41917 0.0146394
\(46\) 0 0
\(47\) 309.261 0.959793 0.479897 0.877325i \(-0.340674\pi\)
0.479897 + 0.877325i \(0.340674\pi\)
\(48\) 0 0
\(49\) 695.779 2.02851
\(50\) 0 0
\(51\) 91.1465 0.250256
\(52\) 0 0
\(53\) −26.9125 −0.0697494 −0.0348747 0.999392i \(-0.511103\pi\)
−0.0348747 + 0.999392i \(0.511103\pi\)
\(54\) 0 0
\(55\) −16.2529 −0.0398461
\(56\) 0 0
\(57\) 388.251 0.902194
\(58\) 0 0
\(59\) 214.256 0.472775 0.236388 0.971659i \(-0.424036\pi\)
0.236388 + 0.971659i \(0.424036\pi\)
\(60\) 0 0
\(61\) −140.744 −0.295417 −0.147708 0.989031i \(-0.547190\pi\)
−0.147708 + 0.989031i \(0.547190\pi\)
\(62\) 0 0
\(63\) −56.2844 −0.112558
\(64\) 0 0
\(65\) −36.3417 −0.0693482
\(66\) 0 0
\(67\) 676.207 1.23301 0.616507 0.787350i \(-0.288547\pi\)
0.616507 + 0.787350i \(0.288547\pi\)
\(68\) 0 0
\(69\) 896.440 1.56404
\(70\) 0 0
\(71\) 966.999 1.61636 0.808181 0.588935i \(-0.200452\pi\)
0.808181 + 0.588935i \(0.200452\pi\)
\(72\) 0 0
\(73\) 639.471 1.02527 0.512633 0.858608i \(-0.328670\pi\)
0.512633 + 0.858608i \(0.328670\pi\)
\(74\) 0 0
\(75\) −635.861 −0.978973
\(76\) 0 0
\(77\) 207.003 0.306366
\(78\) 0 0
\(79\) −1319.92 −1.87978 −0.939892 0.341473i \(-0.889074\pi\)
−0.939892 + 0.341473i \(0.889074\pi\)
\(80\) 0 0
\(81\) −773.101 −1.06050
\(82\) 0 0
\(83\) 1418.37 1.87575 0.937873 0.346979i \(-0.112793\pi\)
0.937873 + 0.346979i \(0.112793\pi\)
\(84\) 0 0
\(85\) 43.0193 0.0548953
\(86\) 0 0
\(87\) 607.876 0.749093
\(88\) 0 0
\(89\) −387.832 −0.461912 −0.230956 0.972964i \(-0.574185\pi\)
−0.230956 + 0.972964i \(0.574185\pi\)
\(90\) 0 0
\(91\) 462.862 0.533200
\(92\) 0 0
\(93\) 855.467 0.953847
\(94\) 0 0
\(95\) 183.247 0.197902
\(96\) 0 0
\(97\) −359.767 −0.376585 −0.188293 0.982113i \(-0.560295\pi\)
−0.188293 + 0.982113i \(0.560295\pi\)
\(98\) 0 0
\(99\) −11.2161 −0.0113864
\(100\) 0 0
\(101\) 1266.61 1.24785 0.623924 0.781485i \(-0.285537\pi\)
0.623924 + 0.781485i \(0.285537\pi\)
\(102\) 0 0
\(103\) 600.773 0.574718 0.287359 0.957823i \(-0.407223\pi\)
0.287359 + 0.957823i \(0.407223\pi\)
\(104\) 0 0
\(105\) −437.288 −0.406428
\(106\) 0 0
\(107\) −2133.76 −1.92784 −0.963918 0.266199i \(-0.914232\pi\)
−0.963918 + 0.266199i \(0.914232\pi\)
\(108\) 0 0
\(109\) −1558.38 −1.36941 −0.684703 0.728822i \(-0.740068\pi\)
−0.684703 + 0.728822i \(0.740068\pi\)
\(110\) 0 0
\(111\) 212.281 0.181521
\(112\) 0 0
\(113\) 411.282 0.342391 0.171196 0.985237i \(-0.445237\pi\)
0.171196 + 0.985237i \(0.445237\pi\)
\(114\) 0 0
\(115\) 423.102 0.343082
\(116\) 0 0
\(117\) −25.0794 −0.0198170
\(118\) 0 0
\(119\) −547.912 −0.422075
\(120\) 0 0
\(121\) −1289.75 −0.969008
\(122\) 0 0
\(123\) 2038.11 1.49407
\(124\) 0 0
\(125\) −616.433 −0.441083
\(126\) 0 0
\(127\) 1538.23 1.07477 0.537385 0.843337i \(-0.319412\pi\)
0.537385 + 0.843337i \(0.319412\pi\)
\(128\) 0 0
\(129\) −1034.26 −0.705902
\(130\) 0 0
\(131\) 1967.91 1.31250 0.656248 0.754546i \(-0.272143\pi\)
0.656248 + 0.754546i \(0.272143\pi\)
\(132\) 0 0
\(133\) −2333.90 −1.52162
\(134\) 0 0
\(135\) −342.634 −0.218439
\(136\) 0 0
\(137\) 633.040 0.394776 0.197388 0.980325i \(-0.436754\pi\)
0.197388 + 0.980325i \(0.436754\pi\)
\(138\) 0 0
\(139\) 984.854 0.600966 0.300483 0.953787i \(-0.402852\pi\)
0.300483 + 0.953787i \(0.402852\pi\)
\(140\) 0 0
\(141\) 1658.12 0.990346
\(142\) 0 0
\(143\) 92.2370 0.0539388
\(144\) 0 0
\(145\) 286.905 0.164318
\(146\) 0 0
\(147\) 3730.46 2.09308
\(148\) 0 0
\(149\) −1818.23 −0.999701 −0.499850 0.866112i \(-0.666612\pi\)
−0.499850 + 0.866112i \(0.666612\pi\)
\(150\) 0 0
\(151\) 950.482 0.512246 0.256123 0.966644i \(-0.417555\pi\)
0.256123 + 0.966644i \(0.417555\pi\)
\(152\) 0 0
\(153\) 29.6876 0.0156869
\(154\) 0 0
\(155\) 403.763 0.209233
\(156\) 0 0
\(157\) −3661.42 −1.86123 −0.930616 0.365996i \(-0.880728\pi\)
−0.930616 + 0.365996i \(0.880728\pi\)
\(158\) 0 0
\(159\) −144.293 −0.0719697
\(160\) 0 0
\(161\) −5388.79 −2.63787
\(162\) 0 0
\(163\) −2121.63 −1.01950 −0.509752 0.860322i \(-0.670263\pi\)
−0.509752 + 0.860322i \(0.670263\pi\)
\(164\) 0 0
\(165\) −87.1407 −0.0411145
\(166\) 0 0
\(167\) −1235.39 −0.572438 −0.286219 0.958164i \(-0.592398\pi\)
−0.286219 + 0.958164i \(0.592398\pi\)
\(168\) 0 0
\(169\) −1990.76 −0.906125
\(170\) 0 0
\(171\) 126.458 0.0565527
\(172\) 0 0
\(173\) 3633.55 1.59684 0.798420 0.602100i \(-0.205669\pi\)
0.798420 + 0.602100i \(0.205669\pi\)
\(174\) 0 0
\(175\) 3822.37 1.65111
\(176\) 0 0
\(177\) 1148.75 0.487825
\(178\) 0 0
\(179\) 1419.17 0.592592 0.296296 0.955096i \(-0.404248\pi\)
0.296296 + 0.955096i \(0.404248\pi\)
\(180\) 0 0
\(181\) −2145.55 −0.881092 −0.440546 0.897730i \(-0.645215\pi\)
−0.440546 + 0.897730i \(0.645215\pi\)
\(182\) 0 0
\(183\) −754.608 −0.304821
\(184\) 0 0
\(185\) 100.192 0.0398178
\(186\) 0 0
\(187\) −109.185 −0.0426974
\(188\) 0 0
\(189\) 4363.92 1.67952
\(190\) 0 0
\(191\) 2578.53 0.976837 0.488418 0.872610i \(-0.337574\pi\)
0.488418 + 0.872610i \(0.337574\pi\)
\(192\) 0 0
\(193\) 3786.54 1.41223 0.706116 0.708096i \(-0.250446\pi\)
0.706116 + 0.708096i \(0.250446\pi\)
\(194\) 0 0
\(195\) −194.848 −0.0715558
\(196\) 0 0
\(197\) −3492.69 −1.26317 −0.631583 0.775308i \(-0.717595\pi\)
−0.631583 + 0.775308i \(0.717595\pi\)
\(198\) 0 0
\(199\) −502.370 −0.178955 −0.0894775 0.995989i \(-0.528520\pi\)
−0.0894775 + 0.995989i \(0.528520\pi\)
\(200\) 0 0
\(201\) 3625.53 1.27226
\(202\) 0 0
\(203\) −3654.14 −1.26340
\(204\) 0 0
\(205\) 961.949 0.327734
\(206\) 0 0
\(207\) 291.982 0.0980394
\(208\) 0 0
\(209\) −465.089 −0.153928
\(210\) 0 0
\(211\) −1564.72 −0.510522 −0.255261 0.966872i \(-0.582161\pi\)
−0.255261 + 0.966872i \(0.582161\pi\)
\(212\) 0 0
\(213\) 5184.62 1.66781
\(214\) 0 0
\(215\) −488.150 −0.154844
\(216\) 0 0
\(217\) −5142.49 −1.60873
\(218\) 0 0
\(219\) 3428.56 1.05790
\(220\) 0 0
\(221\) −244.140 −0.0743106
\(222\) 0 0
\(223\) 5154.57 1.54787 0.773936 0.633264i \(-0.218285\pi\)
0.773936 + 0.633264i \(0.218285\pi\)
\(224\) 0 0
\(225\) −207.108 −0.0613654
\(226\) 0 0
\(227\) 4590.10 1.34210 0.671048 0.741414i \(-0.265844\pi\)
0.671048 + 0.741414i \(0.265844\pi\)
\(228\) 0 0
\(229\) 457.844 0.132119 0.0660593 0.997816i \(-0.478957\pi\)
0.0660593 + 0.997816i \(0.478957\pi\)
\(230\) 0 0
\(231\) 1109.86 0.316118
\(232\) 0 0
\(233\) 5051.77 1.42040 0.710199 0.704001i \(-0.248605\pi\)
0.710199 + 0.704001i \(0.248605\pi\)
\(234\) 0 0
\(235\) 782.599 0.217239
\(236\) 0 0
\(237\) −7076.84 −1.93962
\(238\) 0 0
\(239\) 3812.18 1.03175 0.515877 0.856663i \(-0.327466\pi\)
0.515877 + 0.856663i \(0.327466\pi\)
\(240\) 0 0
\(241\) −4221.39 −1.12831 −0.564157 0.825667i \(-0.690799\pi\)
−0.564157 + 0.825667i \(0.690799\pi\)
\(242\) 0 0
\(243\) −489.254 −0.129159
\(244\) 0 0
\(245\) 1760.70 0.459131
\(246\) 0 0
\(247\) −1039.95 −0.267896
\(248\) 0 0
\(249\) 7604.70 1.93546
\(250\) 0 0
\(251\) 1487.11 0.373966 0.186983 0.982363i \(-0.440129\pi\)
0.186983 + 0.982363i \(0.440129\pi\)
\(252\) 0 0
\(253\) −1073.85 −0.266848
\(254\) 0 0
\(255\) 230.651 0.0566428
\(256\) 0 0
\(257\) 6590.66 1.59966 0.799832 0.600223i \(-0.204922\pi\)
0.799832 + 0.600223i \(0.204922\pi\)
\(258\) 0 0
\(259\) −1276.09 −0.306148
\(260\) 0 0
\(261\) 197.993 0.0469558
\(262\) 0 0
\(263\) 171.136 0.0401244 0.0200622 0.999799i \(-0.493614\pi\)
0.0200622 + 0.999799i \(0.493614\pi\)
\(264\) 0 0
\(265\) −68.1035 −0.0157870
\(266\) 0 0
\(267\) −2079.39 −0.476616
\(268\) 0 0
\(269\) −1676.61 −0.380018 −0.190009 0.981782i \(-0.560852\pi\)
−0.190009 + 0.981782i \(0.560852\pi\)
\(270\) 0 0
\(271\) 3667.65 0.822118 0.411059 0.911609i \(-0.365159\pi\)
0.411059 + 0.911609i \(0.365159\pi\)
\(272\) 0 0
\(273\) 2481.66 0.550173
\(274\) 0 0
\(275\) 761.704 0.167027
\(276\) 0 0
\(277\) −6145.42 −1.33301 −0.666503 0.745502i \(-0.732210\pi\)
−0.666503 + 0.745502i \(0.732210\pi\)
\(278\) 0 0
\(279\) 278.637 0.0597905
\(280\) 0 0
\(281\) 1380.54 0.293082 0.146541 0.989205i \(-0.453186\pi\)
0.146541 + 0.989205i \(0.453186\pi\)
\(282\) 0 0
\(283\) −1028.78 −0.216095 −0.108047 0.994146i \(-0.534460\pi\)
−0.108047 + 0.994146i \(0.534460\pi\)
\(284\) 0 0
\(285\) 982.488 0.204202
\(286\) 0 0
\(287\) −12251.8 −2.51986
\(288\) 0 0
\(289\) 289.000 0.0588235
\(290\) 0 0
\(291\) −1928.91 −0.388573
\(292\) 0 0
\(293\) −1889.19 −0.376682 −0.188341 0.982104i \(-0.560311\pi\)
−0.188341 + 0.982104i \(0.560311\pi\)
\(294\) 0 0
\(295\) 542.185 0.107008
\(296\) 0 0
\(297\) 869.622 0.169901
\(298\) 0 0
\(299\) −2401.16 −0.464423
\(300\) 0 0
\(301\) 6217.27 1.19056
\(302\) 0 0
\(303\) 6791.02 1.28757
\(304\) 0 0
\(305\) −356.160 −0.0668645
\(306\) 0 0
\(307\) −6219.66 −1.15627 −0.578135 0.815941i \(-0.696219\pi\)
−0.578135 + 0.815941i \(0.696219\pi\)
\(308\) 0 0
\(309\) 3221.08 0.593013
\(310\) 0 0
\(311\) 5349.83 0.975437 0.487718 0.873001i \(-0.337829\pi\)
0.487718 + 0.873001i \(0.337829\pi\)
\(312\) 0 0
\(313\) 7696.97 1.38996 0.694982 0.719027i \(-0.255412\pi\)
0.694982 + 0.719027i \(0.255412\pi\)
\(314\) 0 0
\(315\) −142.430 −0.0254763
\(316\) 0 0
\(317\) 7760.26 1.37495 0.687475 0.726208i \(-0.258719\pi\)
0.687475 + 0.726208i \(0.258719\pi\)
\(318\) 0 0
\(319\) −728.179 −0.127806
\(320\) 0 0
\(321\) −11440.3 −1.98920
\(322\) 0 0
\(323\) 1231.03 0.212064
\(324\) 0 0
\(325\) 1703.18 0.290694
\(326\) 0 0
\(327\) −8355.33 −1.41300
\(328\) 0 0
\(329\) −9967.50 −1.67029
\(330\) 0 0
\(331\) −1005.22 −0.166923 −0.0834617 0.996511i \(-0.526598\pi\)
−0.0834617 + 0.996511i \(0.526598\pi\)
\(332\) 0 0
\(333\) 69.1427 0.0113784
\(334\) 0 0
\(335\) 1711.18 0.279079
\(336\) 0 0
\(337\) −10322.2 −1.66851 −0.834255 0.551379i \(-0.814102\pi\)
−0.834255 + 0.551379i \(0.814102\pi\)
\(338\) 0 0
\(339\) 2205.12 0.353290
\(340\) 0 0
\(341\) −1024.77 −0.162740
\(342\) 0 0
\(343\) −11370.1 −1.78987
\(344\) 0 0
\(345\) 2268.49 0.354003
\(346\) 0 0
\(347\) 8743.53 1.35267 0.676337 0.736593i \(-0.263567\pi\)
0.676337 + 0.736593i \(0.263567\pi\)
\(348\) 0 0
\(349\) 1613.43 0.247464 0.123732 0.992316i \(-0.460514\pi\)
0.123732 + 0.992316i \(0.460514\pi\)
\(350\) 0 0
\(351\) 1944.49 0.295696
\(352\) 0 0
\(353\) −3318.12 −0.500299 −0.250150 0.968207i \(-0.580480\pi\)
−0.250150 + 0.968207i \(0.580480\pi\)
\(354\) 0 0
\(355\) 2447.04 0.365846
\(356\) 0 0
\(357\) −2937.66 −0.435511
\(358\) 0 0
\(359\) −9734.63 −1.43113 −0.715563 0.698549i \(-0.753830\pi\)
−0.715563 + 0.698549i \(0.753830\pi\)
\(360\) 0 0
\(361\) −1615.25 −0.235493
\(362\) 0 0
\(363\) −6915.07 −0.999854
\(364\) 0 0
\(365\) 1618.21 0.232058
\(366\) 0 0
\(367\) 6534.42 0.929411 0.464706 0.885465i \(-0.346160\pi\)
0.464706 + 0.885465i \(0.346160\pi\)
\(368\) 0 0
\(369\) 663.840 0.0936535
\(370\) 0 0
\(371\) 867.393 0.121382
\(372\) 0 0
\(373\) −4737.49 −0.657635 −0.328818 0.944393i \(-0.606650\pi\)
−0.328818 + 0.944393i \(0.606650\pi\)
\(374\) 0 0
\(375\) −3305.04 −0.455124
\(376\) 0 0
\(377\) −1628.22 −0.222434
\(378\) 0 0
\(379\) 3837.29 0.520074 0.260037 0.965599i \(-0.416265\pi\)
0.260037 + 0.965599i \(0.416265\pi\)
\(380\) 0 0
\(381\) 8247.31 1.10898
\(382\) 0 0
\(383\) −1600.29 −0.213501 −0.106751 0.994286i \(-0.534045\pi\)
−0.106751 + 0.994286i \(0.534045\pi\)
\(384\) 0 0
\(385\) 523.831 0.0693426
\(386\) 0 0
\(387\) −336.871 −0.0442484
\(388\) 0 0
\(389\) 1468.12 0.191353 0.0956767 0.995412i \(-0.469498\pi\)
0.0956767 + 0.995412i \(0.469498\pi\)
\(390\) 0 0
\(391\) 2842.36 0.367632
\(392\) 0 0
\(393\) 10551.1 1.35428
\(394\) 0 0
\(395\) −3340.13 −0.425469
\(396\) 0 0
\(397\) 6070.98 0.767490 0.383745 0.923439i \(-0.374634\pi\)
0.383745 + 0.923439i \(0.374634\pi\)
\(398\) 0 0
\(399\) −12513.4 −1.57005
\(400\) 0 0
\(401\) 4834.50 0.602054 0.301027 0.953616i \(-0.402671\pi\)
0.301027 + 0.953616i \(0.402671\pi\)
\(402\) 0 0
\(403\) −2291.41 −0.283234
\(404\) 0 0
\(405\) −1956.37 −0.240032
\(406\) 0 0
\(407\) −254.293 −0.0309701
\(408\) 0 0
\(409\) 880.103 0.106402 0.0532008 0.998584i \(-0.483058\pi\)
0.0532008 + 0.998584i \(0.483058\pi\)
\(410\) 0 0
\(411\) 3394.08 0.407342
\(412\) 0 0
\(413\) −6905.49 −0.822753
\(414\) 0 0
\(415\) 3589.27 0.424555
\(416\) 0 0
\(417\) 5280.35 0.620096
\(418\) 0 0
\(419\) −11301.2 −1.31766 −0.658828 0.752293i \(-0.728948\pi\)
−0.658828 + 0.752293i \(0.728948\pi\)
\(420\) 0 0
\(421\) −8699.65 −1.00711 −0.503557 0.863962i \(-0.667976\pi\)
−0.503557 + 0.863962i \(0.667976\pi\)
\(422\) 0 0
\(423\) 540.071 0.0620783
\(424\) 0 0
\(425\) −2016.14 −0.230111
\(426\) 0 0
\(427\) 4536.19 0.514103
\(428\) 0 0
\(429\) 494.534 0.0556558
\(430\) 0 0
\(431\) −9355.08 −1.04552 −0.522759 0.852481i \(-0.675097\pi\)
−0.522759 + 0.852481i \(0.675097\pi\)
\(432\) 0 0
\(433\) 1577.31 0.175060 0.0875298 0.996162i \(-0.472103\pi\)
0.0875298 + 0.996162i \(0.472103\pi\)
\(434\) 0 0
\(435\) 1538.26 0.169549
\(436\) 0 0
\(437\) 12107.4 1.32535
\(438\) 0 0
\(439\) 3598.01 0.391170 0.195585 0.980687i \(-0.437339\pi\)
0.195585 + 0.980687i \(0.437339\pi\)
\(440\) 0 0
\(441\) 1215.06 0.131202
\(442\) 0 0
\(443\) 3503.46 0.375744 0.187872 0.982194i \(-0.439841\pi\)
0.187872 + 0.982194i \(0.439841\pi\)
\(444\) 0 0
\(445\) −981.429 −0.104549
\(446\) 0 0
\(447\) −9748.56 −1.03152
\(448\) 0 0
\(449\) 5763.75 0.605809 0.302905 0.953021i \(-0.402044\pi\)
0.302905 + 0.953021i \(0.402044\pi\)
\(450\) 0 0
\(451\) −2441.47 −0.254910
\(452\) 0 0
\(453\) 5096.07 0.528552
\(454\) 0 0
\(455\) 1171.30 0.120684
\(456\) 0 0
\(457\) 520.746 0.0533030 0.0266515 0.999645i \(-0.491516\pi\)
0.0266515 + 0.999645i \(0.491516\pi\)
\(458\) 0 0
\(459\) −2301.78 −0.234070
\(460\) 0 0
\(461\) 19646.7 1.98490 0.992451 0.122641i \(-0.0391363\pi\)
0.992451 + 0.122641i \(0.0391363\pi\)
\(462\) 0 0
\(463\) 9452.09 0.948760 0.474380 0.880320i \(-0.342672\pi\)
0.474380 + 0.880320i \(0.342672\pi\)
\(464\) 0 0
\(465\) 2164.80 0.215893
\(466\) 0 0
\(467\) 16753.0 1.66004 0.830020 0.557734i \(-0.188329\pi\)
0.830020 + 0.557734i \(0.188329\pi\)
\(468\) 0 0
\(469\) −21794.2 −2.14576
\(470\) 0 0
\(471\) −19631.0 −1.92048
\(472\) 0 0
\(473\) 1238.95 0.120437
\(474\) 0 0
\(475\) −8588.01 −0.829568
\(476\) 0 0
\(477\) −46.9981 −0.00451131
\(478\) 0 0
\(479\) −8142.90 −0.776740 −0.388370 0.921503i \(-0.626962\pi\)
−0.388370 + 0.921503i \(0.626962\pi\)
\(480\) 0 0
\(481\) −568.605 −0.0539005
\(482\) 0 0
\(483\) −28892.3 −2.72184
\(484\) 0 0
\(485\) −910.407 −0.0852360
\(486\) 0 0
\(487\) 10669.6 0.992780 0.496390 0.868100i \(-0.334659\pi\)
0.496390 + 0.868100i \(0.334659\pi\)
\(488\) 0 0
\(489\) −11375.3 −1.05196
\(490\) 0 0
\(491\) −20917.7 −1.92262 −0.961308 0.275475i \(-0.911165\pi\)
−0.961308 + 0.275475i \(0.911165\pi\)
\(492\) 0 0
\(493\) 1927.40 0.176077
\(494\) 0 0
\(495\) −28.3829 −0.00257720
\(496\) 0 0
\(497\) −31166.5 −2.81289
\(498\) 0 0
\(499\) 2264.43 0.203146 0.101573 0.994828i \(-0.467612\pi\)
0.101573 + 0.994828i \(0.467612\pi\)
\(500\) 0 0
\(501\) −6623.60 −0.590660
\(502\) 0 0
\(503\) 10397.1 0.921641 0.460820 0.887493i \(-0.347555\pi\)
0.460820 + 0.887493i \(0.347555\pi\)
\(504\) 0 0
\(505\) 3205.23 0.282437
\(506\) 0 0
\(507\) −10673.6 −0.934969
\(508\) 0 0
\(509\) −8496.07 −0.739846 −0.369923 0.929062i \(-0.620616\pi\)
−0.369923 + 0.929062i \(0.620616\pi\)
\(510\) 0 0
\(511\) −20610.2 −1.78423
\(512\) 0 0
\(513\) −9804.76 −0.843841
\(514\) 0 0
\(515\) 1520.29 0.130081
\(516\) 0 0
\(517\) −1986.28 −0.168968
\(518\) 0 0
\(519\) 19481.5 1.64767
\(520\) 0 0
\(521\) −8994.22 −0.756322 −0.378161 0.925740i \(-0.623443\pi\)
−0.378161 + 0.925740i \(0.623443\pi\)
\(522\) 0 0
\(523\) 5072.24 0.424080 0.212040 0.977261i \(-0.431989\pi\)
0.212040 + 0.977261i \(0.431989\pi\)
\(524\) 0 0
\(525\) 20493.9 1.70367
\(526\) 0 0
\(527\) 2712.45 0.224205
\(528\) 0 0
\(529\) 15788.0 1.29761
\(530\) 0 0
\(531\) 374.161 0.0305786
\(532\) 0 0
\(533\) −5459.18 −0.443646
\(534\) 0 0
\(535\) −5399.59 −0.436345
\(536\) 0 0
\(537\) 7608.99 0.611456
\(538\) 0 0
\(539\) −4468.75 −0.357111
\(540\) 0 0
\(541\) −17722.4 −1.40840 −0.704202 0.710000i \(-0.748695\pi\)
−0.704202 + 0.710000i \(0.748695\pi\)
\(542\) 0 0
\(543\) −11503.5 −0.909140
\(544\) 0 0
\(545\) −3943.55 −0.309951
\(546\) 0 0
\(547\) −282.133 −0.0220533 −0.0110266 0.999939i \(-0.503510\pi\)
−0.0110266 + 0.999939i \(0.503510\pi\)
\(548\) 0 0
\(549\) −245.786 −0.0191072
\(550\) 0 0
\(551\) 8210.03 0.634771
\(552\) 0 0
\(553\) 42541.2 3.27131
\(554\) 0 0
\(555\) 537.188 0.0410853
\(556\) 0 0
\(557\) −13294.8 −1.01134 −0.505672 0.862726i \(-0.668755\pi\)
−0.505672 + 0.862726i \(0.668755\pi\)
\(558\) 0 0
\(559\) 2770.31 0.209609
\(560\) 0 0
\(561\) −585.403 −0.0440566
\(562\) 0 0
\(563\) 1632.96 0.122240 0.0611198 0.998130i \(-0.480533\pi\)
0.0611198 + 0.998130i \(0.480533\pi\)
\(564\) 0 0
\(565\) 1040.77 0.0774965
\(566\) 0 0
\(567\) 24917.1 1.84554
\(568\) 0 0
\(569\) −8267.89 −0.609153 −0.304577 0.952488i \(-0.598515\pi\)
−0.304577 + 0.952488i \(0.598515\pi\)
\(570\) 0 0
\(571\) −2750.70 −0.201599 −0.100800 0.994907i \(-0.532140\pi\)
−0.100800 + 0.994907i \(0.532140\pi\)
\(572\) 0 0
\(573\) 13824.9 1.00793
\(574\) 0 0
\(575\) −19829.0 −1.43813
\(576\) 0 0
\(577\) −18366.0 −1.32510 −0.662552 0.749016i \(-0.730527\pi\)
−0.662552 + 0.749016i \(0.730527\pi\)
\(578\) 0 0
\(579\) 20301.7 1.45719
\(580\) 0 0
\(581\) −45714.4 −3.26429
\(582\) 0 0
\(583\) 172.850 0.0122791
\(584\) 0 0
\(585\) −63.4646 −0.00448536
\(586\) 0 0
\(587\) −19167.2 −1.34772 −0.673862 0.738857i \(-0.735366\pi\)
−0.673862 + 0.738857i \(0.735366\pi\)
\(588\) 0 0
\(589\) 11554.0 0.808277
\(590\) 0 0
\(591\) −18726.3 −1.30338
\(592\) 0 0
\(593\) −16857.3 −1.16736 −0.583680 0.811984i \(-0.698388\pi\)
−0.583680 + 0.811984i \(0.698388\pi\)
\(594\) 0 0
\(595\) −1386.52 −0.0955322
\(596\) 0 0
\(597\) −2693.49 −0.184652
\(598\) 0 0
\(599\) −23326.3 −1.59113 −0.795565 0.605869i \(-0.792826\pi\)
−0.795565 + 0.605869i \(0.792826\pi\)
\(600\) 0 0
\(601\) 1305.09 0.0885783 0.0442891 0.999019i \(-0.485898\pi\)
0.0442891 + 0.999019i \(0.485898\pi\)
\(602\) 0 0
\(603\) 1180.88 0.0797499
\(604\) 0 0
\(605\) −3263.78 −0.219325
\(606\) 0 0
\(607\) −10225.7 −0.683769 −0.341884 0.939742i \(-0.611065\pi\)
−0.341884 + 0.939742i \(0.611065\pi\)
\(608\) 0 0
\(609\) −19591.9 −1.30362
\(610\) 0 0
\(611\) −4441.35 −0.294072
\(612\) 0 0
\(613\) −2611.06 −0.172039 −0.0860194 0.996293i \(-0.527415\pi\)
−0.0860194 + 0.996293i \(0.527415\pi\)
\(614\) 0 0
\(615\) 5157.55 0.338167
\(616\) 0 0
\(617\) 8225.49 0.536703 0.268351 0.963321i \(-0.413521\pi\)
0.268351 + 0.963321i \(0.413521\pi\)
\(618\) 0 0
\(619\) 2517.19 0.163448 0.0817242 0.996655i \(-0.473957\pi\)
0.0817242 + 0.996655i \(0.473957\pi\)
\(620\) 0 0
\(621\) −22638.4 −1.46288
\(622\) 0 0
\(623\) 12499.9 0.803847
\(624\) 0 0
\(625\) 13264.6 0.848936
\(626\) 0 0
\(627\) −2493.60 −0.158828
\(628\) 0 0
\(629\) 673.083 0.0426671
\(630\) 0 0
\(631\) −4769.93 −0.300932 −0.150466 0.988615i \(-0.548077\pi\)
−0.150466 + 0.988615i \(0.548077\pi\)
\(632\) 0 0
\(633\) −8389.36 −0.526773
\(634\) 0 0
\(635\) 3892.57 0.243263
\(636\) 0 0
\(637\) −9992.21 −0.621516
\(638\) 0 0
\(639\) 1688.70 0.104544
\(640\) 0 0
\(641\) −276.323 −0.0170267 −0.00851335 0.999964i \(-0.502710\pi\)
−0.00851335 + 0.999964i \(0.502710\pi\)
\(642\) 0 0
\(643\) −19925.0 −1.22203 −0.611016 0.791618i \(-0.709239\pi\)
−0.611016 + 0.791618i \(0.709239\pi\)
\(644\) 0 0
\(645\) −2617.24 −0.159773
\(646\) 0 0
\(647\) 9664.69 0.587262 0.293631 0.955919i \(-0.405136\pi\)
0.293631 + 0.955919i \(0.405136\pi\)
\(648\) 0 0
\(649\) −1376.09 −0.0832302
\(650\) 0 0
\(651\) −27571.8 −1.65994
\(652\) 0 0
\(653\) −23304.6 −1.39660 −0.698301 0.715805i \(-0.746060\pi\)
−0.698301 + 0.715805i \(0.746060\pi\)
\(654\) 0 0
\(655\) 4979.89 0.297069
\(656\) 0 0
\(657\) 1116.73 0.0663130
\(658\) 0 0
\(659\) −17048.8 −1.00778 −0.503890 0.863768i \(-0.668098\pi\)
−0.503890 + 0.863768i \(0.668098\pi\)
\(660\) 0 0
\(661\) 19310.8 1.13631 0.568157 0.822920i \(-0.307657\pi\)
0.568157 + 0.822920i \(0.307657\pi\)
\(662\) 0 0
\(663\) −1308.97 −0.0766761
\(664\) 0 0
\(665\) −5906.06 −0.344402
\(666\) 0 0
\(667\) 18956.3 1.10044
\(668\) 0 0
\(669\) 27636.5 1.59714
\(670\) 0 0
\(671\) 903.952 0.0520069
\(672\) 0 0
\(673\) 11322.2 0.648496 0.324248 0.945972i \(-0.394889\pi\)
0.324248 + 0.945972i \(0.394889\pi\)
\(674\) 0 0
\(675\) 16057.8 0.915654
\(676\) 0 0
\(677\) 5130.90 0.291280 0.145640 0.989338i \(-0.453476\pi\)
0.145640 + 0.989338i \(0.453476\pi\)
\(678\) 0 0
\(679\) 11595.3 0.655357
\(680\) 0 0
\(681\) 24610.1 1.38482
\(682\) 0 0
\(683\) 19250.8 1.07849 0.539247 0.842148i \(-0.318709\pi\)
0.539247 + 0.842148i \(0.318709\pi\)
\(684\) 0 0
\(685\) 1601.94 0.0893532
\(686\) 0 0
\(687\) 2454.76 0.136324
\(688\) 0 0
\(689\) 386.496 0.0213706
\(690\) 0 0
\(691\) −9816.53 −0.540432 −0.270216 0.962800i \(-0.587095\pi\)
−0.270216 + 0.962800i \(0.587095\pi\)
\(692\) 0 0
\(693\) 361.495 0.0198154
\(694\) 0 0
\(695\) 2492.22 0.136022
\(696\) 0 0
\(697\) 6462.29 0.351186
\(698\) 0 0
\(699\) 27085.4 1.46561
\(700\) 0 0
\(701\) 12994.6 0.700142 0.350071 0.936723i \(-0.386158\pi\)
0.350071 + 0.936723i \(0.386158\pi\)
\(702\) 0 0
\(703\) 2867.09 0.153818
\(704\) 0 0
\(705\) 4195.95 0.224154
\(706\) 0 0
\(707\) −40823.1 −2.17158
\(708\) 0 0
\(709\) 29277.4 1.55083 0.775414 0.631454i \(-0.217541\pi\)
0.775414 + 0.631454i \(0.217541\pi\)
\(710\) 0 0
\(711\) −2305.02 −0.121582
\(712\) 0 0
\(713\) 26677.3 1.40123
\(714\) 0 0
\(715\) 233.410 0.0122085
\(716\) 0 0
\(717\) 20439.2 1.06460
\(718\) 0 0
\(719\) −14651.1 −0.759934 −0.379967 0.925000i \(-0.624065\pi\)
−0.379967 + 0.925000i \(0.624065\pi\)
\(720\) 0 0
\(721\) −19363.0 −1.00016
\(722\) 0 0
\(723\) −22633.2 −1.16423
\(724\) 0 0
\(725\) −13446.0 −0.688791
\(726\) 0 0
\(727\) 15552.0 0.793385 0.396692 0.917952i \(-0.370158\pi\)
0.396692 + 0.917952i \(0.370158\pi\)
\(728\) 0 0
\(729\) 18250.6 0.927225
\(730\) 0 0
\(731\) −3279.34 −0.165925
\(732\) 0 0
\(733\) 21433.2 1.08002 0.540008 0.841660i \(-0.318421\pi\)
0.540008 + 0.841660i \(0.318421\pi\)
\(734\) 0 0
\(735\) 9440.11 0.473747
\(736\) 0 0
\(737\) −4343.05 −0.217067
\(738\) 0 0
\(739\) 3561.65 0.177290 0.0886450 0.996063i \(-0.471746\pi\)
0.0886450 + 0.996063i \(0.471746\pi\)
\(740\) 0 0
\(741\) −5575.74 −0.276424
\(742\) 0 0
\(743\) 8804.42 0.434728 0.217364 0.976091i \(-0.430254\pi\)
0.217364 + 0.976091i \(0.430254\pi\)
\(744\) 0 0
\(745\) −4601.13 −0.226271
\(746\) 0 0
\(747\) 2476.95 0.121321
\(748\) 0 0
\(749\) 68771.3 3.35494
\(750\) 0 0
\(751\) 16398.8 0.796806 0.398403 0.917210i \(-0.369564\pi\)
0.398403 + 0.917210i \(0.369564\pi\)
\(752\) 0 0
\(753\) 7973.22 0.385870
\(754\) 0 0
\(755\) 2405.24 0.115941
\(756\) 0 0
\(757\) −1256.33 −0.0603199 −0.0301599 0.999545i \(-0.509602\pi\)
−0.0301599 + 0.999545i \(0.509602\pi\)
\(758\) 0 0
\(759\) −5757.53 −0.275343
\(760\) 0 0
\(761\) 3988.92 0.190011 0.0950053 0.995477i \(-0.469713\pi\)
0.0950053 + 0.995477i \(0.469713\pi\)
\(762\) 0 0
\(763\) 50226.6 2.38313
\(764\) 0 0
\(765\) 75.1260 0.00355057
\(766\) 0 0
\(767\) −3076.97 −0.144854
\(768\) 0 0
\(769\) −14908.0 −0.699087 −0.349543 0.936920i \(-0.613663\pi\)
−0.349543 + 0.936920i \(0.613663\pi\)
\(770\) 0 0
\(771\) 35336.2 1.65059
\(772\) 0 0
\(773\) 10280.4 0.478343 0.239171 0.970977i \(-0.423124\pi\)
0.239171 + 0.970977i \(0.423124\pi\)
\(774\) 0 0
\(775\) −18922.7 −0.877063
\(776\) 0 0
\(777\) −6841.84 −0.315894
\(778\) 0 0
\(779\) 27527.0 1.26605
\(780\) 0 0
\(781\) −6210.71 −0.284554
\(782\) 0 0
\(783\) −15351.1 −0.700643
\(784\) 0 0
\(785\) −9265.42 −0.421270
\(786\) 0 0
\(787\) 35038.6 1.58703 0.793514 0.608552i \(-0.208249\pi\)
0.793514 + 0.608552i \(0.208249\pi\)
\(788\) 0 0
\(789\) 917.557 0.0414016
\(790\) 0 0
\(791\) −13255.7 −0.595850
\(792\) 0 0
\(793\) 2021.25 0.0905130
\(794\) 0 0
\(795\) −365.141 −0.0162896
\(796\) 0 0
\(797\) −17732.3 −0.788092 −0.394046 0.919091i \(-0.628925\pi\)
−0.394046 + 0.919091i \(0.628925\pi\)
\(798\) 0 0
\(799\) 5257.43 0.232784
\(800\) 0 0
\(801\) −677.283 −0.0298759
\(802\) 0 0
\(803\) −4107.10 −0.180494
\(804\) 0 0
\(805\) −13636.6 −0.597053
\(806\) 0 0
\(807\) −8989.25 −0.392115
\(808\) 0 0
\(809\) 21264.4 0.924123 0.462062 0.886848i \(-0.347110\pi\)
0.462062 + 0.886848i \(0.347110\pi\)
\(810\) 0 0
\(811\) 18812.1 0.814527 0.407263 0.913311i \(-0.366483\pi\)
0.407263 + 0.913311i \(0.366483\pi\)
\(812\) 0 0
\(813\) 19664.3 0.848288
\(814\) 0 0
\(815\) −5368.90 −0.230754
\(816\) 0 0
\(817\) −13968.8 −0.598172
\(818\) 0 0
\(819\) 808.310 0.0344867
\(820\) 0 0
\(821\) −15392.7 −0.654336 −0.327168 0.944966i \(-0.606094\pi\)
−0.327168 + 0.944966i \(0.606094\pi\)
\(822\) 0 0
\(823\) −10506.1 −0.444981 −0.222491 0.974935i \(-0.571419\pi\)
−0.222491 + 0.974935i \(0.571419\pi\)
\(824\) 0 0
\(825\) 4083.92 0.172344
\(826\) 0 0
\(827\) −6255.99 −0.263050 −0.131525 0.991313i \(-0.541987\pi\)
−0.131525 + 0.991313i \(0.541987\pi\)
\(828\) 0 0
\(829\) 39592.2 1.65874 0.829370 0.558700i \(-0.188700\pi\)
0.829370 + 0.558700i \(0.188700\pi\)
\(830\) 0 0
\(831\) −32949.1 −1.37544
\(832\) 0 0
\(833\) 11828.2 0.491986
\(834\) 0 0
\(835\) −3126.21 −0.129565
\(836\) 0 0
\(837\) −21603.7 −0.892154
\(838\) 0 0
\(839\) −5161.09 −0.212372 −0.106186 0.994346i \(-0.533864\pi\)
−0.106186 + 0.994346i \(0.533864\pi\)
\(840\) 0 0
\(841\) −11534.7 −0.472949
\(842\) 0 0
\(843\) 7401.83 0.302411
\(844\) 0 0
\(845\) −5037.71 −0.205092
\(846\) 0 0
\(847\) 41568.7 1.68633
\(848\) 0 0
\(849\) −5515.88 −0.222974
\(850\) 0 0
\(851\) 6619.88 0.266659
\(852\) 0 0
\(853\) 29219.7 1.17288 0.586438 0.809994i \(-0.300530\pi\)
0.586438 + 0.809994i \(0.300530\pi\)
\(854\) 0 0
\(855\) 320.009 0.0128001
\(856\) 0 0
\(857\) 16277.5 0.648810 0.324405 0.945918i \(-0.394836\pi\)
0.324405 + 0.945918i \(0.394836\pi\)
\(858\) 0 0
\(859\) −33890.2 −1.34612 −0.673062 0.739586i \(-0.735021\pi\)
−0.673062 + 0.739586i \(0.735021\pi\)
\(860\) 0 0
\(861\) −65688.6 −2.60007
\(862\) 0 0
\(863\) 24932.1 0.983429 0.491714 0.870757i \(-0.336370\pi\)
0.491714 + 0.870757i \(0.336370\pi\)
\(864\) 0 0
\(865\) 9194.87 0.361428
\(866\) 0 0
\(867\) 1549.49 0.0606960
\(868\) 0 0
\(869\) 8477.41 0.330928
\(870\) 0 0
\(871\) −9711.14 −0.377784
\(872\) 0 0
\(873\) −628.271 −0.0243571
\(874\) 0 0
\(875\) 19867.7 0.767600
\(876\) 0 0
\(877\) 17205.0 0.662452 0.331226 0.943551i \(-0.392538\pi\)
0.331226 + 0.943551i \(0.392538\pi\)
\(878\) 0 0
\(879\) −10129.0 −0.388673
\(880\) 0 0
\(881\) 34861.7 1.33317 0.666583 0.745431i \(-0.267756\pi\)
0.666583 + 0.745431i \(0.267756\pi\)
\(882\) 0 0
\(883\) −30546.3 −1.16417 −0.582087 0.813126i \(-0.697764\pi\)
−0.582087 + 0.813126i \(0.697764\pi\)
\(884\) 0 0
\(885\) 2906.96 0.110414
\(886\) 0 0
\(887\) 8771.17 0.332026 0.166013 0.986124i \(-0.446911\pi\)
0.166013 + 0.986124i \(0.446911\pi\)
\(888\) 0 0
\(889\) −49577.3 −1.87038
\(890\) 0 0
\(891\) 4965.37 0.186696
\(892\) 0 0
\(893\) 22394.7 0.839206
\(894\) 0 0
\(895\) 3591.29 0.134127
\(896\) 0 0
\(897\) −12873.9 −0.479207
\(898\) 0 0
\(899\) 18089.9 0.671114
\(900\) 0 0
\(901\) −457.513 −0.0169167
\(902\) 0 0
\(903\) 33334.3 1.22845
\(904\) 0 0
\(905\) −5429.43 −0.199426
\(906\) 0 0
\(907\) 2257.34 0.0826393 0.0413196 0.999146i \(-0.486844\pi\)
0.0413196 + 0.999146i \(0.486844\pi\)
\(908\) 0 0
\(909\) 2211.92 0.0807095
\(910\) 0 0
\(911\) 49374.3 1.79566 0.897829 0.440344i \(-0.145143\pi\)
0.897829 + 0.440344i \(0.145143\pi\)
\(912\) 0 0
\(913\) −9109.74 −0.330217
\(914\) 0 0
\(915\) −1909.57 −0.0689929
\(916\) 0 0
\(917\) −63425.8 −2.28408
\(918\) 0 0
\(919\) 35777.0 1.28420 0.642098 0.766623i \(-0.278064\pi\)
0.642098 + 0.766623i \(0.278064\pi\)
\(920\) 0 0
\(921\) −33347.1 −1.19308
\(922\) 0 0
\(923\) −13887.3 −0.495238
\(924\) 0 0
\(925\) −4695.60 −0.166909
\(926\) 0 0
\(927\) 1049.15 0.0371721
\(928\) 0 0
\(929\) 8342.11 0.294613 0.147307 0.989091i \(-0.452940\pi\)
0.147307 + 0.989091i \(0.452940\pi\)
\(930\) 0 0
\(931\) 50384.0 1.77365
\(932\) 0 0
\(933\) 28683.4 1.00649
\(934\) 0 0
\(935\) −276.299 −0.00966410
\(936\) 0 0
\(937\) 11057.0 0.385502 0.192751 0.981248i \(-0.438259\pi\)
0.192751 + 0.981248i \(0.438259\pi\)
\(938\) 0 0
\(939\) 41267.8 1.43421
\(940\) 0 0
\(941\) −7354.58 −0.254785 −0.127392 0.991852i \(-0.540661\pi\)
−0.127392 + 0.991852i \(0.540661\pi\)
\(942\) 0 0
\(943\) 63557.6 2.19482
\(944\) 0 0
\(945\) 11043.1 0.380141
\(946\) 0 0
\(947\) −18516.1 −0.635368 −0.317684 0.948197i \(-0.602905\pi\)
−0.317684 + 0.948197i \(0.602905\pi\)
\(948\) 0 0
\(949\) −9183.56 −0.314132
\(950\) 0 0
\(951\) 41607.1 1.41872
\(952\) 0 0
\(953\) 17617.6 0.598835 0.299418 0.954122i \(-0.403208\pi\)
0.299418 + 0.954122i \(0.403208\pi\)
\(954\) 0 0
\(955\) 6525.10 0.221096
\(956\) 0 0
\(957\) −3904.18 −0.131875
\(958\) 0 0
\(959\) −20402.9 −0.687013
\(960\) 0 0
\(961\) −4333.00 −0.145447
\(962\) 0 0
\(963\) −3726.25 −0.124690
\(964\) 0 0
\(965\) 9582.02 0.319644
\(966\) 0 0
\(967\) −24466.2 −0.813629 −0.406815 0.913511i \(-0.633361\pi\)
−0.406815 + 0.913511i \(0.633361\pi\)
\(968\) 0 0
\(969\) 6600.26 0.218814
\(970\) 0 0
\(971\) 27002.9 0.892445 0.446222 0.894922i \(-0.352769\pi\)
0.446222 + 0.894922i \(0.352769\pi\)
\(972\) 0 0
\(973\) −31741.9 −1.04584
\(974\) 0 0
\(975\) 9131.72 0.299948
\(976\) 0 0
\(977\) −57325.0 −1.87716 −0.938582 0.345057i \(-0.887860\pi\)
−0.938582 + 0.345057i \(0.887860\pi\)
\(978\) 0 0
\(979\) 2490.92 0.0813177
\(980\) 0 0
\(981\) −2721.44 −0.0885717
\(982\) 0 0
\(983\) −50052.2 −1.62403 −0.812013 0.583640i \(-0.801628\pi\)
−0.812013 + 0.583640i \(0.801628\pi\)
\(984\) 0 0
\(985\) −8838.42 −0.285904
\(986\) 0 0
\(987\) −53441.3 −1.72346
\(988\) 0 0
\(989\) −32252.8 −1.03699
\(990\) 0 0
\(991\) 35053.6 1.12363 0.561813 0.827264i \(-0.310104\pi\)
0.561813 + 0.827264i \(0.310104\pi\)
\(992\) 0 0
\(993\) −5389.53 −0.172237
\(994\) 0 0
\(995\) −1271.27 −0.0405046
\(996\) 0 0
\(997\) −44154.7 −1.40260 −0.701300 0.712866i \(-0.747397\pi\)
−0.701300 + 0.712866i \(0.747397\pi\)
\(998\) 0 0
\(999\) −5360.87 −0.169780
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 1088.4.a.w.1.2 3
4.3 odd 2 1088.4.a.t.1.2 3
8.3 odd 2 68.4.a.b.1.2 3
8.5 even 2 272.4.a.i.1.2 3
24.5 odd 2 2448.4.a.ba.1.3 3
24.11 even 2 612.4.a.g.1.3 3
40.3 even 4 1700.4.e.d.749.4 6
40.19 odd 2 1700.4.a.d.1.2 3
40.27 even 4 1700.4.e.d.749.3 6
136.67 odd 2 1156.4.a.g.1.2 3
136.115 odd 4 1156.4.b.e.577.4 6
136.123 odd 4 1156.4.b.e.577.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.4.a.b.1.2 3 8.3 odd 2
272.4.a.i.1.2 3 8.5 even 2
612.4.a.g.1.3 3 24.11 even 2
1088.4.a.t.1.2 3 4.3 odd 2
1088.4.a.w.1.2 3 1.1 even 1 trivial
1156.4.a.g.1.2 3 136.67 odd 2
1156.4.b.e.577.3 6 136.123 odd 4
1156.4.b.e.577.4 6 136.115 odd 4
1700.4.a.d.1.2 3 40.19 odd 2
1700.4.e.d.749.3 6 40.27 even 4
1700.4.e.d.749.4 6 40.3 even 4
2448.4.a.ba.1.3 3 24.5 odd 2